url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
https://www.nickzom.org/blog/category/chemistry/ | ## How to Calculate and Solve for Mass, Volume and Linear Momentum | The Calculator Encyclopedia
The image above represents a linear momentum.
To compute for the linear momentum, two essential parameters are needed and these parameters are mass (m) and velocity (v).
The formula for calculating linear momentum:
p = mv
Where;
p = Momentum
m = Mass
v = Velocity
Let’s solve an example;
Find the linear momentum of a mass of 44 and a velocity of 38.
This implies that;
m = Mass = 44
v = Velocity = 38
p = mv
p = 44 x 38
p = 1672
Therefore, the linear momentum is 1672 Kgm/s.
Calculating the Mass when Linear Momentum and Velocity is Given.
m = p / v
Where;
m = Mass
p = Momentum
v = Velocity
Let’s solve an example;
Find the mass with a linear momentum of 320 and a velocity of 80.
This implies that;
p = Momentum = 320
v = Velocity = 80
m = p / v
m = 320 / 80
m = 4
Therefore, the mass is 4 kg.
## How to Calculate and Solve for the Quantity of Charge, Electrochemical Equivalence of a Substance and Mass of an Element in Electrolysis | Nickzom Calculator
The image above represents the mass of an element.
To compute for the mass of an element, two essential parameters are needed and these parameters are Electrochemical Equivalence of the Substance (Z) and quantity of charge (Q).
The formula for calculating mass of an element:
M = ZQ
Where;
M = Mass of the element
Z = Electrochemical Equivalence of the Substance
Q = Quantity of Charge
Let’s solve an example;
Find the mass of an element when the Quantity of charge is 28 and Electrochemical Equivalence of the Substance is 32.
This implies that;
Z = Electrochemical Equivalence of the Substance = 32
Q = Quantity of Charge = 28
M = ZQ
M = 32 x 28
M = 896
Therefore, the mass of an element is 896 kg.
Calculating the Electrochemical Equivalence of the Substance when the Mass of an Element and Quantity of Charge is Given.
Z = M / Q
Where;
Z = Electrochemical Equivalence of the Substance
M = Mass of the element
Q = Quantity of Charge
Let’s solve an example;
Find the Electrochemical Equivalence of the Substance when the Quantity of charge is 12 and mass of an element is 120.
This implies that;
M = Mass of the element = 120
Q = Quantity of Charge = 12
Z = M / Q
Z = 120 / 12
Z = 10
Therefore, the Electrochemical Equivalence of the Substance is 10.
## How to Calculate and Solve for the Current, Time and Quantity of Charge of an Electrolysis | The Calculator Encyclopedia
The image above represents the quantity of charge.
To compute for the quantity of charge, two essential parameters are needed and these parameters are current (I) and time (T).
The formula for calculating the quantity of charge:
Q = It
Where;
Q = Quantity of charge
I = Current
T = Time
Let’s solve an example;
Find the quantity of charge with a current of 24 and time of 12.
This implies that;
I = Current = 24
T = Time = 12
Q = It
Q = 24 x 12
Q = 288
Therefore, the quantity of charge is 288 Coulombs (C).
Calculating the Current (I) using the Quantity of Charge and Time.
I = Q / t
Where;
I = Current
Q = Quantity of charge
T = Time
Let’s solve an example;
Given that the quantity of charge is 240 with a time of 14. Find the Current?
This implies that;
Q = Quantity of charge = 240
T = Time = 14
I = Q / t
I = 240 / 14
I = 17.14
Therefore, the current is 17.14 ampere.
## How to Calculate and Solve for Van’t Hoff Factor, Ebullioscopic Constant, Molality and Boiling Point Elevation | The Calculator Encyclopedia
The image above represents boiling point elevation.
To compute for the boiling point elevation, three parameters are needed and these parameters are Van’t Hoff’s Factor (i), ebullioscopic constant (Kb) and Molality.
The formula for calculating boiling point elevation:
δTb = iKb x Molality
Where;
δTb = boiling point elevation
i = Van’t Hoff’s Factor
Kb = ebullioscopic constant
Molality
Let’s solve an example;
Find the boiling point elevation when the Van’t Hoff’s Factor is 42, ebullioscopic constant is 60 and molality of 180.
This implies that;
i = Van’t Hoff’s Factor = 42
Kb = ebullioscopic constant = 60
Molality = 180
δTb = iKb x Molality
δTb = (42)(60) x 180
δTb = (2520) x 180
δTb = 453600
Therefore, the boiling point elevation is 453600 °C m-1.
Calculating the Molality using the Boiling Point Elevation, Van’t Hoff’s Factor and Ebullioscopic Constant.
Molality = δTb / iKb
Where;
Molality
δTb = boiling point elevation
i = Van’t Hoff’s Factor
Kb = ebullioscopic constant
Let’s solve an example;
Find the molality with a boiling point elevation of 120 and a van’t hoff’s factor of 32 with a ebullioscopic constant of 12.
This implies that;
δTb = boiling point elevation = 120
i = Van’t Hoff’s Factor = 32
Kb = ebullioscopic constant = 12
Molality = δTb / iKb
Molality = 120 / 384
Molality = 0.3125
Therefore, the molality is 0.3125.
## How to Calculate and Solve for Temperature, Number of Moles, Volume, Van’t Hoff Factor and Osmotic Pressure | The Calculator Encyclopedia
The image above represents the osmotic pressure.
To compute for the osmotic pressure, five parameters are needed and these parameters are Ideal Gas Constant (R)Temperature in Kelvin (T), Number of Moles (n), Volume (V) and Van’t Hoff’s Factor (i).
The formula for calculating osmotic pressure:
π = i nRTV
Where;
π = osmotic pressure
n = number of moles
R = ideal gas constant
T = temperature in Kelvin
i = Van’t Hoff’s Factor
V = Volume
Let’s solve an example;
Find the osmotic pressure when the ideal gas constant is 0.08206 with a temperature in kelvin of 120, number of moles is 32, a volume of 48 and a van’t hoff’s factor of 24.
This implies that;
n = number of moles = 32
R = ideal gas constant = 0.08206
T = temperature in Kelvin = 120
i = Van’t Hoff’s Factor = 24
V = Volume = 48
π = i nRTV
π = 24 32 x 0.08206 x 12048
π = (24) (315.110)(48)
π = (24)(6.5647)
π = 157.5
Therefore, the osmotic pressure is 157.5 atm.
Calculating the Van’t Hoff’s Factor using the Osmotic Pressure, Number of Moles, Temperature in Kelvin, Ideal Gas Constant and Volume.
i = / nRT
Where;
i = Van’t Hoff’s Factor
π = osmotic pressure
V = Volume
n = number of moles
R = ideal gas constant
T = temperature in Kelvin
Let’s solve an example;
Find the Van’t Hoff’s Factor when the osmotic pressure is 220, volume of 50, temperature in kelvin of 180 and number of moles of 60. (R = 0.08206)
This implies that;
π = osmotic pressure = 220
V = Volume = 50
n = number of moles = 60
R = ideal gas constant = 0.08206
T = temperature in Kelvin = 180
i = / nRT
i = 50 x 220 / 60 x 0.08206 x 180
i = 11000 / 866.808
i = 12.69
Therefore, the Van’t Hoff’s Factor is 12.69.
Calculating the Volume using the Osmotic Pressure, Number of Moles, Temperature in Kelvin, Ideal Gas Constant and Van’t Hoff’s Factor.
V = i (nRT) / π
Where;
V = Volume
i = Van’t Hoff’s Factor
π = osmotic pressure
n = number of moles
R = ideal gas constant
T = temperature in Kelvin
Let’s solve an example;
Find the volume when the osmotic pressure is 280, Van’t Hoff’s Factor of 40, temperature in kelvin of 90 and number of moles of 70. (R = 0.08206)
This implies that;
i = Van’t Hoff’s Factor = 40
π = osmotic pressure = 280
n = number of moles = 70
R = ideal gas constant = 0.08206
T = temperature in Kelvin = 90
V = i (nRT) / π
V = 40 (70 x 0.08206 x 90) / 280
V = 40 (516.978) / 280
V = 20679.12 / 280
V = 73.854
Therefore, the volume is 73.854.
## How to Calculate and Solve for Van’t Hoff’s Factor, Cryoscopic Constant, Molality and Freezing Point Depression | Nickzom Calculator
The image above represents the freezing point depression.
To compute for the freezing point depression, three essential parameters are needed and these parameters are Van’t Hoff’s Factor (i), cryoscopic constant (Kf) and molality.
The formula for calculating freezing point depression:
δTf = iKf x Molality
Where;
δTf = Freezing point depression
i = Van’t Hoff’s Factor
Kf = cryoscopic constant
Molality
Let’s solve an example;
Find the freezing point depression when the van’t hoff’s factor is 12, cryoscopic constant is 21 with a molality of 16.
δTf = iKf x Molality
δTf = (12 x 21) x 16
δTf = 252 x 16
δTf = 4032
Therefore, the freezing point depression is 4032 °C m-1.
## How to Calculate and Solve for Mass, Volume and Density | The Calculator Encyclopedia
The image above represents density.
To compute for the density, two essential parameters are needed and these parameters are mass (m) and volume (v).
The formula for calculating density:
Density = mass / volume
Let’s solve an example;
Given that the volume is 20 m³ with a mass of 240 kg. Find the density?
This implies that;
Volume = 20
Mass = 240
Density = mass / volume
Density = 240 / 20
Density = 12
Therefore, the density is 12 Kg/m³.
Calculating the Mass when the Density and Volume is Given.
Mass = Volume x Density
Let’s solve an example;
With a density of 90 kg/m³ and a volume of 15 m³, Find the mass?
This implies that;
Density = 90
Volume = 15
Mass = Volume x Density
Mass = 15 x 90
Mass = 1350
Therefore, the mass is 1350 kg.
## How to Calculate and Solve for Percentage Yield, Actual Yield and Theoretical Yield in Chemistry | The Calculator Encyclopedia
The image above represents percentage yield.
To compute the percentage yield, two essential parameters are needed and these parameters are actual yield and theoretical yield.
The formula for calculating percentage yield:
%Yield = (Actual YieldTheoretical Yield) x 100
Let’s solve an example;
Find the percentage yield when the actual yield is 12 with a theoretical yield of 26.
This implies that;
Actual yield = 12
Theoretical yield = 26
%Yield = (Actual YieldTheoretical Yield) x 100
%Yield = (1226) x 100
%Yield = 0.46 x 100
%Yield = 46.15
Therefore, the percentage yield is 46.15.
Calculating the Actual Yield when the Percentage Yield and the Theoretical Yield is Given.
Actual yield = %yield x theoretical yield / 100
Let’s solve an example;
Find the actual yield when the %yield is 52 with a theoretical yield of 10.
This implies that;
%yield = 52
Theoretical yield = 10
Actual yield = %yield x theoretical yield / 100
Actual yield = 52 x 10 / 100
Actual yield = 520 / 100
Actual yield = 5.2
Therefore, the actual yield is 5.2.
## How to Calculate and Solve for the Molar Concentration, Molar Mass and Mass Concentration in Chemistry | The Calculator Encyclopedia
The image above represents the mass concentration.
To compute for mass concentration, two essential parameters are needed and these are molar concentration (c) and molar mass (M).
ρ = c x M
Where;
ρ = mass concentration
c = molar concentration
M = molar mass
Let’s solve an example;
Find the mass concentration when the molar concentration is 24 and molar mass is 15.
This implies that;
c = molar concentration = 24
M = molar mass = 15
ρ = c x M
ρ = 24 x 15
ρ = 360
Therefore, the mass concentration is 360 Kg/dm³.
Calculating the Molar Concentration when the Mass concentration and Molar Mass.
c = ρ / M
Where;
c = molar concentration
ρ = mass concentration
M = molar mass
Let’s solve an example;
Find the molar concentration when the mass concentration is 120 with a molar mass of 40.
This implies that;
ρ = mass concentration = 120
M = molar mass = 40
c = ρ / M
c = 120 / 40
c = 3
Therefore, the molar concentration is 3 mol/L.
## How to Calculate and Solve for the Mass, Volume and Mass Concentration in Chemistry | The Calculator Encyclopedia
The image above represents the mass concentration.
To compute the mass concentration, two essential parameters are needed and these parameters are mass (m) and volume (V).
The formula for calculating mass concentration:
ρ = mV
Where;
ρ = Mass concentration
m = Mass
V = Volume
Let’s solve an example;
Find the mass concentration when the mass is 8 kg with a volume of 24 dm³.
This implies that;
m = Mass = 8
V = Volume = 24
ρ = mV
ρ = 824
ρ = 0.33
Therefore, the mass concentration is 0.33 Kg/dm³.
Calculating the Mass when the Mass Concentration and the Volume is Given.
m = Vρ
Where;
m = Mass
ρ = Mass concentration
V = Volume
Let’s solve an example;
Find the mass when the mass concentration is 12 kg/dm³ with a volume of 7 dm³.
This implies that;
ρ = Mass concentration = 12
V = Volume = 7
m = Vρ
m = 12 x 7
m = 84
Therefore, the mass concentration is 84 kg. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9201937317848206, "perplexity": 3660.4616474343516}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684226.55/warc/CC-MAIN-20191018154409-20191018181909-00314.warc.gz"} |
http://math.stackexchange.com/questions/30687/the-square-roots-of-different-primes-are-linearly-independent-over-the-field-of | # The square roots of different primes are linearly independent over the field of rationals
I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals.
I've tried to solve the problem using elementary algebra and also using the theory of field extensions, without success. To prove linear independence of two primes is easy but then my problems arise. I would be very thankful for an answer to this question.
-
I need the square roots of prime numbers – user8465 Apr 3 '11 at 17:45
Also, see this: qchu.wordpress.com/2009/07/02/… – Ehsan M. Kermani Apr 3 '11 at 18:14
This also comes up in T..'s answer here: math.stackexchange.com/questions/6244/… – Jonas Meyer Apr 3 '11 at 18:41
@J.M.: after Yuval's hint to the mathforum I'd like to mention, that the text of the question is 100% identical with that of the mathforum of 1996, and the fact that neither its reference was given nor anything about the existing answers there was mentioned I assume a) this is not a real question (also there was no followup interaction of "user8465") , and (see the recent meta thread on spam) b) maybe not even a real person asking but possibly an automated transfer of a somehow mathematically sounding text. Maybe that method of spam has been refined recently... – Gottfried Helms Oct 11 '11 at 9:16
Here is a simple proof from one of my old sci.math posts, followed by reviews of a few papers.
THEOREM $\$ Let $\rm\:Q\:$ be a field with $2 \ne 0,\:$ and $\rm\ L = Q(S)\$ be an extension of $\rm\:Q\:$ generated by $\rm\: n\:$ square roots $\rm\ S = \{ \sqrt{a}, \sqrt{b},\ldots \}$ of elts $\rm\ a,b,\:\ldots \in Q\:.\:$ If every nonempty subset of $\rm\:S\:$ has product $\rm\not\in Q\:$ then each successive adjunction $\rm\ Q(\sqrt{a}),\ Q(\sqrt{a},\sqrt{b}),\:\ldots$ doubles degree over $\rm Q,\:$ so, in total, $\rm\: [L:Q] = 2^n\:.\:$ So the $\rm 2^n$ subproducts of the product of $\rm\:S\:$ are a basis of $\rm\:L\:$ over $\rm\:Q\:.$
Proof $\$ By induction on the tower height $\rm\:n =$ number of root adjunctions. The Lemma below implies $\rm\ [1, \sqrt{a}\:]\ [1, \sqrt{b}\:]\ =\ [1, \sqrt{a}, \sqrt{b}, \sqrt{ab}\:]\$ is a $\rm\:Q$-vector space basis of $\rm\: Q(\sqrt{a}, \sqrt{b})\$ iff $\ 1\$ is the only basis element in $\rm\:Q\:.\:$ We must lift this to $\rm\: n > 2:\ [1, \sqrt{a}\:]\ [1, \sqrt{b}\:]\ [1, \sqrt{c}\:]\:\cdots$ ($\rm 2^n\:$ elts)
$\rm n = 1:\ L = Q(\sqrt{a})\$ so $\rm\:[L:Q] = 2,\:$ since $\rm\:\sqrt{a}\not\in Q\:$ by hypothesis.
$\rm n > 1:\ L = K(\sqrt{a},\sqrt{b})\:,\ K\$ of height $\rm\:n-2\:.\:$ By induction $\rm\:[K:Q]\ =\ 2^{\:n-2}\$ so we need only show $\rm\: [L:K] = 4,\:$ since then $\rm\:[L:Q] = [L:K]\ [K:Q] = 4\cdot 2^{n-2} = 2^n\:.\:$ The lemma below shows $\rm\:[L:K] = 4\:$ if $\rm\ r = \sqrt{a},\ \sqrt{b},\ \sqrt{a\:b}\$ all $\rm\not\in K,\:$ true by induction on $\rm\:K(r)\:$ of height $\rm\:n-1\:$ shows $\rm\:[K(r):K] = 2\:$ $\Rightarrow$ $\rm\:r\not\in K\:.\quad$ QED
LEMMA $\rm\ \ [K(\sqrt{a},\sqrt{b}) : K] = 4\$ if $\rm\ \sqrt{a},\ \sqrt{b},\ \sqrt{a\:b}\$ all $\rm\not\in K\:$ and $\rm\: 2 \ne 0\:$ in $\rm\:K\:.$
Proof $\ \$ Let $\rm\ L = K(\sqrt{b})\:.\:$ $\rm\: [L:K] = 2\:$ by $\rm\:\sqrt{b} \not\in K,\:$ so it suffices to show $\rm\: [L(\sqrt{a}):L] = 2\:.\:$ This fails only if $\rm\:\sqrt{a} \in L = K(\sqrt{b})$ $\:\Rightarrow\:$ $\rm \sqrt{a}\ =\ r + s\ \sqrt{b}\$ for $\rm\ r,s\in K,\:$ which is false, because squaring yields $\rm\:(1):\ \ a\ =\ r^2 + b\ s^2 + 2\:r\:s\ \sqrt{b}\:,\:$ which is contra to hypotheses as follows:
$\rm\qquad\qquad rs \ne 0\ \ \Rightarrow\ \ \sqrt{b}\ \in\ K\ \$ by solving $(1)$ for $\rm\sqrt{b}\:,\:$ using $\rm\:2 \ne 0$
$\rm\qquad\qquad\ s = 0\ \ \Rightarrow\ \ \ \sqrt{a}\ \in\ K\ \$ via $\rm\ \sqrt{a}\ =\ r + s\ \sqrt{b}\ =\ r \in K$
$\rm\qquad\qquad\ r = 0\ \ \Rightarrow\ \ \sqrt{a\:b}\in K\ \$ via $\rm\ \sqrt{a}\ =\ s\ \sqrt{b}\:,\: \$times $\rm\:\sqrt{b}\quad\quad$ QED
In the classical case $\rm\:Q\:$ is the field of rationals and the square roots have radicands being distinct primes. Here it is quite familiar that a product of any nonempty subset of them is irrational since, over a UFD, a product of coprime elements is a square iff each factor is a square (mod units). Hence the classical case satisfies the theorem's hypotheses.
Elementary proofs like that above are often credited to Besicovitch (see below). But I have not seen his paper so I cannot say for sure whether or not Besicovic's proof is essentially the same as above. Finally, see the papers reviewed below for some stronger results.
2,33f 10.0X
Besicovitch, A. S.
On the linear independence of fractional powers of integers.
J. London Math. Soc. 15 (1940). 3-6.
Let $\ a_i = b_i\ p_i,\ i=1,\ldots s\:,\:$ where the $p_i$ are $s$ different primes and the $b_i$ positive integers not divisible by any of them. The author proves by an inductive argument that, if $x_j$ are positive real roots of $x^{n_j} - a_j = 0,\ j=1,...,s ,$ and $P(x_1,...,x_s)$ is a polynomial with rational coefficients and of degree not greater than $n_j - 1$ with respect to $x_j,$ then $P(x_1,...,x_s)$ can vanish only if all its coefficients vanish. $\quad$ Reviewed by W. Feller.
15,404e 10.0X
Mordell, L. J.
On the linear independence of algebraic numbers.
Pacific J. Math. 3 (1953). 625-630.
Let $K$ be an algebraic number field and $x_1,\ldots,x_s$ roots of the equations $\ x_i^{n_i} = a_i\ (i=1,2,...,s)$ and suppose that (1) $K$ and all $x_i$ are real, or (2) $K$ includes all the $n_i$ th roots of unity, i.e. $K(x_i)$ is a Kummer field. The following theorem is proved. A polynomial $P(x_1,...,x_s)$ with coefficients in $K$ and of degrees in $x_i$, less than $n_i$ for $i=1,2,\ldots s$, can vanish only if all its coefficients vanish, provided that the algebraic number field $K$ is such that there exists no relation of the form $\ x_1^{m_1}\ x_2^{m_2}\:\cdots\: x_s^{m_s} = a$, where $a$ is a number in $K$ unless $\ m_i \equiv 0 \mod n_i\ (i=1,2,...,s)$. When $K$ is of the second type, the theorem was proved earlier by Hasse [Klassenkorpertheorie, Marburg, 1933, pp. 187--195] by help of Galois groups. When $K$ is of the first type and $K$ also the rational number field and the $a_i$ integers, the theorem was proved by Besicovitch in an elementary way. The author here uses a proof analogous to that used by Besicovitch [J. London Math. Soc. 15b, 3--6 (1940) these Rev. 2, 33]. $\quad$ Reviewed by H. Bergstrom.
46 #1760 12A99
Siegel, Carl Ludwig
Algebraische Abhaengigkeit von Wurzeln. (German)
Acta Arith. 21 (1972), 59-64.
Two nonzero real numbers are said to be equivalent with respect to a real field $R$ if their ratio belongs to $R$. Each real number $r \ne 0$ determines a class $[r]$ under this equivalence relation, and these classes form a multiplicative abelian group $G$ with identity element $[1]$. If $r_1,\dots,r_h$ are nonzero real numbers such that $r_i^{n_i}\in R$ for some positive integers $n_i\ (i=1,...,h)$, denote by $G(r_1,...,r_h) = G_h$ the subgroup of $G$ generated by $[r_1],\dots,[r_h]$ and by $R(r_1,...,r_h) = R_h$ the algebraic extension field of $R = R_0$ obtained by the adjunction of $r_1,...,r_h$. The central problem considered in this paper is to determine the degree and find a basis of $R_h$ over $R$. Special cases of this problem have been considered earlier by A. S. Besicovitch [J. London Math. Soc. 15 (1940), 3-6; MR 2, 33] and by L. J. Mordell [Pacific J. Math. 3 (1953), 625-630; MR 15, 404]. The principal result of this paper is the following theorem: the degree of $R_h$ with respect to $R_{h-1}$ is equal to the index $j$ of $G_{h-1}$ in $G_h$, and the powers $r_i^t\ (t=0,1,...,j-1)$ form a basis of $R_h$ over $R_{h-1}$. Several interesting applications and examples of this result are discussed. $\quad$ Reviewed by H. S. Butts
-
This is top-notch! – The Chaz 2.0 Apr 3 '11 at 20:59
@BillDubuque: Yes, a truly excellent posting! – paul garrett Dec 22 '11 at 23:56
Thanks! I just came across this theorem in an article related to rigidity of matrices and was at wits end how to prove it, until Qiaochu Yuan gave me a link to this post. – Jalaj Jan 30 '12 at 2:26
I don't understand a small part in the proof, can you please explain it ? a.what is the meaning of the notation $[1,\sqrt(a)][1,\sqrt(b)]$ ? b. I got to "true by induction on $K(r)$", I don't understand the use of the induction hypothesis (I understand it sais $[K:Q]=2^{n-2}$, but thats all I get from the induction hypothesis...). – Belgi Jul 9 '12 at 20:36
@BillDubuque: I read this couple more times and given it some thouht, I agree that $\sqrt{a},\sqrt{b}$ are not in $K$ by the hypothesis that the degree of the tower with hight $n-1$ is $2^{n-1}$ but I can't figure why $\sqrt{ab}$ is not in $K$ (it is not clear by the induction hypothesis since $\sqrt{ab}$ is not adjoined in any step...). I would be greatfull if you can explain this part of the proof, it is very interesting! – Belgi Jul 10 '12 at 10:31
Iurie Boreico presents several Olympiad-style proofs of this fact in the Harvard College Mathematics Review. I give a somewhat more sophisticated proof in this blog post.
The source of the sophistication is interesting. For any particular finite set of primes, there is a completely elementary proof which is found by finding a suitable prime witness $q$ relative to which all but one of the primes is a quadratic residue. But in the above I use quadratic reciprocity and Dirichlet's theorem to show that $q$ always exists in general. (I am actually not sure if Dirichlet's theorem is necessary here.)
-
The first link seems to be broken now. Can anyone provide a copy of it? – Leullame Apr 4 at 23:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9527263045310974, "perplexity": 225.8021596772622}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042986148.56/warc/CC-MAIN-20150728002306-00288-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/sequence-of-ratios-of-primes-and-integers.160831/ | # Sequence of ratios of primes and integers
1. Mar 15, 2007
### Dragonfall
I am fairly certain that $$\frac{n}{p_n}$$ is not monotone for any n, but I can't give a proof of it without assuming something at least as strong as the twin prime conjecture. I was wondering if anyone has some advice to prove this using known methods?
2. Mar 15, 2007
### mathman
Statement is confusing - what is variable (not n from what you said)?
3. Mar 15, 2007
### Dragonfall
The variable is n. I mean no tail of the sequence is monotone.
4. Mar 20, 2007
### Dragonfall
Any ideas at all? I'm drawing dead here.
5. Mar 21, 2007
### CRGreathouse
I'm still not sure what you mean, exactly.
6. Mar 21, 2007
### Dragonfall
The tail of a sequence $$a_n$$ is the subsequence $$(a_n)_{n>N}$$ for some N. A sequence may not be monotone for the first N numbers, but the tail of the sequence might be monotone. If $$\frac{n}{p_n}$$ is eventually monotone, then the alternating series test shows that $$\sum_n(-1)^n\frac{n}{p_n}$$ converges. I believe that the series converges, but I don't think the sequence is eventually monotone, mainly because I think the twin prime conjecture is true.
Last edited: Mar 21, 2007
7. Mar 22, 2007
### Gib Z
I still have no idea what you mean, but $$\sum_{n=1}^{\infty} \frac{n}{p_n}$$ diverges if that helps.
8. Mar 22, 2007
### Dragonfall
Ok I don't know how I can possibly make it clearer. Look at this sequence:
2, 3, 5, 3, 5, 6, 7, 6, 4, 2, 1/11, 1/12, 1/13, 1/14, 1/15, 1/16, 1/17, 1/18, ...
This sequence is monotone decreasing from the 11th entry onwards. IS THE SAME TRUE FOR $$\frac{n}{p_n}$$? I don't think so, but it is not obvious either way.
9. Mar 22, 2007
### tehno
10. Mar 22, 2007
### Dragonfall
All I can conclude from PNT is that $$\lim\frac{n}{p_n}=0$$.
11. Mar 22, 2007
### tehno
Dragonfall,I just realized what was your question in a first place...
I think it's darn difficult to prove that!
Need to know of distribution of primes for every finite segment [n,n+k] of natural numbers.Looks intractible at first glance.Sorry.
12. Mar 22, 2007
No, it doesn't.
13. Mar 22, 2007
### Moo Of Doom
Why not? $\lim_{n\to \infty}\frac{n}{p_n}=0$ by the PNT, so wouldn't eventual monotonicity be sufficient for convergence?
Anyway, let's just see what we have.
No tail is monotone if for all natural numbers N, there is an n>N such that
$$\frac{n}{p_n} < \frac{n+1}{p_{n+1}}$$.
This is equivalent to
$$np_{n+1}<np_n+p_n$$
$$p_{n+1}-p_n<\frac{p_n}{n}$$
You don't need the twin primes conjecture here. It suffices to have this lemma:
There exists a natural number k such that there are infinitely many consecutive primes whose difference is less than k.
Even this is stronger than you need, but much, much weaker than the twin primes conjecture.
Last edited: Mar 22, 2007
14. Mar 22, 2007
My deepest apologies, I somehow had forgotten the correct statement of the alternating series test.
Man do I feel stupid.
15. Mar 22, 2007
### Dragonfall
Thanks, and you're right, something like "there exists infinitely many n such that $$p_{n+1}-p_n<\log n$$" is approximately what I need.
16. Mar 23, 2007
### Moo Of Doom
Don't feel stupid. Stuff like that happens to me all the time.
Anyway, I did some looking, and I found this paper, which states that if the Elliot-Halberstam Conjecture is true, then there are infinitely many consecutive primes differing by less than 16.
I can't seem to find any similar results that don't depend on unproven conjectures, so proof might not be easy.
A proven result that might be useful is that ${\lim \inf}_{n \to \infty} \frac{p_{n+1}-p_n}{\log{p_n}}=0$ (found in the same paper).
17. Mar 23, 2007
### Dragonfall
I'm reading through it now, thanks. An immediate corollary is that $$\lim\inf\sqrt{p_{n+1}}-\sqrt{p_n}=0$$, which brings us infinitesimally closer to Andrica's conjecture (I haven't had time to think it through, it's just the first thing that popped into my head).
18. Mar 24, 2007
### Moo Of Doom
After a bit of tinkering and smoothing out some holes in the logic, I've come up with a proof of your conjecture using ${\lim \inf}_{n \to \infty} \frac{p_{n+1}-p_n}{\log{p_n}}=0$ as a lemma. I'll post it if you want to see.
19. Mar 24, 2007
### Dragonfall
I'll try and figure it out myself. If I get stuck maybe I'll crack and ask for your proof.
20. Mar 24, 2007
### Moo Of Doom
Fair enough. That's why I asked.
If you do manage to prove it, I'd love to see your proof. Please do post it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9203450083732605, "perplexity": 765.7679988772184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718311.12/warc/CC-MAIN-20161020183838-00313-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/73223/left-aligned-multi-line-equations-and-newenvironment/85702 | # Left aligned multi line equations and \newenvironment
I'm building a table of integrals, for my own use. I need simple things: equations must be left aligned, sometimes I need alignment on the equal symbol and to save space sometimes I need 2 equations on the same line.
\begin{fleqn}[0pt]
\setlength{\jot}{6pt}
\begin{align*}
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F(x) + C \\
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F(x) + C \\
&\int f(x) dx = F(x) + C &&\int f(x) dx = F(x) + C \\
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F_1(x) + C = \\
\end{align*}
\end{fleqn}
First I'm not satisfied, I can't set the point of alignment with 2 equations in one line and I can't align the equations on the equal sign. Most environment (tabular, tabbed, split,...) do only one job and when I try to mix them I always get some errors.
Second I would like to create a new environment with
\newenvironment{mathtable}
{ \begin{fleqn}[0pt] \setlength{\jot}{6pt} \begin{align*} }
{ \end{align*} \end{fleqn} }
But it show me an error, it seems I can't use align inside newenvironment. Did a lot of research, used the environ package, used \csname and so on, but I don't know how to build this environment.
-
The amsmath documentation documents that in an environment definition you need to use the \align \endalign forms not \begin{align} \end{align} – David Carlisle Sep 19 '12 at 16:43
But then I have a generic error on \begin{fleqn}... – ColOfAbRiX Sep 20 '12 at 10:39
it's possible that the aligned environment of amsmath might be helpful, since it's meant to be embedded within another environment. – barbara beeton Sep 24 '12 at 14:26
There is no fleqn environment. – egreg Oct 13 '12 at 15:39
@egreg: There is a fleqn environment with \usepackage{nccmath}. – Mafra Dec 6 '12 at 0:00
You give too little information for giving advice about the first point.
For what concerns the second point, here's how you can define your mathtable environment:
\documentclass{article}
\usepackage{nccmath}
\newenvironment{mathtable}
{\fleqn[0pt] \setlength{\jot}{6pt} \csname align*\endcsname}
{\csname endalign*\endcsname\endfleqn}
\begin{document}
\begin{mathtable}
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F(x) + C \\
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F(x) + C \\
&\int f(x) dx = F(x) + C &&\int f(x) dx = F(x) + C \\
&\int f(x) \ldots \mbox{long expression here} \ldots dx = F_1(x) + C = \\
One can't use \begin{foo} and \end{foo} in the definition of a new environment, when foo is the name of an amsmath alignment environment. So one has to use the forms \foo and \endfoo in their places (or \csname foo*\endcsname and \csname endfoo*\endcsname for the *-variants). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.992461621761322, "perplexity": 3272.277445075071}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375095668.34/warc/CC-MAIN-20150627031815-00099-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/determine-whether-the-sequences-are-increasing.196009/ | Determine whether the sequences are increasing
1. Nov 4, 2007
will_lansing
1. The problem statement, all variables and given/known data
Determine whether the sequences are increasing, decreasing, or not monotonic.
1) an= $$\frac{\sqrt{n+2}}{4n+2}$$
2) an=$$\frac{1}{4n+2}$$
3) an=$$\frac{cosn}{2^{n}}$$
4) an=$$\frac{n-2}{n+2}$$
2. Relevant equations
3. The attempt at a solution
I thought that the number 1 and 2 were decreasing because $$a_{n}$$ $$\geq$$ $$a_{n+1}$$ in both cases
#3 is increasing because $$a_{n}$$$$\leq$$ $$a_{n+1}$$
#4 is monotonic because if n=3 then its 0
for number #1 I compared it to $$a_{n+1}$$=$$\frac{\sqrt{n+3}}{4n+6}$$
and i found that it was smaller than an so it was decreasing
i did this procedure for the rest of them and i still get the wrong answer | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9566724896430969, "perplexity": 788.1137786369492}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720845.92/warc/CC-MAIN-20161020183840-00449-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/142847/summation-of-floor-function | # summation of floor function
I need to calculate $$\sum\limits_{k=0}^{\lfloor(b-a)/a\rfloor}\left\lfloor\frac{a+ka-1}{2} \right\rfloor$$ For example for $a=3$ and $b=7$ we have $$\left\lfloor\frac{2}{2}\right\rfloor+\left\lfloor\frac{5}{2}\right\rfloor=1+2=3$$ Can a general answer be reached?
-
Also posted to MO, mathoverflow.net/questions/96388/summation-of-floor-function, where it will undoubtedly be closed very soon. – Gerry Myerson May 8 '12 at 23:41
You want $$\sum_{a\le ka\le b}\left\lfloor\frac{ka-1}2\right\rfloor=\sum_{k=1}^m\left\lfloor\frac{ka-1}2\right\rfloor\;,$$ where $m=\lfloor b/a\rfloor$. If $a$ is even, $ka-1$ is always odd, and $$\left\lfloor\frac{ka-1}2\right\rfloor=\frac{ka}2-1\;,$$ so \begin{align*} \sum_{k=1}^m\left\lfloor\frac{ka-1}2\right\rfloor&=\sum_{k=1}^m\left(\frac{ka}2-1\right)\\ &=\frac{a}2\sum_{k=1}^mk-m\\ &=\frac{am(m+1)}4-m\;. \end{align*}
If $a$ is odd,
$$\left\lfloor\frac{ka-1}2\right\rfloor=\begin{cases} \frac{ka}2-1,&\text{if }k\text{ is even}\\\\ \frac{ka-1}2=\left(\frac{ka}2-1\right)+\frac12,&\text{if }k\text{ is odd}\;. \end{cases}$$
Let $c$ be the number of odd integers in $\{1,\dots,m\}$; then
\begin{align*} \sum_{k=1}^m\left\lfloor\frac{ka-1}2\right\rfloor&=\frac{c}2+\sum_{k=1}^m\left(\frac{ka}2-1\right)\\ &=\frac{c}2+\frac{am(m+1)}4-m\;. \end{align*}
Finally, $c=\lceil m/2\rceil$, so
$$\sum_{k=1}^m\left\lfloor\frac{ka-1}2\right\rfloor=\begin{cases} \frac{am(m+1)}4-m,&\text{if }a\text{ is even}\\\\ \frac{\lceil m/2\rceil}2+\frac{am(m+1)}4-m,&\text{if }a\text{ is odd}\;. \end{cases}$$
-
Here are some ideas that should enable you to answer the question. We have $$\left\lfloor \frac{x}{2} \right\rfloor = \begin{cases} \frac{x}{2} & \text{if } x \text{ is even}, \\ \frac{x-1}{2} & \text{if } x \text{ is odd}. \end{cases}$$ Therefore $$\sum_{k=0}^{\lfloor (b-a)/a \rfloor} \left\lfloor \frac{a+ka-1}{2} \right\rfloor = \sum_{k=0}^{\lfloor (b-a)/a \rfloor} \frac{a+ka-1}{2} - \sum_{\substack{k=0\\k\text{ odd}}}^{\lfloor (b-a)/a \rfloor} \frac{1}{2}.$$ The first sum you can calculate using a formula. The second sum depends on the number of odd integers in the given range.
-
$S=\sum\limits_{k=0}^{\lfloor(b-a)/a\rfloor}\left\lfloor\dfrac{a+ka-1}{2} \right\rfloor=\sum\limits_{k=1}^{\lfloor\frac{b}{a}\rfloor}\left\lfloor\dfrac{ka-1}{2}\right\rfloor$
If $\quad a=2c\quad$ then
$S=\sum\limits_{k=1}^{\left\lfloor\frac{b}{2c}\right\rfloor}\left\lfloor kc -\dfrac{1}{2}\right\rfloor=\sum\limits_{k=1}^{\left\lfloor\frac{b}{2c}\right\rfloor}(kc -1)=\dfrac{c}{2}\left\lfloor\dfrac{b}{2c}\right\rfloor\left(\left\lfloor\dfrac{b}{2c}\right\rfloor+1\right)-\left\lfloor\dfrac{b}{2c}\right\rfloor$
If $\quad a=2c+1\quad$ then
$S=\sum\limits_{k=1}^{\left\lfloor\frac{b}{2c+1}\right\rfloor}\left\lfloor kc +\dfrac{k-1}{2}\right\rfloor=c\sum\limits_{k=1}^{\left\lfloor\frac{b}{2c+1}\right\rfloor}k +\sum\limits_{k=0}^{\left\lfloor\frac{b}{2c+1}\right\rfloor-1}\left\lfloor\dfrac{k}{2}\right\rfloor=\dfrac{c}{2}\left\lfloor\dfrac{b}{2c+1}\right\rfloor\left(\left\lfloor\dfrac{b}{2c+1}\right\rfloor+1\right)+\left\lfloor\dfrac{\left\lfloor\frac{b}{2c+1}\right\rfloor-1}{2}\right\rfloor\left\lfloor\dfrac{\left\lfloor\frac{b}{2c+1}\right\rfloor}{2}\right\rfloor$
Combine two formular in one, we get:
$\boxed{S=\dfrac{1}{2}\left\lfloor\dfrac{a}{2}\right\rfloor\left\lfloor\dfrac{b}{a}\right\rfloor\left(\left\lfloor\dfrac{b}{a}\right\rfloor+1\right)+\left(\left\lfloor\dfrac{b-a}{2a}\right\rfloor\left\lfloor\dfrac{b}{2a}\right\rfloor+\left\lfloor\dfrac{b}{a}\right\rfloor\right)\left(a-2\left\lfloor\dfrac{a}{2}\right\rfloor\right)-\left\lfloor\dfrac{b}{a}\right\rfloor}$
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.959639847278595, "perplexity": 620.1853019247958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246633512.41/warc/CC-MAIN-20150417045713-00115-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://blog.industrialguide.co.in/2021/02/ohms-law.html | ### Translate
Ohm's law states that
"The current through a conductor between two points is directly proportional to the potential difference or voltage across the two points, and inversely proportional to the resistance between them".
Ohm's law can be expressed as
I = U / R (1)
where
I = current (ampere, A)
U = electrical potential (volts, V)
R = resistance (ohms, Ω)
#### Example - Ohm's law
12 volt battery supplies power to a resistance of 18 ohms. The current in the elctrical circuit can be calculated as
I = (12 volts) / (18 ohm)
= 0.67 ampere
### Equivalent Expressions of Ohm's Law
Ohm's law (1) can also be expressed as
U = R I (2)
or
R = U / I (3)
#### Example - Electric Circuit Resistance
A current of 1 ampere is flowing through a 230 V electric circuit. From the diagram above this indicates resistance
R ≈ 220 Ω
This can alternatively be calculated with Ohm's law
R = (230 V) / (1 A)
= 230 Ω
#### Example - Ohm's Law and Multiples and Submultiples
Currents, voltages and resistances in electric circuits may often be very small or very large - so multiples and submultiples are often used.
The voltage required applied to a 3.3 kΩ resistor to generate a current of 20 mA can be calculated as
U = (3.3 kΩ) (1000 Ω/kΩ) (20 mA) (10-3 A/mA)
= 66 V
#### Electric Resistance Nomogram
The default values in the nomogram above indicates 230 volts, resistance 24 ohm and current 10 amps.
### Power
Electric power can be expressed as
P = U I
= R I2
= U2 / R (4)
where
P = electrical power (watts, W)
#### Example - Power Consumed
The power consumed in the 12V electrical circuit above can be calculated as
P = (12 volts)2 / (18 ohm)
= 8 W
#### Example - Power and Electrical Resistance
100 W electric light bulb is connected to a 230 V supply. The current flowing can be calculated by reorganizing (4) to
I = P / U
= (100 W) / (230 V)
= 0.43 ampere
The resistance can be calculated by reorganizing (4) to
R = U/ P
= (230 V)/ (100 W)
= 529 Ω
#### Electric Power Nomogram
This nomogram can be used to estimate power vs. voltage and ampere.
The default values in the nomogram above indicates 240 volts, resistance 10 amps and power 2.4 kW for DC or single phase AC - and 4 kW for three phase AC. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8948845863342285, "perplexity": 3859.73079559416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334579.46/warc/CC-MAIN-20220925132046-20220925162046-00219.warc.gz"} |
http://www.powertransformerdesign.net/2012/03/constant-voltage-transformer-basics-and.html | ### CONSTANT VOLTAGE TRANSFORMER BASICS AND TUTORIALS
CONSTANT VOLTAGE TRANSFORMER BASIC INFORMATION
What Is a Constant-Voltage Transformer?
A well-known solution for electrical “noise” in industrial plants has been the constant-voltage transformer, or CVT.
The typical components of a CVT are shown in Figure 2.8.2. The magnetic shunt on the central core has the following effects on the core’s reluctance. It reduces the reluctance of the core.
This can be thought of as introducing more resistance in parallel to an existing resistance. The magnetic shunt in the CVT design allows the portion of the core below the magnetic shunt to become saturated while the upper portion of the core remains unsaturated.
This condition occurs because of the presence of the air-gap between the magnetic shunt and the core limbs. Air has a much higher reluctance than the iron core.
Therefore, most of the flux passes through the lower portion of the core, as shown by the thick lines in Figure 2.8.2.
In terms of an electrical analogy, this configuration can be thought of as two resistances of unequal values in parallel.
The smaller resistance carries the larger current, and the larger resistance carries the smaller current.
The CVT is designed such that:
• The lower portion of the central limb is saturated under normal operating conditions, and the secondary and the resonating windings operate in the nonlinear portion of the flux-current curve.
• Because of saturation in the central limb, the voltage in the secondary winding is not linearly related to the voltage in the primary winding.
There is consonance between the resonating winding on the saturated core and the capacitor. This arrangement acts as a tank circuit, drawing power from the primary. This results in sustained, regulated
oscillations at the secondary with the applied line frequency. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8217666149139404, "perplexity": 1226.3199231840576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814249.56/warc/CC-MAIN-20180222180516-20180222200516-00672.warc.gz"} |
https://physics.stackexchange.com/questions/372467/is-the-non-trivial-topology-on-the-torus-reflected-on-the-bloch-sphere | # Is the non-trivial topology on the torus reflected on the Bloch sphere?
Almost every text on topological insulators have the Bloch sphere example of a two level system showing the non triviality of the bundle of an eigenvector over the sphere: we can't define an eigenstate over the whole Bloch sphere, instead of this, we must construct two local trivializations, namely
$$\psi_{-}^{S}(\vec{n}) = \left(\begin{matrix} -\sin\frac{\theta}{2} \\ e^{i\varphi} \cos\frac{\theta}{2} \end{matrix}\right) \quad \psi_{-}^{N}(\vec{n}) = \left(\begin{matrix} - e^{-i\varphi}\sin\frac{\theta}{2} \\ \cos\frac{\theta}{2} \end{matrix}\right)$$
defined in the south and north hemispheres respectively to avoid the obstruction (to Stokes theorem). We can easily compute the Chern number via Berry curvature on the sphere on any of the states. But we are getting the Chern number integrating over the SPHERE, and not over the Brillouin TORUS. Does this result reflect directly the non-triviality of the fiber bundle over the Brillouin TORUS? (I think getting the pullback of the map from the sphere to the torus and using the chain rule may suffice to prove this, but I am not sure it's enough for a proof). Will we get the same result?
Aside: when in the Bloch sphere we choose the trivializations, we are defining a transition function like
$$t_{NS} = e^{i\varphi}$$
that applies to the fiber space, with that definition, when going from the north to the south hemisphere. Is that $\varphi$ the berry phase? Quantitatively or qualitatively?
I'll give you in the following a rather detailed answer, but let me first, shortly state the answers to your questions:
1. The Chern number of the eigenvector bundle over the torus can be evaluated by integration over the sphere, the integrand will indeed be the monopole Berry curvature. However the integration region will not in general be a single sweep of the surface of the sphere, because the map from $$T^2$$ (the Brillouin zone) to $$S^2$$ may wind the sphere several times.
2. The transition functions are rather related to the Chern number. Their winding number, i.e., the number of times they wind the equator is equal to the Chern number.
3. Concerning the question paused in the title: The topology of all principal $$U(1)$$ bundles over the$$2$$-torus $$T^2$$ is determined solely by the Chern number. (This result is not general ; for example it is not true for the torus $$T^3$$ because it is of a higher dimension)
Details:
The Berry phase of a nondegenerate state is a holonomy of a principal $$U(1)$$ bundle over the parameter space $$M$$. There can be many topologically inequivalent $$U(1)$$ bundles over a given parameter space corresponding to inequivalent quantum systems. Thus the Berry phase allows a classification of parametrized quantum systems based upon the classification of principal bundles. This point of view was noticed by: Bohm, Boya, Mostafazadeh and Rudolph.
The classification theorem of principal bundles asserts the existence of a universal principal bundle $$U(1)\rightarrow\eta\rightarrow B$$, such that any principal $$U(1)$$ bundle $$\lambda$$ over the parameter space is the pullback of which under some map $$f$$. (Please see Nash and Sen for a more detailed explanation of the classification theory of principal bundles.)
$$\begin{array}{ccc} \lambda& \overset{f^*}\leftarrow & \eta\\ \downarrow & & \downarrow \\ \mathcal{M} & \overset{f}\rightarrow & B \end{array}$$
The classification theory of principal bundles specifies a base space (the classifying space) and a total space of the universal bundle depending only on the structure group (in our case U(1)). In the case of a nondegenerate state of a Hamiltonian unconstrained by any anti-unitary symmetry, the classifying space is known to be the infinite dimensional complex projective space $$B = \mathbb{C}P^{\infty}$$. (For a good account of complex projective spaces , please see chapter 4 in Bengtsson and Życzkowski . Also, the following exposition of John Baez of classifying spaces is very useful and contains some more explanation of the infinite dimensional case $$\mathbb{C}P^{\infty}$$ .
$$\mathbb{C}P(\infty)$$ is the space of all one dimensional projectors, and the map $$f$$ from the parameter space to the universal base space is:
$$P: \mathcal{M}\overset{P}\rightarrow \mathbb{C}P^{\infty}$$ $$\mathcal{M} \ni m \mapsto P(m) = |u(m)\rangle\langle u(m)|\in \mathbb{C}P^{\infty}$$
Where $$|u(m)\rangle$$ is the state of the system. (The bundle $$\lambda$$ is sometimes called the Berry bundle, and when the state is an eigenstate of a Hamiltonian, Synonyms: the eigenbundle or the spectral bundle).
The construction of the Berry phase stems from the existence of a universal $$U(1)$$ connection over the infinite dimensional projective space whose curvature is the Fubini-Study form:
$$dA = \frac{1}{2i}\mathrm{Tr} (P dP \wedge dP)$$
The pulled back Berry connection $$P^*(A) = A(m)$$ is the Berry connection whose holonomy is the Berry phase on $$\mathcal{M}$$ and the integral of its curvature on $$M$$ is the first Chen class of $$\lambda$$.
The infinite dimensional projective space $$\mathbb{C}P^{\infty}$$ is the direct limit of a series of inclusions:
$$\mathbb{C}P^{1} \subset \mathbb{C}P^{2} . . . \subset \mathbb{C}P^{\infty}$$
In the case of when the parameter space is two dimensional, it is sufficient to approximate the classifying space by its first component namely $$\mathbb{C}P^{1} = S^2$$ and consider maps to the space of one dimensional projectors in two dimensions namely $$S^2$$, and consider the bundle map:
$$\begin{array}{ccc} \lambda& \overset{P^*}\leftarrow & \eta\\ \downarrow & & \downarrow \\ \mathcal{M} & \overset{P}\rightarrow & S^2 \end{array}$$
(Please see for example, Viennot, for a more detailed expposition of the finite dimensional approximations of the classifying spaces) Here, it is very easy to write the formula for the one dimensional projector map:
$$P(m) = \frac{1}{2} \begin{bmatrix} 1-\cos \theta(m) & \sin \theta(m) e^{i\phi(m)}\\ \sin \theta(m) e^{-i\phi(m)} & 1+\cos \theta(m) \end{bmatrix}$$ This projector gives rise to the pull back to $$\mathcal{M}$$ of well-known magnetic monopole Berry connection:
$$A_{\pm} = \frac{1\mp\cos \theta(m) }{\sin \theta(m) } d\phi(m)$$
whose Berry curvature is proportional to the area element of the sphere
$$F = \frac{1}{2} \sin \theta(m) d\theta(m) d\phi(m)$$
The integration is performed along a path in the manifold $$\mathcal{M}$$, thus the Berry phase corresponding to the path $$\Gamma_M$$ is given by:
$$\phi_B = \int_{\Gamma_M} A(m)$$
It can be pulled back to the two sphere (by change of the integration variable), but in this case we need to integrate on the image of the path $$\Gamma = P(\Gamma_M)$$:
$$\phi_B = \int_{\Gamma} A$$
The same is true for the Chern class:
$$c = \int_{\mathcal{M}} F(m) = \int_{P(\mathcal{M})} F$$
The integration region may wind the two sphere several times and the Chern number will be equal to a multiple of the charge of the monopole.
For the second question, we observe that:
\begin{align*} \int_{P(\mathcal{M})} F & = \int_{P(\mathcal{M}) \cap S^1} \left (A_{+}-A_{-}\right ) &=\int_{P(\mathcal{M}) \cap S^1} d \phi &= \frac{1}{i}\int_{P(\mathcal{M}) \cap S^1} g^{-1}d g \end{align*} Where $$S^1$$ is the equator and $$g= e^{i \phi}$$. The last term is the winding number of the mapping $$g$$. It is a one dimensional Wess-Zumino-Witten term.
• Thanks for such a complete answer. I think I need to get deeper in classification theory to fully understand it, however, let me see if I get the general idea. We have a non-trivial topology if the map from the torus covers the whole sphere at least one time. If we integrate over the sphere, the result is the same as if that map covered only one time the sphere completely (because this is the integration surface). So it shows if there is non-trivial topology, but the Chern number computed does not match in general. I see it now. – vbarcelo Dec 13 '17 at 12:16
• ... So, can it, the Chern number, be viewed as the times we need to apply the transition function to cover the whole manifold? And it is the times we add the corresponding Berry phase to the wave-function? And, for the last eq. you wrote, I think there is a missing i in the last term. – vbarcelo Dec 13 '17 at 12:22
• I think that your statements are correct, but let me emphasize: In order to find the Chern number for a system parametrized by the torus, we need to perform the integration on the torus. We can make a transformation to the sphere, because due to the classification theorem, the integrand will always have the form of the monopole curvature on the sphere. But in this case we must take into account the winding of the map over the sphere; once we do so, we will obtain the same Chern number that we would have obtained by integration on the torus. – David Bar Moshe Dec 14 '17 at 8:32
• cont. All we did is a variable transformation that does not affect the result. We can compute the Chern number also, by computing the integral of the transition functions over the equator; here we must be also careful to the number of windings. There is no need to add the Berry phase to the wave functions (and it will not change the result, since the Chern number is gauge invariant). Thank you for the correction. – David Bar Moshe Dec 14 '17 at 8:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 47, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9845688939094543, "perplexity": 223.13291671688546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657151197.83/warc/CC-MAIN-20200714181325-20200714211325-00192.warc.gz"} |
https://arxiv.org/abs/math/0612469 | math
(what is this?)
(what is this?)
# Title: G_2 and the "Rolling Distribution"
Authors: Gil Bor (CIMAT), Richard Montgomery (University of California, Santa Cruz)
Abstract: Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group SO_3 with the two-sphere and its distribution enjoys an obvious symmetry group; the product of two SO_3's, one for each sphere. But if the ratio of radii of the spheres is 1:3 and if the distribution is lifted to the universal cover S^3 \times S^2 of M, then the symmetry group becomes much larger: the split real form of the Lie group G_2. This fact goes back to Cartan in a sense, and can be found in a paper by Bryant and Hsu. We prove this fact through two explicit constructions, relying on the theory of roots and weights for the Lie algebra of G_2, and on its 7-dimensional representation.
Comments: 27 pages. four figures Subjects: Differential Geometry (math.DG); Optimization and Control (math.OC) Cite as: arXiv:math/0612469 [math.DG] (or arXiv:math/0612469v1 [math.DG] for this version)
## Submission history
From: Richard Montgomery [view email]
[v1] Mon, 18 Dec 2006 05:00:15 GMT (49kb) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.834582507610321, "perplexity": 1409.084251107314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424909.50/warc/CC-MAIN-20170724182233-20170724202233-00139.warc.gz"} |
http://www.math.gatech.edu/seminars-colloquia/series/pde-seminar/rafael-de-la-llave-20121120 | Quasi-periodic solutions for some ill-posed Hamiltonian evolution equations
Series:
PDE Seminar
Tuesday, November 20, 2012 - 15:05
1 hour (actually 50 minutes)
Location:
Skiles 006
,
Georgia Tech
Organizer:
We prove an a-posteriori KAM theorem which applies to some ill-posed Hamiltonian equations. We show that given an approximate solution of an invariance equation which also satisfies some non-degeneracy conditions, there is a true solution nearby. Furthermore, the solution is "whiskered" in the sense that it has stable and unstable directions. We do not assume that the equation defines an evolution equation. Some examples are the Boussinesq equation (and system) and the elliptic equations in cylindrical domains. This is joint work with Y. Sire. Related work with E. Fontich and Y. Sire. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9805160760879517, "perplexity": 1170.8210396602296}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823255.12/warc/CC-MAIN-20171019065335-20171019085335-00526.warc.gz"} |
http://krystalguo.com/?tag=linear-algebra | # Why are Hermitian matrices diagonalizable?
“Since we are working with a Hermitian matrix, we may take an eigenbasis of the space …”
“Wait, sorry, why are Hermitian matrices diagonalizable, again?”
“Umm … it’s not quick to explain.”
This exchange happens often when I give talks about spectra of graphs and digraphs in Bojan’s graph theory meeting. I know several ways to prove that Hermitian matrices are diagonalizable, but I couldn’t think of a simple, succinct statement which gives the right intuitive thing. So, in this post, I’d like to arrive at such a one-liner.
### Definitions
We consider $n\times n$ matrices with entries in $\mathbb{C}$. A matrix $H$ is Hermitian if $H^* = H$, where $H^* = \bar{H}^T$ is the conjugate transpose of $H$. A matrix $H$ is diagonalizable if $H$ is similar to a diagonal matrix: i.e. there exist invertible matrix $P$ such that $P^{-1} H P = D$ where $D$ is a diagonal matrix.
An eigenvalue of $H$ is $\lambda$ such that there exist $v \in \mathbb{C}^n$ such that $$Hv = \lambda v$$ and $v$ is said to be an eigenvector of $H$. The characteristic polynomial of $H$ is $$\phi(H,t) = \det(tI – H).$$ If $\lambda$ is a number such that $\phi(H,\lambda) = 0$, then $\lambda I – H$ has a non-trivial kernel and so there exists a vector $v$ such that $(\lambda I – H)v = 0$. Thus, every root of $\phi(H,t)$ is an eigenvalue of $H$. However, the roots of the characteristic polynomial are not the same as the multiset of eigenvalues because there is a question of multiplicity.
### Geometric vs algebraic multiplicities
The geometric multiplicity, $m_g(\lambda)$, of an eigenvalue $\lambda$ of $H$ is the dimension of the subspace of $\mathbb{C}^n$ generated by all eigenvectors $H$ with eigenvalue $\lambda$ (this space is called the eigenspace of $\lambda$). The algebraic multiplicity, $m_a(\lambda)$, is the multiplicity of $\lambda$ as a root of $\phi(H,t)$.
Proposition. A Hermitian matrix $H$ is diagonalizable if and only if $m_a(\lambda) = m_g(\lambda)$ for each eigenvalue $\lambda$ of $H$.
Proof. Suppose $H$ is a $n\times n$ Hermitian matrix.
“$\Leftarrow$” It is easy to see that the characteristic polynomial has degree $n$ and hence $n$ roots. Since the algebraic and geometric multiplicities agree, we see that $H$ has $n$ orthonormal eigenvectors (details left as an exercise) which can be used to form the columns of a matrix $P$. Then $P^*P = I$ and $P^*HP$ is a diagonal matrix with the eigenvalues of $H$ on the diagonal.
“$\Rightarrow$” Let $P$ be a matrix such that $P^{-1}HP = D$, where $D$ is a diagonal matrix. Then $$HP =PD$$ and we may consider the $j$th column of both sides. On the left side, we get $Hv$ where $v$ is the $j$th column of $P$. On the right side, we get $D_{jj}v$, and so $v$ is an eigenvector of $H$ with eigenvalue $D_{jj}$. Since $H$ and $D$ are similar, $$\phi(H,t) = \phi(D,t) = \prod_{j = 1}^n (t-D_{jj}).$$ Thus the multiset $\{D_{jj}\}_{j=1}^n$ is the set of eigenvalues of $H$ with both geometric and with algebraic multiplicities. $\Box$
### Diagonalizability
We will now show that Hermitian matrices are diagonalizable by showing that every eigenvalue has the same algebraic and geometric multiplicities.
Theorem. If $H$ is a Hermitian matrix with eigenvalue $\lambda$, then $m_g(\lambda) = m_a(\lambda)$.
Proof. We take $H$ to be a $n\times n$ Hermitian matrix and $\lambda$ an eigenvalue of $H$. We proceed by induction on $n$.
If $m_a(\lambda) = 1$, we are done, since there must be exactly one eigenvector of $\lambda$. We may assume $a = m_a(\lambda) >1$. Let $x$ be an eigenvector of $H$ such that $Hx = \lambda x$. We may assume that $x$ is normalized, i.e. $x^*x = 1$. We may extend $\{x\}$ to an orthonormal basis of $\mathbb{C}^n$, say $\{x, v_2, \ldots, v_n\}$.
Let $V= \langle v_2, \ldots, v_n \rangle = \langle x \rangle ^{\perp}$. We may consider $\mathbb{C}^n$ as the direct sum $\langle x \rangle \oplus V$. Let $v \in V$. Observe that $$x^*(Hv) = x^*H^*v= (Hx)^*v = (\lambda x)^*v = \bar{\lambda} x^*v = 0.$$ Thus $Hv \in V$ and $V$ is a $H$-invariant subspace of $\mathbb{C}^n$.
Let $P$ be the unitary matrix with $\{x, v_2, \ldots, v_n\}$ as its columns. The above gives that $$P^* HP = \left( \begin{array}{cccc} \lambda & 0 & \cdots & 0 \\ 0 & & & \\ \vdots & & B & \\0 &&& \end{array} \right)$$ and $B$ is Hermitian since the left side is. We see that $$\phi(H,t) = \phi(P^* HP, t) = (t-\lambda)\phi(B,t).$$ Thus, $\lambda$ is an eigenvalue of $B$ with algebraic and geometric multiplicity $a-1$ (by induction) and $B$ has pair-wise orthogonal eigenvectors $x’_2, \ldots, x’_a$ of $\lambda$. For $j = 2, \ldots, a$, let $x_j = P\left( \begin{array}{c} 0 \\ x’_j \end{array} \right) P^*$. It is easy to see that $x, x_2, \ldots, x_a$ are pair-wise orthogonal eigenvectors of $H$ with eigenvalue $\lambda$, which proves the theorem. $\Box$
### The Take-away
There are many (mostly equivalent ways) to show this; we could have used induction to prove $H$ is diagonalizable, without consider geometric vs algebraic multiplicities, we could have proved the decomposition into Jordan blocks, or we could have proven the spectral decomposition theorem.
The crux of the proof is that, when $H$ is Hermitian, the vector space $W^{\perp}$ is $H$-invariant when $W$ is. In our proof, this allowed us to, colloquially speaking, keep pulling eigenvectors out of $\mathbb{C}^n$. In general, given an $H$-invariant subspace $W$ of $\mathbb{C}^n$, we can consider the action of $H$ (by multiplication on the left) on $W$ and find the minimal polynomial of $H$ over $W$. If $\psi_1$ and $\psi_2$ are the minimal polynomials of $H$ over $W$ and $W^{\perp}$, respectively, then $\phi(H,t) = \psi_1 \psi_2$.
Intuitively, a Hermitian matrix $H$ is diagonalizable because we can break $\mathbb{C}^n$ into $H$-invariant, pairwise orthogonal, subspaces and diagonalize $H$ over each subspace.
To see that this a property that is not true of all matrices, consider the following matrix: $$N = \left(\begin{array}{cccc} 0 & 1 & 1 & 1 \\ 0& 0& 1 & 1 \\ 0& 0 & 0 & 1 \\ 0&0&0&0 \end{array} \right).$$ For graph theorists, this is the adjacency matrix of a transitive tournament on $4$ vertices. It is also a nilpotent matrix; that is, $N^4 = 0$. The characteristic polynomial of $N$ is $t^4$ and $N$ has $0$ as an eigenvalue with algebraic multiplicity $4$. However, $N$ has only one linearly independent eigenvector $e_1$ (the elementary basis vector, $(1\,\, 0\,\, 0\,\, 0)^T$).
Here, $\langle e_1 \rangle$ is an eigenspace of $N$ and hence $N$-invariant. Observe that $e_2 \in \langle e_1 \rangle^{\perp}$ but $Ne_2 = e_1$ and so $\langle e_1 \rangle^{\perp}$ is certainly not $N$-invariant.
###### References
Roman, Steven. Advanced Linear Algebra. (Springer Graduate Texts in Mathematics, Vol. 135)
Prasolov, V. V. Problems and Theorems in Linear Algebra. (AMS Translations of Mathematical Monographs, Vol. 134) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9952746629714966, "perplexity": 73.81303853165475}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320593.91/warc/CC-MAIN-20170625221343-20170626001343-00362.warc.gz"} |
https://mathoverflow.net/questions/295516/do-2n-1-equiv1-pmod-n-and-n-1-2-prime-imply-n-prime | # Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?
Do $2^{n-1}\equiv1\pmod n$ and $(n-1)/2$ prime imply $n$ prime?
Equivalently: Does $n$ being a Fermat pseudoprimes to base 2 (OEIS A001567) imply that $(n-1)/2$ is composite? That holds for all $n<2^{64}$, based on Jan Feitsma's table.
Motivation is a simplification in the search of safe primes as used in cryptography.
Progress so far: Pocklington's theorem states that if $q>\sqrt n-1$ is a prime dividing $n-1$, and $a^{n-1}\equiv1\pmod n$, then $n$ is prime or $\gcd(a^{(n-1)/q},n)\ne1$. Applying this for $a=2$, it comes that any counterexample $n$ to the propositions would be a multiple of $3$.
The question then boils down to: do $6k+1$ prime imply $4^{6k+1}\not\equiv1\pmod{4k+1}$ ?
We have $2^{\varphi(n)}\equiv 1 \pmod n$, thus $2^k\equiv 1$, where $k=\text{gcd}(\varphi(n),n-1)$. Note that $k$ is even, since both $n-1$ and $\varphi(n)$ are even. If $n=2p+1$ for prime $p=(n-1)/2$, then even divisor of $n-1$ is either 2 or $2p$. If $k=2$, we get $n|2^2-1=3$; if $k=2p=n-1$, we get $\varphi(n)\geqslant n-1$, thus $n$ is prime.
• Introducing $\varphi(n)$ was nice! I made a detailed version there. – fgrieu Mar 18 '18 at 20:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9952758550643921, "perplexity": 95.66888106209943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739046.14/warc/CC-MAIN-20200813132415-20200813162415-00212.warc.gz"} |
https://www.science.gov/topicpages/r/regulated+massive+star.html | #### Sample records for regulated massive star
1. Massive Stars
NASA Astrophysics Data System (ADS)
Livio, Mario; Villaver, Eva
2009-11-01
Participants; Preface Mario Livio and Eva Villaver; 1. High-mass star formation by gravitational collapse of massive cores M. R. Krumholz; 2. Observations of massive star formation N. A. Patel; 3. Massive star formation in the Galactic center D. F. Figer; 4. An X-ray tour of massive star-forming regions with Chandra L. K. Townsley; 5. Massive stars: feedback effects in the local universe M. S. Oey and C. J. Clarke; 6. The initial mass function in clusters B. G. Elmegreen; 7. Massive stars and star clusters in the Antennae galaxies B. C. Whitmore; 8. On the binarity of Eta Carinae T. R. Gull; 9. Parameters and winds of hot massive stars R. P. Kudritzki and M. A. Urbaneja; 10. Unraveling the Galaxy to find the first stars J. Tumlinson; 11. Optically observable zero-age main-sequence O stars N. R. Walborn; 12. Metallicity-dependent Wolf-Raynet winds P. A. Crowther; 13. Eruptive mass loss in very massive stars and Population III stars N. Smith; 14. From progenitor to afterlife R. A. Chevalier; 15. Pair-production supernovae: theory and observation E. Scannapieco; 16. Cosmic infrared background and Population III: an overview A. Kashlinsky.
2. SELF-REGULATED SHOCKS IN MASSIVE STAR BINARY SYSTEMS
SciTech Connect
Parkin, E. R.; Sim, S. A. E-mail: [email protected]
2013-04-20
In an early-type, massive star binary system, X-ray bright shocks result from the powerful collision of stellar winds driven by radiation pressure on spectral line transitions. We examine the influence of the X-rays from the wind-wind collision shocks on the radiative driving of the stellar winds using steady-state models that include a parameterized line force with X-ray ionization dependence. Our primary result is that X-ray radiation from the shocks inhibits wind acceleration and can lead to a lower pre-shock velocity, and a correspondingly lower shocked plasma temperature, yet the intrinsic X-ray luminosity of the shocks, L{sub X}, remains largely unaltered, with the exception of a modest increase at small binary separations. Due to the feedback loop between the ionizing X-rays from the shocks and the wind driving, we term this scenario as self-regulated shocks. This effect is found to greatly increase the range of binary separations at which a wind-photosphere collision is likely to occur in systems where the momenta of the two winds are significantly different. Furthermore, the excessive levels of X-ray ionization close to the shocks completely suppress the line force, and we suggest that this may render radiative braking less effective. Comparisons of model results against observations reveal reasonable agreement in terms of log (L{sub X}/L{sub bol}). The inclusion of self-regulated shocks improves the match for kT values in roughly equal wind momenta systems, but there is a systematic offset for systems with unequal wind momenta (if considered to be a wind-photosphere collision).
3. Outflow Feedback Regulated Massive Star Formation in Parsec-Scale Cluster Forming Clumps
SciTech Connect
Wang, Peng; Li, Zhi-Yun; Abel, Tom; Nakamura, Fumitaka; /Niigata U.
2010-02-15
We investigate massive star formation in turbulent, magnetized, parsec-scale clumps of molecular clouds including protostellar outflow feedback using three dimensional numerical simulations of effective resolution 2048{sup 3}. The calculations are carried out using a block structured adaptive mesh refinement code that solves the ideal MHD equations including self-gravity and implements accreting sink particles. We find that, in the absence of regulation by magnetic fields and outflow feedback, massive stars form readily in a turbulent, moderately condensed clump of {approx} 1,600 M{sub {circle_dot}} (containing {approx} 10{sup 2} initial Jeans masses), along with a cluster of hundreds of lower mass stars. The massive stars are fed at high rates by (1) transient dense filaments produced by large-scale turbulent compression at early times, and (2) by the clump-wide global collapse resulting from turbulence decay at late times. In both cases, the bulk of the massive star's mass is supplied from outside a 0.1 pc-sized 'core' that surrounds the star. In our simulation, the massive star is clump-fed rather than core-fed. The need for large-scale feeding makes the massive star formation prone to regulation by outflow feedback, which directly opposes the feeding processes. The outflows reduce the mass accretion rates onto the massive stars by breaking up the dense filaments that feed the massive star formation at early times, and by collectively slowing down the global collapse that fuel the massive star formation at late times. The latter is aided by a moderate magnetic field of strength in the observed range (corresponding to a dimensionless clump mass-to-flux ratio {lambda} {approx} a few); the field allows the outflow momenta to be deposited more efficiently inside the clump. We conclude that the massive star formation in our simulated turbulent, magnetized, parsec-scale clump is outflow-regulated and clump-fed (ORCF for short). An important implication is that the
4. Massive soliton stars
NASA Technical Reports Server (NTRS)
Chiu, Hong-Yee
1990-01-01
The structure of nontopological solutions of Einstein field equations as proposed by Friedberg, Lee, and Pang (1987) is examined. This analysis incorporates finite temperature effects and pair creation. Quarks are assumed to be the only species that exist in interior of soliton stars. The possibility of primordial creation of soliton stars in the incomplete decay of the degenerate vacuum in early universe is explored. Because of dominance of pair creation inside soliton stars, the luminosity of soliton stars is not determined by its radiative transfer characteristics, and the surface temperature of soliton stars can be the same as its interior temperature. It is possible that soliton stars are intense X-ray radiators at large distances. Soliton stars are nearly 100 percent efficient energy converters, converting the rest energy of baryons entering the interior into radiation. It is possible that a sizable number of baryons may also be trapped inside soliton stars during early epochs of the universe. In addition, if soliton stars exist they could assume the role played by massive black holes in galactic centers.
5. Massive soliton stars
SciTech Connect
Chiu, Hongyee )
1990-05-01
The structure of nontopological solutions of Einstein field equations as proposed by Friedberg, Lee, and Pang (1987) is examined. This analysis incorporates finite temperature effects and pair creation. Quarks are assumed to be the only species that exist in interior of soliton stars. The possibility of primordial creation of soliton stars in the incomplete decay of the degenerate vacuum in early universe is explored. Because of dominance of pair creation inside soliton stars, the luminosity of soliton stars is not determined by its radiative transfer characteristics, and the surface temperature of soliton stars can be the same as its interior temperature. It is possible that soliton stars are intense X-ray radiators at large distances. Soliton stars are nearly 100 percent efficient energy converters, converting the rest energy of baryons entering the interior into radiation. It is possible that a sizable number of baryons may also be trapped inside soliton stars during early epochs of the universe. In addition, if soliton stars exist they could assume the role played by massive black holes in galactic centers. 27 refs.
6. Constraining massive star evolution from massive clusters
NASA Astrophysics Data System (ADS)
Chene, Andre-Nicolas; Herve, Anthony; Martins, Fabrice; Bouret, Jean-Claude; Borissova, Jordanka; Ramirez, Sebastian; Kurtev, Radostin; Kumar, Nanda; Amigo, Pia; Fierro, Celia
2013-06-01
The exact evolution of massive stars is not accurately known at present. The general trend is that stars with masses above 40 - 60 Mo go from O-type stars to H-rich WN stars, and Luminous Blue Variables (?), before turning into H-poor WN stars and finally WC stars. At lower masses, the H-rich WN and LBV phases are replaced by a blue and a red supergiant phases, respectively. However, what are the details of such evolutionary sequences? The study of massive clusters is a golden opportunity to establish this. Indeed, the turn-off mass of massive clusters can be directly translated into the mass, and hence the nature, of the progenitors of their evolved objects contents. So far, only the Arches, Quintuplet, NGC3603, NGC2244 and central clusters have been studied this way. But 6 newly discovered heavily-obscured clusters in the large survey â"VISTA Variables in the Via Lactea" (VVV) have been found to have Wolf-Rayet stars as well as blue and/or red supergiants, together with many main sequence OB stars. This poster presents our efforts to model the massive star components of these clusters using CMFGEN, bringing new blocks to the pavement of massive stellar evolution and more than doubling the number of clusters in which such evolutionary sequence are established.
7. The evolution of massive stars
NASA Technical Reports Server (NTRS)
1982-01-01
The hypotheses underlying theoretical studies of the evolution of massive model stars with and without mass loss are summarized. The evolutionary tracks followed by the models across theoretical Hertzsprung-Russell (HR) diagrams are compared with the observed distribution of B stars in an HR diagram. The pulsational properties of models of massive star are also described.
8. Mass loss of massive stars
NASA Astrophysics Data System (ADS)
Martins, F.
2015-12-01
In this contribution we review the properties of the winds of massive stars. We focus on OB stars, red supergiants, Luminous Blue Variables (LBVs) and Wolf-Rayet stars. For each type of star, we summarize the main wind properties and we give a brief description of the physical mechanism(s) responsible for mass loss.
9. Fragmentation in massive star formation.
PubMed
Beuther, Henrik; Schilke, Peter
2004-02-20
Studies of evolved massive stars indicate that they form in a clustered mode. During the earliest evolutionary stages, these regions are embedded within their natal cores. Here we present high-spatial-resolution interferometric dust continuum observations disentangling the cluster-like structure of a young massive star-forming region. The derived protocluster mass distribution is consistent with the stellar initial mass function. Thus, fragmentation of the initial massive cores may determine the initial mass function and the masses of the final stars. This implies that stars of all masses can form via accretion processes, and coalescence of intermediate-mass protostars appears not to be necessary.
10. Self-similar fragmentation regulated by magnetic fields in a region forming massive stars.
PubMed
Li, Hua-bai; Yuen, Ka Ho; Otto, Frank; Leung, Po Kin; Sridharan, T K; Zhang, Qizhou; Liu, Hauyu; Tang, Ya-Wen; Qiu, Keping
2015-04-23
Most molecular clouds are filamentary or elongated. For those forming low-mass stars (<8 solar masses), the competition between self-gravity and turbulent pressure along the dynamically dominant intercloud magnetic field (10 to 100 parsecs) shapes the clouds to be elongated either perpendicularly or parallel to the fields. A recent study also suggested that on the scales of 0.1 to 0.01 parsecs, such fields are dynamically important within cloud cores forming massive stars (>8 solar masses). But whether the core field morphologies are inherited from the intercloud medium or governed by cloud turbulence is unknown, as is the effect of magnetic fields on cloud fragmentation at scales of 10 to 0.1 parsecs. Here we report magnetic-field maps inferred from polarimetric observations of NGC 6334, a region forming massive stars, on the 100 to 0.01 parsec scale. NGC 6334 hosts young star-forming sites where fields are not severely affected by stellar feedback, and their directions do not change much over the entire scale range. This means that the fields are dynamically important. The ordered fields lead to a self-similar gas fragmentation: at all scales, there exist elongated gas structures nearly perpendicular to the fields. Many gas elongations have density peaks near the ends, which symmetrically pinch the fields. The field strength is proportional to the 0.4th power of the density, which is an indication of anisotropic gas contractions along the field. We conclude that magnetic fields have a crucial role in the fragmentation of NGC 6334.
11. Massive star clusters in galaxies.
PubMed
Harris, William E
2010-02-28
The ensemble of all star clusters in a galaxy constitutes its star cluster system. In this review, the focus of the discussion is on the ability of star clusters, particularly the systems of old massive globular clusters (GCs), to mark the early evolutionary history of galaxies. I review current themes and key findings in GC research, and highlight some of the outstanding questions that are emerging from recent work.
12. Formation of Massive Stars in Massive Young Clusters
NASA Astrophysics Data System (ADS)
Zinnecker, H.
2004-12-01
There are two scenarios for the formation of massive stars: the accretion'' and the coalescence'' scenario. Here we discuss the conditions for coalescence (mergers) to occur in very dense young star clusters. We also ask whether the observed multiplicity of tight massive stars in young clusters is consistent with failed mergers and tidal capture. Finally, we propose some ideas for the origin of many massive stars in the heart of the 30 Doradus cluster and other extragalactic starburst clusters. We believe that all massive star formation is triggered and propose a 4-stage process of massive star birth in dense clusters.
13. Properties of Massive Stars in VVV Clusters
NASA Astrophysics Data System (ADS)
Hervé, A.; Martins, F.; Chené, A.-N.; Bouret, J.-C.; Borrissova, J.
2015-12-01
The evolution of massive stars is only partly understood. Observational constraints can be obtained from the study of massive stars located in young massive clusters. The ESO Public Survey VISTA Variables in the Via Lactea (VVV) discovered several new clusters hosting massive stars (Borrissova et al. [1]). We derive the stellar parameters of all targets as well as surface abundances for a subset of them. For the cluster with the largest number of objects, we establish firmly that the WN and WC stars were initially more massive than the O stars still present in the cluster.
14. Massive Stars in Interactive Binaries
NASA Astrophysics Data System (ADS)
St.-Louis, Nicole; Moffat, Anthony F. J.
Massive stars start their lives above a mass of ~8 time solar, finally exploding after a few million years as core-collapse or pair-production supernovae. Above ~15 solar masses, they also spend most of their lives driving especially strong, hot winds due to their extreme luminosities. All of these aspects dominate the ecology of the Universe, from element enrichment to stirring up and ionizing the interstellar medium. But when they occur in close pairs or groups separated by less than a parsec, the interaction of massive stars can lead to various exotic phenomena which would not be seen if there were no binaries. These depend on the actual separation, and going from wie to close including colliding winds (with non-thermal radio emission and Wolf-Rayet dust spirals), cluster dynamics, X-ray binaries, Roche-lobe overflow (with inverse mass-ratios and rapid spin up), collisions, merging, rejuventation and massive blue stragglers, black-hole formation, runaways and gamma-ray bursts. Also, one wonders whether the fact that a massive star is in a binary affects its parameters compared to its isolated equivalent. These proceedings deal with all of these phenomena, plus binary statistics and determination of general physical properties of massive stars, that would not be possible with their single cousins. The 77 articles published in these proceedings, all based on oral talks, vary from broad revies to the lates developments in the field. About a third of the time was spent in open discussion of all participants, both for ~5 minutes after each talk and 8 half-hour long general dialogues, all audio-recorded, transcribed and only moderately edited to yield a real flavour of the meeting. The candid information in these discussions is sometimes more revealing than the article(s) that preceded them and also provide entertaining reading. The book is suitable for researchers and graduate students interested in stellar astrophysics and in various physical processes involved when
15. PRISM Polarimetry of Massive Stars
NASA Astrophysics Data System (ADS)
Kerkstra, Brennan; Lomax, Jamie R.; Bjorkman, Karen S.; Bjorkman, Jon Eric; Skiff, Brian; Covey, Kevin R.; Wisniewski, John P.
2016-01-01
We present the early results from our long-term, multi-epoch filter polarization survey of massive stars in and around young Galactic clusters. These BVRI polarization data were obtained using the PRISM instrument mounted on the 1.8m Perkins Telescope at Lowell Observatory. We first detail the creation of our new semi-automated polarization data reduction pipeline that we developed to process these data. Next, we present our analysis of the instrumental polarization properties of the PRISM instrument, via observations of polarized and unpolarized standard stars. Finally, we present early results on the total and intrinsic polarization behavior of several isolated, previously suggested classical Be stars, and discuss these results in the context of the larger project.BK acknowledges support from a NSF/REU at the University of Oklahoma. This program was also supported by NSF-AST 11411563, 1412110, and 1412135.
16. GALAXY FORMATION WITH SELF-CONSISTENTLY MODELED STARS AND MASSIVE BLACK HOLES. I. FEEDBACK-REGULATED STAR FORMATION AND BLACK HOLE GROWTH
SciTech Connect
Kim, Ji-hoon; Abel, Tom; Wise, John H.; Alvarez, Marcelo A.
2011-09-01
There is mounting evidence for the coevolution of galaxies and their embedded massive black holes (MBHs) in a hierarchical structure formation paradigm. To tackle the nonlinear processes of galaxy-MBH interaction, we describe a self-consistent numerical framework which incorporates both galaxies and MBHs. The high-resolution adaptive mesh refinement (AMR) code Enzo is modified to model the formation and feedback of molecular clouds at their characteristic scale of 15.2 pc and the accretion of gas onto an MBH. Two major channels of MBH feedback, radiative feedback (X-ray photons followed through full three-dimensional adaptive ray tracing) and mechanical feedback (bipolar jets resolved in high-resolution AMR), are employed. We investigate the coevolution of a 9.2 x 10{sup 11} M{sub sun} galactic halo and its 10{sup 5} M{sub sun} embedded MBH at redshift 3 in a cosmological {Lambda}CDM simulation. The MBH feedback heats the surrounding interstellar medium (ISM) up to 10{sup 6} K through photoionization and Compton heating and locally suppresses star formation in the galactic inner core. The feedback considerably changes the stellar distribution there. This new channel of feedback from a slowly growing MBH is particularly interesting because it is only locally dominant and does not require the heating of gas globally on the disk. The MBH also self-regulates its growth by keeping the surrounding ISM hot for an extended period of time.
17. Galaxy Formation with Self-Consistently Modeled Stars and Massive Black Holes. I: Feedback-Regulated Star Formation and Black Hole Growth
SciTech Connect
Kim, Ji-hoon; Wise, John H.; Alvarez, Marcelo A.; Abel, Tom; /KIPAC, Menlo Park /Stanford U., Phys. Dept.
2011-11-04
There is mounting evidence for the coevolution of galaxies and their embedded massive black holes (MBHs) in a hierarchical structure formation paradigm. To tackle the nonlinear processes of galaxy-MBH interaction, we describe a self-consistent numerical framework which incorporates both galaxies and MBHs. The high-resolution adaptive mesh refinement (AMR) code Enzo is modified to model the formation and feedback of molecular clouds at their characteristic scale of 15.2 pc and the accretion of gas onto an MBH. Two major channels of MBH feedback, radiative feedback (X-ray photons followed through full three-dimensional adaptive ray tracing) and mechanical feedback (bipolar jets resolved in high-resolution AMR), are employed. We investigate the coevolution of a 9.2 x 10{sup 11} M {circle_dot} galactic halo and its 10{sup 5} {circle_dot} M embedded MBH at redshift 3 in a cosmological CDM simulation. The MBH feedback heats the surrounding interstellar medium (ISM) up to 10{sup 6} K through photoionization and Compton heating and locally suppresses star formation in the galactic inner core. The feedback considerably changes the stellar distribution there. This new channel of feedback from a slowly growing MBH is particularly interesting because it is only locally dominant and does not require the heating of gas globally on the disk. The MBH also self-regulates its growth by keeping the surrounding ISM hot for an extended period of time.
18. Physics of Mass Loss in Massive Stars
NASA Astrophysics Data System (ADS)
Puls, Joachim; Sundqvist, Jon O.; Markova, Nevena
2015-01-01
We review potential mass-loss mechanisms in the various evolutionary stages of massive stars, from the well-known line-driven winds of O-stars and BA-supergiants to the less-understood winds of Red Supergiants. We discuss optically thick winds from Wolf-Rayet stars and Very Massive Stars, and the hypothesis of porosity-moderated, continuum-driven mass loss from stars formally exceeding the Eddington limit, which might explain the giant outbursts from Luminous Blue Variables. We finish this review with a glance on the impact of rapid rotation, magnetic fields and small-scale inhomogeneities in line-driven winds.
19. Massive Stars: Input Physics and Stellar Models
NASA Astrophysics Data System (ADS)
El Eid, M. F.; The, L.-S.; Meyer, B. S.
2009-10-01
We present a general overview of the structure and evolution of massive stars of masses ≥12 M ⊙ during their pre-supernova stages. We think it is worth reviewing this topic owing to the crucial role of massive stars in astrophysics, especially in the evolution of galaxies and the universe. We have performed several test computations with the aim to analyze and discuss many physical uncertainties still encountered in massive-star evolution. In particular, we explore the effects of mass loss, convection, rotation, 12C( α, γ)16O reaction and initial metallicity. We also compare and analyze the similarities and differences among various works and ours. Finally, we present useful comments on the nucleosynthesis from massive stars concerning the s-process and the yields for 26Al and 60Fe.
20. How Massive Single Stars End Their Life
NASA Technical Reports Server (NTRS)
Heger, A.; Fryer, C. L.; Woosley, S. E.; Langer, N.; Hartmann, D. H.
2003-01-01
How massive stars die-what sort of explosion and remnant each produces-depends chiefly on the masses of their helium cores and hydrogen envelopes at death. For single stars, stellar winds are the only means of mass loss, and these are a function of the metallicity of the star. We discuss how metallicity, and a simplified prescription for its effect on mass loss, affects the evolution and final fate of massive stars. We map, as a function of mass and metallicity, where black holes and neutron stars are likely to form and where different types of supernovae are produced. Integrating over an initial mass function, we derive the relative populations as a function of metallicity. Provided that single stars rotate rapidly enough at death, we speculate on stellar populations that might produce gamma-ray bursts and jet-driven supernovae.
1. Circumstellar bubble created by two massive stars
NASA Astrophysics Data System (ADS)
Meliani, Z.; van Marle, A. J.; Marcowith, A.
2013-11-01
The massive stars are formed in clusters then numerical models of wind-blown bubble should evolve bubble created by several stars. Aims. We develop a two-dimensional (2D) model of the circumstellar bubble created by two massive stars, a 40 M_{odot} star and a 25 M_{odot} star, and follow its evolution with MPI-AMRVAC hydrodynamics code until the end of the stellar evolution and he supernova explosion of each star. The stars are separated by approximately 16 pc and surrounded by a cold medium with a density of 20 particles per cm3. The simulations showed that the evolution of a wind-blown bubble created by two stars deviates from that of the bubbles around single stars. In particular, once one of the stars has exploded, the bubble is too large for the wind of the remaining star to maintain and the outer shell starts to disintegrate. The lack of thermal pressure inside the bubble also changes the behavior of circumstellar features close to the remaining star. The supernovae are contained inside the bubble, which reflects part of the energy back into the circumstellar medium.
2. Olivier Chesneau's Work on Massive Stars
NASA Astrophysics Data System (ADS)
Millour, F.
2015-12-01
Olivier Chesneau challenged several fields of observational stellar astrophysics with bright ideas and an impressive amount of work to make them real in the span of his career, from his first paper on P Cygni in 2000, up to his last one on V838 Mon in 2014. He was using all the so-called high-angular resolution techniques since it helped his science to be made, namely study in details the inner structure of the environments around stars, be it small mass (AGBs), more massive (supergiant stars), or explosives (Novae). I will focus here on his work on massive stars.
3. Triggered star formation in the environment of young massive stars
NASA Astrophysics Data System (ADS)
Gritschneder, Matthias; Naab, T.; Heitsch, F.; Burkert, A.
Recent observations with the Spitzer Space Telescope show clear evidence that star formation takes place in the surrounding of young massive O-type stars, which are shaping their environment due to their powerful radiation and stellar winds. In this work we investigate the effect of ionising radiation of massive stars on the ambient interstellar medium (ISM): In particular we want to examine whether the UV-radiation of O-type stars can lead to the observed pillar-like structures and can trigger star formation. We developed a new implementation, based on a parallel Smooth Particle Hydrodynamics code (VINE), that allows an efficient treatment of the effect of ionising radiation from massive stars on their turbulent gaseous environment. Here we present first results at very high resolution. We show that ionising radiation can trigger the collapse of an otherwise stable molecular cloud. The arising structures resemble observed structures (e.g. the pillars of creation in the Eagle Nebula (M16) or the Horsehead Nebula B33). Including the effect of gravitation we find small regions that can be identified as formation places of individual stars. We conclude that ionising radiation from massive stars alone can trigger substantial star formation in molecular clouds.
4. Heavy element abundances and massive star formation
NASA Technical Reports Server (NTRS)
Wang, Boqi; Silk, Joseph
1993-01-01
The determination of the stellar initial mass function (IMF) remains a great challenge in astronomy. In the solar neighborhood, the IMF is reasonable well determined for stellar masses from about 0.1 to 60 solar mass. However, outside the solar neighborhood, the IMF is poorly known. Among those frequently discussed arguments favoring a different IMF outside the solar neighborhood are the estimated time to consume the remaining gas in spiral galaxies, and the high rate of forming massive stars in starburst galaxies. An interesting question then is whether there may be an independent way of testing possible variations in the IMF. Indeed, the heavy elements in the interstellar medium are mostly synthesized in massive stars, so increasing, or decreasing, the fraction of massive stars naturally leads to a variation in the heavy element yield, and thus, the metallicity. The observed abundance should severely constrain any deviations of the IMF from the locally determined IMF. We focus on element oxygen, which is the most abundant heavy element in the interstellar medium. Oxygen is ejected only by massive stars that can become Type 1 supernovae, and the oxygen abundance is, therefore, a sensitive function of the fraction of massive stars in the IMF. Adopting oxygen enables us to avoid uncertainties in Type 1 supernovae. We use the nucleosynthesis results to calculate the oxygen yield for given IMF. We then calculate the oxygen abundance in the interstellar medium assuming instantaneous recycling of oxygen.
5. Modeling populations of rotationally mixed massive stars
NASA Astrophysics Data System (ADS)
Brott, I.
2011-02-01
Massive stars can be considered as cosmic engines. With their high luminosities, strong stellar winds and violent deaths they drive the evolution of galaxies through-out the history of the universe. Despite the importance of massive stars, their evolution is still poorly understood. Two major issues have plagued evolutionary models of massive stars until today: mixing and mass loss On the main sequence, the effects of mass loss remain limited in the considered mass and metallicity range, this thesis concentrates on the role of mixing in massive stars. This thesis approaches this problem just on the cross road between observations and simulations. The main question: Do evolutionary models of single stars, accounting for the effects of rotation, reproduce the observed properties of real stars. In particular we are interested if the evolutionary models can reproduce the surface abundance changes during the main-sequence phase. To constrain our models we build a population synthesis model for the sample of the VLT-FLAMES Survey of Massive stars, for which star-formation history and rotational velocity distribution are well constrained. We consider the four main regions of the Hunter diagram. Nitrogen un-enriched slow rotators and nitrogen enriched fast rotators that are predicted by theory. Nitrogen enriched slow rotators and nitrogen unenriched fast rotators that are not predicted by our model. We conclude that currently these comparisons are not sufficient to verify the theory of rotational mixing. Physical processes in addition to rotational mixing appear necessary to explain the stars in the later two regions. The chapters of this Thesis have been published in the following Journals: Ch. 2: Rotating Massive Main-Sequence Stars I: Grids of Evolutionary Models and Isochrones'', I. Brott, S. E. de Mink, M. Cantiello, N. Langer, A. de Koter, C. J. Evans, I. Hunter, C. Trundle, J.S. Vink submitted to Astronomy & Astrop hysics Ch. 3: The VLT-FLAMES Survey of Massive
6. Eruptive outflow phases of massive stars
NASA Astrophysics Data System (ADS)
Smith, Nathan
2011-07-01
I review recent progress on understanding eruptions of unstable massive stars, with particular attention to the diversity of observed behavior in extragalatic optical transient sources that are generally associated with giant eruptions of luminous blue variables (LBVs). These eruptions are thought to represent key mass loss episodes in the lives of massive stars. I discuss the possibility of dormant LBVs and implications for the duration of the greater LBV phase and its role in stellar evolution. These eruptive variables show a wide range of peak luminosity, decay time, expansion speeds, and progenitor luminosity, and in some cases they have been observed to suffer multiple eruptions. This broadens our view of massive star eruptions compared to prototypical sources like Eta Carinae, and provides important clues for the nature of the outbursts. I will also review and discuss some implications about the possible physical mechanisms involved, although the cause of the eruptions is not yet understood.
7. Massive Stars in the Quintuplet Cluster
NASA Astrophysics Data System (ADS)
Figer, Donald F.; McLean, Ian S.; Morris, Mark
1999-03-01
We present near-infrared photometry and K-band spectra of newly identified massive stars in the Quintuplet cluster, one of the three massive clusters projected within 50 pc of the Galactic center. We find that the cluster contains a variety of massive stars, including more unambiguously identified Wolf-Rayet stars than any cluster in the Galaxy, and over a dozen stars in earlier stages of evolution, i.e., luminous blue variables (LBVs), Ofpe/WN9, and OB supergiants. One newly identified star is the second luminous blue variable in the cluster, after the Pistol star.'' Although we are unable to provide certain spectral classifications for the five enigmatic Quintuplet-proper members, we tentatively propose that they are extremely dusty versions of the WC stars found elsewhere in the cluster and similar to the dozen or so known examples in the Galaxy. Although the cluster parameters are uncertain because of photometric errors and uncertainties in stellar models, i.e., extrapolating initial masses and estimating ionizing fluxes, we have the following conclusions. Given the evolutionary stages of the identified stars, the cluster appears to be about 4+/-1 Myr old, assuming coeval formation. The total mass in observed stars is ~103 Msolar, and the implied mass is ~104 Msolar, assuming a lower mass cutoff of 1 Msolar and a Salpeter initial mass function. The implied mass density in stars is greater than or similar to a few thousand Msolar pc-3. The newly identified stars increase the estimated ionizing flux from this cluster by about an order of magnitude with respect to earlier estimates, to 1050.9 photons s-1, or roughly what is required to ionize the nearby Sickle'' H II region (G0.18-0.04). The total luminosity from the massive cluster stars is ~107.5 Lsolar, enough to account for the heating of the nearby molecular cloud, M0.20-0.033. We propose a picture that integrates most of the major features in this part of the sky, excepting the nonthermal filaments. We
8. The initial conditions of massive star evolution
NASA Astrophysics Data System (ADS)
Sana, Hugues
2016-07-01
Massive stars are some of the most energetic building blocks of galaxies. They are the progenitors of supernovae and of neutrons stars and black holes, the coallescence of which is one of the most likely detectable sources of gravitational waves. Yet their formation remains poorly understood. As a consequence, the mechanisms that set initial parameters such as rotation rates, multiplicity and orbital distributions are also ill constrained. These quantities are however critical as they affect the internal mixing, the rate and nature of the interactions, the stars final fates and their end-of-life products. In this presentation, I will review existing and new observations that allow us to better constraints these parameters, hence the initial conditions for massive star evolution.
9. Comments on the Evolution of Massive Stars
NASA Astrophysics Data System (ADS)
El Eid, M. F.; The, L.-S.; Meyer, B. S.
We describe in a brief form present results we have obtained from a careful and up to date study of the evolution of massive stars including their advanced evolutionary phases beyond the oxygen burning phase. We describe the effects of mass loss, treatment of convection in inhomogeneous stellar layers and the rate of the 12C(α,γ)16O reaction on the properties of stellar models in the interesting case of a 25 M⊙ star of solar-like initial metallicity.
10. Massive stars. A chemical signature of first-generation very massive stars.
PubMed
Aoki, W; Tominaga, N; Beers, T C; Honda, S; Lee, Y S
2014-08-22
Numerical simulations of structure formation in the early universe predict the formation of some fraction of stars with several hundred solar masses. No clear evidence of supernovae from such very massive stars has, however, yet been found in the chemical compositions of Milky Way stars. We report on an analysis of a very metal-poor star SDSS J001820.5-093939.2, which possesses elemental-abundance ratios that differ significantly from any previously known star. This star exhibits low [α-element Fe] ratios and large contrasts between the abundances of odd and even element pairs, such as scandium/titanium and cobalt/nickel. Such features have been predicted by nucleosynthesis models for supernovae of stars more than 140 times as massive as the Sun, suggesting that the mass distribution of first-generation stars might extend to 100 solar masses or larger. PMID:25146286
11. Massive stars. A chemical signature of first-generation very massive stars.
PubMed
Aoki, W; Tominaga, N; Beers, T C; Honda, S; Lee, Y S
2014-08-22
Numerical simulations of structure formation in the early universe predict the formation of some fraction of stars with several hundred solar masses. No clear evidence of supernovae from such very massive stars has, however, yet been found in the chemical compositions of Milky Way stars. We report on an analysis of a very metal-poor star SDSS J001820.5-093939.2, which possesses elemental-abundance ratios that differ significantly from any previously known star. This star exhibits low [α-element Fe] ratios and large contrasts between the abundances of odd and even element pairs, such as scandium/titanium and cobalt/nickel. Such features have been predicted by nucleosynthesis models for supernovae of stars more than 140 times as massive as the Sun, suggesting that the mass distribution of first-generation stars might extend to 100 solar masses or larger.
12. Massive stars in the galaxies of the Local Group
NASA Astrophysics Data System (ADS)
Massey, Philip
2013-07-01
The star-forming galaxies of the Local Group act as our laboratories for testing massive star evolutionary models. In this review, I briefly summarize what we believe we know about massive star evolution, and the connection between OB stars, Luminous Blue Variables, yellow supergiants, red supergiants, and Wolf-Rayet stars. The difficulties and recent successes in identifying these various types of massive stars in the neighboring galaxies of the Local Group will be discussed.
13. Towards Realistic Modeling of Massive Star Clusters
NASA Astrophysics Data System (ADS)
Gnedin, O.; Li, H.
2016-06-01
Cosmological simulations of galaxy formation are rapidly advancing towards smaller scales. Current models can now resolve giant molecular clouds in galaxies and predict basic properties of star clusters forming within them. I will describe new theoretical simulations of the formation of the Milky Way throughout cosmic time, with the adaptive mesh refinement code ART. However, many challenges - physical and numerical - still remain. I will discuss how observations of massive star clusters and star forming regions can help us overcome some of them. Video of the talk is available at https://goo.gl/ZoZOfX
14. Some correlations for massive MS stars.
NASA Astrophysics Data System (ADS)
Angelov, T.
1994-11-01
Criteria are derived for estimating the values of photospheric density, of the core mass and of the energy-generation rate for massive main-sequence stars. Based on the observational material concerning M, L and Te the demarcation lines are determined for the domain of values expected for these quantities in cor relation with measured Mb.
15. Probing Massive Star Cluster Formation with ALMA
NASA Astrophysics Data System (ADS)
Johnson, Kelsey
2015-08-01
Observationally constraining the physical conditions that give rise to massive star clusters has been a long-standing challenge. Now with the ALMA Observatory coming on-line, we can finally begin to probe the birth environments of massive clusters in a variety of galaxies with sufficient angular resolution. In this talk I will give an overview of ALMA observations of galaxies in which candidate proto-super star cluster molecular clouds have been identified. These new data probe the physical conditions that give rise to super star clusters, providing information on their densities, pressures, and temperatures. In particular, the observations indicate that these clouds may be subject to external pressures of P/k > 108 K cm-3, which is consistent with the prevalence of optically observed adolescent super star clusters in interacting galaxy systems and other high pressure environments. ALMA observations also enable an assessement of the molecular cloud chemical abundances in the regions surrounding super star clusters. Molecular clouds associated with existing super star clusters are strongly correlated with HCO+ emission, but appear to have relatively low ratio of CO/HCO+ emission compared to other clouds, indicating that the super star clusters are impacting the molecular abundances in their vicinity.
16. Formation and Assembly of Massive Star Clusters
NASA Astrophysics Data System (ADS)
McMillan, Stephen
The formation of stars and star clusters is a major unresolved problem in astrophysics. It is central to modeling stellar populations and understanding galaxy luminosity distributions in cosmological models. Young massive clusters are major components of starburst galaxies, while globular clusters are cornerstones of the cosmic distance scale and represent vital laboratories for studies of stellar dynamics and stellar evolution. Yet how these clusters form and how rapidly and efficiently they expel their natal gas remain unclear, as do the consequences of this gas expulsion for cluster structure and survival. Also unclear is how the properties of low-mass clusters, which form from small-scale instabilities in galactic disks and inform much of our understanding of cluster formation and star-formation efficiency, differ from those of more massive clusters, which probably formed in starburst events driven by fast accretion at high redshift, or colliding gas flows in merging galaxies. Modeling cluster formation requires simulating many simultaneous physical processes, placing stringent demands on both software and hardware. Simulations of galaxies evolving in cosmological contexts usually lack the numerical resolution to simulate star formation in detail. They do not include detailed treatments of important physical effects such as magnetic fields, radiation pressure, ionization, and supernova feedback. Simulations of smaller clusters include these effects, but fall far short of the mass of even single young globular clusters. With major advances in computing power and software, we can now directly address this problem. We propose to model the formation of massive star clusters by integrating the FLASH adaptive mesh refinement magnetohydrodynamics (MHD) code into the Astrophysical Multi-purpose Software Environment (AMUSE) framework, to work with existing stellar-dynamical and stellar evolution modules in AMUSE. All software will be freely distributed on-line, allowing
17. Evolution and Nucleosynthesis of Very Massive Stars
NASA Astrophysics Data System (ADS)
Hirschi, Raphael
In this chapter, after a brief introduction and overview of stellar evolution, we discuss the evolution and nucleosynthesis of very massive stars (VMS: M > 100 M_{odot } ) in the context of recent stellar evolution model calculations. This chapter covers the following aspects: general properties, evolution of surface properties, late central evolution, and nucleosynthesis including their dependence on metallicity, mass loss and rotation. Since very massive stars have very large convective cores during the main-sequence phase, their evolution is not so much affected by rotational mixing, but more by mass loss through stellar winds. Their evolution is never far from a homogeneous evolution even without rotational mixing. All VMS at metallicities close to solar end their life as WC(-WO) type Wolf-Rayet stars. Due to very important mass loss through stellar winds, these stars may have luminosities during the advanced phases of their evolution similar to stars with initial masses between 60 and 120 M_{odot } . A distinctive feature which may be used to disentangle Wolf-Rayet stars originating from VMS from those originating from lower initial masses is the enhanced abundances of neon and magnesium at the surface of WC stars. At solar metallicity, mass loss is so strong that even if a star is born with several hundred solar masses, it will end its life with less than 50 M_{odot } (using current mass loss prescriptions). At the metallicity of the LMC and lower, on the other hand, mass loss is weaker and might enable stars to undergo pair-instability supernovae.
18. MASSIVE INFANT STARS ROCK THEIR CRADLE
NASA Technical Reports Server (NTRS)
2002-01-01
Extremely intense radiation from newly born, ultra-bright stars has blown a glowing spherical bubble in the nebula N83B, also known as NGC 1748. A new NASA Hubble Space Telescope image has helped to decipher the complex interplay of gas and radiation of a star-forming region in a nearby galaxy. The image graphically illustrates just how these massive stars sculpt their environment by generating powerful winds that alter the shape of the parent gaseous nebula. These processes are also seen in our Milky Way in regions like the Orion Nebula. The Hubble telescope is famous for its contribution to our knowledge about star formation in very distant galaxies. Although most of the stars in the Universe were born several billions of years ago, when the Universe was young, star formation still continues today. This new Hubble image shows a very compact star-forming region in a small part of one of our neighboring galaxies - the Large Magellanic Cloud. This galaxy lies only 165,000 light-years from our Milky Way and can easily be seen with the naked eye from the Southern Hemisphere. Young, massive, ultra-bright stars are seen here just as they are born and emerge from the shelter of their pre-natal molecular cloud. Catching these hefty stars at their birthplace is not as easy as it may seem. Their high mass means that the young stars evolve very rapidly and are hard to find at this critical stage. Furthermore, they spend a good fraction of their youth hidden from view, shrouded by large quantities of dust in a molecular cloud. The only chance is to observe them just as they start to emerge from their cocoon - and then only with very high-resolution telescopes. Astronomers from France, the U.S., and Germany have used Hubble to study the fascinating interplay between gas, dust, and radiation from the newly born stars in this nebula. Its peculiar and turbulent structure has been revealed for the first time. This high-resolution study has also uncovered several individual stars
19. MASSIVE STAR FORMATION IN NGC 2074
SciTech Connect
Fleener, Christine E.; Chu, Y.-H.; Gruendl, Robert A.; Payne, James T.; Chen, C.-H. Rosie
2010-01-15
Spitzer observations of the Large Magellanic Cloud (LMC) have revealed a large population of young stellar objects (YSOs), but complementary high-resolution images in the optical or near-IR wavelengths are still needed to resolve the multiplicity and immediate environments of the YSOs. The Hubble Space Telescope imaged the star-forming region NGC 2074 in the LMC during its 100,000th orbit, providing an opportunity to more closely examine the YSOs and their environments in this region. We have studied the 10 YSO candidates identified from Spitzer observations, confirming their nature and determining their physical parameters by modeling their spectral energy distributions. The majority of the YSOs and central stars of ultracompact H II regions in NGC 2074 have masses consistent with spectral types of early B to late O. The co-existence of massive early-type O stars and the less massive YSOs indicates that their formation may have started at a similar time, a few 10{sup 5} yr ago. NGC 2074 provides an opportunity to study the evolution of massive stars at their infancy.
20. The Evolution and Stability of Massive Stars
NASA Astrophysics Data System (ADS)
Shiode, Joshua Hajime
Massive stars are the ultimate source for nearly all the elements necessary for life. The first stars forge these elements from the sparse set of ingredients supplied by the Big Bang, and distribute enriched ashes throughout their galactic homes via their winds and explosive deaths. Subsequent generations follow suit, assembling from the enriched ashes of their predecessors. Over the last several decades, the astrophysics community has developed a sophisticated theoretical picture of the evolution of these stars, but it remains an incomplete accounting of the rich set of observations. Using state of the art models of massive stars, I have investigated the internal processes taking place throughout the life-cycles of stars spanning those from the first generation ("Population III") to the present-day ("Population I"). I will argue that early-generation stars were not highly unstable to perturbations, contrary to a host of past investigations, if a correct accounting is made for the viscous effect of convection. For later generations, those with near solar metallicity, I find that this very same convection may excite gravity-mode oscillations that produce observable brightness variations at the stellar surface when the stars are near the main sequence. If confirmed with modern high-precision monitoring experiments, like Kepler and CoRoT, the properties of observed gravity modes in massive stars could provide a direct probe of the poorly constrained physics of gravity mode excitation by convection. Finally, jumping forward in stellar evolutionary time, I propose and explore an entirely new mechanism to explain the giant eruptions observed and inferred to occur during the final phases of massive stellar evolution. This mechanism taps into the vast nuclear fusion luminosity, and accompanying convective luminosity, in the stellar core to excite waves capable of carrying a super-Eddington luminosity out to the stellar envelope. This energy transfer from the core to the
1. Instability considerations for massive star eruptions
SciTech Connect
Guzik, J. A.
2004-01-01
We propose a mechanism to explain the observed properties of the giant eruptions of 'supernova imposters' such as {eta} Car and P Cyg. This mechanism must be episodic, generate a large amount of energy, and be very deep-seated, in order to lift about 10 solar masses out of the deep gravitational potential well of these massive evolved stars. We suggest that nonradial gravity mode oscillations capable of existing in the core grow slowly to sufficient amplitude to cause an episode of mixing. This mixing generates a burst of nuclear energy deep in the star that is responsible for the observed large mass ejection and bolometric magnitude increase.
2. Very Massive Stars in the local Universe
NASA Astrophysics Data System (ADS)
Vink, Jorick S.; Heger, Alexander; Krumholz, Mark R.; Puls, Joachim; Banerjee, S.; Castro, N.; Chen, K.-J.; Chenè, A.-N.; Crowther, P. A.; Daminelli, A.; Gräfener, G.; Groh, J. H.; Hamann, W.-R.; Heap, S.; Herrero, A.; Kaper, L.; Najarro, F.; Oskinova, L. M.; Roman-Lopes, A.; Rosen, A.; Sander, A.; Shirazi, M.; Sugawara, Y.; Tramper, F.; Vanbeveren, D.; Voss, R.; Wofford, A.; Zhang, Y.
2015-03-01
Recent studies have claimed the existence of very massive stars (VMS) up to 300 M ⊙ in the local Universe. As this finding may represent a paradigm shift for the canonical stellar upper-mass limit of 150 M ⊙, it is timely to discuss the status of the data, as well as the far-reaching implications of such objects. We held a Joint Discussion at the General Assembly in Beijing to discuss (i) the determination of the current masses of the most massive stars, (ii) the formation of VMS, (iii) their mass loss, and (iv) their evolution and final fate. The prime aim was to reach broad consensus between observers and theorists on how to identify and quantify the dominant physical processes.
3. Speckle Interferometry of Massive and Cluster Stars
NASA Astrophysics Data System (ADS)
Mason, Brian; Hartkopf, William I.; Gies, Douglas R.; Henry, Todd J.; Tokovinin, Andrei A.
2006-02-01
Conducted on NOAO 4-m telescopes in 1994, the first speckle survey of O stars (Mason et al. 1998) had success far in excess of our expectations. In addition to the frequently cited multiplicity analysis, many of the new systems which were first resolved in this paper are of significant astrophysical importance. Now, some ten years after the original survey, we propose to re-investigate all systems analyzed before (N=195). Improvements in detector technology will allow for the detection of companions missed before as well as systems which may have been closer than the resolution limit in 1994. We will also make a first high-resolution inspection of the additional O stars (N=108) in the recent Galactic O Star Catalog of Maiz- Apellaniz & Walborn (2004). Further, we propose to investigate several additional samples of interesting objects, including 15 accessible Galactic WR stars from the speckle survey of Hartkopf et al. (1999), 16 massive, hot stars with separations which would indicate their applicability for mass determinations (for fully detached O stars masses are presently known for only twelve pairs), and 56 multiple stars for a study of their co- planarity statistics.
4. Speckle Interferometry of Massive and Cluster Stars
NASA Astrophysics Data System (ADS)
Mason, Brian; Hartkopf, William I.; Gies, Douglas R.; Henry, Todd J.; Torres, Guillermo
2005-08-01
Conducted on NOAO 4-m telescopes in 1994, the first speckle survey of O stars (Mason et al. 1998) had success far in excess of our expectations. In addition to the frequently cited multiplicity analysis, many of the new systems which were first resolved in this paper are of significant astrophysical importance. To date, this paper has resulted in 86 citations in the refereed literature. Now, some ten years after the original survey, we propose to re-investigate all systems analyzed before (N=98) as well as make a first high-resolution inspection of the additional O stars (N=62) in the recent Galactic O Star Catalog of Maiz-Apellaniz & Walborn (2004). In addition, we propose to investigate several additional samples of interesting objects, including 10 accessible Galactic WR stars from the speckle survey of Hartkopf et al. (1999), 16 massive, hot stars with separations which would indicate their applicability for mass determinations (for fully detached O stars, we have only twelve mass determinations), 92 members of the Hyades and Pleiades clusters to complement RV studies of these clusters, and 197 Hyades & Pleiades stars, reobserved from the 1991 lists (Mason et al. 1993a,b).
5. Wolf-Rayet stars from Very Massive Stars
NASA Astrophysics Data System (ADS)
Yusof, Norhasliza
2015-01-01
Many studies focused on very massive stars (VMS) within the framework of Pop. III stars, because this is where they were thought to be abundant. In this work, we focus on the evolution of VMS in the local universe following the discovery of VMS in the R136 cluster in the Large Magellanic Cloud (LMC). We computed grids of VMS evolutionary tracks in the range 120-500 M ⊙ with solar, LMC and Small Magellanic Cloud metallicities. All models end their lives as Wolf-Rayet (WR) stars of the WC (or WO) type. We discuss the evolution and fate of VMS around solar metallicity with particular focus on the WR phase. For example, we show that a distinctive feature that may be used to disentangle Wolf-Rayet stars originating from VMS from those originating from lower initial masses is the enhanced abundances of Ne and Mg at the surface of WC stars.
6. The Massive Star Population in M101
NASA Astrophysics Data System (ADS)
Grammer, Skyler H.
An increasing number of non-terminal giant eruptions are being observed by modern supernova and transient surveys. Very little is known about the origin of these giant eruptions and their progenitors which are presumably very-massive, evolved stars such as luminous blue variables, hypergiants, and supergiants. Motivated by the small number of progenitors positively associated with these giant eruptions, we have begun a survey of the luminous and evolved massive star populations in several nearby galaxies. We aim to identify the likely progenitors of the giant eruptions, study the spatial variations in the stellar populations, and examine the relationship between massive star populations and their environment. The work presented here is focused on stellar populations in the relatively nearby, giant, spiral galaxy M101 from sixteen archival BVI HST/ACS images. We create a catalog of stars in the direction to M101 with photometric errors < 10% for V < 24.5 and 50% completeness down to V ˜ 26.5 even in regions of high stellar crowding. Using color and magnitude criteria we have identified candidate luminous OB type stars and blue supergiants, yellow supergiants, and red supergiants for future observation. We examine their spatial distributions across the face of M101 and find that the ratio of blue to red supergiants decreases by two orders of magnitude over the radial extent. From our catalog, we derive the star formation history (SFH) for the stellar populations in five 2' wide annuli by fitting the color-magnitude diagrams. Binning the SFH into time frames corresponding to populations traced by Halpha, far ultraviolet (FUV), and near ultraviolet (NUV) emission, we show that the fraction of stellar populations young enough to contribute in Halpha is 15% " 35% in the inner regions, compared to less than 5% in the outer regions. This provides a sufficient explanation for the lack of Halpha emission at large radii. We also model the blue to red supergiant ratio in our
7. An Unstable Truth: How Massive Stars get their Mass
NASA Astrophysics Data System (ADS)
Rosen, Anna L.; Krumholz, Mark R.; McKee, Christopher F.; Klein, Richard I.
2016-08-01
The pressure exerted by massive stars' radiation fields is an important mechanism regulating their formation. Detailed simulation of massive star formation therefore requires an accurate treatment of radiation. However, all published simulations have either used a diffusion approximation of limited validity; have only been able to simulate a single star fixed in space, thereby suppressing potentially-important instabilities; or did not provide adequate resolution at locations where instabilities may develop. To remedy this we have developed a new, highly accurate radiation algorithm that properly treats the absorption of the direct radiation field from stars and the re-emission and processing by interstellar dust. We use our new tool to perform three-dimensional radiation-hydrodynamic simulations of the collapse of massive pre-stellar cores with laminar and turbulent initial conditions and properly resolve regions where we expect instabilities to grow. We find that mass is channeled to the stellar system via gravitational and Rayleigh-Taylor (RT) instabilities, in agreement with previous results using stars capable of moving, but in disagreement with methods where the star is held fixed or with simulations that do not adequately resolve the development of RT instabilities. For laminar initial conditions, proper treatment of the direct radiation field produces later onset of instability, but does not suppress it entirely provided the edges of radiation-dominated bubbles are adequately resolved. Instabilities arise immediately for turbulent pre-stellar cores because the initial turbulence seeds the instabilities. Our results suggest that RT features should be present around accreting massive stars throughout their formation.
8. Proper Motions of Massive Stars in 30 Doradus
NASA Astrophysics Data System (ADS)
Lennon, Daniel
2013-10-01
We propose an ambitious proper motion survey of massive stars in the 30 Doradus region of the Large Magellanic Cloud using the unique capabilities of HST. We will derive the directions of motion of massive runaway stars, searching in particular for massive stars which have been ejected from the central very massive cluster R136. These data will be combined with radial velocities from the VLT-FLAMES Survey of the Tarantula Nebula and with atmospheric analyses and stellar evolution models to constrain their origins. We will also search for very young isolated massive stars to test models of single-star formation. This work is highly relevant to star formation, cluster dynamics, the origin of field WR stars and GRBs, the creation of very massive stars by runaway mergers, and the possible formation of intermediate-mass black holes.
9. Magnetic fields and massive star formation
SciTech Connect
Zhang, Qizhou; Keto, Eric; Ho, Paul T. P.; Ching, Tao-Chung; Chen, How-Huan; Qiu, Keping; Girart, Josep M.; Juárez, Carmen; Liu, Hauyu; Tang, Ya-Wen; Koch, Patrick M.; Rao, Ramprasad; Lai, Shih-Ping; Li, Zhi-Yun; Frau, Pau; Li, Hua-Bai; Padovani, Marco; Bontemps, Sylvain
2014-09-10
Massive stars (M > 8 M {sub ☉}) typically form in parsec-scale molecular clumps that collapse and fragment, leading to the birth of a cluster of stellar objects. We investigate the role of magnetic fields in this process through dust polarization at 870 μm obtained with the Submillimeter Array (SMA). The SMA observations reveal polarization at scales of ≲0.1 pc. The polarization pattern in these objects ranges from ordered hour-glass configurations to more chaotic distributions. By comparing the SMA data with the single dish data at parsec scales, we found that magnetic fields at dense core scales are either aligned within 40° of or perpendicular to the parsec-scale magnetic fields. This finding indicates that magnetic fields play an important role during the collapse and fragmentation of massive molecular clumps and the formation of dense cores. We further compare magnetic fields in dense cores with the major axis of molecular outflows. Despite a limited number of outflows, we found that the outflow axis appears to be randomly oriented with respect to the magnetic field in the core. This result suggests that at the scale of accretion disks (≲ 10{sup 3} AU), angular momentum and dynamic interactions possibly due to close binary or multiple systems dominate over magnetic fields. With this unprecedentedly large sample of massive clumps, we argue on a statistical basis that magnetic fields play an important role during the formation of dense cores at spatial scales of 0.01-0.1 pc in the context of massive star and cluster star formation.
10. The simultaneous formation of massive stars and stellar clusters
NASA Astrophysics Data System (ADS)
Smith, Rowan J.; Longmore, Steven; Bonnell, Ian
2009-12-01
We show that massive stars and stellar clusters are formed simultaneously, the global evolution of the forming cluster is what allows the central stars to become massive. We predict that massive star-forming clumps, such as those observed in Motte et al., contract and grow in mass leading to the formation of massive stars. This occurs as mass is continually channelled from large radii on to the central protostars, which can become massive through accretion. Using smoothed particle hydrodynamic simulations of massive star-forming clumps in a giant molecular cloud, we show that clumps are initially diffuse and filamentary, and become more concentrated as they collapse. Simulated interferometry observations of our data provide an explanation as to why young massive star-forming regions show more substructure than older ones. The most massive stars in our model are found within the most bound cluster. Most of the mass accreted by the massive stars was originally distributed throughout the clump at low densities and was later funnelled to the star due to global infall. Even with radiative feedback no massive pre-stellar cores are formed. The original cores are of intermediate mass and gain their additional mass in the protostellar stage. We also find that cores which form low-mass stars exist within the volume from which the high-mass stars accrete, but are largely unaffected by this process.
11. The Chandra Carina Complex Project: Massive Stars
NASA Astrophysics Data System (ADS)
Gagne, Marc; Townsley, L.; Corcoran, M.; Cohen, D.; Dickerson, K.; Oskinova, L.; Naze, Y.; Broos, P.; Chandra Carina Complex Project
2010-03-01
The Great Nebula in Carina is a superb site to study the violent massive star formation and feedback that typifies giant HII regions and starburst galaxies. We have combined 20 deep, new Chandra ACIS-I pointings with two existing ACIS-I fields to map over one square degree of the Carina complex. A state-of-the-art source detection algorithm has been used to create a catalog of 14,368 x-ray sources, the great majority with counterparts at near- and mid-infrared wavelengths. Carina contains the largest catalogued population of OB stars within 3 kpc, including many known binaries. In this paper, we report on the 130 x-ray detected OB and Wolf-Rayet Stars in the Carina complex. We use their x-ray spectra and light curves to categorize their x-ray emission. Not surprisingly, we find that the known OB and WolfRayet binaries have hard x-ray spectra and high Lx/Lbol strongly suggesting colling wind shocks. Most of the single OB stars have generally lower shock temperatures and lower Lx/Lbol, suggesting wind shocks embedded in the wind. About a dozen of the apparently single OB stars have harder x-ray spectra, and some time variability, possibly suggesting magnetically confined wind shocks, or flaring T Tauri companions.
12. Massive Star Formation: The Role of Disks
NASA Astrophysics Data System (ADS)
Fallscheer, Cassandra L.; Beuther, H.; Sauter, J.; Wolf, S.; Zhang, Q.; Keto, E.; Sridharan, T. K.
2011-01-01
We have obtained multiple data sets from the SMA, PdBI, and IRAM 30m telescope of the Infrared Dark Cloud IRDC18223-3, the High-Mass Protostellar Object IRAS18151-1208, and the hot core source IRAS18507+0121 in order to search for clues regarding the role of rotation and disks in high mass star formation. These three objects allow us to compare the central-most regions surrounding the embedded continuum source at three different evolutionary stages of the formation process. Toward all three regions we see rotational or elongated structures perpendicular to the molecular outflows. Similarities and differences in the evolutionary sequence are discussed in the context of core and disk evolution. We have also carried out continuum and line radiative transfer modeling of the disk-like structures. Having a more complete picture of the evolutionary process that a massive star experiences will contribute significantly to the future of massive star formation research. Support for this project comes from the Deutsche Forschungsgemeinschaft and the International Max-Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg.
13. Radiative ablation of disks around massive stars
NASA Astrophysics Data System (ADS)
Kee, Nathaniel Dylan
Hot, massive stars (spectral types O and B) have extreme luminosities (10. 4 -10. 6 L?) that drive strong stellar winds through UV line-scattering.Some massive stars also have disks, formed by either decretion from the star (as in the rapidly rotating "Classical Be stars"), or accretion during the star's formation. This dissertation examines the role of stellar radiation in driving (ablating) material away from these circumstellar disks. A key result is that the observed month to year decay of Classical Be disks can be explained by line-driven ablation without, as previously done, appealing to anomalously strong viscous diffusion. Moreover, the higher luminosity of O stars leads to ablation of optically thin disks on dynamical timescales of order a day, providing a natural explanation for the lack of observed Oe stars. In addition to the destruction of Be disks, this dissertation also introduces a model for their formation by coupling observationally inferred non-radial pulsation modes and rapid stellar rotation to launch material into orbiting Keplerian disks of Be-like densities. In contrast to such Be decretion disks, star-forming accretion disks are much denser and so are generally optically thick to continuum processes. To circumvent the computational challenges associated with radiation hydrodynamics through optically thick media, we develop an approximate method for treating continuum absorption in the limit of geometrically thin disks. The comparison of ablation with and without continuum absorption shows that accounting for disk optical thickness leads to less than a 50% reduction in ablation rate, implying that ablation rate depends mainly on stellar properties like luminosity. Finally, we discuss the role of "thin-shell mixing" in reducing X-rays from colliding wind binaries. Laminar, adiabatic shocks produce well understood X-ray emission, but the emission from radiatively cooled shocks is more complex due to thin-shell instabilities. The parameter
14. Evolution and fate of very massive stars
NASA Astrophysics Data System (ADS)
Yusof, Norhasliza; Hirschi, Raphael; Meynet, Georges; Crowther, Paul A.; Ekström, Sylvia; Frischknecht, Urs; Georgy, Cyril; Abu Kassim, Hasan; Schnurr, Olivier
2013-08-01
There is observational evidence that supports the existence of very massive stars (VMS) in the local universe. First, VMS (Mini ≲ 320 M⊙) have been observed in the Large Magellanic Clouds (LMC). Secondly, there are observed supernovae (SNe) that bear the characteristics of pair creation supernovae (PCSNe, also referred to as pair instability SN) which have VMS as progenitors. The most promising candidate to date is SN 2007bi. In order to investigate the evolution and fate of nearby VMS, we calculated a new grid of models for such objects, for solar, LMC and Small Magellanic Clouds (SMC) metallicities, which covers the initial mass range from 120 to 500 M⊙. Both rotating and non-rotating models were calculated using the GENEVA stellar evolution code and evolved until at least the end of helium burning and for most models until oxygen burning. Since VMS have very large convective cores during the main-sequence phase, their evolution is not so much affected by rotational mixing, but more by mass loss through stellar winds. Their evolution is never far from a homogeneous evolution even without rotational mixing. All the VMS, at all the metallicities studied here, end their life as WC(WO)-type Wolf-Rayet stars. Because of very important mass losses through stellar winds, these stars may have luminosities during the advanced phases of their evolution similar to stars with initial masses between 60 and 120 M⊙. A distinctive feature which may be used to disentangle Wolf-Rayet stars originating from VMS from those originating from lower initial masses would be the enhanced abundances of Ne and Mg at the surface of WC stars. This feature is however not always apparent depending on the history of mass loss. At solar metallicity, none of our models is expected to explode as a PCSN. At the metallicity of the LMC, only stars more massive than 300 M⊙ are expected to explode as PCSNe. At the SMC metallicity, the mass range for the PCSN progenitors is much larger and
15. Formation of massive stars by growing accretion
NASA Astrophysics Data System (ADS)
Maeder, Andre
There are at present three scenarios for the formation of massive star. 1) The classical scenario of constant mass pre-Main Sequence (MS) evolution on the Kelvin-Helmholtz timescale. 2) The coalescence scenario, with merging of intermediate mass protostars. 3) The accretion scenario. The various arguments for and against these scenarios are briefly reviewed. We examine the pre-MS evolution of accreting stars for constant accretion rates and for accretion rates which are growing with the stellar masses. The location of the birthlines in the HRD and the lifetimes support accretion rates growing fastly with the stellar masses. Remarkably the dependence found is similar to that of the mass outflows from UC HII regions observed by Churchwell (1999) and Henning et al. (2000). The accretion scenario also leads to a new concept for the maximum stellar mass.
16. Hot, Massive Stars in I Zw 18
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Lindler, D.; Malumuth, E.
2011-01-01
I Zw 18 is one of the most primitive blue, compact dwarf galaxies. The ionized gas in I Zw 18 has a low oxygen abundance (O approx.1/30 Osun) and nitrogen abundance (N-1/100 Nsun) (Pequignot 2008). We have obtained a far-UV spectrum of the northwest massive star cluster of I Zw 18 using Hubble's Cosmic Origins Spectrograph (COS). The spectrum is compatible with continuous star-formation over the past approx.10 Myr, and a very low metallicity, log Z/Zsun 1.7, although the stellar surface may be enhanced in carbon. Stellar wind lines are very weak, and the edge velocity of wind lines is very low (approx.250 km/s).
17. Energetic Supernovae of Very Massive Primordial Stars
NASA Astrophysics Data System (ADS)
Chen, Ke-Jung; Woosley, Stan
2015-08-01
Current models of the formation of the first stars in the universe suggest that these stars were very massive, having a typical mass scale of hundreds of solar masses. Some of them would die as pair instability supernovae (PSNe) which might be the biggest explosions of the universe. We present the results from multidimensional numerical studies of PSNe with a new radiation-hydrodynamics code, CASTRO and with realistic nuclear reaction networks. We simulate the fluid instabilities that occur in multiple spatial dimensions and discuss how the resulting mixing affects the explosion, mixing, and nucleosynthesis of these supernovae. Our simulations provide useful predictions for the observational signatures of PSNe, which might soon be examined by the James Webb Space Telescope.
18. Binary interaction dominates the evolution of massive stars.
PubMed
Sana, H; de Mink, S E; de Koter, A; Langer, N; Evans, C J; Gieles, M; Gosset, E; Izzard, R G; Le Bouquin, J-B; Schneider, F R N
2012-07-27
The presence of a nearby companion alters the evolution of massive stars in binary systems, leading to phenomena such as stellar mergers, x-ray binaries, and gamma-ray bursts. Unambiguous constraints on the fraction of massive stars affected by binary interaction were lacking. We simultaneously measured all relevant binary characteristics in a sample of Galactic massive O stars and quantified the frequency and nature of binary interactions. More than 70% of all massive stars will exchange mass with a companion, leading to a binary merger in one-third of the cases. These numbers greatly exceed previous estimates and imply that binary interaction dominates the evolution of massive stars, with implications for populations of massive stars and their supernovae. PMID:22837522
19. Binary interaction dominates the evolution of massive stars.
PubMed
Sana, H; de Mink, S E; de Koter, A; Langer, N; Evans, C J; Gieles, M; Gosset, E; Izzard, R G; Le Bouquin, J-B; Schneider, F R N
2012-07-27
The presence of a nearby companion alters the evolution of massive stars in binary systems, leading to phenomena such as stellar mergers, x-ray binaries, and gamma-ray bursts. Unambiguous constraints on the fraction of massive stars affected by binary interaction were lacking. We simultaneously measured all relevant binary characteristics in a sample of Galactic massive O stars and quantified the frequency and nature of binary interactions. More than 70% of all massive stars will exchange mass with a companion, leading to a binary merger in one-third of the cases. These numbers greatly exceed previous estimates and imply that binary interaction dominates the evolution of massive stars, with implications for populations of massive stars and their supernovae.
20. Massive stars in their death throes.
PubMed
Eldridge, John J
2008-12-13
The study of the stars that explode as supernovae used to be a forensic study, working backwards from the remnants of the star. This changed in 1987 when the first progenitor star was identified in pre-explosion images. Currently, there are eight detected progenitors with another 21 non-detections, for which only a limit on the pre-explosion luminosity can be placed. This new avenue of supernova research has led to many interesting conclusions, most importantly that the progenitors of the most common supernovae, type IIP, are red supergiants, as theory has long predicted. However, no progenitors have been detected thus far for the hydrogen-free type Ib/c supernovae, which, given the expected progenitors, is an unlikely result. Also, observations have begun to show evidence that luminous blue variables, which are among the most massive stars, may directly explode as supernovae. These results contradict the current stellar evolution theory. This suggests that we may need to update our understanding.
1. Limiting Accretion onto Massive Stars by Fragmentation-Induced Starvation
SciTech Connect
Peters, Thomas; Klessen, Ralf S.; Mac Low, Mordecai-Mark; Banerjee, Robi; /ZAH, Heidelberg
2010-08-25
Massive stars influence their surroundings through radiation, winds, and supernova explosions far out of proportion to their small numbers. However, the physical processes that initiate and govern the birth of massive stars remain poorly understood. Two widely discussed models are monolithic collapse of molecular cloud cores and competitive accretion. To learn more about massive star formation, we perform simulations of the collapse of rotating, massive, cloud cores including radiative heating by both non-ionizing and ionizing radiation using the FLASH adaptive mesh refinement code. These simulations show fragmentation from gravitational instability in the enormously dense accretion flows required to build up massive stars. Secondary stars form rapidly in these flows and accrete mass that would have otherwise been consumed by the massive star in the center, in a process that we term fragmentation-induced starvation. This explains why massive stars are usually found as members of high-order stellar systems that themselves belong to large clusters containing stars of all masses. The radiative heating does not prevent fragmentation, but does lead to a higher Jeans mass, resulting in fewer and more massive stars than would form without the heating. This mechanism reproduces the observed relation between the total stellar mass in the cluster and the mass of the largest star. It predicts strong clumping and filamentary structure in the center of collapsing cores, as has recently been observed. We speculate that a similar mechanism will act during primordial star formation.
2. LIMITING ACCRETION ONTO MASSIVE STARS BY FRAGMENTATION-INDUCED STARVATION
SciTech Connect
Peters, Thomas; Klessen, Ralf S.; Banerjee, Robi; Low, Mordecai-Mark Mac
2010-12-10
Massive stars influence their surroundings through radiation, winds, and supernova explosions far out of proportion to their small numbers. However, the physical processes that initiate and govern the birth of massive stars remain poorly understood. Two widely discussed models are monolithic collapse of molecular cloud cores and competitive accretion. To learn more about massive star formation, we perform and analyze simulations of the collapse of rotating, massive, cloud cores including radiative heating by both non-ionizing and ionizing radiation using the FLASH adaptive-mesh refinement code. These simulations show fragmentation from gravitational instability in the enormously dense accretion flows required to build up massive stars. Secondary stars form rapidly in these flows and accrete mass that would have otherwise been consumed by the massive star in the center, in a process that we term fragmentation-induced starvation. This explains why massive stars are usually found as members of high-order stellar systems that themselves belong to large clusters containing stars of all masses. The radiative heating does not prevent fragmentation, but does lead to a higher Jeans mass, resulting in fewer and more massive stars than would form without the heating. This mechanism reproduces the observed relation between the total stellar mass in the cluster and the mass of the largest star. It predicts strong clumping and filamentary structure in the center of collapsing cores, as has recently been observed. We speculate that a similar mechanism will act during primordial star formation.
3. First Circumstellar Disk around a Massive Star
NASA Astrophysics Data System (ADS)
1998-06-01
Observations with an infrared-sensitive instrument at the ESO 3.6-m telescope at La Silla have for the first time shown the presence of a disk around a hot and massive star, known as G339.88-1.26 . Until now, disks have only been found around less massive stars. Planets are formed in such disks. The new discovery may thus have important implications for our understanding of the formation of planetary systems around stars. TIMMI observations Observations at mid-infrared wavelengths were carried out in July 1997 by Bringfried Stecklum (Landessternwarte Thüringen, Tautenburg, Germany) and Hans-Ulrich Käufl (ESO), using the TIMMI instrument at the ESO 3.6-m telescope. Additional measurements were carried out in March 1998. TIMMI ( T hermal I nfrared M ulti M ode I nstrument) is a general-purpose camera spectrometer operating at a wavelength of 10 µm. To reach sufficient sensitivity, the camera must be cooled to approx. -260 o C, i.e. a few degrees above the absolute minimum, by use of liquid Helium. Astronomical objects whose temperatures are between -120 o C and 300 o C radiate most of their energy at this wavelength. In addition, dust and haze that are absolutely impenetrable for light visible to the human eye, are often found to be nearly transparent at this wavelength. This is why fire-fighters now use similar equipment to look through smoke. G339.88-1.26: A very special object ESO PR Photo 22a/98 ESO PR Photo 22a/98 [JPEG, 800k] This image is a true-color composite of near-infrared observations of the sky region around the radio source G339.88-1.26 with the ESO/MPI 2.2-m telescope at La Silla. In this image, the visible colors red, green and blue have been used to represent the infrared filters J, H and K (at 1.25, 1.63 and 2.2 µm wavelength, respectively). No object is visible at the position of the radio source, even at these near-infrared wavelengths. A dark band of absorbing dust is clearly visible, exactly at the position of the object (indicated by an
4. Evolutionary Connections Between RSGs and Other Massive Stars
NASA Astrophysics Data System (ADS)
Smith, Nathan
2015-08-01
Red supergiants are an important mass-loss phase near the end of a massive star's life, but there are many other evolved mass-losing stars that populate the HR Diagram, and not all massive stars will pass through a red supergiant phase. This talk will provide an overview of other types of massive stars and how they relate to red supergiants. Mass loss by red supergiant winds will be weighed against the mass loss of other massive stars in terms of their contribution to pre-supernova evolution, focussing on trends with initial mass and metallicity. Moreover, some other evolved massive stars have already been RSG or will be in the future, and circumstellar material is an important clue in this regard. Last, the diversity of different supernova explosions, their circumstellar material, and statistics of SN types provide important constraints on the role of RSGs in the latest phases of evolution and mass loss.
5. Instability Considerations for Massive Star Eruptions
NASA Astrophysics Data System (ADS)
Guzik, J. A.
2005-09-01
We propose a mechanism to explain the observed properties of the giant eruptions of supernova imposters' such as η Car and P Cyg. This mechanism must be episodic, generate a large amount of energy, and be deep seated, in order to lift about 10 solar masses out of the deep gravitational potential well of these massive evolved stars. We suggest that nonradial gravity mode oscillations capable of existing in the core near the hydrogen-burning shell grow slowly to an amplitude sufficient to cause an episode of mixing of hydrogen-rich material downward into hotter denser layers. This mixing generates a burst of nuclear energy production that is responsible for the observed mass ejection and bolometric magnitude increase.
6. X-ray emission of hot massive stars
NASA Astrophysics Data System (ADS)
Oskinova, L.
2014-07-01
Massive hot stars are important cosmic engines that severely influence their environment by powerful stellar wind and strong ionizing radiation. Modern observations of X-ray emission from massive stars provide deep insight into the structure and dynamics of their winds and allow to study the very hot gas in wind blown bubbles. I will review the recent findings on X-ray emission from OB and Wolf-Rayet stars and massive star clusters. While our knowledge about the X-ray emission from massive stars is increasing, a small fraction of massive stars that have strong magnetic fields are often unusual in their X-ray light. Massive star clusters provide an excellent opportunity to study stellar feedback and the hot gas filling the intracluster medium. The most massive stars are often binaries where the stellar winds collide and produce X-ray or even gamma-ray radiation. Finally, I will discuss the progress towards an unified view of stellar winds in single stars and in high mass X-ray binaries.
7. YOUNG STELLAR GROUPS AND THEIR MOST MASSIVE STARS
SciTech Connect
Kirk, Helen; Myers, Philip C.
2011-02-01
We analyze the masses and spatial distributions of 14 young stellar groups in Taurus, Lupus3, ChaI, and IC348. These nearby groups, which typically contain 20-40 members, have membership catalogs complete to {approx}0.02 M{sub sun}, and are sufficiently young that their locations should be similar to where they formed. These groups show five properties seen in clusters having many more stars and much greater surface density of stars: (1) a broad range of masses, (2) a concentration of the most massive star toward the center of the group, (3) an association of the most massive star with a high surface density of lower mass stars, (4) a correlation of the mass of the most massive star with the total mass of the group, and (5) the distribution of a large fraction of the mass in a small fraction of the stars.
8. WHAT SETS THE INITIAL ROTATION RATES OF MASSIVE STARS?
SciTech Connect
Rosen, Anna L.; Krumholz, Mark R.; Ramirez-Ruiz, Enrico
2012-04-01
The physical mechanisms that set the initial rotation rates in massive stars are a crucial unknown in current star formation theory. Observations of young, massive stars provide evidence that they form in a similar fashion to their low-mass counterparts. The magnetic coupling between a star and its accretion disk may be sufficient to spin down low-mass pre-main-sequence (PMS) stars to well below breakup at the end stage of their formation when the accretion rate is low. However, we show that these magnetic torques are insufficient to spin down massive PMS stars due to their short formation times and high accretion rates. We develop a model for the angular momentum evolution of stars over a wide range in mass, considering both magnetic and gravitational torques. We find that magnetic torques are unable to spin down either low-mass or high-mass stars during the main accretion phase, and that massive stars cannot be spun down significantly by magnetic torques during the end stage of their formation either. Spin-down occurs only if massive stars' disk lifetimes are substantially longer or their magnetic fields are much stronger than current observations suggest.
9. The Unevolved Massive Star Content of the Magellanic Clouds
NASA Astrophysics Data System (ADS)
Massey, Philip
2012-10-01
The Magellanic Clouds offer a unique astrophysical laboratory where we can actually obtain an unbiased estimate of the number of unevolved massive stars above a certain mass. Comparing this number with the {known} number of evolved massive stars, such as Wolf-Rayets, yellow supergiants, and red supergiants, provides a hiterto unavailable test of massive star evolutionary theory. We are engaged in a long-term {5 year} effort to characterize the massive star content of select OB associations of the SMC and LMC using spectroscopy with the Magellan 6.5-m telescopes. Here we are asking for a short { 1 sec} SNAPshot of each of 23 OB associations in the F225W filter. These HST data will provide a crucial complement to our ground based data, allowing us to concentrate on the early and mid O-type stars with our spectroscopy, and to recognize close doubles that would otherwise be unrecognized from the ground.
10. On stars, galaxies and black holes in massive bigravity
SciTech Connect
Enander, Jonas; Mörtsell, Edvard E-mail: [email protected]
2015-11-01
In this paper we study the phenomenology of stars and galaxies in massive bigravity. We give parameter conditions for the existence of viable star solutions when the radius of the star is much smaller than the Compton wavelength of the graviton. If these parameter conditions are not met, we constrain the ratio between the coupling constants of the two metrics, in order to give viable conditions for e.g. neutron stars. For galaxies, we put constraints on both the Compton wavelength of the graviton and the conformal factor and coupling constants of the two metrics. The relationship between black holes and stars, and whether the former can be formed from the latter, is discussed. We argue that the different asymptotic structure of stars and black holes makes it unlikely that black holes form from the gravitational collapse of stars in massive bigravity.
11. The Prevalence and Impact of Wolf-Rayet Stars in Emerging Massive Star Clusters
NASA Astrophysics Data System (ADS)
Sokal, Kimberly R.; Johnson, Kelsey E.; Indebetouw, Rémy; Massey, Philip
2016-08-01
We investigate Wolf-Rayet (WR) stars as a source of feedback contributing to the removal of natal material in the early evolution of massive star clusters. Despite previous work suggesting that massive star clusters clear out their natal material before the massive stars evolve into the WR phase, WR stars have been detected in several emerging massive star clusters. These detections suggest that the timescale for clusters to emerge can be at least as long as the time required to produce WR stars (a few million years), and could also indicate that WR stars may be providing the tipping point in the combined feedback processes that drive a massive star cluster to emerge. We explore the potential overlap between the emerging phase and the WR phase with an observational survey to search for WR stars in emerging massive star clusters hosting WR stars. We select candidate emerging massive star clusters from known radio continuum sources with thermal emission and obtain optical spectra with the 4 m Mayall Telescope at Kitt Peak National Observatory and the 6.5 m MMT.4 We identify 21 sources with significantly detected WR signatures, which we term “emerging WR clusters.” WR features are detected in ˜50% of the radio-selected sample, and thus we find that WR stars are commonly present in currently emerging massive star clusters. The observed extinctions and ages suggest that clusters without WR detections remain embedded for longer periods of time, and may indicate that WR stars can aid, and therefore accelerate, the emergence process.
12. The Prevalence and Impact of Wolf–Rayet Stars in Emerging Massive Star Clusters
NASA Astrophysics Data System (ADS)
Sokal, Kimberly R.; Johnson, Kelsey E.; Indebetouw, Rémy; Massey, Philip
2016-08-01
We investigate Wolf–Rayet (WR) stars as a source of feedback contributing to the removal of natal material in the early evolution of massive star clusters. Despite previous work suggesting that massive star clusters clear out their natal material before the massive stars evolve into the WR phase, WR stars have been detected in several emerging massive star clusters. These detections suggest that the timescale for clusters to emerge can be at least as long as the time required to produce WR stars (a few million years), and could also indicate that WR stars may be providing the tipping point in the combined feedback processes that drive a massive star cluster to emerge. We explore the potential overlap between the emerging phase and the WR phase with an observational survey to search for WR stars in emerging massive star clusters hosting WR stars. We select candidate emerging massive star clusters from known radio continuum sources with thermal emission and obtain optical spectra with the 4 m Mayall Telescope at Kitt Peak National Observatory and the 6.5 m MMT.4 We identify 21 sources with significantly detected WR signatures, which we term “emerging WR clusters.” WR features are detected in ˜50% of the radio-selected sample, and thus we find that WR stars are commonly present in currently emerging massive star clusters. The observed extinctions and ages suggest that clusters without WR detections remain embedded for longer periods of time, and may indicate that WR stars can aid, and therefore accelerate, the emergence process.
13. MASSIVE STARS IN THE Cl 1813-178 CLUSTER: AN EPISODE OF MASSIVE STAR FORMATION IN THE W33 COMPLEX
SciTech Connect
Messineo, Maria; Davies, Ben; Figer, Donald F.; Trombley, Christine; Kudritzki, R. P.; Valenti, Elena; Najarro, F.; Michael Rich, R.
2011-05-20
Young massive (M > 10{sup 4} M{sub sun}) stellar clusters are a good laboratory to study the evolution of massive stars. Only a dozen of such clusters are known in the Galaxy. Here, we report about a new young massive stellar cluster in the Milky Way. Near-infrared medium-resolution spectroscopy with UIST on the UKIRT telescope and NIRSPEC on the Keck telescope, and X-ray observations with the Chandra and XMM satellites, of the Cl 1813-178 cluster confirm a large number of massive stars. We detected 1 red supergiant, 2 Wolf-Rayet stars, 1 candidate luminous blue variable, 2 OIf, and 19 OB stars. Among the latter, twelve are likely supergiants, four giants, and the faintest three dwarf stars. We detected post-main-sequence stars with masses between 25 and 100 M{sub sun}. A population with age of 4-4.5 Myr and a mass of {approx}10, 000 M{sub sun} can reproduce such a mixture of massive evolved stars. This massive stellar cluster is the first detection of a cluster in the W33 complex. Six supernova remnants and several other candidate clusters are found in the direction of the same complex.
14. The massive star population of Cygnus OB2
NASA Astrophysics Data System (ADS)
Wright, Nicholas J.; Drew, Janet E.; Mohr-Smith, Michael
2015-05-01
We have compiled a significantly updated and comprehensive census of massive stars in the nearby Cygnus OB2 association by gathering and homogenizing data from across the literature. The census contains 169 primary OB stars, including 52 O-type stars and 3 Wolf-Rayet stars. Spectral types and photometry are used to place the stars in a Hertzsprung-Russell diagram, which is compared to both non-rotating and rotating stellar evolution models, from which stellar masses and ages are calculated. The star formation history and mass function of the association are assessed, and both are found to be heavily influenced by the evolution of the most massive stars to their end states. We find that the mass function of the most massive stars is consistent with a universal' power-law slope of Γ = 1.3. The age distribution inferred from stellar evolutionary models with rotation and the mass function suggest the majority of star formation occurred more or less continuously between 1 and 7 Myr ago, in agreement with studies of low- and intermediate-mass stars in the association. We identify a nearby young pulsar and runaway O-type star that may have originated in Cyg OB2 and suggest that the association has already seen its first supernova. Finally we use the census and mass function to calculate the total mass of the association of 16 500^{+3800}_{-2800} M⊙, at the low end, but consistent with, previous estimates of the total mass of Cyg OB2. Despite this Cyg OB2 is still one of the most massive groups of young stars known in our Galaxy making it a prime target for studies of star formation on the largest scales.
15. The Final Stages of Massive Star Evolution and Their Supernovae
NASA Astrophysics Data System (ADS)
Heger, Alexander
In this chapter I discuss the final stages in the evolution of massive stars - stars that are massive enough to burn nuclear fuel all the way to iron group elements in their core. The core eventually collapses to form a neutron star or a black hole when electron captures and photo-disintegration reduce the pressure support to an extent that it no longer can hold up against gravity. The late burning stages of massive stars are a rich subject by themselves, and in them many of the heavy elements in the universe are first generated. The late evolution of massive stars strongly depends on their mass, and hence can be significantly effected by mass loss due to stellar winds and episodic mass loss events - a critical ingredient that we do not know as well as we would like. If the star loses all the hydrogen envelope, a Type I supernova results, if it does not, a Type II supernova is observed. Whether the star makes neutron star or a black hole, or a neutron star at first and a black hole later, and how fast they spin largely affects the energetics and asymmetry of the observed supernova explosion. Beyond photon-based astronomy, other than the sun, a supernova (SN 1987) has been the only object in the sky we ever observed in neutrinos, and supernovae may also be the first thing we will ever see in gravitational wave detectors like LIGO. I conclude this chapter reviewing the deaths of the most massive stars and of Population III stars.
16. Infrared galaxies - Evolutionary stages of massive star formation
NASA Technical Reports Server (NTRS)
Harwit, M.; Pacini, F.
1975-01-01
We cite evidence which indicates that infrared galaxies may represent evolutionary stages during which a large number of massive stars are being formed. The lifetimes of these stars would be rather short (1-10 million years), and the resulting supernova explosions could account for the level of nonthermal activity which often accompanies the thermal infrared emission.
17. Massive binary stars and self-enrichment of globular clusters
NASA Astrophysics Data System (ADS)
Izzard, R. G.; de Mink, S. E.; Pols, O. R.; Langer, N.; Sana, H.; de Koter, A.
~Globular clusters contain many stars with surface abundance patterns indicating contributions from hydrogen burning products, as seen in the anti-correlated elemental abundances of e.g. sodium and oxygen, and magnesium and aluminium. Multiple generations of stars can explain this phenomenon, with the second generation forming from a mixture of pristine gas and ejecta from the first generation. We show that massive binary stars may be a source of much of the material that makes this second generation of stars. Mass transfer in binaries is often non-conservative and the ejected matter moves slowly enough that it can remain inside a globular cluster and remain available for subsequent star formation. Recent studies show that there are more short-period massive binaries than previously thought, hence also more stars that interact and eject nuclear-processed material.
18. Dynamical ejections of massive stars from young star clusters under diverse initial conditions
NASA Astrophysics Data System (ADS)
Oh, Seungkyung; Kroupa, Pavel
2016-05-01
We study the effects that initial conditions of star clusters and their massive star population have on dynamical ejections of massive stars from star clusters up to an age of 3 Myr. We use a large set of direct N-body calculations for moderately massive star clusters (Mecl ≈ 103.5 M⊙). We vary the initial conditions of the calculations, such as the initial half-mass radius of the clusters, initial binary populations for massive stars and initial mass segregation. We find that the initial density is the most influential parameter for the ejection fraction of the massive systems. The clusters with an initial half-mass radius rh(0) of 0.1 (0.3) pc can eject up to 50% (30)% of their O-star systems on average, while initially larger (rh(0) = 0.8 pc) clusters, that is, lower density clusters, eject hardly any OB stars (at most ≈ 4.5%). When the binaries are composed of two stars of similar mass, the ejections are most effective. Most of the models show that the average ejection fraction decreases with decreasing stellar mass. For clusters that are efficient at ejecting O stars, the mass function of the ejected stars is top-heavy compared to the given initial mass function (IMF), while the mass function of stars that remain in the cluster becomes slightly steeper (top-light) than the IMF. The top-light mass functions of stars in 3 Myr old clusters in our N-body models agree well with the mean mass function of young intermediate-mass clusters in M 31, as reported previously. This implies that the IMF of the observed young clusters is the canonical IMF. We show that the multiplicity fraction of the ejected massive stars can be as high as ≈ 60%, that massive high-order multiple systems can be dynamically ejected, and that high-order multiples become common especially in the cluster. We also discuss binary populations of the ejected massive systems. Clusters that are initially not mass-segregated begin ejecting massive stars after a time delay that is caused by mass
19. Magnetically regulated fragmentation of a massive, dense, and turbulent clump
NASA Astrophysics Data System (ADS)
Fontani, F.; Commerçon, B.; Giannetti, A.; Beltrán, M. T.; Sánchez-Monge, A.; Testi, L.; Brand, J.; Caselli, P.; Cesaroni, R.; Dodson, R.; Longmore, S.; Rioja, M.; Tan, J. C.; Walmsley, C. M.
2016-09-01
Massive stars, multiple stellar systems, and clusters are born of the gravitational collapse of massive, dense, gaseous clumps, and the way these systems form strongly depends on how the parent clump fragments into cores during collapse. Numerical simulations show that magnetic fields may be the key ingredient in regulating fragmentation. Here we present ALMA observations at ~ 0.25'' resolution of the thermal dust continuum emission at ~ 278 GHz towards a turbulent, dense, and massive clump, IRAS 16061-5048c1, in a very early evolutionary stage. The ALMA image shows that the clump has fragmented into many cores along a filamentary structure. We find that the number, the total mass, and the spatial distribution of the fragments are consistent with fragmentation dominated by a strong magnetic field. Our observations support the theoretical prediction that the magnetic field plays a dominant role in the fragmentation process of massive turbulent clumps.
20. Instabilities in the Envelopes and Winds of Very Massive Stars
NASA Astrophysics Data System (ADS)
Owocki, Stanley P.
The high luminosity of Very Massive Stars (VMS) means that radiative forces play an important, dynamical role both in the structure and stability of their stellar envelope, and in driving strong stellar-wind mass loss. Focusing on the interplay of radiative flux and opacity, with emphasis on key distinctions between continuum vs. line opacity, this chapter reviews instabilities in the envelopes and winds of VMS. Specifically, we discuss how: (1) the iron opacity bump can induce an extensive inflation of the stellar envelope; (2) the density dependence of mean opacity leads to strange mode instabilities in the outer envelope; (3) desaturation of line-opacity by acceleration of near-surface layers initiates and sustains a line-driven stellar wind outflow; (4) an associated line-deshadowing instability leads to extensive small-scale structure in the outer regions of such line-driven winds; (5) a star with super-Eddington luminosity can develop extensive atmospheric structure from photon bubble instabilities, or from stagnation of flow that exceeds the "photon tiring" limit; (6) the associated porosity leads to a reduction in opacity that can regulate the extreme mass loss of such continuum-driven winds. Two overall themes are the potential links of such instabilities to Luminous Blue Variable (LBV) stars, and the potential role of radiation forces in establishing the upper mass limit of VMS.
1. Formation and Evolution of Massive Stars: Current Surveys
NASA Astrophysics Data System (ADS)
de Koter, A.
2016-10-01
The advent of multi-object spectrographs on 8-10 m class telescopes has provided the opportunity to perform detailed atmospheric analysis of samples of several hundreds of massive stars, prior studies being limited to several tens of objects at most. These analyses have highlighted some serious problems regarding our understanding of massive-star evolution. A central theme in the findings is the prominent role of multiplicity, with the majority of high-mass stars being in close binary systems of which the components will interact at some point in their lives.
2. VLT-Flames Tarantula Survey and Multiplicity of Massive Stars
NASA Astrophysics Data System (ADS)
Sana, H.
2013-06-01
The VLT-Flames Tarantula Survey (VFTS) has obtained optical spectroscopy of over 800 OB and Wolf-Rayet stars in the 30 Doradus region with the aim to investigate a number of questions regarding the formation, evolution and final fate of the most massive stars and the dynamics of the region. In this presentation, I will review some of the most important results obtained by the VFTS so far. The multi-epoch strategy was designed to identify spectroscopic binaries, and I will describe the binary properties in the 30 Dor region in the broader context of our knowledge of the multiplicity of massive stars.
3. Eccentricity boost of stars around shrinking massive black hole binaries
NASA Astrophysics Data System (ADS)
Iwasa, Mao; Seto, Naoki
2016-06-01
Based on a simple geometrical approach, we analyze the evolution of the Kozai-Lidov mechanism for stars around shrinking massive black hole binaries on circular orbits. We find that, due to a peculiar bifurcation pattern induced by the Newtonian potential of stellar clusters, the orbit of stars could become highly eccentric. This transition occurs abruptly for stars with small initial eccentricities. The approach presented in this paper may be useful for studying the Kozai-Lidov mechanism in various astrophysical contexts.
4. The evolutionary tracks of young massive star clusters
SciTech Connect
Pfalzner, S.; Steinhausen, M.; Vincke, K.; Menten, K.; Parmentier, G.
2014-10-20
Stars mostly form in groups consisting of a few dozen to several ten thousand members. For 30 years, theoretical models have provided a basic concept of how such star clusters form and develop: they originate from the gas and dust of collapsing molecular clouds. The conversion from gas to stars being incomplete, the leftover gas is expelled, leading to cluster expansion and stars becoming unbound. Observationally, a direct confirmation of this process has proved elusive, which is attributed to the diversity of the properties of forming clusters. Here we take into account that the true cluster masses and sizes are masked, initially by the surface density of the background and later by the still present unbound stars. Based on the recent observational finding that in a given star-forming region the star formation efficiency depends on the local density of the gas, we use an analytical approach combined with N-body simulations to reveal evolutionary tracks for young massive clusters covering the first 10 Myr. Just like the Hertzsprung-Russell diagram is a measure for the evolution of stars, these tracks provide equivalent information for clusters. Like stars, massive clusters form and develop faster than their lower-mass counterparts, explaining why so few massive cluster progenitors are found.
5. The Deaths of Very Massive Stars
NASA Astrophysics Data System (ADS)
Woosley, Stan. E.; Heger, Alexander
The theory underlying the evolution and death of stars heavier than 10 M⊙ on the main sequence is reviewed with an emphasis upon stars much heavier than 30 M⊙. These are stars that, in the absence of substantial mass loss, are expected to either produce black holes when they die, or, for helium cores heavier than about 35 M⊙, encounter the pair instability. A wide variety of outcomes is possible depending upon the initial composition of the star, its rotation rate, and the physics used to model its evolution. These stars can produce some of the brightest supernovae in the universe, but also some of the faintest. They can make gamma-ray bursts or collapse without a whimper. Their nucleosynthesis can range from just CNO to a broad range of elements up to the iron group. Though rare nowadays, they probably played a disproportionate role in shaping the evolution of the universe following the formation of its first stars.
6. Discovery of X-ray pulsations from a massive star.
PubMed
Oskinova, Lidia M; Nazé, Yael; Todt, Helge; Huenemoerder, David P; Ignace, Richard; Hubrig, Swetlana; Hamann, Wolf-Rainer
2014-01-01
X-ray emission from stars much more massive than the Sun was discovered only 35 years ago. Such stars drive fast stellar winds where shocks can develop, and it is commonly assumed that the X-rays emerge from the shock-heated plasma. Many massive stars additionally pulsate. However, hitherto it was neither theoretically predicted nor observed that these pulsations would affect their X-ray emission. All X-ray pulsars known so far are associated with degenerate objects, either neutron stars or white dwarfs. Here we report the discovery of pulsating X-rays from a non-degenerate object, the massive B-type star ξ(1) CMa. This star is a variable of β Cep-type and has a strong magnetic field. Our observations with the X-ray Multi-Mirror (XMM-Newton) telescope reveal X-ray pulsations with the same period as the fundamental stellar oscillations. This discovery challenges our understanding of stellar winds from massive stars, their X-ray emission and their magnetism. PMID:24892504
7. Searching for Massive Star Clusters around Luminous Blue Variables
NASA Astrophysics Data System (ADS)
Stensland, Jared; Edwards, M. L.; Mikles, V. J.
2011-01-01
We present a method to search for the massive birth clusters of Luminous Blue Variables (LBVs). Using theoretical absolute magnitudes of early-type stars, we calculated expected color and magnitude limits for candidate massive stars at the distance and reddening of the Pistol Star and FMM 362 in the Quintuplet. We then applied these cuts to stars found in the 2MASS catalog surrounding the LBVs. By using a well-characterized cluster, we were able to confirm the method's effectiveness and determine the color and magnitude criteria that eliminated the highest number of false candidates while recovering the largest number of known massive cluster members. We then calculated and applied similar cuts to stars within a 1pc radius of WRA 751 to confirm its cluster, Teutsch 143a, discovered by Pasquali et al (2006) and later investigated by Froebrich et al (2008). We used our method to select 22 strong candidate massive cluster stars, 18 medium confidence candidates and 39 weak candidates, categorized based on their colors and magnitudes. These stars are prime candidates for follow-up spectroscopy to determine their spectral types and confirm cluster membership. We plan to apply a similar method to other LBVs without known birth clusters.
8. Discovery of X-ray pulsations from a massive star.
PubMed
Oskinova, Lidia M; Nazé, Yael; Todt, Helge; Huenemoerder, David P; Ignace, Richard; Hubrig, Swetlana; Hamann, Wolf-Rainer
2014-06-03
X-ray emission from stars much more massive than the Sun was discovered only 35 years ago. Such stars drive fast stellar winds where shocks can develop, and it is commonly assumed that the X-rays emerge from the shock-heated plasma. Many massive stars additionally pulsate. However, hitherto it was neither theoretically predicted nor observed that these pulsations would affect their X-ray emission. All X-ray pulsars known so far are associated with degenerate objects, either neutron stars or white dwarfs. Here we report the discovery of pulsating X-rays from a non-degenerate object, the massive B-type star ξ(1) CMa. This star is a variable of β Cep-type and has a strong magnetic field. Our observations with the X-ray Multi-Mirror (XMM-Newton) telescope reveal X-ray pulsations with the same period as the fundamental stellar oscillations. This discovery challenges our understanding of stellar winds from massive stars, their X-ray emission and their magnetism.
9. Massive star evolution: luminous blue variables as unexpected supernova progenitors
NASA Astrophysics Data System (ADS)
Groh, J. H.; Meynet, G.; Ekström, S.
2013-02-01
Stars more massive than about 8 M⊙ end their lives as a supernova (SN), an event of fundamental importance Universe-wide. Theoretically, these stars have been expected to be either at the red supergiant, blue supergiant, or Wolf-Rayet stage before the explosion. We performed coupled stellar evolution and atmospheric modeling of stars with initial masses between 20 M⊙ and 120 M⊙. We found that the 20 M⊙ and 25 M⊙ rotating models, before exploding as SN, have spectra that do not resemble any of the aforementioned classes of massive stars. Rather, they have remarkable similarities with rare, unstable massive stars known as luminous blue variables (LBV). While observations show that some SNe seem to have had LBVs as progenitors, no theoretical model had yet predicted that a star could explode at this stage. Our models provide theoretical support for relatively low-luminosity LBVs exploding as SN in the framework of single stellar evolution. This is a significant shift in paradigm, meaning that a fraction of LBVs could be the end stage of massive star evolution, rather than a transitory evolutionary phase. We suggest that type IIb SN could have LBV as progenitors, and a prime example could be SN 2008ax.
10. Photon Bubbles in Young Massive Stars
NASA Astrophysics Data System (ADS)
Turner, N. J.; Yorke, H. W.; Socrates, A.; Blaes, O. M.
2004-12-01
Spectroscopic studies indicate that gas in the photospheres of young O stars moves at speeds up to the sound speed. We show, using two-dimensional radiation MHD calculations and results from a local linear analysis, that the motions may be due to photon bubble instability if young O stars have magnetic fields.
11. INTERNAL GRAVITY WAVES IN MASSIVE STARS: ANGULAR MOMENTUM TRANSPORT
SciTech Connect
Rogers, T. M.; Lin, D. N. C.; McElwaine, J. N.; Lau, H. H. B. E-mail: [email protected] E-mail: [email protected]
2013-07-20
We present numerical simulations of internal gravity waves (IGW) in a star with a convective core and extended radiative envelope. We report on amplitudes, spectra, dissipation, and consequent angular momentum transport by such waves. We find that these waves are generated efficiently and transport angular momentum on short timescales over large distances. We show that, as in Earth's atmosphere, IGW drive equatorial flows which change magnitude and direction on short timescales. These results have profound consequences for the observational inferences of massive stars, as well as their long term angular momentum evolution. We suggest IGW angular momentum transport may explain many observational mysteries, such as: the misalignment of hot Jupiters around hot stars, the Be class of stars, Ni enrichment anomalies in massive stars, and the non-synchronous orbits of interacting binaries.
12. Magnetic Fields in Massive Stars, Their Winds, and Their Nebulae
NASA Astrophysics Data System (ADS)
Walder, Rolf; Folini, Doris; Meynet, Georges
2012-05-01
Massive stars are crucial building blocks of galaxies and the universe, as production sites of heavy elements and as stirring agents and energy providers through stellar winds and supernovae. The field of magnetic massive stars has seen tremendous progress in recent years. Different perspectives—ranging from direct field measurements over dynamo theory and stellar evolution to colliding winds and the stellar environment—fruitfully combine into a most interesting and still evolving overall picture, which we attempt to review here. Zeeman signatures leave no doubt that at least some O- and early B-type stars have a surface magnetic field. Indirect evidence, especially non-thermal radio emission from colliding winds, suggests many more. The emerging picture for massive stars shows similarities with results from intermediate mass stars, for which much more data are available. Observations are often compatible with a dipole or low order multi-pole field of about 1 kG (O-stars) or 300 G to 30 kG (Ap/Bp stars). Weak and unordered fields have been detected in the O-star ζ Ori A and in Vega, the first normal A-type star with a magnetic field. Theory offers essentially two explanations for the origin of the observed surface fields: fossil fields, particularly for strong and ordered fields, or different dynamo mechanisms, preferentially for less ordered fields. Numerical simulations yield the first concrete stable (fossil) field configuration, but give contradictory results as to whether dynamo action in the radiative envelope of massive main sequence stars is possible. Internal magnetic fields, which may not even show up at the stellar surface, affect stellar evolution as they lead to a more uniform rotation, with more slowly rotating cores and faster surface rotation. Surface metallicities may become enhanced, thus affecting the mass-loss rates.
13. Massive Stars and Their Possible Impacts in Globular Clusters
NASA Astrophysics Data System (ADS)
Decressin, Thibaut
2012-05-01
Globular clusters exhibit peculiar chemical patterns where Fe and heavy elements abundances stay constant inside a given cluster while light elements (Li to Al) show strong star-to-star variations. This peculiar chemical pattern can be explained by self-pollution of the intracluster gas occurring in the early evolution of clusters. Here I present the possible strong impact of fast rotating massive stars on clusters evolution. First providing they rotate initially fast enough, these stars can reach the break-up velocity during the main sequence and matter will be ejected from the equator at low velocity. Rotation-induced mixing will also bring matter from the convective core to the surface. From this ejected matter loaded in H-burning material a second generation of stars will born. The chemical pattern of these second generation stars are similar to the one observed for stars in globular cluster with abundance anomalies in light elements. Then during the explosion as supernovae the massive stars will also clear the cluster of the remaining gas. One important feature of globular clusters observed today is that 50 to 80% of the low mass stars still evolving in the cluster are second generation starts whereas, with a standard IMF, these stars should be at most 10% of the cluster stars. This strong discrepancy can be solved if the proto-globular clusters were more massive (up to a factor 20-30) and mass-segregated during their formation. In this case a strong loss of first generation stars occupying the outer part of the cluster is possible through the dynamical history of the cluster.
14. Astronomers Gain Important Insight on How Massive Stars Form
NASA Astrophysics Data System (ADS)
2006-09-01
Astronomers using the National Science Foundation's Very Large Array (VLA) radio telescope have discovered key evidence that may help them figure out how very massive stars can form. Young Star Graphic Artist's Conception of Young Star Showing Motions Detected in G24 A1: (1) Infall toward torus, (2) Rotation and (3) outflow. CREDIT: Bill Saxton, NRAO/AUI/NSF Click on image for larger graphic file (JPEG, 129K) "We think we know how stars like the Sun are formed, but there are major problems in determining how a star 10 times more massive than the Sun can accumulate that much mass. The new observations with the VLA have provided important clues to resolving that mystery," said Maria Teresa Beltran, of the University of Barcelona in Spain. Beltran and other astronomers from Italy and Hawaii studied a young, massive star called G24 A1 about 25,000 light-years from Earth. This object is about 20 times more massive than the Sun. The scientists reported their findings in the September 28 issue of the journal Nature. Stars form when giant interstellar clouds of gas and dust collapse gravitationally, compacting the material into what becomes the star. While astronomers believe they understand this process reasonably well for smaller stars, the theoretical framework ran into a hitch with larger stars. "When a star gets up to about eight times the mass of the Sun, it pours out enough light and other radiation to stop the further infall of material," Beltran explained. "We know there are many stars bigger than that, so the question is, how do they get that much mass?" One idea is that infalling matter forms a disk whirling around the star. With most of the radiation escaping without hitting the disk, material can continue to fall into the star from the disk. According to this model, some material will be flung outward along the rotation axis of the disk into powerful outflows. "If this model is correct, there should be material falling inward, rushing outward and rotating
15. RCW 108: Massive Young Stars Trigger Stellar Birth
NASA Technical Reports Server (NTRS)
2008-01-01
RCW 108 is a region where stars are actively forming within the Milky Way galaxy about 4,000 light years from Earth. This is a complicated region that contains young star clusters, including one that is deeply embedded in a cloud of molecular hydrogen. By using data from different telescopes, astronomers determined that star birth in this region is being triggered by the effect of nearby, massive young stars.
This image is a composite of X-ray data from NASA's Chandra X-ray Observatory (blue) and infrared emission detected by NASA's Spitzer Space Telescope (red and orange). More than 400 X-ray sources were identified in Chandra's observations of RCW 108. About 90 percent of these X-ray sources are thought to be part of the cluster and not stars that lie in the field-of-view either behind or in front of it. Many of the stars in RCW 108 are experiencing the violent flaring seen in other young star-forming regions such as the Orion nebula. Gas and dust blocks much of the X-rays from the juvenile stars located in the center of the image, explaining the relative dearth of Chandra sources in this part of the image.
The Spitzer data show the location of the embedded star cluster, which appears as the bright knot of red and orange just to the left of the center of the image. Some stars from a larger cluster, known as NGC 6193, are also visible on the left side of the image. Astronomers think that the dense clouds within RCW 108 are in the process of being destroyed by intense radiation emanating from hot and massive stars in NGC 6193.
Taken together, the Chandra and Spitzer data indicate that there are more massive star candidates than expected in several areas of this image. This suggests that pockets within RCW 108 underwent localized episodes of star formation. Scientists predict that this type of star formation is triggered by the effects of radiation from bright, massive stars such as those in NGC 6193. This radiation may cause the interior of gas
16. Gamma Ray Emission from Chaotic Winds of Massive Stars
NASA Technical Reports Server (NTRS)
White, Richard L.
2000-01-01
The purpose of this proposal was to search for gamma-ray emission from the winds of hot, massive stars. According to our theoretical calculations, shocks in the winds of massive stars accelerate particles to high energies. The high-energy particles emit synchrotron radio emission (observed by ground-based radio telescopes) and high-energy gamma-ray emission that we predicted should be detectable by the EGRET instrument on the Compton Gamma Ray Observatory between 100 MeV and a few GeV. We obtained EGRET from phases 1, 2, and 3 of the Cygnus OB2 association, a cluster of massive, young stars, to search for this gamma-ray emission. The data products and analysis show a source consistent with the position of Cyg OB2 with approximately the predicted count rate and spectrum.
17. Light element production by low energy nuclei from massive stars
NASA Technical Reports Server (NTRS)
Vangioni-Flam, E.; Casse, M.; Ramaty, R.
1997-01-01
The Orion complex is a source of gamma rays attributed to the de-excitation of fast carbon and oxygen nuclei excited through interactions with ambient hydrogen and helium. This has consequences for the production and evolution of light isotopes in the Galaxy, as massive stars appear as prolific sources of C-O rich low energy nuclei. The different stages of massive star evolution are considered in relation to the acceleration of nuclei to moderate energies. It is concluded that the low energy nuclear component originating from massive stars plays a larger role than the usual Galactic cosmic rays in shaping the evolution of Li-6, Be-9, B-10 and B-11, especially in the early Galactic evolution. The enhancement of the B-11/B-10 ratio observed in meteorites and in the interstellar medium is attributed to the interaction of low energy carbon nuclei with ambient H and to a lesser degree, to neutrino spallation.
18. The Formation Of Massive Stars And The Effects Of Rotation On Star Formation
NASA Astrophysics Data System (ADS)
Maeder, A.
2011-11-01
We first review the current debates about massive star formation over the last decade. Then we concentrate on the accretion scenario, emphasizing the evidences in favor of it. We study the basic properties of the accretion scenario in the spherical case. In the case of massive stars, the free-fall time is longer than the Kelvin-Helmholtz timescale, so that the massive stars in formation reach thermal equilibrium before the accretion is completed. This is why the history of the accretion rates for massive stars is so critical. We derive analytically the typical accretion rates, their upper and lower limits, showing the importance of dust properties. We examine the basic properties of the disk, their luminosity and temperature in the stationary approximation, as well as their various components. The results of some recent numerical models are discussed with a particular attention to the effects that favor accretion on the central body relatively to the case of spherical accretion. These effects strongly influence the final stellar mass resulting from a collapsing clump in a cloud. We also show some properties of the pre-main sequence tracks of massive stars in the Hertzsprung-Russell diagram. During the first part of their evolution up to a mass of about 3M⊙ the forming stars are overluminous, then they are strongly underluminous (with respect to the zero age main sequence) up to a mass of about 10M⊙ until they adjust after a slight overluminosity to the main sequence values. We consider some rotational properties related to star formation. The angular momentum has to be reduced by a factor of about 106 during star formation. Some effects contributing to this reduction have been studied particularly in the case of low- and intermediate-mass stars: disk locking and magnetic braking. We also discuss the case of massive stars and emphasize the effects of the gravity darkening of rotating stars that may favor the accretion from the disk of massive stars in formation.
19. Massive-Star Magnetospheres: Now in 3-D!
NASA Astrophysics Data System (ADS)
Townsend, Richard
Magnetic fields are unexpected in massive stars, due to the absence of a dynamo convection zone beneath their surface layers. Nevertheless, kilogauss-strength, ordered fields were detected in a small subset of these stars over three decades ago, and the intervening years have witnessed the steady expansion of this subset. A distinctive feature of magnetic massive stars is that they harbor magnetospheres --- circumstellar environments where the magnetic field interacts strongly with the star's radiation-driven wind, confining it and channelling it into energetic shocks. A wide range of observational signatures are associated with these magnetospheres, in diagnostics ranging from X-rays all the way through to radio emission. Moreover, these magnetospheres can play an important role in massive-star evolution, by amplifying angular momentum loss in the wind. Recent progress in understanding massive-star magnetospheres has largely been driven by magnetohydrodynamical (MHD) simulations. However, these have been restricted to two- dimensional axisymmetric configurations, with three-dimensional configurations possible only in certain special cases. These restrictions are limiting further progress; we therefore propose to develop completely general three-dimensional models for the magnetospheres of massive stars, on the one hand to understand their observational properties and exploit them as plasma-physics laboratories, and on the other to gain a comprehensive understanding of how they influence the evolution of their host star. For weak- and intermediate-field stars, the models will be based on 3-D MHD simulations using a modified version of the ZEUS-MP code. For strong-field stars, we will extend our existing Rigid Field Hydrodynamics (RFHD) code to handle completely arbitrary field topologies. To explore a putative 'photoionization-moderated mass loss' mechanism for massive-star magnetospheres, we will also further develop a photoionization code we have recently
20. Near-Infrared Mass Loss Diagnostics for Massive Stars
NASA Technical Reports Server (NTRS)
Sonneborn, George; Bouret, J. C.
2010-01-01
Stellar wind mass loss is a key process which modifies surface abundances, luminosities, and other physical properties of hot, massive stars. Furthermore, mass loss has to be understood quantitatively in order to accurately describe and predict massive star evolution. Two urgent problems have been identified that challenge our understanding of line-driven winds, the so-called weak-wind problem and wind clumping. In both cases, mass-loss rates are drastically lower than theoretically expected (up to a factor 1001). Here we study how the expected spectroscopic capabilities of the James Webb Space Telescope (JWST), especially NIRSpec, could be used to significantly improve constraints on wind density structures (clumps) and deep-seated phenomena in stellar winds of massive stars, including OB, Wolf-Rayet and LBV stars. Since the IR continuum of objects with strong winds is formed in the wind, IR lines may sample different depths inside the wind than UV-optical lines and provide new information about the shape of the velocity field and clumping properties. One of the most important applications of IR line diagnostics will be the measurement of mass-loss rates in massive stars with very weak winds by means of the H I Bracket alpha line, which has been identified as one of the most promising diagnostics for this problem.
1. OBSERVATIONAL SIGNATURES OF CONVECTIVELY DRIVEN WAVES IN MASSIVE STARS
SciTech Connect
Aerts, C.; Rogers, T. M.
2015-06-20
We demonstrate observational evidence for the occurrence of convectively driven internal gravity waves (IGWs) in young massive O-type stars observed with high-precision CoRoT space photometry. This evidence results from a comparison between velocity spectra based on two-dimensional hydrodynamical simulations of IGWs in a differentially rotating massive star and the observed spectra. We also show that the velocity spectra caused by IGWs may lead to detectable line-profile variability and explain the occurrence of macroturbulence in the observed line profiles of OB stars. Our findings provide predictions that can readily be tested by including a sample of bright, slowly and rapidly rotating OB-type stars in the scientific program of the K2 mission accompanied by high-precision spectroscopy and their confrontation with multi-dimensional hydrodynamic simulations of IGWs for various masses and ages.
2. Peering to the Heart of Massive Star Birth - V. Highest Priority Massive Protostars
NASA Astrophysics Data System (ADS)
Tan, Jonathan
2015-10-01
As part of an on-going, multi-year program to build up a sample of massive and intermediate-mass protostars that are observed across MIR and FIR bands to test theoretical models of massive star formation, we propose to observe about 15 highest priority massive protostar targets with SOFIA-FORCAST with this Regular Program proposal. Especially the unique 37 micron imaging can help reveal thermal emission from outflow cavities and the relative fluxes from the near and far-facing sides probes the amount of dense gas in the immediate vicinity of the protostar. Core Accretion models generally involve larger quantities of such gas than Competitive Accretion models. We will compare observational results against specific predictions of a grid of radiative transfer simulations developed for the Turbulent Core Model of massive star formation.
3. Herbig Ae/Be stars - Intermediate-mass stars surrounded by massive circumstellar accretion disks
NASA Technical Reports Server (NTRS)
Hillenbrand, Lynne A.; Strom, Stephen E.; Vrba, Frederick J.; Keene, Jocelyn
1992-01-01
The proposition that Herbig Ae/Be stars are young intermediate mass stars surrounded by optically thick accretion disks is explored. From a study of 47 such objects, a subset of 30 stars is identified whose spectral energy distributions can be interpreted convincingly in terms of pre-main sequence stars surrounded by massive optically thick circumstellar accretion disks. Constraints on the physical properties of the disks, such as size, mass, accretion rate, lifetime, and radial structure are derived from the photometric data.
4. Massive Star Formation: Characterising Infall and Outflow in dense cores.
NASA Astrophysics Data System (ADS)
Akhter, Shaila; Cunningham, Maria; Harvey-Smith, Lisa; Jones, Paul Andrew; Purcell, Cormac; Walsh, Andrew John
2015-08-01
Massive stars are some of the most important objects in the Universe, shaping the evolution of galaxies, creating chemical elements, and hence shaping the evolution of the Universe. However, the processes by which they form, and how they shape their environment during their birth processes, are not well understood. We are using NH3 data from the "The H2O Southern Galactic Plane Survey" (HOPS) to define the positions of dense cores/clumps of gas in the southern Galactic plane that are likely to form stars. Due to its effective critical density, NH3 can detect massive star forming regions effectively compared to other tracers. We did a comparative study with different methods for finding clumps and found Fellwalker as the best. We found ~ 10% of the star forming clumps with multiple components and ~ 90% clumps with single component along the line of sight. Then, using data from the "The Millimetre Astronomy Legacy Team 90 GHz" (MALT90) survey, we search for the presence of infall and outflow associated with these cores. We will subsequently use the "3D Molecular Line Radiative Transfer Code" (MOLLIE) to constrain properties of the infall and outflow, such as velocity and mass flow. The aim of the project is to determine how common infall and outflow are in star forming cores, hence providing valuable constraints on the timescales and physical process involved in massive star formation.
5. Dynamical Models for High-Energy Emission from Massive Stars
NASA Astrophysics Data System (ADS)
Owocki, Stanley %FAA(University of Delaware)
Massive stars are prominent sources of X-rays and gamma-rays detected by both targeted and survey observations from orbiting telescopes like Chandra, XMM/Newton, RXTE, and Fermi. Such high-energy emissions represent key probes of the dynamics of massive-star mass loss, and their penetration through many magnitudes of visible interstellar extinction makes them effective beacons of massive stars in distant reaches of the Galaxy, and in young, active star-forming regions. The project proposed here will develop a comprehensive theoretical framework for interpreting both surveys and targeted observations of high-energy emission from massive stars. It will build on our team's extensive experience in both theoretical models and observational analyses for three key types of emission mechanisms in the stellar wind outflows of these stars, namely: 1) Embedded Wind Shocks (EWS) arising from internal instabilities in the wind driving; 2) shocks in Colliding Wind Binary (CWB) systems; and 3) High-Mass X-ray Binaries (HMXB) systems with interaction between massive-star wind with a compact companion (neutron star or black hole). Taking advantage of commonalities in the treatment of radiative driving, hydrodynamics, shock heating and cooling, and radiation transport, we will develop radiation hydrodynamical models for the key observational signatures like energy distribution, emission line spectrum, and variability, with an emphasis on how these can be used in affiliated analyses of both surveys like the recent Chandra mapping of the Carina association, and targeted observations of galactic X-ray and gamma-ray sources associated with each of the above specific model types. The promises of new clumping-insensitive diagnostics of mass loss rates, and the connection to mass transfer and binarity, all have broad relevance for understanding the origin, evolution, and fate of massive stars, in concert with elements of NASA's Strategic Subgoal 3D. Building on our team's expertise, the
6. On the evolution and explosion of massive stars
SciTech Connect
Limongi, Marco; Chieffi, Alessandro
2008-05-21
We review our recent progresses on the presupernova evolution of massive stars in the range 11-120 M{sub {center_dot}} of solar metallicity. Special attention will be devoted to the effect of the mass loss rate during the Wolf-Rayet stages in determining the structure and the physical properties of the star prior the supernova explosion. We also discuss the explosive yields and the initial mass-remnant mass relation in the framework of the kinetic bomb induced explosion and hence the contribution of these stars to the global chemical enrichment of the interstellar medium.
7. Massive Star Clusters in Dwarf Galaxies
NASA Astrophysics Data System (ADS)
Larsen, Soeren
2015-08-01
Dwarf galaxies are often characterized by very high globular cluster specific frequencies, in some cases exceeding that of the Milky Way by a factor of 100 or more. Moreover, the GCs are typically much more metal-poor than the bulk of the field stars, so that a substantial fraction (up to 20-25% or more) of all metal-poor stars in some dwarf galaxies are associated with GCs. The metal-poor components of these galaxies thus represent an extreme case of the "specific frequency problem". In this talk I will review the current status of our understanding of GC systems in dwarf galaxies. Particular emphasis will be placed on the implications of the high GC specific frequencies for the amount of mass loss the clusters could have experienced and the constraints this provides on theories for the origin of multiple populations in globular clusters.
8. The Brief Lives of Massive Stars as Witnessed by Interferometry
NASA Astrophysics Data System (ADS)
Hummel, C.
2014-09-01
Massive stars present the newest and perhaps most challenging opportunity for long baseline interferometry to excel. Large distances require high angular resolution both to study the means of accreting enough mass in a short time and to split new-born multiples into their components for the determination of their fundamental parameters. Dust obscuration of young stellar objects requires interferometry in the mid-infrared, while post-main-sequence stellar phases require high-precision measurements to challenge stellar evolution models. I will summarize my recent work on modeling mid-IR observations of a massive YSO in NGC 3603, and on the derivation of masses and luminosities of a massive hot supergiant star in another star-forming region in Orion. Challenges presented themselves when constraining the geometry of a hypothetical accretion disk as well as obtaining spectroscopy matching the interferometric precision when working with only a few photospheric lines. As a rapidly evolving application of interferometry, massive stars have a bright future.
9. The spectroscopic Hertzsprung-Russell diagram of Galactic massive stars
NASA Astrophysics Data System (ADS)
Castro, N.; Fossati, L.; Langer, N.; Simón-Díaz, S.; Schneider, F. R. N.; Izzard, R. G.
2014-10-01
The distribution of stars in the Hertzsprung-Russell diagram narrates their evolutionary history and directly assesses their properties. Placing stars in this diagram however requires the knowledge of their distances and interstellar extinctions, which are often poorly known for Galactic stars. The spectroscopic Hertzsprung-Russell diagram (sHRD) tells similar evolutionary tales, but is independent of distance and extinction measurements. Based on spectroscopically derived effective temperatures and gravities of almost 600 stars, we derive for the first time the observational distribution of Galactic massive stars in the sHRD. While biases and statistical limitations in the data prevent detailed quantitative conclusions at this time, we see several clear qualitative trends. By comparing the observational sHRD with different state-of-the-art stellar evolutionary predictions, we conclude that convective core overshooting may be mass-dependent and, at high mass (≳15 M⊙), stronger than previously thought. Furthermore, we find evidence for an empirical upper limit in the sHRD for stars with Teff between 10 000 and 32 000 K and, a strikingly large number of objects below this line. This over-density may be due to inflation expanding envelopes in massive main-sequence stars near the Eddington limit. Appendix A is available in electronic form at http://www.aanda.org
10. Massive Stars in Colliding Wind Systems: the GLAST Perspective
SciTech Connect
Reimer, Anita; Reimer, Olaf; /Stanford U., HEPL /KIPAC, Menlo Park
2011-11-29
Colliding winds of massive stars in binary systems are considered as candidate sites of high-energy non-thermal photon emission. They are already among the suggested counterparts for a few individual unidentified EGRET sources, but may constitute a detectable source population for the GLAST observatory. The present work investigates such population study of massive colliding wind systems at high-energy gamma-rays. Based on the recent detailed model (Reimer et al. 2006) for non-thermal photon production in prime candidate systems, we unveil the expected characteristics of this source class in the observables accessible at LAT energies. Combining the broadband emission model with the presently cataloged distribution of such systems and their individual parameters allows us to conclude on the expected maximum number of LAT-detections among massive stars in colliding wind binary systems.
11. Massive star formation in 100,000 years from turbulent and pressurized molecular clouds.
PubMed
McKee, Christopher F; Tan, Jonathan C
2002-03-01
Massive stars (with mass m* > 8 solar masses Mmiddle dot in circle) are fundamental to the evolution of galaxies, because they produce heavy elements, inject energy into the interstellar medium, and possibly regulate the star formation rate. The individual star formation time, t*f, determines the accretion rate of the star; the value of the former quantity is currently uncertain by many orders of magnitude, leading to other astrophysical questions. For example, the variation of t*f with stellar mass dictates whether massive stars can form simultaneously with low-mass stars in clusters. Here we show that t*f is determined by the conditions in the star's natal cloud, and is typically about 105yr. The corresponding mass accretion rate depends on the pressure within the cloud--which we relate to the gas surface density--and on both the instantaneous and final stellar masses. Characteristic accretion rates are sufficient to overcome radiation pressure from about 100M middle dot in circle protostars, while simultaneously driving intense bipolar gas outflows. The weak dependence of t*f on the final mass of the star allows high- and low-mass star formation to occur nearly simultaneously in clusters.
12. Eta Carinae in the Context of the Most Massive Stars
NASA Technical Reports Server (NTRS)
Gull, Theodore R.; Damineli, Augusto
2009-01-01
Eta Car, with its historical outbursts, visible ejecta and massive, variable winds, continues to challenge both observers and modelers. In just the past five years over 100 papers have been published on this fascinating object. We now know it to be a massive binary system with a 5.54-year period. In January 2009, Car underwent one of its periodic low-states, associated with periastron passage of the two massive stars. This event was monitored by an intensive multi-wavelength campaign ranging from -rays to radio. A large amount of data was collected to test a number of evolving models including 3-D models of the massive interacting winds. August 2009 was an excellent time for observers and theorists to come together and review the accumulated studies, as have occurred in four meetings since 1998 devoted to Eta Car. Indeed, Car behaved both predictably and unpredictably during this most recent periastron, spurring timely discussions. Coincidently, WR140 also passed through periastron in early 2009. It, too, is a intensively studied massive interacting binary. Comparison of its properties, as well as the properties of other massive stars, with those of Eta Car is very instructive. These well-known examples of evolved massive binary systems provide many clues as to the fate of the most massive stars. What are the effects of the interacting winds, of individual stellar rotation, and of the circumstellar material on what we see as hypernovae/supernovae? We hope to learn. Topics discussed in this 1.5 day Joint Discussion were: Car: the 2009.0 event: Monitoring campaigns in X-rays, optical, radio, interferometry WR140 and HD5980: similarities and differences to Car LBVs and Eta Carinae: What is the relationship? Massive binary systems, wind interactions and 3-D modeling Shapes of the Homunculus & Little Homunculus: what do we learn about mass ejection? Massive stars: the connection to supernovae, hypernovae and gamma ray bursters Where do we go from here? (future
13. The Role of Rotation in the Evolution of Massive Stars
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Lanz, Thierry M.
2003-01-01
Recent evolutionary models of massive stars predict important effects of rotation including: increasing the rate of mass loss; lowering the effective gravity; altering the evolutionary track on the Hertzsprung-Russel Diagram (HRD); extending the main-sequence phase (both on the HR diagram and in time); and mixing of CNO-processed elements up to the stellar surface. Observations suggest that rotation is a more important factor at lower metallicities because of higher initial rotational velocities and weaker winds. This makes the Small Magellanic Cloud (SMC), a low-metallicity galaxy (Z=0.2 solar Z), an excellent environment for discerning the role of rotation in massive stars. We report on a FUSE+STIS+optical spectral analysis of 17 O-type stars in the SMC, where we found an enormous range in N abundances. Three stars in the sample have the same (low) CN abundances as the nebular material out of which they formed, namely C=0.085 solar C and N=0.034 solar N. However, more than half show N approx. solar N, an enrichment factor of 30X! Such unexpectedly high levels of N have ramifications for the evolution of massive stars including precursors to supernovae. They also raise questions about the sources of nitrogen in the early universe. This study was supported in part by grants from NASA's ADP, HST GO-7437, and FUSE B134.
14. The Role of Rotation in the Evolution of Massive Stars
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Lanz, Thierry M.
2002-01-01
Recent evolutionary models of massive stars predict important effects of rotation including: increasing the rate of mass-loss; lowering the effective gravity; altering the evolutionary track on the HRD; extending the main-sequence phase (both on the HR diagram and in time); and mixing of CNO-processed elements up to the stellar surface. Observations suggest that rotation is a more important factor at lower metallicities because of higher initial rotational velocities and weaker winds. This makes the SMC, a low-metallicity galaxy (Z= 0.2 solar Z), an excellent environment for discerning the role of rotation in massive stars. We report on a FUSE + STIS + optical spectral analysis of 17 O-type stars in the SMC, where we found an enormous range in N abundances. Three stars in the sample have the same (low) CN abundances as the nebular material out of which they formed, namely C = 0.085 solar C and N = 0.034 solar N. However, more than half show N approx. solar N, an enrichment factor of 30X! Such unexpectedly high levels of N have ramifications for the evolution of massive stars including precursors to supernovae. They also raise questions about the sources of nitrogen in the early universe.
15. Late stages of massive star evolution and nucleosynthesis
SciTech Connect
Nomoto, Ken'ichi; Hashimoto, Masa-aki
1986-01-01
The evolution of massive stars in the mass range of 8 to 25 M solar mass is reviewed. The effect of electron degeneracy on the gravothermal nature of stars is discussed. Depending on the stellar mass, the stars form three types of cores, namely, non-degenerate, semi-degenerate, and strongly degenerate cores. The evolution for these cases is quite distinct from each other and leads to the three different types of final fate. It is suggested that our helium star model, which is equivalent to a 25 M solar mass star, will form a relatively small mass iron core despite the faster /sup 12/C(..cap alpha..,..gamma..)/sup 16/O reaction. 50 refs., 21 figs.
16. Interferometric Radio Observations of the Interactive Winds of Massive Stars
NASA Astrophysics Data System (ADS)
Brookes, Diane Patricia
2016-06-01
Massive stars have very strong stellar winds which interact with their environment. This work has involved the study of these interactive winds at radio and other wavelengths. Radio observations have been made of the massive runaway star BD+43 3654 and its bow shock which is interacting with the inter-stellar medium. These observations, together with archive data at other wavelengths, have revealed stratified dust and turbulent gas in this interaction zone. Further radio studies have been undertaken of the interaction zones of the colliding winds of massive binary systems. Observations of the colliding wind binary WR 147 at 5GHz have revealed a curved collision zone, suggestive of simple interactive models. Measurements of the flux from the Wolf-Rayet component of this massive binary system has allowed a mass-loss rate to be derived and though the companion O-star is not detected, an upper flux limit has allowed upper limits on the mass-loss rate and limits on the terminal velocity to be inferred. Also revealed is a curious ’bridge’ feature previously observed in WR 147 which occurs between the two binary components. One mechanism is suggested to explain this anomalous feature, the ionising flux of one binary component, the O-star, may be ionising the wind of the other, the WR component. Modelling of the ionisation structure of the stellar winds has been undertaken to verify that this may be occurring. Radio observations of massive stars made at low-frequency have produced detections of WR 147 and the brighter colliding wind binary, WR 146. These detections have allowed modelling of the non-thermal emission in order to deduce where the non-thermal absorption turn-over occurs in these systems. The resultant modelling has illustrated that these colliding wind regions are complex, with multiple absorption regions best describing their nature.
17. UH cosmic rays: Possible origin in massive stars
NASA Technical Reports Server (NTRS)
Wefel, J. P.; Schramm, D. N.; Blake, J. B.
1977-01-01
The origin of the Z greater than 28, ultraheavy, cosmic rays in supernova explosions of massive stars is considered. For Z greater than 70, the UH data is dominated by an r-process source distribution, but for the elements just beyond iron, 29 or = Z less than 36, the data cannot be explained by any single process of nucleosynthesis. This problem is solved naturally in a massive star model by secondary neutron capture reactions occuring during core helium burning and during explosive carbon burning. Interstellar propagation calculations were performed with these episodes of synthesis as source distributions, and the results offer an explanation for the current UH cosmic-ray data. The heavy element synthesis during explosive carbon burning is reexamined using more realistic initial conditions given by the post-helium-burning configuration of the star. Effects of preferential acceleration are considered, and experimental tests are discussed.
18. Rb and Zr abundances in massive Galactic AGB stars revisited
NASA Astrophysics Data System (ADS)
Pérez-Mesa, V.; Zamora, O.; García-Hernández, D. A.; Plez, B.; Manchado, A.; Karakas, A. I.; Lugaro, M.
2016-07-01
We report new abundances of Rb and Zr in a sample of massive Galactic asymptotic giant branch (AGB) stars that were previously studied with hydrostatic models by using more realistic dynamical model atmospheres. We use a modified version of the spectral synthesis code Turbospectrum, and consider the presence of a circumstellar envelope and a radial wind in the modelling of these Galactic AGB stars. The Rb and Zr are determined from the 7800 Å Rb I resonant line and the 6474 Å ZrO bandhead, respectively, and they are compared with the AGB nucleosynthesis theoretical predictions. The derived Rb abundances are much lower (∼⃒1-2 dex) with the new dynamical models, while the Zr abundances, however, are closer to the hydrostatic values. The new model atmospheres can help to resolve the problem of the mismatch between the observations and the nucleosynthesis theoretical predictions of massive AGB stars.
19. EQUATION OF STATE FOR MASSIVE NEUTRON STARS
SciTech Connect
Katayama, Tetsuya; Saito, Koichi; Miyatsu, Tsuyoshi
2012-12-15
Using the relativistic Hartree-Fock approximation, we investigate the properties of neutron-star matter in detail. In the present calculation, we consider not only the tensor coupling of vector mesons to octet baryons and the form factors at interaction vertices but also the change in the internal (quark) structure of baryons in dense matter. The relativistic Hartree-Fock calculations are performed in two ways: one with coupling constants determined by SU(6) (quark model) symmetry and the other with coupling constants based on SU(3) (flavor) symmetry. For the latter case, we use the latest Nijmegen (ESC08) model. Then, it is very remarkable that the particle composition of the core matter in SU(3) symmetry is completely different from that in SU(6) symmetry. In SU(6) symmetry, all octet baryons appear in the density region below {approx}1.2 fm{sup -3}, while in the ESC08 model only the {Xi}{sup -} hyperon is produced. Furthermore, the medium modification of the internal baryon structure hardens the equation of state for the core matter. Taking all these effects into account, we can obtain the maximum neutron-star mass which is consistent with the recently observed mass, 1.97 {+-} 0.04 M{sub Sun} (PSR J1614-2230). We therefore conclude that the extension from SU(6) symmetry to SU(3) symmetry in meson-baryon couplings and the internal baryon structure variation in matter considerably enhance the mass of neutron stars. Furthermore, the effects of the form factor at the vertex and the Fock contribution, including the tensor coupling due to vector mesons, are indispensable for describing the core matter.
20. Evolution of massive single stars with rotation
NASA Astrophysics Data System (ADS)
Meynet, Georges
2015-08-01
After a brief recall of the physics of rotation, we shall discuss how this physics can be implemented in stellar evolution codes and what are the main calibration processes allowing to constrain some poorly known parameters associated with the description of the turbulence. Models with and without magnetic fields will be discussed. Stellar models predictions will be confronted with observed features. Consequences for the origin of various stellar populations, as red and blue supergiants and Wolf-Rayet stars, of various types of core collapse supernovae will be presented.
1. One of the most massive stars in the Galaxy may have formed in isolation
NASA Astrophysics Data System (ADS)
Oskinova, L. M.; Steinke, M.; Hamann, W.-R.; Sander, A.; Todt, H.; Liermann, A.
2013-12-01
Very massive stars, 100 times heavier than the sun, are rare. It is not yet known whether such stars can form in isolation or only in star clusters. The answer to this question is of fundamental importance. The central region of our Galaxy is ideal for investigating very massive stars and clusters located in the same environment. We used archival infrared images to investigate the surroundings of apparently isolated massive stars presently known in the Galactic Centre (GC). We find that two such isolated massive stars display bow shocks and hence may be runaways' from their birthplace. Thus, some isolated massive stars in the GC region might have been born in star clusters known in this region. However, no bow shock is detected around the isolated star WR 102ka (Peony nebula star), which is one of the most massive and luminous stars in the Galaxy. This star is located at the centre of an associated circumstellar nebula. To study whether a star cluster may be hidden' in the surroundings of WR 102ka, to obtain new and better spectra of this star, and to measure its radial velocity, we obtained observations with the integral-field spectrograph SINFONI at the ESO's Very Large Telescope. Our observations confirm that WR 102ka is one of the most massive stars in the Galaxy and reveal that this star is not associated with a star cluster. We suggest that WR 102ka has been born in relative isolation, outside of any massive star cluster.
2. Mid-Infrared Spectroscopy of the Most Massive Stars
NASA Astrophysics Data System (ADS)
Figer, Donald; Najarro, Paco; Stolovy, Susan
2004-09-01
The most massive star that can form is presently defined by observations of a class of very rare stars having inferred initial masses of ~200 solar masses. There are only a few such stars in the Galaxy, including the Pistol Star, FMM362, and LBV 1806-20, the first two being located near the Galactic center, and third located in the disk near W31. Each has only recently been identified as so massive within the past 10 years through the analysis of infrared observations, but they are otherwise too faint, due to extinction, to observe at shorter wavelengths. These stars appear to be very luminous (L>10^6.3 solar luminosities), "blue" (T>10000 K), and variable (delta K~1 mag.), and the Pistol Star has ejected 10 solar masses of material in the past 10000 years. In addition, these stars have near-infrared spectra similar to those of prototypical Luminous Blue Variables, i.e. Eta Car and AG Car. Given their apparent violation of the Humphries-Davidson limit, they are presumably in a short-lived phase of stellar evolution that is often associated with rapid mass-loss through episodic eruptions of their outer atmospheres. We propose to determine the physical properties of these stars and the velocity and ionization structure in their winds by using spectra obtained with the high resolution modes of the Infrared Spectrograph (IRS) on the Spitzer Space Telescope. The 10 to 40 micron wavelength region is ideally suited for accessing a variety of lines from transitions of hydrogen, helium, iron, silicon, sulfur, among others; indeed, through our models, we predict that sufficiently sensitive spectra will yield over 300 spectral lines. In addition, we predict that the mid-infrared continuum will be dominated by free-free emission generated in the thick winds associated with these stars, an effect that should be clearly detectable in the spectra.
3. The brief lives of massive stars as witnessed by interferometry}
NASA Astrophysics Data System (ADS)
Hummel, Christian
2013-06-01
Massive stars present the newest and perhaps most challenging opportunity for long baseline interferometry to excel. Large distances require high angular resolution both to study the means of accreting enough mass in a short time and to split new-born multiples into their components for the determination of their fundamental parameters. Dust obscuration of young stellar objects require interferometry in the infrared, while post-mainsequence stellar phases require high-precision measurements to challenge stellar evolution models. I will summarize our work on a massive YSO in NGC 3603 including modeling mid-IR interferometric observations, as well as recent sub-mm imaging and spectroscopy with APEX. We find some evidence for a disk in the MIR, resolve multiple cores in the sub-mm with emission line spectra untypical for hot cores. I also report on the derivation of masses and luminosities of a massive O-type supergiant (ζ Orionis) in another star forming region in Orion. The small radial velocity semi-amplitudes coupled with few usable (i.e. wind-free) lines have made this work very challenging and forced us to base the mass determination on a photometric distance estimate. As a rapidly evolving application of interferometry, massive stars have a bright future.
4. The Galactic Distribution of Massive Star Formation from the Red MSX Source Survey
NASA Astrophysics Data System (ADS)
Figura, Charles C.; Urquhart, J. S.
2013-01-01
Massive stars inject enormous amounts of energy into their environments in the form of UV radiation and molecular outflows, creating HII regions and enriching local chemistry. These effects provide feedback mechanisms that aid in regulating star formation in the region, and may trigger the formation of subsequent generations of stars. Understanding the mechanics of massive star formation presents an important key to understanding this process and its role in shaping the dynamics of galactic structure. The Red MSX Source (RMS) survey is a multi-wavelength investigation of ~1200 massive young stellar objects (MYSO) and ultra-compact HII (UCHII) regions identified from a sample of colour-selected sources from the Midcourse Space Experiment (MSX) point source catalog and Two Micron All Sky Survey. We present a study of over 900 MYSO and UCHII regions investigated by the RMS survey. We review the methods used to determine distances, and investigate the radial galactocentric distribution of these sources in context with the observed structure of the galaxy. The distribution of MYSO and UCHII regions is found to be spatially correlated with the spiral arms and galactic bar. We examine the radial distribution of MYSOs and UCHII regions and find variations in the star formation rate between the inner and outer Galaxy and discuss the implications for star formation throughout the galactic disc.
5. Circumstellar medium around rotating massive stars at solar metallicity
NASA Astrophysics Data System (ADS)
Georgy, Cyril; Walder, Rolf; Folini, Doris; Bykov, Andrei; Marcowith, Alexandre; Favre, Jean M.
2013-11-01
Aims: Observations show nebulae around some massive stars but not around others. If observed, their chemical composition is far from homogeneous. Our goal is to put these observational features into the context of the evolution of massive stars and their circumstellar medium (CSM) and, more generally, to quantify the role of massive stars for the chemical and dynamical evolution of the ISM. Methods: Using the A-MAZE code, we perform 2d-axisymmetric hydrodynamical simulations of the evolution of the CSM, shaped by stellar winds, for a whole grid of massive stellar models from 15 to 120 M⊙ and following the stellar evolution from the zero-age main-sequence to the time of supernova explosion. In addition to the usual quantities, we also follow five chemical species: H, He, C, N, and O. Results: We show how various quantities evolve as a function of time: size of the bubble, position of the wind termination shock, chemical composition of the bubble, etc. The chemical composition of the bubble changes considerably compared to the initial composition, particularly during the red-supergiant (RSG) and Wolf-Rayet (WR) phases. In some extreme cases, the inner region of the bubble can be completely depleted in hydrogen and nitrogen, and is mainly composed of carbon, helium, and oxygen. We argue why the bubble typically expands at a lower rate than predicted by self-similarity theory. In particular, the size of the bubble is very sensitive to the density of the ISM, decreasing by a factor of ~2.5 for each additional dex in ISM density. The bubble size also decreases with the metallicity of the central star, because low-metallicity stars have weaker winds. Our models qualitatively fit the observations of WR ejecta nebulae.
6. Tidal capture of stars by a massive black hole
NASA Technical Reports Server (NTRS)
Novikov, I. D.; Pethick, C. J.; Polnarev, A. G.
1992-01-01
The processes leading to tidal capture of stars by a massive black hole and the consequences of these processes in a dense stellar cluster are discussed in detail. When the amplitude of a tide and the subsequent oscillations are sufficiently large, the energy deposited in a star after periastron passage and formation of a bound orbit cannot be estimated directly using the linear theory of oscillations of a spherical star, but rather numerical estimates must be used. The evolution of a star after tidal capture is discussed. The maximum ratio R of the cross-section for tidal capture to that for tidal disruption is about 3 for real systems. For the case of a stellar system with an empty capture loss cone, even in the case when the impact parameter for tidal capture only slightly exceeds the impact parameter for direct tidal disruption, tidal capture would be much more important than tidal disruption.
7. The Pistol Star and Unstable Massive Stars at the Galactic Center
NASA Astrophysics Data System (ADS)
Najarro, F.
2005-09-01
We present recent results on quantitative spectroscopic studies of the Pistol Star and other massive stars in the Quintuplet and Arches clusters. Thanks to the impressive evolution of IR detectors and the new generation of line blanketed models for the extended atmospheres of hot stars we are able to accurately derive the physical properties of the massive stars in these clusters. Our analysis of the LBVs in the Quintuplet cluster provides, for the first time, a direct estimate of α-elements and Fe chemical abundances in these objects. Preliminary results point to a slightly enhanced enrichment of α-elements compared to Fe and suggest an initial mass function dominated by massive stars, as found for the Arches cluster. On the other hand, from our analysis of the Arches cluster, we introduce a new method to estimate metallicity in very young clusters based on the N abundance of WNL stars and the theory of evolution of massive stars. Results indicating solar metallicity are presented.
8. Ionizing feedback from massive stars in massive clusters - II. Disruption of bound clusters by photoionization
NASA Astrophysics Data System (ADS)
Dale, J. E.; Ercolano, B.; Bonnell, I. A.
2012-07-01
We present a smoothed particle hydrodynamics parameter study of the dynamical effect of photoionization from O-type stars on star-forming clouds of a range of masses and sizes during the time window before supernovae explode. Our model clouds all have the same degree of turbulent support initially, the ratio of turbulent kinetic energy to gravitational potential energy being set to Ekin/|Epot|= 0.7. We allow the clouds to form stars and study the dynamical effects of the ionizing radiation from the massive stars or clusters born within them. We find that dense filamentary structures and accretion flows limit the quantities of gas that can be ionized, particularly in the higher density clusters. More importantly, the higher escape velocities in our more massive (106 M⊙) clouds prevent the H II regions from sweeping up and expelling significant quantities of gas, so that the most massive clouds are largely dynamically unaffected by ionizing feedback. However, feedback has a profound effect on the lower density 104 and 105 M⊙ clouds in our study, creating vast evacuated bubbles and expelling tens of per cent of the neutral gas in the 3-Myr time-scale before the first supernovae are expected to detonate, resulting in clouds highly porous to both photons and supernova ejecta.
9. Role of Rotation in Massive Stars in the SMC
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Lanz, Thierry
2002-01-01
We report on an analysis of FUSE+STIS+optical spectra of 17 O-type stars in the SMC. We found an enormous range in N abundances. Three stars in the sample have the same (low) CN abundances as the nebular material out of which they formed, namely C=0.08 C(sub circle dot) and N=0.03 N(sub circle dot). However, more than half shows NO, an enrichment factor of 30X! Such a high level of N enrichment cannot be reproduced by current evolutionary models accounting for rotationally induced mixing. It suggests that the sum of CNO nuclei may not be conserved, i.e. massive stars might be producing primary nitrogen. It raises questions concerning the sources of nitrogen in the early universe, presently thought to be almost exclusively intermediate-mass stars. It also raises basic questions about the evolution of massive stars in low-metallicity environments, including the precursors to supernovae.
10. Simulating the Birth of Massive Star Clusters: Is Destruction Inevitable?
NASA Astrophysics Data System (ADS)
Rosen, Anna
2013-10-01
Very early in its operation, the Hubble Space Telescope {HST} opened an entirely new frontier: study of the demographics and properties of star clusters far beyond the Milky Way. However, interpretation of HST's observations has proven difficult, and has led to the development of two conflicting models. One view is that most massive star clusters are disrupted during their infancy by feedback from newly formed stars {i.e., "infant mortality"}, independent of cluster mass or environment. The other model is that most star clusters survive their infancy and are disrupted later by mass-dependent dynamical processes. Since observations at present have failed to discriminate between these views, we propose a theoretical investigation to provide new insight. We will perform radiation-hydrodynamic simulations of the formation of massive star clusters, including for the first time a realistic treatment of the most important stellar feedback processes. These simulations will elucidate the physics of stellar feedback, and allow us to determine whether cluster disruption is mass-dependent or -independent. We will also use our simulations to search for observational diagnostics that can distinguish bound from unbound clusters, and to predict how cluster disruption affects the cluster luminosity function in a variety of galactic environments.
11. Astronomers Discover Most Massive Neutron Star Yet Known
NASA Astrophysics Data System (ADS)
2010-10-01
Astronomers using the National Science Foundation's Green Bank Telescope (GBT) have discovered the most massive neutron star yet found, a discovery with strong and wide-ranging impacts across several fields of physics and astrophysics. "This neutron star is twice as massive as our Sun. This is surprising, and that much mass means that several theoretical models for the internal composition of neutron stars now are ruled out," said Paul Demorest, of the National Radio Astronomy Observatory (NRAO). "This mass measurement also has implications for our understanding of all matter at extremely high densities and many details of nuclear physics," he added. Neutron stars are the superdense "corpses" of massive stars that have exploded as supernovae. With all their mass packed into a sphere the size of a small city, their protons and electrons are crushed together into neutrons. A neutron star can be several times more dense than an atomic nucleus, and a thimbleful of neutron-star material would weigh more than 500 million tons. This tremendous density makes neutron stars an ideal natural "laboratory" for studying the most dense and exotic states of matter known to physics. The scientists used an effect of Albert Einstein's theory of General Relativity to measure the mass of the neutron star and its orbiting companion, a white dwarf star. The neutron star is a pulsar, emitting lighthouse-like beams of radio waves that sweep through space as it rotates. This pulsar, called PSR J1614-2230, spins 317 times per second, and the companion completes an orbit in just under nine days. The pair, some 3,000 light-years distant, are in an orbit seen almost exactly edge-on from Earth. That orientation was the key to making the mass measurement. As the orbit carries the white dwarf directly in front of the pulsar, the radio waves from the pulsar that reach Earth must travel very close to the white dwarf. This close passage causes them to be delayed in their arrival by the distortion of
12. Interaction of massive stars with the interstellar medium
NASA Astrophysics Data System (ADS)
de Geus, E. J.
This paper reviews observations and theory regarding the interaction between massive stars in open clusters and OB associations and the interstellar medium. The results of a systematic study of the gas and dust surrounding a large sample of open clusters are described. Different models for the bubbles blown by stellar winds of O-type stars are discussed, and the effects of subsequent supernova are investigated. The effects of correlated supernovae on the morphology of the interstellar gas and on the communication of the disk with the halo of a galaxy are presented.
13. Massive star-formation in the Trifid nebula
NASA Astrophysics Data System (ADS)
Lefloch, B.; Cernicharo, J.; Perez-Martinez, S.; Cesarsky, D.
1999-03-01
The Trifid nebula is a young galactic HII region where several protostellar sources have been detected using ISO and ground-based telescopes. The sources are massive (17 to 60 0.20em Modot) and are associated with molecular gas condensations at the edges or inside the nebula. They appear to be in an early evolutionary stage and may represent the most recent generation of stars in the Trifid. These sources range from dense apparently still inactive cores to somewhat more evolved sources, undergoing violent mass ejection episodes, including a source which powers an optical jet. these observations suggest that the protostellar sources may have evolved by induced star formation.
14. Massive main-sequence stars evolving at the Eddington limit
NASA Astrophysics Data System (ADS)
Sanyal, D.; Grassitelli, L.; Langer, N.; Bestenlehner, J. M.
2015-08-01
Context. Massive stars play a vital role in the Universe, however, their evolution even on the main-sequence is not yet well understood. Aims: Because of the steep mass-luminosity relation, massive main-sequence stars become extremely luminous. This brings their envelopes very close to the Eddington limit. We analyse stellar evolutionary models in which the Eddington limit is reached and exceeded, explore the rich diversity of physical phenomena that take place in their envelopes, and investigate their observational consequences. Methods: We use published grids of detailed stellar models, computed with a state-of-the-art, one-dimensional hydrodynamic stellar evolution code using LMC composition, to investigate the envelope properties of core hydrogen burning massive stars. Results: We find that the Eddington limit is almost never reached at the stellar surface, even for stars up to 500 M⊙. When we define an appropriate Eddington limit locally in the stellar envelope, we can show that most stars more massive than ~40 M⊙ actually exceed this limit, in particular, in the partial ionisation zones of iron, helium, or hydrogen. While most models adjust their structure such that the local Eddington limit is exceeded at most by a few per cent, our most extreme models do so by a factor of more than seven. We find that the local violation of the Eddington limit has severe consequences for the envelope structure, as it leads to envelope inflation, convection, density inversions, and, possibly to, pulsations. We find that all models with luminosities higher than 4 × 105L⊙, i.e. stars above ~40 M⊙ show inflation, with a radius increase of up to a factor of about 40. We find that the hot edge of the S Dor variability region coincides with a line beyond which our models are inflated by more than a factor of two, indicating a possible connection between S Dor variability and inflation. Furthermore, our coolest models show highly inflated envelopes with masses of up to
15. Relativistic stars in de Rham-Gabadadze-Tolley massive gravity
NASA Astrophysics Data System (ADS)
Katsuragawa, Taishi; Nojiri, Shin'ichi; Odintsov, Sergei D.; Yamazaki, Masashi
2016-06-01
We study relativistic stars in the simplest model of the de Rham-Gabadadze-Tolley massive gravity which describes the massive graviton without a ghost propagating mode. We consider the hydrostatic equilibrium and obtain the modified Tolman-Oppenheimer-Volkoff equation and the constraint equation coming from the potential terms in the gravitational action. We give analytical and numerical results for quark and neutron stars and discuss the deviations compared with general relativity and F (R ) gravity. It is shown that the theory under investigation leads to a small deviation from general relativity in terms of density profiles and mass-radius relation. Nevertheless, such a deviation may be observable in future astrophysical probes.
16. Effects of axions on nucleosynthesis in massive stars
NASA Astrophysics Data System (ADS)
Aoyama, Shohei; Suzuki, Takeru K.
2015-09-01
We investigate the effect of axion cooling on nucleosynthesis in a massive star with 16 M⊙ by a standard stellar evolution calculation. We find that axion cooling suppresses nuclear reactions in carbon, oxygen, and silicon burning phases because of the extraction of the energy. As a result, larger amounts of the already synthesized neon and magnesium remain without being consumed to produce further, heavier elements. Even in the case with axion-photon coupling constant ga γ=10-11 GeV-1 , which is six times smaller than the current upper limit, the amount of neon and magnesium that remain just before the core-collapse supernova explosion is considerably larger than the standard value. This implies that we could give a more stringent constraint on ga γ from the nucleosynthesis of heavy elements in massive stars.
17. THE ROLE OF THE MAGNETOROTATIONAL INSTABILITY IN MASSIVE STARS
SciTech Connect
Wheeler, J. Craig; Kagan, Daniel; Chatzopoulos, Emmanouil
2015-01-20
The magnetorotational instability (MRI) is key to physics in accretion disks and is widely considered to play some role in massive star core collapse. Models of rotating massive stars naturally develop very strong shear at composition boundaries, a necessary condition for MRI instability, and the MRI is subject to triply diffusive destabilizing effects in radiative regions. We have used the MESA stellar evolution code to compute magnetic effects due to the Spruit-Tayler (ST) mechanism and the MRI, separately and together, in a sample of massive star models. We find that the MRI can be active in the later stages of massive star evolution, leading to mixing effects that are not captured in models that neglect the MRI. The MRI and related magnetorotational effects can move models of given zero-age main sequence mass across ''boundaries'' from degenerate CO cores to degenerate O/Ne/Mg cores and from degenerate O/Ne/Mg cores to iron cores, thus affecting the final evolution and the physics of core collapse. The MRI acting alone can slow the rotation of the inner core in general agreement with the observed ''initial'' rotation rates of pulsars. The MRI analysis suggests that localized fields ∼10{sup 12} G may exist at the boundary of the iron core. With both the ST and MRI mechanisms active in the 20 M {sub ☉} model, we find that the helium shell mixes entirely out into the envelope. Enhanced mixing could yield a population of yellow or even blue supergiant supernova progenitors that would not be standard SN IIP.
18. The role of massive stars in young starburst galaxies
NASA Astrophysics Data System (ADS)
Norris, Richard Paul Furber
Starburst galaxies are defined as those galaxies undergoing violent star formation over relatively short periods of time (10 to 100 Myr). These objects may form stellar populations of > 106 Msun, containing massive stars with masses > 100 Msun. Although most starburst galaxies are observed at relatively low redshift, recent evidence suggests that these types of galaxies were far more important in the high redshift past. It is believed that the chemical evolution of the Universe has been strongly influenced by this mode of star formation through the dense winds from massive stars and supernovae ejecta. Our understanding of starbursts is still relatively poor, since most are too distant to be resolved. We can gain some understanding of starbursts indirectly through the modelling of associated nebulae via the calculation of theoretical spectral energy distributions (SEDs) and photoionization modelling. This technique heavily relies upon the accuracy of the predicted far UV continuum of the massive star population. This thesis presents a new grid of SEDs for O stars, early B supergiants and Wolf-Rayet stars which have been incorporated into the evolutionary synthesis code Starburst99 (Leitherer et al. 1999). A total of 285 expanding, non-LTE, line-blanketed model atmospheres have been calculated to replace old, inaccurate LTE models for O stars, and pure helium, unblanketed models for W-R stars. These new grids cover five metallicities and the wind parameters are scaled with metallicity. We find that the new models yield significantly less ionizing flux below the He 0 ionizing edge at early phases and as a consequence, nebular He II lambda4686 will not be observable in young starbursts. We use the photoionization code CLOUDY to test the accuracy of the predicted ionizing fluxes from our new models. We find that they are in much better agreement with observed optical and IR nebular line diagnostics than any previous models. The new W-R atmospheres are used in
19. X-RAY EMISSION FROM MAGNETIC MASSIVE STARS
SciTech Connect
Nazé, Yaël; Petit, Véronique; Rinbrand, Melanie; Owocki, Stan; Cohen, David; Ud-Doula, Asif; Wade, Gregg A.
2014-11-01
Magnetically confined winds of early-type stars are expected to be sources of bright and hard X-rays. To clarify the systematics of the observed X-ray properties, we have analyzed a large series of Chandra and XMM-Newton observations, corresponding to all available exposures of known massive magnetic stars (over 100 exposures covering ∼60% of stars compiled in the catalog of Petit et al.). We show that the X-ray luminosity is strongly correlated with the stellar wind mass-loss rate, with a power-law form that is slightly steeper than linear for the majority of the less luminous, lower- M-dot B stars and flattens for the more luminous, higher- M-dot O stars. As the winds are radiatively driven, these scalings can be equivalently written as relations with the bolometric luminosity. The observed X-ray luminosities, and their trend with mass-loss rates, are well reproduced by new MHD models, although a few overluminous stars (mostly rapidly rotating objects) exist. No relation is found between other X-ray properties (plasma temperature, absorption) and stellar or magnetic parameters, contrary to expectations (e.g., higher temperature for stronger mass-loss rate). This suggests that the main driver for the plasma properties is different from the main determinant of the X-ray luminosity. Finally, variations of the X-ray hardnesses and luminosities, in phase with the stellar rotation period, are detected for some objects and they suggest that some temperature stratification exists in massive stars' magnetospheres.
20. Chemical Evolution of Collapsing Clouds in Massive Star Formation
NASA Astrophysics Data System (ADS)
Oman, Kris; Doty, S.; Krumholz, M.
2011-01-01
The process of massive star formation is not well understood. Recent work in large scale radiation hydrodynamical simulations have strongly suggested that radiation pressure can play an important role in opening cavities through which energy can be released, thus avoiding the problems of high radiation pressure supressing massive star formation. As a result, this pressure valve allows for the direct accretion of matter, and formation of massive stars. While these models include significant microphysics, it is important that predictions be made that allow the models to be compared with observations. Toward that end, we have undertaken a study of the chemistry in one of these collapsing cloud models. The chemical model involves the application of a large gas-phase and grain surface chemistry to the dynamical structure, including the effects of density, temperature, and radiation field. We present maps of H2, CO, and other molecular abundances as functions of space and time, as well as consider the resulting observational consequences of these results.
1. Kinematics of a Massive Star Cluster in Formation
NASA Astrophysics Data System (ADS)
Tan, Jonathan
2014-10-01
We propose to measure the proper motion stellar kinematics of a massive (~10^4Msun), forming proto-star-cluster to test basic theoretical models of formation. This will be the first time such a measurement has been performed. It requires HST-WFC3/IR and is beyond the practical capabilities of ground-based adaptive optics (AO) observations. In contrast to previously-studied massive, young (<10 Myr-old), already-formed clusters, such as NGC3603, Westerlund 1 or the Arches, our target protocluster, G286.21+0.17 (hereafter G286), is still gas-dominated and undergoing active star formation. It has been carefully selected from a complete survey of ~300 dense molecular gas clumps in a 120 sq. deg. region of the Galactic plane. The cluster is also relatively nearby (~2.5 kpc), but not too close that it would span a prohibitively large angular area or suffer from significant saturation problems. Such massive systems are rare and indeed we are unaware of any equivalent, early-stage (i.e., gas dominated) cluster that is closer. Given the depth of its gravitational potential based on its mass and size, the expected proper motions of many independent sub-clusters of stars are detectable at the ~5 sigma level over a 2-year baseline and global contraction of the cluster can be seen if it is happening even at just ~10% of the free-fall rate.
2. Young and intermediate-age massive star clusters.
PubMed
Larsen, Søren S
2010-02-28
An overview of our current understanding of the formation and evolution of star clusters is given, with the main emphasis on high-mass clusters. Clusters form deeply embedded within dense clouds of molecular gas. Left-over gas is cleared within a few million years and, depending on the efficiency of star formation, the clusters may disperse almost immediately or remain gravitationally bound. Current evidence suggests that a small percentage of star formation occurs in clusters that remain bound, although it is not yet clear whether this fraction is truly universal. Internal two-body relaxation and external shocks will lead to further, gradual dissolution on time scales of up to a few hundred million years for low-mass open clusters in the Milky Way, while the most massive clusters (>10(5) M(o)) have lifetimes comparable to or exceeding the age of the Universe. The low-mass end of the initial cluster mass function is well approximated by a power-law distribution, dN/dM proportional to M(-2), but there is mounting evidence that quiescent spiral discs form relatively few clusters with masses M > 2 x 10(5) M(o). In starburst galaxies and old globular cluster systems, this limit appears to be higher, at least several x10(6) M(o). The difference is likely related to the higher gas densities and pressures in starburst galaxies, which allow denser, more massive giant molecular clouds to form. Low-mass clusters may thus trace star formation quite universally, while the more long-lived, massive clusters appear to form preferentially in the context of violent star formation.
3. Young and intermediate-age massive star clusters.
PubMed
Larsen, Søren S
2010-02-28
An overview of our current understanding of the formation and evolution of star clusters is given, with the main emphasis on high-mass clusters. Clusters form deeply embedded within dense clouds of molecular gas. Left-over gas is cleared within a few million years and, depending on the efficiency of star formation, the clusters may disperse almost immediately or remain gravitationally bound. Current evidence suggests that a small percentage of star formation occurs in clusters that remain bound, although it is not yet clear whether this fraction is truly universal. Internal two-body relaxation and external shocks will lead to further, gradual dissolution on time scales of up to a few hundred million years for low-mass open clusters in the Milky Way, while the most massive clusters (>10(5) M(o)) have lifetimes comparable to or exceeding the age of the Universe. The low-mass end of the initial cluster mass function is well approximated by a power-law distribution, dN/dM proportional to M(-2), but there is mounting evidence that quiescent spiral discs form relatively few clusters with masses M > 2 x 10(5) M(o). In starburst galaxies and old globular cluster systems, this limit appears to be higher, at least several x10(6) M(o). The difference is likely related to the higher gas densities and pressures in starburst galaxies, which allow denser, more massive giant molecular clouds to form. Low-mass clusters may thus trace star formation quite universally, while the more long-lived, massive clusters appear to form preferentially in the context of violent star formation. PMID:20083510
4. NEARBY MASSIVE STAR CLUSTER YIELDS INSIGHTS INTO EARLY UNIVERSE
NASA Technical Reports Server (NTRS)
2002-01-01
A NASA Hubble Space Telescope 'family portrait' of young, ultra-bright stars nested in their embryonic cloud of glowing gases. The celestial maternity ward, called N81, is located 200,000 light-years away in the Small Magellanic Cloud (SMC), a small irregular satellite galaxy of our Milky Way. Hubble's exquisite resolution allows astronomers to pinpoint 50 separate stars tightly packed in the nebula's core within a 10 light-year diameter - slightly more than twice the distance between earth and the nearest star to our sun. The closest pair of stars is only 1/3 of a light-year apart (0.3 arcseconds in the sky). This furious rate of mass loss from these super-hot stars is evident in the Hubble picture that reveals dramatic shapes sculpted in the nebula's wall of glowing gases by violent stellar winds and shock waves. A pair of bright stars in the center of the nebula is pouring out most of the ultraviolet radiation to make the nebula glow. Just above them, a small dark knot is all that's left of the cold cloud of molecular hydrogen and dust the stars were born from. Dark absorption lanes of residual dust trisect the nebula. The nebula offers a unique opportunity for a close-up glimpse at the 'firestorm' accompanying the birth of extremely massive stars, each blazing with the brilliance of 300,000 of our suns. Such galactic fireworks were much more common billions of years ago in the early universe, when most star formation took place. The 'natural-color' view was assembled from separate images taken with the Wide Field and Planetary Camera 2, in ultraviolet light and two narrow emission lines of ionized Hydrogen (H-alpha, H-beta). The picture was taken on September 4, 1997. Credit: Mohammad Heydari-Malayeri (Paris Observatory, France), NASA/ESA
5. Fallback and Black Hole Production in Massive Stars
SciTech Connect
Zhang, Wei-Qun; Woosley, S.E.; Heger, A.; /UC, Santa Cruz /Los Alamos
2007-01-08
The compact remnants of core collapse supernovae--neutron stars and black holes--have properties that reflect both the structure of their stellar progenitors and the physics of the explosion. In particular, the masses of these remnants are sensitive to the density structure of the presupernova star and to the explosion energy. To a considerable extent, the final mass is determined by the ''fallback'', during the explosion, of matter that initially moves outwards, yet ultimately fails to escape. We consider here the simulated explosion of a large number of massive stars (10 to 100 M{sub {circle_dot}}) of Population I (solar metallicity) and III (zero metallicity), and find systematic differences in the remnant mass distributions. As pointed out by Chevalier (1989), supernovae in more compact progenitor stars have stronger reverse shocks and experience more fallback. For Population III stars above about 25 M{sub {circle_dot}} and explosion energies less than 1.5 x 10{sup 51} erg, black holes are a common outcome, with masses that increase monotonically with increasing main sequence mass up to a maximum hole mass of about 35 M{sub {circle_dot}}. If such stars produce primary nitrogen, however, their black holes are systematically smaller. For modern supernovae with nearly solar metallicity, black hole production is much less frequent and the typical masses, which depend sensitively on explosion energy, are smaller. We explore the neutron star initial mass function for both populations and, for reasonable assumptions about the initial mass cut of the explosion, find good agreement with the average of observed masses of neutron stars in binaries. We also find evidence for a bimodal distribution of neutron star masses with a spike around 1.2 M{sub {circle_dot}} (gravitational mass) and a broader distribution peaked around 1.4 M{sub {circle_dot}}.
6. Nucleosynthesis of Short-lived Radioactivities in Massive Stars
NASA Technical Reports Server (NTRS)
Meyer, B. S.
2004-01-01
A leading model for the source of many of the short-lived radioactivities in the early solar nebula is direct incorporation from a massive star [1]. A recent and promising incarnation of this model includes an injection mass cut, which is a boundary between the stellar ejecta that become incorporated into the solar cloud and those ejecta that do not [2-4]. This model also includes a delay time between ejection from the star and incorporation into early solar system solid bodies. While largely successful, this model requires further validation and comparison against data. Such evaluation becomes easier if we have a better sense of the nature of the synthesis of the various radioactivities in the star. That is the goal of this brief abstract.
7. Massive stars and expanding shells within the violent interstellar medium
NASA Astrophysics Data System (ADS)
Thilker, David Allan
Massive stars have a tremendous impact on their surroundings, largely due to a prodigious production rate of Lyman continuum photons and their inevitable termination in a supernova explosion. A single OB star may ionize a sufficiently luminous HII region to remain detectable out to distances of many Mpc. By concentrating the mechanical power of many high mass stars in a limited volume over a short time period, OB associations are known to produce large expanding bubbles in the interstellar medium (ISM). Aperture synthesis observations of HI in nearby galaxies clearly reveal the bubbly character of the diffuse ISM and highlight the connection with massive stars. In this dissertation I close the loop between theory and observations regarding massive stars, their incipient HII regions, and related expanding shells, all in the hope of learning more about the diffuse ISM. The research described herein has three main components: (1)object recognition in the context of HI datacubes and hydrodynamic shell models, (2)automated photometry of HII regions in crowded narrowband images, and (3)population synthesis modeling of stellar clusters and expanding shells in disk galaxies. I have created efficient procedures for conducting a census of HI superbubbles and young massive star clusters in nearby galaxies, plus a modeling framework allowing one to check these databases for relative agreement. My population synthesis algorithm predicts ensemble characteristics: of a disk-galaxy shell population, given details of the stellar cluster formation process and global properties of the galaxy in question. My automated HI object recognition method has made possible the Las Cruces/Dwingeloo Supershell Survey (LCDSS) of 21 nearby disk galaxies. In this dissertation I present early LCDSS results for NGC 300, NGC 2403, M81, and M101. Furthermore, I demonstrate the technique for photometry of HII regions by analyzing a small sample of 11 prominent spirals. The photometric measurements are
8. H II REGIONS: WITNESSES TO MASSIVE STAR FORMATION
SciTech Connect
Peters, Thomas; Banerjee, Robi; Klessen, Ralf S.; Low, Mordecai-Mark Mac; Galvan-Madrid, Roberto; Keto, Eric R.
2010-03-10
We describe the first three-dimensional simulation of the gravitational collapse of a massive, rotating molecular cloud that includes heating by both non-ionizing and ionizing radiation. These models were performed with the FLASH code, incorporating a hybrid, long characteristic, ray-tracing technique. We find that as the first protostars gain sufficient mass to ionize the accretion flow, their H II regions are initially gravitationally trapped, but soon begin to rapidly fluctuate between trapped and extended states, in agreement with observations. Over time, the same ultracompact H II region can expand anisotropically, contract again, and take on any of the observed morphological classes. In their extended phases, expanding H II regions drive bipolar neutral outflows characteristic of high-mass star formation. The total lifetime of H II regions is given by the global accretion timescale, rather than their short internal sound-crossing time. This explains the observed number statistics. The pressure of the hot, ionized gas does not terminate accretion. Instead, the final stellar mass is set by fragmentation-induced starvation. Local gravitational instabilities in the accretion flow lead to the build-up of a small cluster of stars, all with relatively high masses due to heating from accretion radiation. These companions subsequently compete with the initial high-mass star for the same common gas reservoir and limit its mass growth. This is in contrast to the classical competitive accretion model, where the massive stars are never hindered in growth by the low-mass stars in the cluster. Our findings show that the most significant differences between the formation of low-mass and high-mass stars are all explained as the result of rapid accretion within a dense, gravitationally unstable, ionized flow.
9. H II Regions: Witnesses to Massive Star Formation
NASA Astrophysics Data System (ADS)
Peters, Thomas; Banerjee, Robi; Klessen, Ralf S.; Mac Low, Mordecai-Mark; Galván-Madrid, Roberto; Keto, Eric R.
2010-03-01
We describe the first three-dimensional simulation of the gravitational collapse of a massive, rotating molecular cloud that includes heating by both non-ionizing and ionizing radiation. These models were performed with the FLASH code, incorporating a hybrid, long characteristic, ray-tracing technique. We find that as the first protostars gain sufficient mass to ionize the accretion flow, their H II regions are initially gravitationally trapped, but soon begin to rapidly fluctuate between trapped and extended states, in agreement with observations. Over time, the same ultracompact H II region can expand anisotropically, contract again, and take on any of the observed morphological classes. In their extended phases, expanding H II regions drive bipolar neutral outflows characteristic of high-mass star formation. The total lifetime of H II regions is given by the global accretion timescale, rather than their short internal sound-crossing time. This explains the observed number statistics. The pressure of the hot, ionized gas does not terminate accretion. Instead, the final stellar mass is set by fragmentation-induced starvation. Local gravitational instabilities in the accretion flow lead to the build-up of a small cluster of stars, all with relatively high masses due to heating from accretion radiation. These companions subsequently compete with the initial high-mass star for the same common gas reservoir and limit its mass growth. This is in contrast to the classical competitive accretion model, where the massive stars are never hindered in growth by the low-mass stars in the cluster. Our findings show that the most significant differences between the formation of low-mass and high-mass stars are all explained as the result of rapid accretion within a dense, gravitationally unstable, ionized flow.
10. Massive star formation by accretion. I. Disc accretion
NASA Astrophysics Data System (ADS)
Haemmerlé, L.; Eggenberger, P.; Meynet, G.; Maeder, A.; Charbonnel, C.
2016-01-01
Context. Massive stars likely form by accretion and the evolutionary track of an accreting forming star corresponds to what is called the birthline in the Hertzsprung-Russell (HR) diagram. The shape of this birthline is quite sensitive to the evolution of the entropy in the accreting star. Aims: We first study the reasons why some birthlines published in past years present different behaviours for a given accretion rate. We then revisit the question of the accretion rate, which allows us to understand the distribution of the observed pre-main-sequence (pre-MS) stars in the HR diagram. Finally, we identify the conditions needed to obtain a large inflation of the star along its pre-MS evolution that may push the birthline towards the Hayashi line in the upper part of the HR diagram. Methods: We present new pre-MS models including accretion at various rates and for different initial structures of the accreting core. We compare them with previously published equivalent models. From the observed upper envelope of pre-MS stars in the HR diagram, we deduce the accretion law that best matches the accretion history of most of the intermediate-mass stars. Results: In the numerical computation of the time derivative of the entropy, some treatment leads to an artificial loss of entropy and thus reduces the inflation that the accreting star undergoes along the birthline. In the case of cold disc accretion, the existence of a significant swelling during the accretion phase, which leads to radii ≳ 100 R⊙ and brings the star back to the red part of the HR diagram, depends sensitively on the initial conditions. For an accretion rate of 10-3M⊙ yr-1, only models starting from a core with a significant radiative region evolve back to the red part of the HR diagram. We also obtain that, in order to reproduce the observed upper envelope of pre-MS stars in the HR diagram with an accretion law deduced from the observed mass outflows in ultra-compact HII regions, the fraction of the
11. Complete Stellar Models: Spectral and Interior Evolution of Massive Stars
NASA Astrophysics Data System (ADS)
Schaerer, Daniel
1995-08-01
This thesis work presents the first "complete stellar models" for massive stars, which consistently treat the stellar interior, the atmosphere, and the stellar winds. This approach allows to simultaneously predict basic stellar parameters (luminosity, radii, temperatures), nucleosynthesis (abundances), as well as the detailed emergent spectrum through the relevant evolutionary phases (corresponding to OB, LBV and Wolf--Rayet stars). On the other hand, our modelling including the stellar winds also allows to study the influence of the outer layers on the stellar structure and evolution. Conceptually the thesis is divided in two main parts. In the first part we construct the first non-LTE line blanketed hydrodynamic models of spherically expanding atmospheres of hot stars. The entire domain from the optically thick photosphere out to the terminal velocity of the wind is treated. We discuss in detail the effects of line blanketing on the atmospheric structure and on the predicted spectrum. We study the influence of the hydrodynamic structure on the profiles of both photospheric and wind lines. Our results also show that for precise determinations of stellar parameters and abundances of hot luminous stars, the use of plane parallel models may lead to systematic errors. In the second part we develop the "complete stellar models" (CoStar). As a first application we study the main sequence (MS) interior and spectral evolution of massive stars at solar metallicity. The evolutionary tracks and the interior evolution are found to be basically unchanged by the realistic treatment of the outer layers. The main CoStar predictions presented and discussed for the MS are the following: 1. Ejected mass of the most important elements. Deposition of wind momentum and mechanical energy 2. Estimates of mass loss rates due to radiation pressure including multiple scattering and line overlap 3. Continuous spectral energy distribution (EUV to IR) and ionising fluxes 4. UBVRIJHKLMN
12. Effects of Ionization Feedback in Massive Star Formation
NASA Astrophysics Data System (ADS)
Peters, Thomas; Banerjee, R.; Klessen, R. S.; Mac Low, M.
2009-01-01
We present 3D high-resolution radiation-hydrodynamical simulations of massive star formation. We model the collapse of a massive molecular cloud core forming a high-mass star in its center. We use a version of the FLASH code that has been extended by including sink particles which are a source of both ionizing and non-ionizing radiation. The sink particles evolve according to a prestellar model which determines the stellar and accretion luminosities. Radiation transfer is done using the hybrid characteristics raytracing approach on the adaptive mesh developed by Rijkhorst et al. (2006). The radiative transfer module has been augmented to allow simulations with arbitrarily high resolution. Our highest resolution models resolve the disk scale height by at least 16 zones. Opacities for non-ionizing radiation have been added to account for the accretion heating, which is expected to be strong at the initial stage of star formation and believed to prevent fragmentation. Studies of collapsing massive cores show the formation of a gravitationally highly unstable disk. The accretion heating is not strong enough to suppress this instability. The ionizing radiation builds up an H II region around the protostar, which destroys the accretion disk close to it. We describe preliminary results, with a focus on how long the H II region remains confined by the accretion flow, and whether it can ever cut off accretion entirely. Thomas Peters acknowledges support from a Kade Fellowship for his visit to the American Museum of Natural History. He is a fellow of the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg and the Heidelberg Graduate School of Fundamental Physics. We also thank the DFG for support via the Emmy Noether Grant BA 3607/1 and the individual grant KL1358/5.
13. Ionizing feedback from massive stars in massive clusters - III. Disruption of partially unbound clouds
NASA Astrophysics Data System (ADS)
Dale, J. E.; Ercolano, B.; Bonnell, I. A.
2013-03-01
We extend our previous smoothed particle hydrodynamics parameter study of the effects of photoionization from O-stars on star-forming clouds to include initially unbound clouds. We generate a set of model clouds in the mass range 104-106 M⊙ with initial virial ratios Ekin/Epot = 2.3, allow them to form stars and study the impact of the photoionizing radiation produced by the massive stars. We find that, on the 3 Myr time-scale before supernovae are expected to begin detonating, the fraction of mass expelled by ionizing feedback is a very strong function of the cloud escape velocities. High-mass clouds are largely unaffected dynamically, while low-mass clouds have large fractions of their gas reserves expelled on this time-scale. However, the fractions of stellar mass unbound are modest and significant portions of the unbound stars are so only because the clouds themselves are initially partially unbound. We find that ionization is much more able to create well-cleared bubbles in the unbound clouds, owing to their intrinsic expansion, but that the presence of such bubbles does not necessarily indicate that a given cloud has been strongly influenced by feedback. We also find, in common with the bound clouds from our earlier work, that many of the systems simulated here are highly porous to photons and supernova ejecta, and that most of them will likely survive their first supernova explosions.
14. Induced massive star formation in the trifid nebula?
PubMed
Cernicharo; Lefloch; Cox; Cesarsky; Esteban; Yusef-Zadeh; Mendez; Acosta-Pulido; Garcia Lopez RJ; Heras
1998-10-16
The Trifid nebula is a young (10(5) years) galactic HII region where several protostellar sources have been detected with the infrared space observatory. The sources are massive (17 to 60 solar masses) and are associated with molecular gas condensations at the edges or inside the nebula. They appear to be in an early evolutionary stage and may represent the most recent generation of stars in the Trifid. These sources range from dense, apparently still inactive cores to more evolved sources, undergoing violent mass ejection episodes, including a source that powers an optical jet. These observations suggest that the protostellar sources may have evolved by induced star formation in the Trifid nebula. PMID:9774270
15. The evolution of rotating very massive stars with LMC composition
NASA Astrophysics Data System (ADS)
Köhler, K.; Langer, N.; de Koter, A.; de Mink, S. E.; Crowther, P. A.; Evans, C. J.; Gräfener, G.; Sana, H.; Sanyal, D.; Schneider, F. R. N.; Vink, J. S.
2015-01-01
Context. With growing evidence for the existence of very massive stars at subsolar metallicity, there is an increased need for corresponding stellar evolution models. Aims: We present a dense model grid with a tailored input chemical composition appropriate for the Large Magellanic Cloud (LMC). Methods: We use a one-dimensional hydrodynamic stellar evolution code, which accounts for rotation, transport of angular momentum by magnetic fields, and stellar wind mass loss to compute our detailed models. We calculate stellar evolution models with initial masses from 70 to 500 M⊙ and with initial surface rotational velocities from 0 to 550 km s-1, covering the core-hydrogen burning phase of evolution. Results: We find our rapid rotators to be strongly influenced by rotationally induced mixing of helium, with quasi-chemically homogeneous evolution occurring for the fastest rotating models. Above 160 M⊙, homogeneous evolution is also established through mass loss, producing pure helium stars at core hydrogen exhaustion independent of the initial rotation rate. Surface nitrogen enrichment is also found for slower rotators, even for stars that lose only a small fraction of their initial mass. For models above ~150 M⊙ at zero age, and for models in the whole considered mass range later on, we find a considerable envelope inflation due to the proximity of these models to their Eddington limit. This leads to a maximum ZAMS surface temperature of ~56 000 K, at ~180 M⊙, and to an evolution of stars in the mass range 50 M⊙...100 M⊙ to the regime of luminous blue variables in the Hertzsprung-Russell diagram with high internal Eddington factors. Inflation also leads to decreasing surface temperatures during the chemically homogeneous evolution of stars above ~180 M⊙. Conclusions: The cool surface temperatures due to the envelope inflation in our models lead to an enhanced mass loss, which prevents stars at LMC metallicity from evolving into pair-instability supernovae
16. How Very Massive Metal Free Stars Start Cosmological Reionization
SciTech Connect
Wise, John H.; Abel, Tom
2007-11-07
The initial conditions and relevant physics for the formation of the earliest galaxies are well specified in the concordance cosmology. Using ab initio cosmological Eulerian adaptive mesh refinement radiation hydrodynamical calculations, we discuss how very massive stars start the process of cosmological reionization. The models include non-equilibrium primordial gas chemistry and cooling processes and accurate radiation transport in the Case B approximation using adaptively ray traced photon packages, retaining the time derivative in the transport equation. Supernova feedback is modeled by thermal explosions triggered at parsec scales. All calculations resolve the local Jeans length by at least 16 grid cells at all times and as such cover a spatial dynamic range of {approx}10{sup 6}. These first sources of reionization are highly intermittent and anisotropic and first photoionize the small scales voids surrounding the halos they form in, rather than the dense filaments they are! embedded in. As the merging objects form larger, dwarf sized galaxies, the escape fraction of UV radiation decreases and the H II regions only break out on some sides of the galaxies making them even more anisotropic. In three cases, SN blast waves induce star formation in overdense regions that were formed earlier from ionization front instabilities. These stars form tens of parsecs away from the center of their parent DM halo. Approximately 5 ionizing photons are needed per sustained ionization when star formation in 10{sup 6} M{sub {circle_dot}} halos are dominant in the calculation. As the halos become larger than {approx}10{sup 7} M{sub {circle_dot}}, the ionizing photon escape fraction decreases, which in turn increases the number of photons per ionization to 15--50, in calculations with stellar feedback only. Supernova feedback in these more massive halos creates a more diffuse medium, allowing the stellar radiation to escape more easily and maintaining the ratio of 5 ionizing
17. PROTOSTELLAR OUTFLOWS AND RADIATIVE FEEDBACK FROM MASSIVE STARS
SciTech Connect
Kuiper, Rolf; Yorke, Harold W.; Turner, Neal J. E-mail: [email protected]
2015-02-20
We carry out radiation hydrodynamical simulations of the formation of massive stars in the super-Eddington regime including both their radiative feedback and protostellar outflows. The calculations start from a prestellar core of dusty gas and continue until the star stops growing. The accretion ends when the remnants of the core are ejected, mostly by the force of the direct stellar radiation in the polar direction and elsewhere by the reradiated thermal infrared radiation. How long the accretion persists depends on whether the protostellar outflows are present. We set the mass outflow rate to 1% of the stellar sink particle's accretion rate. The outflows open a bipolar cavity extending to the core's outer edge, through which the thermal radiation readily escapes. The radiative flux is funneled into the polar directions while the core's collapse proceeds near the equator. The outflow thus extends the ''flashlight effect'', or anisotropic radiation field, found in previous studies from the few hundred AU scale of the circumstellar disk up to the 0.1 parsec scale of the core. The core's flashlight effect allows core gas to accrete on the disk for longer, in the same way that the disk's flashlight effect allows disk gas to accrete on the star for longer. Thus although the protostellar outflows remove material near the core's poles, causing slower stellar growth over the first few free-fall times, they also enable accretion to go on longer in our calculations. The outflows ultimately lead to stars of somewhat higher mass.
18. Pair instability supernovae of very massive population III stars
SciTech Connect
Chen, Ke-Jung; Woosley, Stan; Heger, Alexander; Almgren, Ann; Whalen, Daniel J.
2014-09-01
Numerical studies of primordial star formation suggest that the first stars in the universe may have been very massive. Stellar models indicate that non-rotating Population III stars with initial masses of 140-260 M {sub ☉} die as highly energetic pair-instability supernovae. We present new two-dimensional simulations of primordial pair-instability supernovae done with the CASTRO code. Our simulations begin at earlier times than previous multidimensional models, at the onset of core contraction, to capture any dynamical instabilities that may be seeded by core contraction and explosive burning. Such instabilities could enhance explosive yields by mixing hot ash with fuel, thereby accelerating nuclear burning, and affect the spectra of the supernova by dredging up heavy elements from greater depths in the star at early times. Our grid of models includes both blue supergiants and red supergiants over the range in progenitor mass expected for these events. We find that fluid instabilities driven by oxygen and helium burning arise at the upper and lower boundaries of the oxygen shell ∼20-100 s after core bounce. Instabilities driven by burning freeze out after the SN shock exits the helium core. As the shock later propagates through the hydrogen envelope, a strong reverse shock forms that drives the growth of Rayleigh-Taylor instabilities. In red supergiant progenitors, the amplitudes of these instabilities are sufficient to mix the supernova ejecta.
19. Pair Instability Supernovae of Very Massive Population III Stars
NASA Astrophysics Data System (ADS)
Chen, Ke-Jung; Heger, Alexander; Woosley, Stan; Almgren, Ann; Whalen, Daniel J.
2014-09-01
Numerical studies of primordial star formation suggest that the first stars in the universe may have been very massive. Stellar models indicate that non-rotating Population III stars with initial masses of 140-260 M ⊙ die as highly energetic pair-instability supernovae. We present new two-dimensional simulations of primordial pair-instability supernovae done with the CASTRO code. Our simulations begin at earlier times than previous multidimensional models, at the onset of core contraction, to capture any dynamical instabilities that may be seeded by core contraction and explosive burning. Such instabilities could enhance explosive yields by mixing hot ash with fuel, thereby accelerating nuclear burning, and affect the spectra of the supernova by dredging up heavy elements from greater depths in the star at early times. Our grid of models includes both blue supergiants and red supergiants over the range in progenitor mass expected for these events. We find that fluid instabilities driven by oxygen and helium burning arise at the upper and lower boundaries of the oxygen shell ~20-100 s after core bounce. Instabilities driven by burning freeze out after the SN shock exits the helium core. As the shock later propagates through the hydrogen envelope, a strong reverse shock forms that drives the growth of Rayleigh-Taylor instabilities. In red supergiant progenitors, the amplitudes of these instabilities are sufficient to mix the supernova ejecta.
20. Massive stars on the verge of exploding: the properties of oxygen sequence Wolf-Rayet stars
NASA Astrophysics Data System (ADS)
Tramper, F.; Straal, S. M.; Sanyal, D.; Sana, H.; de Koter, A.; Gräfener, G.; Langer, N.; Vink, J. S.; de Mink, S. E.; Kaper, L.
2015-09-01
Context. Oxygen sequence Wolf-Rayet (WO) stars are a very rare stage in the evolution of massive stars. Their spectra show strong emission lines of helium-burning products, in particular highly ionized carbon and oxygen. The properties of WO stars can be used to provide unique constraints on the (post-)helium burning evolution of massive stars, and their remaining lifetimes and the expected properties of their supernovae. Aims: We aim to homogeneously analyze the currently known presumed-single WO stars to obtain the key stellar and outflow properties and to constrain their evolutionary state. Methods: We use the line-blanketed non-local thermal equilibrium atmosphere code cmfgen to model the X-Shooter spectra of the WO stars and to deduce the atmospheric parameters. We calculate dedicated evolutionary models to determine the evolutionary state of the stars. Results: The WO stars have extremely high temperatures that range from 150 kK to 210 kK, and very low surface helium mass fractions that range from 44% down to 14%. Their properties can be reproduced by evolutionary models with helium zero-age main sequence masses of MHe,ini = 15-25 M⊙ that exhibit a fairly strong (a few times 10-5M⊙ yr-1), homogeneous (fc> 0.3) stellar wind. Conclusions: WO stars represent the final evolutionary stage of stars with estimated initial masses of Mini = 40-60 M⊙. They are post core-helium burning and predicted to explode as type Ic supernovae within a few thousand years. Based on observations obtained at the European Southern Observatory under program IDs 091.C-0934 and 093.D-0591.Appendices are available in electronic form at http://www.aanda.org
1. Properties of massive stars in four clusters of the VVV survey
NASA Astrophysics Data System (ADS)
Hervé, A.; Martins, F.; Chené, A.-N.; Bouret, J.-C.; Borissova, J.
2016-05-01
The evolution of massive stars is only partly understood. Observational constraints can be obtained from the study of massive stars located in young massive clusters. The ESO Public Survey "VISTA Variables in the Vía Lácteá (VVV)" discovered several new clusters hosting massive stars. We present an analysis of massive stars in four of these new clusters. Our aim is to provide constraints on stellar evolution and to better understand the relation between different types of massive stars. We use the radiative transfer code CMFGEN to analyse K-band spectra of twelve stars with spectral types ranging from O and B to WN and WC. We derive the stellar parameters of all targets as well as surface abundances for a subset of them. In the Hertzsprung-Russell diagram, the Wolf-Rayet stars are more luminous or hotter than the O stars. From the log(C/N)-log(C/He) diagram, we show quantitatively that WN stars are more chemically evolved than O stars, WC stars being more evolved than WN stars. Mass loss rates among Wolf-Rayet stars are a factor of 10 larger than for O stars, in agreement with previous findings.
2. A minimum column density of 1 g cm(-2) for massive star formation.
PubMed
Krumholz, Mark R; McKee, Christopher F
2008-02-28
Massive stars are very rare, but their extreme luminosities make them both the only type of young star we can observe in distant galaxies and the dominant energy sources in the Universe today. They form rarely because efficient radiative cooling keeps most star--forming gas clouds close to isothermal as they collapse, and this favours fragmentation into stars of one solar mass or lower. Heating of a cloud by accreting low-mass stars within it can prevent fragmentation and allow formation of massive stars, but the necessary properties for a cloud to form massive stars-and therefore where massive stars form in a galaxy--have not yet been determined. Here we show that only clouds with column densities of at least 1 g cm(-2) can avoid fragmentation and form massive stars. This threshold, and the environmental variation of the stellar initial mass function that it implies, naturally explain the characteristic column densities associated with massive star clusters and the difference between the radial profiles of Halpha and ultraviolet emission in galactic disks. The existence of a threshold also implies that the initial mass function should show detectable variation with environment within the Galaxy, that the characteristic column densities of clusters containing massive stars should vary between galaxies, and that star formation rates in some galactic environments may have been systematically underestimated.
3. Combining magnetic and seismic studies to constrain processes in massive stars
NASA Astrophysics Data System (ADS)
Neiner, Coralie; Degroote, Pieter; Coste, Blanche; Briquet, Maryline; Mathis, Stéphane
2014-08-01
The presence of pulsations influences the local parameters at the surface of massive stars and thus it modifies the Zeeman magnetic signatures. Therefore it makes the characterisation of a magnetic field in pulsating stars more difficult and the characterisation of pulsations is thus required for the study of magnetic massive stars. Conversely, the presence of a magnetic field can inhibit differential rotation and mixing in massive stars and thus provides important constraints for seismic modelling based on pulsation studies. As a consequence, it is necessary to combine spectropolarimetric and seismic studies for all massive classical pulsators. Below we show examples of such combined studies and the interplay between physical processes.
4. Dynamic Star Formation in the Massive DR21 Filament
SciTech Connect
Schneider, N.; Csengeri, T.; Bontemps, S.; Motte, F.; Simon, R.; Hennebelle, P.; Federrath, C.; Klessen, R.; /ZAH, Heidelberg /KIPAC, Menlo Park
2010-08-25
The formation of massive stars is a highly complex process in which it is unclear whether the star-forming gas is in global gravitational collapse or an equilibrium state supported by turbulence and/or magnetic fields. By studying one of the most massive and dense star-forming regions in the Galaxy at a distance of less than 3 kpc, i.e. the filament containing the well-known sources DR21 and DR21(OH), we attempt to obtain observational evidence to help us to discriminate between these two views. We use molecular line data from our {sup 13}CO 1 {yields} 0, CS 2 {yields} 1, and N{sub 2}H{sup +} 1 {yields} 0 survey of the Cygnus X region obtained with the FCRAO and CO, CS, HCO{sup +}, N{sub 2}H{sup +}, and H{sub 2}CO data obtained with the IRAM 30m telescope. We observe a complex velocity field and velocity dispersion in the DR21 filament in which regions of the highest column-density, i.e., dense cores, have a lower velocity dispersion than the surrounding gas and velocity gradients that are not (only) due to rotation. Infall signatures in optically thick line profiles of HCO{sup +} and {sup 12}CO are observed along and across the whole DR21 filament. By modelling the observed spectra, we obtain a typical infall speed of {approx}0.6 km s{sup -1} and mass accretion rates of the order of a few 10{sup -3} M{sub {circle_dot}} yr{sup -1} for the two main clumps constituting the filament. These massive clumps (4900 and 3300 M{sub {circle_dot}} at densities of around 10{sup 5} cm{sup -3} within 1 pc diameter) are both gravitationally contracting. The more massive of the clumps, DR21(OH), is connected to a sub-filament, apparently 'falling' onto the clump. This filament runs parallel to the magnetic field. Conclusions. All observed kinematic features in the DR21 filament (velocity field, velocity dispersion, and infall), its filamentary morphology, and the existence of (a) sub-filament(s) can be explained if the DR21 filament was formed by the convergence of flows on large
5. The role of low-mass star clusters in forming the massive stars in DR 21
NASA Astrophysics Data System (ADS)
Rivilla, V. M.; Jiménez-Serra, I.; Martín-Pintado, J.; Sanz-Forcada, J.
2014-01-01
We have studied the young low-mass pre-main sequence (PMS) stellar population associated with the massive star-forming region DR 21 by using archival X-ray Chandra observations and by complementing them with existing optical and infrared (IR) surveys. The Chandra observations have revealed for the first time a new highly extincted population of PMS low-mass stars previously missed in observations at other wavelengths. The X-ray population exhibits three main stellar density peaks, coincident with the massive star-forming regions, being the DR 21 core the main peak. The cross-correlated X-ray/IR sample exhibits a radial Spokes-like' stellar filamentary structure that extends from the DR 21 core towards the northeast. The near-IR data reveal a centrally peaked structure for the extinction, which exhibits its maximum in the DR 21 core and gradually decreases with the distance to the N-S cloud axis and to the cluster centre. We find evidence of a global mass segregation in the full low-mass stellar cluster, and of a stellar age segregation, with the youngest stars still embedded in the N-S cloud, and more evolved stars more spatially distributed. The results are consistent with the scenario where an elongated overall potential well created by the full low-mass stellar cluster funnels gas through filaments feeding stellar formation. Besides the full gravitational well, smaller scale local potential wells created by dense stellar sub-clusters of low-mass stars are privileged in the competition for the gas of the common reservoir, allowing the formation of massive stars. We also discuss the possibility that a stellar collision in the very dense stellar cluster revealed by Chandra in the DR 21 core is the origin of the large-scale and highly energetic outflow arising from this region.
6. Mass Loss and Pre-SN Evolution of Massive Stars
NASA Astrophysics Data System (ADS)
Smith, N.
2010-06-01
I review the role that mass loss plays in the pre-SN evolution of massive stars in a variety of different scenarios, and what observable effect it may have on the resulting SN. The amount of mass lost, its speed, and how soon before core collapse the material is removed can have a dramatic effect on the resulting SN light curve and spectrum. Massive stars trek across the HR diagram as they evolve, and the SN can look very different depending on where along this path core collapse occurs; it may not depend solely on initial mass. The most extreme pre-SN mass ejections in massive luminous blue variables (LBVs) have recently (and surprisingly) been linked to the very luminous Type IIn supernovae with circumstellar interaction that dominates the spectrum and enhances the visual luminosity. In some cases these objects require strong LBV-like shell ejections in the decades immediately before a SN. Strong winds or episodic mass loss of luminous red supergiants (RSGs) and yellow hypergiants may also lead to less extreme Type IIn events. Post-RSG blue supergiants like SN 1987A's progenitor and lower-luminosity LBVs like HD 168625 are also candidates for Type II SNe with visible circumstellar material. Finally, progenitors that successfully shed their H envelopes (either through LBV eruptions, strong winds, or binary mass transfer) die as Type Ib or Ic supernovae, and some of these also show evidence for immediate pre-SN shell ejections. Many of the potential progenitors of Types Ib, Ic, IIn, IIb, and II-L overlap in their range of probable initial mass, and I will point to some open questions about how they fit together in the context of stellar evolution, and the roles of mass loss and initial mass in determining their relative rates.
7. How Very Massive Metal-Free Stars Start Cosmological Reionization
NASA Technical Reports Server (NTRS)
Wise, John H.; Abel, Tom
2008-01-01
The initial conditions and relevant physics for the formation of the earliest galaxies are well specified in the concordance cosmology. Using ab initio cosmological Eulerian adaptive mesh refinement radiation hydrodynamical calculations, we discuss how very massive stars start the process of cosmological reionization. The models include nonequilibrium primordial gas chemistry and cooling processes and accurate radiation transport in the case B approximation using adaptively ray-traced photon packages, retaining the time derivative in the transport equation. Supernova feedback is modeled by thermal explosions triggered at parsec scales. All calculations resolve the local Jeans length by at least 16 grid cells at all times and as such cover a spatial dynamic range of approx.10(exp 6). These first sources of reionization are highly intermittent and anisotropic and first photoionize the small-scale voids surrounding the halos they form in, rather than the dense filaments they are embedded in. As the merging objects form larger, dwarf-sized galaxies, the escape fraction of UV radiation decreases and the H II regions only break out on some sides of the galaxies, making them even more anisotropic. In three cases, SN blast waves induce star formation in overdense regions that were formed earlier from ionization front instabilities. These stars form tens of parsecs away from the center of their parent DM halo. Approximately five ionizing photons are needed per sustained ionization when star formation in 10(exp 6) stellar Mass halos is dominant in the calculation. As the halos become larger than approx.10(exp 7) Stellar Mass, the ionizing photon escape fraction decreases, which in turn increases the number of photons per ionization to 15-50, in calculations with stellar feedback only. Radiative feedback decreases clumping factors by 25% when compared to simulations without star formation and increases the average temperature of ionized gas to values between 3000 and 10,000 K.
8. JET FORMATION FROM MASSIVE YOUNG STARS: MAGNETOHYDRODYNAMICS VERSUS RADIATION PRESSURE
SciTech Connect
Vaidya, Bhargav; Porth, Oliver; Fendt, Christian; Beuther, Henrik E-mail: [email protected]
2011-11-20
Observations indicate that outflows from massive young stars are more collimated during their early evolution compared to later stages. Our paper investigates various physical processes that impact the outflow dynamics, i.e., its acceleration and collimation. We perform axisymmetric magnetohydrodynamic (MHD) simulations particularly considering the radiation pressure exerted by the star and the disk. We have modified the PLUTO code to include radiative forces in the line-driving approximation. We launch the outflow from the innermost disk region (r < 50 AU) by magnetocentrifugal acceleration. In order to disentangle MHD effects from radiative forces, we start the simulation in pure MHD and later switch on the radiation force. We perform a parameter study considering different stellar masses (thus luminosity), magnetic flux, and line-force strength. For our reference simulation-assuming a 30 M{sub Sun} star-we find substantial de-collimation of 35% due to radiation forces. The opening angle increases from 20 Degree-Sign to 32 Degree-Sign for stellar masses from 20 M{sub Sun} to 60 M{sub Sun }. A small change in the line-force parameter {alpha} from 0.60 to 0.55 changes the opening angle by {approx}8 Degree-Sign . We find that it is mainly the stellar radiation that affects the jet dynamics. Unless the disk extends very close to the star, its force is too small to have much impact. Essentially, our parameter runs with different stellar masses can be understood as a proxy for the time evolution of the star-outflow system. Thus, we have shown that when the stellar mass (thus luminosity) increases with age, the outflows become less collimated.
9. The chemical composition of Galactic ring nebulae around massive stars
NASA Astrophysics Data System (ADS)
Esteban, C.; Mesa-Delgado, A.; Morisset, C.; García-Rojas, J.
2016-08-01
We present deep spectra of ring nebulae associated with Wolf-Rayet (WR) and O-type stars: NGC 6888, G2.4+1.4, RCW 58, S 308, NGC 7635 and RCW 52. The data have been taken with the 10m Gran Telescopio Canarias and the 6.5m Clay Telescope. We extract spectra of several apertures in some of the objects. We derive C2+ and O2+ abundances from faint recombination lines in NGC 6888 and NGC 7635, permitting to derive their C/H and C/O ratios and estimate the abundance discrepancy factor (ADF) of O2+. The ADFs are larger than the typical ones of normal H II regions but similar to those found in the ionized gas of star-forming dwarf galaxies. We find that chemical abundances are rather homogeneous in the nebulae where we have spectra of several apertures: NGC 6888, NGC 7635 and G2.4+1.4. We obtain very high values of electron temperature in a peripheral zone of NGC 6888, finding that shock excitation can reproduce its spectral properties. We find that all the objects associated with WR stars show N enrichment. Some of them also show He enrichment and O deficiency as well as a lower Ne/O than expected, this may indicate the strong action of the ON and NeNa cycles. We have compared the chemical composition of NGC 6888, G2.4+1.4, RCW 58 and S 308 with the nucleosynthesis predicted by stellar evolution models of massive stars. We find that non-rotational models of stars of initial masses between 25 and 40 M⊙ seem to reproduce the observed abundance ratios of most of the nebulae.
10. Massive Young Star Clusters in M33: Stochastic Star Formation Ruled Out
NASA Astrophysics Data System (ADS)
González-Lópezlira, R. A.; Pflamm-Altenburg, J.; Kroupa, P.
2014-09-01
It is widely accepted that the distribution function of the masses of young star clusters is universal and can be purely interpreted as a probability density distribution function with a constant upper mass limit. As a result of this picture, the masses of the most massive objects would be exclusively determined by the size of the sample. Conversely we show, with very high confidence, that the masses of the most massive young (< 10 Myr) star clusters in the flocculent galaxy M33 decrease with increasing galactocentric radius, in contradiction with a constant shape and upper mass limit of the cluster mass function. Moreover, by comparing the radial distributions of gas surface densities and highest cluster masses, we find that M_{max} ∝ Σ_{gas, total}^{3.8 ± 0.3}, M_{max} ∝ Σ_{H_2}^{1.2± 0.1} and M_{max} ∝ Σ_{SFR}^{0.9 ± 0.1}. Hence, in M33 we can rule out stochastic star formation. The change of the maximum cluster mass there must be due to physical causes, i.e., very massive star clusters may require special physical conditions, like high gas surface densities, in order to form.
11. Massive star formation within the Leo 'primordial' ring.
PubMed
Thilker, David A; Donovan, Jennifer; Schiminovich, David; Bianchi, Luciana; Boissier, Samuel; de Paz, Armando Gil; Madore, Barry F; Martin, D Christopher; Seibert, Mark
2009-02-19
Few intergalactic, plausibly primordial clouds of neutral atomic hydrogen (H(i)) have been found in the local Universe, suggesting that such structures have either dispersed, become ionized or produced a stellar population on gigayear timescales. The Leo ring, a massive (M(H(i)) approximately 1.8 x 10(9)M[symbol: see text], M[symbol: see text] denoting the solar mass), 200-kpc-wide structure orbiting the galaxies M105 and NGC 3384 with a 4-Gyr period, is a candidate primordial cloud. Despite repeated atttempts, it has previously been seen only from H i emission, suggesting the absence of a stellar population. Here we report the detection of ultraviolet light from gaseous substructures of the Leo ring, which we attribute to recent massive star formation. The ultraviolet colour of the detected complexes is blue, implying the onset of a burst of star formation or continuous star formation of moderate (approximately 10(8)-yr) duration. Measured ultraviolet-visible photometry favours models with low metallicity (Z approximately Z[symbol: see text]/50-Z[symbol: see text]/5, Z[symbol: see text] denoting the solar metallicity), that is, a low proportion of elements heavier than helium, although spectroscopic confirmation is needed. We speculate that the complexes are dwarf galaxies observed during their formation, but distinguished by their lack of a dark matter component. In this regard, they resemble tidal dwarf galaxies, although without the enrichment preceding tidal stripping. If structures like the Leo ring were common in the early Universe, they may have produced a large, yet undetected, population of faint, metal-poor, halo-lacking dwarf galaxies.
12. The Very Massive Star Content of the Nuclear Star Clusters in NGC 5253
NASA Astrophysics Data System (ADS)
Smith, L. J.; Crowther, P. A.; Calzetti, D.; Sidoli, F.
2016-05-01
The blue compact dwarf galaxy NGC 5253 hosts a very young starburst containing twin nuclear star clusters, separated by a projected distance of 5 pc. One cluster (#5) coincides with the peak of the Hα emission and the other (#11) with a massive ultracompact H ii region. A recent analysis of these clusters shows that they have a photometric age of 1 ± 1 Myr, in apparent contradiction with the age of 3-5 Myr inferred from the presence of Wolf-Rayet features in the cluster #5 spectrum. We examine Hubble Space Telescope ultraviolet and Very Large Telescope optical spectroscopy of #5 and show that the stellar features arise from very massive stars (VMSs), with masses greater than 100 M ⊙, at an age of 1-2 Myr. We further show that the very high ionizing flux from the nuclear clusters can only be explained if VMSs are present. We investigate the origin of the observed nitrogen enrichment in the circumcluster ionized gas and find that the excess N can be produced by massive rotating stars within the first 1 Myr. We find similarities between the NGC 5253 cluster spectrum and those of metal-poor, high-redshift galaxies. We discuss the presence of VMSs in young, star-forming galaxies at high redshift; these should be detected in rest-frame UV spectra to be obtained with the James Webb Space Telescope. We emphasize that population synthesis models with upper mass cutoffs greater than 100 M ⊙ are crucial for future studies of young massive star clusters at all redshifts.
13. The Very Massive Star Content of the Nuclear Star Clusters in NGC 5253
NASA Astrophysics Data System (ADS)
Smith, L. J.; Crowther, P. A.; Calzetti, D.; Sidoli, F.
2016-05-01
The blue compact dwarf galaxy NGC 5253 hosts a very young starburst containing twin nuclear star clusters, separated by a projected distance of 5 pc. One cluster (#5) coincides with the peak of the Hα emission and the other (#11) with a massive ultracompact H ii region. A recent analysis of these clusters shows that they have a photometric age of 1 ± 1 Myr, in apparent contradiction with the age of 3–5 Myr inferred from the presence of Wolf-Rayet features in the cluster #5 spectrum. We examine Hubble Space Telescope ultraviolet and Very Large Telescope optical spectroscopy of #5 and show that the stellar features arise from very massive stars (VMSs), with masses greater than 100 M ⊙, at an age of 1–2 Myr. We further show that the very high ionizing flux from the nuclear clusters can only be explained if VMSs are present. We investigate the origin of the observed nitrogen enrichment in the circumcluster ionized gas and find that the excess N can be produced by massive rotating stars within the first 1 Myr. We find similarities between the NGC 5253 cluster spectrum and those of metal-poor, high-redshift galaxies. We discuss the presence of VMSs in young, star-forming galaxies at high redshift; these should be detected in rest-frame UV spectra to be obtained with the James Webb Space Telescope. We emphasize that population synthesis models with upper mass cutoffs greater than 100 M ⊙ are crucial for future studies of young massive star clusters at all redshifts.
14. Resolved photometry of extragalactic young massive star clusters
NASA Astrophysics Data System (ADS)
Larsen, S. S.; de Mink, S. E.; Eldridge, J. J.; Langer, N.; Bastian, N.; Seth, A.; Smith, L. J.; Brodie, J.; Efremov, Yu. N.
2011-08-01
Aims: We present colour-magnitude diagrams (CMDs) of young massive star clusters in several galaxies located well beyond the Local Group. The richness of these clusters allows us to obtain large samples of post-main sequence stars and test how well the observed CMDs are reproduced by canonical stellar isochrones. Methods: We use imaging of seven clusters in the galaxies NGC 1313, NGC 1569, NGC 1705, NGC 5236 and NGC 7793 obtained with the Advanced Camera for Surveys on board the Hubble Space Telescope and carry out PSF-fitting photometry of individual stars in the clusters. The clusters have ages in the range ~(5-50) × 106 years and masses of ~105 M⊙-106 M⊙. Although crowding prevents us from obtaining photometry in the inner regions of the clusters, we are still able to measure up to 30-100 supergiant stars in each of the richest clusters. The resulting CMDs and luminosity functions are compared with photometry of artificially generated clusters, designed to reproduce the photometric errors and completeness as realistically as possible. Results: In agreement with previous studies, our CMDs show no clear gap between the H-burning main sequence and the He-burning supergiant stars, contrary to predictions by common stellar isochrones. In general, the isochrones also fail to match the observed number ratios of red-to-blue supergiant stars, although the difficulty of separating blue supergiants from the main sequence complicates this comparison. In several cases we observe a large spread (1-2 mag) in the luminosities of the supergiant stars that cannot be accounted for by observational errors. We find that this spread can be reproduced by including an age spread of ~(10-30) × 106 years in the models. However, age spreads cannot fully account for the observed morphology of the CMDs and other processes, such as the evolution of interacting binary stars, may also play a role. Conclusions: Colour-magnitude diagrams can be successfully obtained for massive star
15. The growth of massive stars via stellar collisions in ensemble star clusters
NASA Astrophysics Data System (ADS)
Fujii, M. S.; Portegies Zwart, S.
2013-04-01
Recent simulations and observations suggest that star clusters form via the assembling of smaller subclusters. Because of their short relaxation time, subclusters experience core collapse much earlier than virialized solo clusters, which have similar properties of the merger remnant of the assembling clusters. As a consequence, it seems that the assembling clusters result in efficient multiple collisions of stars in the cluster core. We performed a series of N-body simulations of ensemble and solitary clusters including stellar collisions and found that the efficiency of multiple collisions between stars is suppressed if subclusters assemble after they experience core collapse individually. In this case, subclusters form their own multiple collision stars which experienced a few collisions, but they fail to collide with each other after their host subclusters assemble. The multiple collision stars scatter each other and escape, and furthermore the central density of the remnant clusters had already been depleted for the stars to experience more collisions. On the other hand, if subclusters assemble before they experience core collapse, the multiple collisions of stars proceed efficiently in the remnant cluster, and the collision products are more massive than virialized solo clusters and comparable in mass to cold solo clusters.
16. Massive stars and miniature robots: today's research and tomorrow's technologies
NASA Astrophysics Data System (ADS)
Taylor, William David
2013-03-01
This thesis documents the reduction of the VLT-FLAMES Tarantula Survey (VFTS) data set, whilst also describing the analysis for one of the serendipitous discoveries: the massive binary R139. This high-mass binary will provide an excellent future calibration point for stellar models, in part as it seems to defy certain expectations about its evolution. Out with the VFTS, a search for binary companions around a trio of B-type supergiants is presented. These stars are surrounded by nebulae that closely resemble the triple-ring structure associated with the poorly-understood SN1987A. Do these stars share a similar evolutionary fate? While strong evidence is found for periodic pulsations in one of the stars, there appears to be no indication of a short-period binary companion suggested in the literature. Gathering observations from a wide range of environments builds a fuller picture of massive stars, but the samples remain somewhat limited. The coming generation of extremely large telescopes will open new regions for studies like the VFTS. Fully utilising these remarkable telescopes will require many new technologies, and this thesis presents one such development project. For adaptive-optics corrected, multi-object instruments it will be necessary to position small pick-off mirrors in the telescope¿s focal plane to select the sub-fields on the sky. This could be most efficiently achieved if the mirrors were self-propelled, which has led to a miniature robot project called MAPS - the Micro Autonomous Positioning System. A number of robots have been built with a footprint of only 30 x 30mm. These wirelessly-controlled robots draw their power from the floor on which they operate and have shown the potential to be positioned to an accuracy of tens of microns. This thesis details much of the early design work and testing of the robots, and also the development of the camera imaging system used to determine the position of the robots. The MAPS project is ongoing and a
17. X-ray diagnostics of massive star winds
NASA Astrophysics Data System (ADS)
Oskinova, Lidia M.
2016-09-01
Nearly all types of massive stars with radiatively driven stellar winds are X-ray sources that can be observed by the presently operating powerful X-ray telescopes. In this review I briefly address recent advances in our understanding of stellar winds obtained from X-ray observations. X-rays may strongly influence the dynamics of weak winds of main sequence B-type stars. X-ray pulsations were detected in a β Cep type variable giving evidence of tight photosphere-wind connections. The winds of OB dwarfs with subtypes later than O9V may be predominantly in a hot phase, and X-ray observations offer the best window for their studies. The X-ray properties of OB supergiants are largely determined by the effects of radiative transfer in their clumped stellar winds. The recently suggested method to directly measure mass-loss rates of O stars by fitting the shapes of X-ray emission lines is considered but its validity cannot be confirmed. To obtain robust quantitative information on stellar wind parameters from X-ray spectroscopy, a multiwavelength analysis by means of stellar atmosphere models is required. Independent groups are now performing such analyses with encouraging results. Joint analyses of optical, UV, and X-ray spectra of OB supergiants yield consistent mass-loss rates. Depending on the adopted clumping parameters, the empirically derived mass-loss rates are a factor of a few smaller or comparable to those predicted by standard recipes (Vink et al., 2001). All sufficiently studied O stars display variable X-ray emission that might be related to corotating interaction regions in their winds. In the latest stages of stellar evolution, single red supergiants (RSG) and luminous blue variable (LBV) stars do not emit observable amounts of X-rays. On the other hand, nearly all types of Wolf-Rayet (WR) stars are X-ray sources. X-ray spectroscopy allows a sensitive probe of WR wind abundances and opacities.
18. Modeling and analysing massive star spectra: recent advances
NASA Astrophysics Data System (ADS)
Hamann, Wolf-Rainer; Todt, Helge; Sander, Andreas; Hainich, Rainer; Shenar, Tomer; Oskinova, Lidia
2013-06-01
Depending on their mass-loss rate, the spectra of massive stars are more or less formed in the expanding parts of their atmosphere, i.e. in the stellar wind. Over decades we have developed a sophisticated non-LTE code for modeling such spectra adequately. Originally, the "Potsdam WR PoWR" code aimed at Wolf-Rayet stars with their emission-line dominated spectra. Meanwhile we have added a more detailed treatment of the lower, nearly static parts of the atmosphere, including pressure broadening of lines. This extends the applicability of the models to spectra showing both, photospheric absorption lines and stellar wind features, e.g. from O and B-type stars. The ionizing effect of X-rays, which are intrinsically produced in stellar winds, can be taken into account. Instead of a one-temperature plasma, a power-law distribution of the X-ray emission measure can be chosen and gives the best fit of the EUV spectral energy distribution. The effect of rotation on the emergent spectrum can be simulated under suitable assumptions on the angular motions in the wind. When clumping is accounted for in the approximation of optically thin structures, this leads to a reduction of empirical mass-loss rates when determined from recombination lines. A more general, but not fully consistent formalism has been incorporated to account for the effect of "macroclumping" on resonance lines. PoWR calculations were also combined with a 3-D Monte Carlo code for resonance line scattering in a structured stellar wind. A formalism has been developed to establish the hydrodynamically consistent solution for radiation-driven winds, including all multiple-scattering effects that are essential e.g. for WR stars, but this branch of the code is not ready yet for routinely use. PoWR models have been used extensively for analyzing WR stars in the Galaxy and the Magellanic Clouds, and for a couple of OB-type stars and LBVs. An increasing number of models is made available via internet.
19. Rotational Signatures of Disks in Massive Star Formation
NASA Astrophysics Data System (ADS)
Fallscheer, Cassandra L.; Beuther, H.; Zhang, Q.; Sridharan, T. K.
2008-03-01
We have obtained multiple data sets from the SMA, PdBI, and IRAM 30m telescope of the massive Infrared Dark Cloud IRDC18223-3 and High-Mass Protostellar Object IRAS18151-1208 in order to look for clues regarding the role of rotation and disks in high mass star formation. Because IRAS 18151-1208 is at a later evolutionary stage than IRDC 18223-3, these two objects allow us to compare the central-most regions surrounding the embedded continuum source at two different periods in the formation process. Toward both regions we see rotational structures perpendicular to molecular outflows. Similarities and differences will be discussed in the context of core and disk evolution.
20. Dynamic star formation in the massive DR21 filament
NASA Astrophysics Data System (ADS)
Schneider, N.; Csengeri, T.; Bontemps, S.; Motte, F.; Simon, R.; Hennebelle, P.; Federrath, C.; Klessen, R.
2010-09-01
Context. The formation of massive stars is a highly complex process in which it is unclear whether the star-forming gas is in global gravitational collapse or an equilibrium state supported by turbulence and/or magnetic fields. In addition, magnetic fields may play a decisive role in the star-formation process since they influence the efficiency of gas infall onto the protostar. Aims: By studying one of the most massive and dense star-forming regions in the Galaxy at a distance of less than 3 kpc, i.e. the filament containing the well-known sources DR21 and DR21(OH), we attempt to obtain observational evidence to help us to discriminate between these two views. Methods: We use molecular line data from our 13CO 1 to 0, CS 2 to 1, and N2H+ 1 to 0 survey of the Cygnus X region obtained with the FCRAO and high-angular resolution observations in isotopomeric lines of CO, CS, HCO+, N2H+, and H2CO, obtained with the IRAM 30 m telescope, to investigate the distribution of the different phases of molecular gas. Gravitational infall is identified by the presence of inverse P Cygni profiles that are detected in optically thick lines, while the optically thinner isotopomers are found to reach a peak in the self-absorption gap. Results: We observe a complex velocity field and velocity dispersion in the DR21 filament in which regions of the highest column-density, i.e., dense cores, have a lower velocity dispersion than the surrounding gas and velocity gradients that are not (only) due to rotation. Infall signatures in optically thick line profiles of HCO+ and 12CO are observed along and across the whole DR21 filament. By modelling the observed spectra, we obtain a typical infall speed of 0.6 km s-1 and mass accretion rates of the order of a few 10-3 M_⊙ yr-1 for the two main clumps constituting the filament. These massive clumps (4900 and 3300 M_⊙ at densities of around 105 cm-3 within 1 pc diameter) are both gravitationally contracting (with free-fall times much shorter
1. Gamma rays from colliding winds of massive stars
NASA Astrophysics Data System (ADS)
Reimer, Anita; Reimer, Olaf; Pohl, Martin
2007-06-01
Colliding winds of massive binaries have long been considered as potential sites of non-thermal high-energy photon production. This is motivated by the detection of non-thermal spectra in the radio band, as well as by correlation studies of yet unidentified EGRET γ-ray sources with source populations appearing in star formation regions. This work re-considers the basic radiative processes and its properties that lead to high energy photon production in long-period massive star systems. We show that Klein Nishina effects as well as the anisotropic nature of the inverse Compton scattering, the dominating leptonic emission process, likely yield spectral and variability signatures in the γ-ray domain at or above the sensitivity of current or upcoming gamma ray instruments like GLAST-LAT. In addition to all relevant radiative losses, we include propagation (such as convection in the stellar wind) as well as photon absorption effects, which a priori can not be neglected. The calculations are applied to WR 140 and WR 147, and predictions for their detectability in the γ-ray regime are provided. Physically similar specimen of their kind like WR 146, WR 137, WR 138, WR 112 and WR 125 may be regarded as candidate sources at GeV energies for near-future γ-ray experiments. Finally, we discuss several aspects relevant for eventually identifying this source class as a γ-ray emitting population. Thereby we utilize our findings on the expected radiative behavior of typical colliding wind binaries in the γ-ray regime as well as its expected spatial distribution on the γ-ray sky.
2. Unravelling the Mystery of Massive Star Birth - All Stars are Born the Same Way
NASA Astrophysics Data System (ADS)
2010-07-01
Astronomers have obtained the first image of a dusty disc closely encircling a massive baby star, providing direct evidence that massive stars form in the same way as their smaller brethren. This discovery, made thanks to a combination of ESO's telescopes, is described in an article in this week's issue of Nature. "Our observations show a disc surrounding an embryonic young, massive star, which is now fully formed," says Stefan Kraus, who led the study. "One can say that the baby is about to hatch!" The team of astronomers looked at an object known by the cryptic name of IRAS 13481-6124. About twenty times the mass of our Sun and five times its radius, the young central star, which is still surrounded by its pre-natal cocoon, is located in the constellation of Centaurus, about 10 000 light-years away. From archival images obtained by the NASA Spitzer Space Telescope as well as from observations done with the APEX 12-metre submillimetre telescope, astronomers discovered the presence of a jet. "Such jets are commonly observed around young low-mass stars and generally indicate the presence of a disc," says Kraus. Circumstellar discs are an essential ingredient in the formation process of low-mass stars such as our Sun. However, it is not known whether such discs are also present during the formation of stars more massive than about ten solar masses, where the strong light emitted might prevent mass falling onto the star. For instance, it has been proposed that massive stars might form when smaller stars merge. In order to discover and understand the properties of this disc, astronomers employed ESO's Very Large Telescope Interferometer (VLTI). By combining light from three of the VLTI's 1.8-metre Auxiliary Telescopes with the AMBER instrument, this facility allows astronomers to see details equivalent to those a telescope with a mirror of 85 metres in diameter would see. The resulting resolution is about 2.4 milliarcseconds, which is equivalent to picking out the head
3. Stability of metal-rich very massive stars
NASA Astrophysics Data System (ADS)
Goodman, J.; White, Christopher J.
2016-02-01
We revisit the stability of very massive non-rotating main-sequence stars at solar metallicity, with the goal of understanding whether radial pulsations set a physical upper limit to stellar mass. Models of up to 938 solar masses are constructed with the MESA code, and their linear stability in the fundamental mode, assumed to be the most dangerous, is analysed with a fully non-adiabatic method. Models above 100 M⊙ have extended tenuous atmospheres (shelves') that affect the stability of the fundamental. Even when positive, this growth rate is small, in agreement with previous results. We argue that small growth rates lead to saturation at small amplitudes that are not dangerous to the star. A mechanism for saturation is demonstrated involving non-linear parametric coupling to short-wavelength g-modes and the damping of the latter by radiative diffusion. The shelves are subject to much more rapidly growing strange modes. This also agrees with previous results but is extended here to higher masses. The strange modes probably saturate via shocks rather than mode coupling but have very small amplitudes in the core, where almost all of the stellar mass resides. Although our stellar models are hydrostatic, the structure of their outer parts suggests that optically thick winds, driven by some combination of radiation pressure, transonic convection, and strange modes, are more likely than pulsation in the fundamental mode to limit the main-sequence lifetime.
4. Modeling Broadband X-Ray Absorption of Massive Star Winds
NASA Technical Reports Server (NTRS)
Leutenegger, Maurice A.; Cohen,David H.; Zsargo, Janos; Martell, Erin M.; MacArthur, James P.; Owocki, Stanley P.; Gagne, Marc; Hillier, D. John
2010-01-01
We present a method for computing the net transition of X-rays emitted by shock-heated plasma distributed throughout a partially optically thick stellar wind from a massive star. We find the transmission by an exact integration of the formal solution, assuming the emitting plasma and absorbing plasma are mixed at a constant mass ratio above some minimum radius, below which there is assumed to be no emission. This model is more realistic than either the slab absorption associated with a corona at the base of the wind or the exospheric approximation that assumes all observed X-rays are emitted without attenuation from above the radius of optical depth unity. Our model is implemented in XSPEC as a pre-calculated table that can be coupled to a user-defined table of the wavelength dependent wind opacity. We provide a default wind opacity model that is more representative of real wind opacities than the commonly used neutral ISM tabulation. Preliminary modeling of Chandra grating data indicates that the X-ray hardness trend of OB stars with spectral subtype cars largely be understood as a wind absorption effect.
5. Studying the nature of runaway stars using Andromeda's massive stellar population
NASA Astrophysics Data System (ADS)
Bulkley, Jordan; Seth, Anil; Johnson, Cliff; Dalcanton, Julianne; Guhathakurta, Raja; Dorman, Claire; Hamren, Katie; Caldwell, Nelson; Williams, Ben
2016-03-01
Theory of the formation of massive stars remains incomplete, the question of the environments required have yet to be answered. An agreement on whether all massive stars must form in cluster type environments, or if isolated formation is viable has yet to be reached. This is further complicated by the presence of runaway stars, stellar objects which have been ejected from their host cluster. Studying the nature of these isolated runaway stars becomes paramount in the larger goal of developing a more comprehensive massive star formation theory. Creating a survey of runaway star candidates is possible thanks to Panchromatic Hubble Andromeda Treasury's UV and optical photometry, and the identified clusters from the Andromeda Project. A first glimpse into the data suggests large body of massive stars are 50 parsecs or more from the closest cluster and roughly half of the entire massive stellar population is found outside of defined cluster boundaries. Additional analysts shows a stark difference between the velocity dispersion of massive stars and appropriately young clusters, the stars exhibiting a inflated dispersion. Using this result in conjunction with artificial clusters and star populations, constrains on the percentage of expected runaway objects can be made.
6. Stellar neutron sources and s-process in massive stars
NASA Astrophysics Data System (ADS)
Talwar, Rashi
The s-process or the slow neutron capture process is a nucleosynthesis process taking place at relatively low neutron densities in stars. It runs along the valley of beta stability since the neutron capture rate is much slower compared to the beta decay rate. The s-process occurs mainly during core helium burning and shell carbon burning phase in massive stars and during thermally pulsing helium burning phase in asymptotic giant-branch stars. The potential stellar neutron source for the s-process is associated with alpha-capture reactions on light nuclei. The capture-reaction rates provide the reaction flow for the build-up of22Ne neutron source during the heliumburning phase in these stars. The low energy 26Mg resonances at stellar energies below 800 keV are predicted to have a critical influence on the alpha-capture rates on 22Ne. Some of these resonances may also correspond to pronounced alpha cluster structure near the alpha-threshold. However, these resonances have remained elusive during direct alpha capture measurements owing to the high Coulomb barrier and background from cosmic rays and beam induced reactions. Hence, in the present work, alpha-inelastic scattering and alpha- transfer measurements have been performed to probe the level structure of 26Mg nucleus in order to determine the 22Ne+alpha-capture rates. Both experiments have been performed using the high-resolution Grand Raiden Spectrometer at the Research Center for Nuclear Physics (RCNP), Osaka, Japan. For the alpha-inelastic scattering measurement, a self-supporting solid 26Mg target was used and for the alpha-transfer study via the (6Li,d) reaction, 22Ne gas enclosed in a gas cell with Aramid windows was used. The reaction products were momentum analysed by the spectrometer and detected at the focal plane equipped with two multi-wire drift chambers and two plastic-scintillation detectors. The focal plane detection system provided information on the position, the angle, the time of flight and
7. Recent results on the connection between massive stars and supernovae
NASA Astrophysics Data System (ADS)
Hillier, D. John
2015-08-01
With the dramatic increase in observational data on supernovae (SNe), SN studies are undergoing a renaissance. It is known that Type II SN IIP arise from the explosion of a red supergiant (RSG). In several cases the RSG is seen in pre-explosion images, but it is absent in post-SN images — unambiguous proof that the RSG has exploded. Surprisingly, all RSG progenitors identified have a mass less than approximately 20 M⊙. To date, there has been no direct detection of the progenitor of a Type Ib or Ic SN. Because their ejecta masses are generally low (3 to 5 M⊙), these SNe are believed to arise from a relatively low mass star in a binary system. Such systems dominate the statistics due to the initial mass function. The broad-lined Ic SNe tend to have higher mass, and some of these may be associated with classic Wolf-Rayet (W-R) stars. Type IIn SNe are a heterogeneous class of SN — they arise when the SN ejecta interacts with preexisting circumstellar material. Their spectra often exhibit narrow emission lines, and they can be particularly luminous due to the efficient conversion of kinetic energy into radiation. The origin of Type IIn SN and their connection to stellar evolution is the subject of fierce debate and controversy. The final class to be discussed are the pair-instability supernovae (PISNe) which arise from a nuclear detonation. PISNe have a distinct chemical signature, and the observational evidence for the existence of this class of SN is ambiguous and controversial. While much progress has been made, it is still difficult to get models of core-collapse SNe to explode from first principles. The problem is inherently 3D and numerous questions remain unanswered. How much material falls back onto the core? What is the nature and extent of mixing in the ejecta? What are the chemical yields? Do all massive stars end their life as a luminous SN?
8. The pre-supernova evolution of massive stars and concomitant nucleosynthesis
SciTech Connect
Meynet, Georges
1998-02-15
After a recall of the main features of massive star evolution, we briefly describe the different nuclear burning phases through which these stars evolve. We discuss determinations of stellar surface abundances which provide some clues to the nuclear processes occuring in massive stars. Finally, we emphasize the role played by mass loss in the process of enrichment of the interstellar medium in newly synthesized elements.
9. The Evolution of Massive Stars: a Selection of Facts and Questions
NASA Astrophysics Data System (ADS)
Vanbeveren, D.
In the present paper we discuss a selection of facts and questions related to observations and evolutionary calculations of massive single stars and massive stars in interacting binaries. We focus on the surface chemical abundances, the role of stellar winds, the early Be-stars, the high mass X-ray binaries and the effects of rotation on stellar evolution. Finally, we present an unconventionally formed object scenario (UFO-scenario) of WR binaries in dense stellar environments.
10. RUNAWAY MASSIVE STARS FROM R136: VFTS 682 IS VERY LIKELY A 'SLOW RUNAWAY'
SciTech Connect
Banerjee, Sambaran; Kroupa, Pavel; Oh, Seungkyung E-mail: [email protected]
2012-02-10
We conduct a theoretical study on the ejection of runaway massive stars from R136-the central massive, starburst cluster in the 30 Doradus complex of the Large Magellanic Cloud. Specifically, we investigate the possibility of the very massive star (VMS) VFTS 682 being a runaway member of R136. Recent observations of the above VMS, by virtue of its isolated location and its moderate peculiar motion, have raised the fundamental question of whether isolated massive star formation is indeed possible. We perform the first realistic N-body computations of fully mass-segregated R136-type star clusters in which all the massive stars are in primordial binary systems. These calculations confirm that the dynamical ejection of a VMS from an R136-like cluster, with kinematic properties similar to those of VFTS 682, is common. Hence, the conjecture of isolated massive star formation is unnecessary to account for this VMS. Our results are also quite consistent with the ejection of 30 Dor 016, another suspected runaway VMS from R136. We further note that during the clusters' evolution, mergers of massive binaries produce a few single stars per cluster with masses significantly exceeding the canonical upper limit of 150 M{sub Sun }. The observations of such single super-canonical stars in R136, therefore, do not imply an initial mass function with an upper limit greatly exceeding the accepted canonical 150 M{sub Sun} limit, as has been suggested recently, and they are consistent with the canonical upper limit.
11. Effects of massive star radiation on circumstellar disks evolution in the Eagle Nebula
NASA Astrophysics Data System (ADS)
Guarcello, Mario
2007-09-01
We will determine the frequency of disk and disk-less stars in the outer regions (relatively poor of massive stars) of the young cluster NGC 6611, with the aim to study the effects of UV flux due to massive stars on the evolution of circumstellar disks around low mass stars. Our previous results for the central region of the cluster show that this effect may be present, but we need to observe stars at larger distance from massive stars. This cluster is particularly well suited for our study, thanks to the irregular spatial distribution of its OB stars. CHANDRA observations are crucial for identifying the disk-less population undetectable with other method.
12. Intermediate-Mass Star-Forming Regions: What are the Most Massive Stars Formed?
NASA Astrophysics Data System (ADS)
Kobulnicky, Chip; Vargas, Carlos; Kerton, Charles; Arvidsson, Kim
2010-08-01
High-mass star formation cannot be viewed as simply a scaled-up version of the paradigm for low-mass star formation. The high-mass regime (M> 10 Msun) appears to require significant differences in cloud fragmentation, accretion, radiation, turbulence, and overall molecular density compared to the low-mass regime. We have identified a sample of intermediate-mass star-forming regions (IM SFRs) hosting embedded clusters that straddle the boundary of these two regimes and can be used to understand the factors that govern the transition between these extremes. Most notable among these factors is the possibility of a critical cloud mass column density that appears to divide high-mass SFRs from IM SFRs. Yet, the very nature of IM SFRs and their stellar content are almost completely unknown, primarily because of the previous difficulty in identifying such objects. We propose HK band spectroscopy of the brightest stellar sources near nine IM SFRs to identify probable members, confirm the IM nature of the most massive stars, and characterize their evolutionary state. Three nights with FLAMINGOS on the 4 m (or equivalent IR spectrograph) will suffice to obtain classification spectra and several spectral diagnostics sensitive to accretion for at least 8-10 stars per object.
13. An Extraordinary Cluster of Massive Young Stars in the Milky Way's Nucleus
NASA Technical Reports Server (NTRS)
Serabyn, E.; Shupe, D.; Figer, D. F.
1998-01-01
The mass distribution of newborn stars is key to the evolution of galaxies, as it determines whether a galaxy's interstellar medium is funneled predominantly into dim, long-lived, low-mass stars, as is the case in normal galactic disks, or into bright, short-lived, massive stars, as is perhaps the case in starburst nuclei.
14. The Wind and Mass-loss Properties of the Most Massive Stars
NASA Astrophysics Data System (ADS)
Bestenlehner, Joachim; Vink, Jorick; Gräfener, Götz; Najarro, Francisco
2013-06-01
Mass-loss rates play an important role in the evolution of massive stars. The initial, present day and the mass at their end of their lifetime is considerable different as a result of mass loss. Different stages of evolution have different mass-loss rates. The understanding of massive star evolution is tightly connected to the understanding of their mass loss properties. In the context of the VLT-Flames Tarantula Survey I will present the results from our spectral analysis of stars in the transition region from O-stars to very massive WN(h)-stars. WN(h)-stars are very young and massive stars which develop already in the earliest stages of their evolution WR-star like winds. For the analysis we used the non-LTE radiative transfer code CMFGEN to investigate the wind and mass-loss properties of these very massive stars. This analysis also tests theoretical predictions which suggest a notable change of the mass-loss behaviour at a certain Eddington factor in the transition region from O to WN(h)-stars (Bestenlehner et al. 2011, Bestenlehner et al. in prep.)
15. WO-Type Wolf-Rayet Stars: the Last Hurrah of the Most Massive Stars?
NASA Astrophysics Data System (ADS)
Massey, Philip
2014-10-01
WO-type Wolf-Rayet (WR) stars are considered the final evolutionary stage of the highest mass stars, immediate precursors to Type Ic (He-poor) core-collapse supernovae. These WO stars are rare, and until recently only 6 were known. Our knowledge about their physical properties is mostly based on a single object, Sand 2 in the LMC. It was the only non-binary WO star both bright and unreddened enough that its FUV and NUV spectra could be obtained by FUSE and HST/FOS. A non-LTE analysis showed that Sand 2 is very hot and its (C+O)/He abundance ratio is higher than that found in WC-type WRs, suggesting it is indeed highly evolved. However, the O VI resonance doublet in the FUV required a considerably cooler temperature (120,000 K) model than did the optical O VI lines (170,000 K). Further, the enhanced chemical abundances did not match the predictions of stellar evolutionary models. Another non-LTE study found a 3x higher (C+O)/He abundance ratio and a cooler temperature. We have recently discovered two other bright, single, and lightly reddened WOs in the LMC, allowing us to take a fresh look at these important objects. Our newly found WOs span a range in excitation type, from WO1 (the highest) to WO4 (the lowest). Sand 2 is intermediate (WO3). We propose to use COS to obtain FUV and NUV data of all three stars for as comprehensive a study as is currently possible. These UV data will be combined with our optical Magellan spectra for a detailed analysis with CMFGEN with the latest atomic data. Knowing the degree of chemical evolution of these WO stars is crucial to determining their evolutionary status, and thus in understanding the final stages of the most massive stars.
16. Massive stars: flare activity due to infalls of comet-like bodies
NASA Astrophysics Data System (ADS)
Ibadov, Subhon; Ibodov, Firuz S.
2015-01-01
Passages of comet-like bodies through the atmosphere/chromosphere of massive stars at velocities more than 600 km/s will be accompanied, due to aerodynamic effects as crushing and flattening, by impulse generation of hot plasma within a relatively very thin layer near the stellar surface/photosphere as well as blast'' shock wave, i.e., impact-generated photospheric stellar/solar flares. Observational manifestations of such high-temperature phenomena will be eruption of the explosive layer's hot plasma, on materials of the star and exploding'' comet nuclei, into the circumstellar environment and variable anomalies in chemical abundances of metal atoms/ions like Fe, Si etc. Interferometric and spectroscopic observations/monitoring of young massive stars with dense protoplanetary discs are of interest for massive stars physics/evolution, including identification of mechanisms for massive stars variability.
17. Recovery from Giant Eruptions in Very Massive Stars
NASA Astrophysics Data System (ADS)
Kashi, Amit; Davidson, Kris; Humphreys, Roberta M.
2016-01-01
We use a hydro-and-radiative-transfer code to explore the behavior of a very massive star (VMS) after a giant eruption—i.e., following a supernova impostor event. Beginning with reasonable models for evolved VMSs with masses of 80 M⊙ and 120 M⊙, we simulate the change of state caused by a giant eruption via two methods that explicitly conserve total energy. (1) Synthetically removing outer layers of mass of a few M⊙ while reducing the energy of the inner layers. (2) Synthetically transferring energy from the core to the outer layers, an operation that automatically causes mass ejection. Our focus is on the aftermath, not the poorly understood eruption itself. Then, using a radiation-hydrodynamic code in 1D with realistic opacities and convection, the interior disequilibrium state is followed for about 200 years. Typically the star develops a ˜400 km s-1 wind with a mass loss rate that begins around 0.1 M⊙ yr-1 and gradually decreases. This outflow is driven by κ-mechanism radial pulsations. The 1D models have regular pulsations but 3D models will probably be more chaotic. In some cases a plateau in the mass-loss rate may persist about 200 years, while other cases are more like η Car which lost >10 M⊙ and then had an abnormal mass loss rate for more than a century after its eruption. In our model, the post-eruption outflow carried more mass than the initial eruption. These simulations constitute a useful preliminary reconnaissance for 3D models which will be far more difficult.
18. UV-selected Young Massive Star Cluster Populations in Nearby Star-forming Galaxies
NASA Astrophysics Data System (ADS)
Smith, Linda J.
2015-08-01
The Legacy ExtraGalactic UV Survey (LEGUS) is an HST Treasury program aimed at the investigation of star-formation and its relationship to environment in nearby galaxies. The results of a UV-selected study of young massive star clusters in a sample of nearby galaxies (< 10 Mpc) using detections based on the WFC3/UVIS F275W filter will be presented. Previous studies have used V or I-band detections and tend to ignore clusters younger than 10 Myr old. This very young population, which represents the most recent cluster-forming event in the LEGUS galaxies will be discussed.This poster is presented on behalf of the LEGUS team (PI Daniela Calzetti).
19. Atomic Physics of Shocked Plasma in Winds of Massive Stars
NASA Technical Reports Server (NTRS)
Leutenegger, Maurice A.; Cohen, David H.; Owocki, Stanley P.
2012-01-01
High resolution diffraction grating spectra of X-ray emission from massive stars obtained with Chandra and XMM-Newton have revolutionized our understanding of their powerful, radiation-driven winds. Emission line shapes and line ratios provide diagnostics on a number of key wind parameters. Modeling of resolved emission line velocity profiles allows us to derive independent constraints on stellar mass-loss rates, leading to downward revisions of a factor of a few from previous measurements. Line ratios in He-like ions strongly constrain the spatial distribution of Xray emitting plasma, confirming the expectations of radiation hydrodynamic simulations that X-ray emission begins moderately close to the stellar surface and extends throughout the wind. Some outstanding questions remain, including the possibility of large optical depths in resonance lines, which is hinted at by differences in line shapes of resonance and intercombination lines from the same ion. Resonance scattering leads to nontrivial radiative transfer effects, and modeling it allows us to place constraints on shock size, density, and velocity structure
20. SUPERSONIC LINE BROADENING WITHIN YOUNG AND MASSIVE SUPER STAR CLUSTERS
SciTech Connect
Tenorio-Tagle, Guillermo; Silich, Sergiy; Wuensch, Richard; Munoz-Tunon, Casiana; Palous, Jan E-mail: [email protected] E-mail: [email protected]
2010-01-10
The origin of supersonic infrared and radio recombination nebular lines often detected in young and massive superstar clusters is discussed. We suggest that these arise from a collection of repressurizing shocks (RSs), acting effectively to re-establish pressure balance within the cluster volume and from the cluster wind which leads to an even broader although much weaker component. The supersonic lines here are shown to occur in clusters that undergo a bimodal hydrodynamic solution, that is within clusters that are above the threshold line in the mechanical luminosity or cluster mass versus the size of the cluster plane. A plethora of RSs is due to frequent and recurrent thermal instabilities that take place within the matter reinserted by stellar winds and supernovae. We show that the maximum speed of the RSs and of the cluster wind are both functions of the temperature reached at the stagnation radius. This temperature depends only on the cluster heating efficiency (eta). Based on our two-dimensional simulations we calculate the line profiles that result from several models and confirm our analytical predictions. From a comparison between the predicted and observed values of the half-width zero intensity of the two line components, we conclude that the thermalization efficiency in young super star clusters above the threshold line must be lower than 20%.
1. Atomic physics of shocked plasma in winds of massive stars
SciTech Connect
Leutenegger, Maurice A.; Cohen, David H.; Owocki, Stanley P.
2012-05-25
High resolution diffraction grating spectra of X-ray emission from massive stars obtained with Chandra and XMM-Newton have revolutionized our understanding of their powerful, radiation-driven winds. Emission line shapes and line ratios provide diagnostics on a number of key wind parameters. Modeling of resolved emission line velocity profiles allows us to derive independent constraints on stellar mass-loss rates, leading to downward revisions of a factor of a few from previous measurements. Line ratios in He-like ions strongly constrain the spatial distribution of Xray emitting plasma, confirming the expectations of radiation hydrodynamic simulations that X-ray emission begins moderately close to the stellar surface and extends throughout the wind. Some outstanding questions remain, including the possibility of large optical depths in resonance lines, which is hinted at by differences in line shapes of resonance and intercombination lines from the same ion. Resonance scattering leads to nontrivial radiative transfer effects, and modeling it allows us to place constraints on shock size, density, and velocity structure.
2. Self-gravitating disc candidates around massive young stars
NASA Astrophysics Data System (ADS)
Forgan, D. H.; Ilee, J. D.; Cyganowski, C. J.; Brogan, C. L.; Hunter, T. R.
2016-11-01
There have been several recent detections of candidate Keplerian discs around massive young protostars. Given the relatively large disc-to-star mass ratios in these systems, and their young ages, it is worth investigating their propensity to becoming self-gravitating. To this end, we compute self-consistent, semi-analytic models of putative self-gravitating discs for five candidate disc systems. Our aim is not to fit exactly the observations, but to demonstrate that the expected dust continuum emission from marginally unstable self-gravitating discs can be quite weak, due to high optical depth at the mid-plane even at millimetre wavelengths. In the best cases, the models produce observable' disc masses within a factor of <1.5 of those observed, with mid-plane dust temperatures comparable to measured temperatures from molecular line emission. We find in two cases that a self-gravitating disc model compares well with observations. If these discs are self-gravitating, they satisfy the conditions for disc fragmentation in their outer regions. These systems may hence have as-yet-unresolved low-mass stellar companions, and are thus promising targets for future high angular resolution observations.
3. Discovering Massive Runaway Stars with Infrared Bowshock Nebulae: Identifying Twelve New Early-Type Stars using SMOG
NASA Astrophysics Data System (ADS)
Chick, William T.; Andrews, Julian E.; Kobulnicky, Henry A.; Povich, Matthew S.; Dale, Daniel A.; Munari, Stephan; Olivier, Grace M.; Schurhammer, Danielle; Sorber, Rebecca L.; Wernke, Heather N.
2016-01-01
Massive O and B type stars are crucial to the evolution of the interstellar medium, dominating the production of ionizing radiation, mechanical energy, and heavy elements. However, due to their short lives and relative scarcity, these stars are some of the least well understood and are difficult to locate outside of large star forming regions. A small but significant fraction of these massive stars have been observed to be high-velocity runaway stars moving rapidly away from their origin. When these stars encounter nebular gas they create characteristic arc-shaped bowshocks of heated compressed dust and gas. Using the distinct infrared emission morphology of the hot dust, these bowshock nebulae are predicted to give the location of the massive early type stars.Visual inspection of 24-micron band images from the Spitzer Mapping of the Outer Galaxy (SMOG) revealed 12 new bowshock nebula candidates. Follow up optical spectroscopy from the Wyoming Infrared Observatory confirmed that all 12 of the associated stellar sources are early-type stars. Combined with related results from visual searches for bowshock nebulae using WISE and Spitzer surveys in the inner Galaxy, we have identified over 85 new early type bowshock supporting stellar sources, a 95% success rate. We conclude that morphological selection of arc-shared infrared nebulae with a symmetrically placed star is an efficient way to discover early type stars.This work is supported by the National Science Foundation under grants AST-1063146 (REU), AST-1411851 (RUI), and AST-1412845.
4. Imprints of fast-rotating massive stars in the Galactic Bulge.
PubMed
Chiappini, Cristina; Frischknecht, Urs; Meynet, Georges; Hirschi, Raphael; Barbuy, Beatriz; Pignatari, Marco; Decressin, Thibaut; Maeder, André
2011-04-28
The first stars that formed after the Big Bang were probably massive, and they provided the Universe with the first elements heavier than helium ('metals'), which were incorporated into low-mass stars that have survived to the present. Eight stars in the oldest globular cluster in the Galaxy, NGC 6522, were found to have surface abundances consistent with the gas from which they formed being enriched by massive stars (that is, with higher α-element/Fe and Eu/Fe ratios than those of the Sun). However, the same stars have anomalously high abundances of Ba and La with respect to Fe, which usually arises through nucleosynthesis in low-mass stars (via the slow-neutron-capture process, or s-process). Recent theory suggests that metal-poor fast-rotating massive stars are able to boost the s-process yields by up to four orders of magnitude, which might provide a solution to this contradiction. Here we report a reanalysis of the earlier spectra, which reveals that Y and Sr are also overabundant with respect to Fe, showing a large scatter similar to that observed in extremely metal-poor stars, whereas C abundances are not enhanced. This pattern is best explained as originating in metal-poor fast-rotating massive stars, which might point to a common property of the first stellar generations and even of the 'first stars'. PMID:21525928
5. Imprints of fast-rotating massive stars in the Galactic Bulge.
PubMed
Chiappini, Cristina; Frischknecht, Urs; Meynet, Georges; Hirschi, Raphael; Barbuy, Beatriz; Pignatari, Marco; Decressin, Thibaut; Maeder, André
2011-04-28
The first stars that formed after the Big Bang were probably massive, and they provided the Universe with the first elements heavier than helium ('metals'), which were incorporated into low-mass stars that have survived to the present. Eight stars in the oldest globular cluster in the Galaxy, NGC 6522, were found to have surface abundances consistent with the gas from which they formed being enriched by massive stars (that is, with higher α-element/Fe and Eu/Fe ratios than those of the Sun). However, the same stars have anomalously high abundances of Ba and La with respect to Fe, which usually arises through nucleosynthesis in low-mass stars (via the slow-neutron-capture process, or s-process). Recent theory suggests that metal-poor fast-rotating massive stars are able to boost the s-process yields by up to four orders of magnitude, which might provide a solution to this contradiction. Here we report a reanalysis of the earlier spectra, which reveals that Y and Sr are also overabundant with respect to Fe, showing a large scatter similar to that observed in extremely metal-poor stars, whereas C abundances are not enhanced. This pattern is best explained as originating in metal-poor fast-rotating massive stars, which might point to a common property of the first stellar generations and even of the 'first stars'.
6. Effects of a new 3-alpha reaction on the s-process in massive stars
SciTech Connect
Kikuch, Yukihiro; Ono, Masaomi; Matsuo, Yasuhide; Hashimoto, Masa-aki; Fujimoto, Shin-ichiro
2012-11-12
Effect of a new 3-alpha reaction rate on the s-process during the evolution of a massive star of 25 solar mass is investigated for the first time, because the s-process in massive stars have been believed to be established with only minor change. We find that the s-process with use of the new rate during the core helium burning is very inefficient compared to the case with the previous 3-alpha rate. However, the difference of the overproduction is found to be largely compensated by the subsequent carbon burning. Since the s-process in massive stars has been attributed so far to the neutron irradiation during core helium burning, our finding reveals for the first time the importance of the carbon burning for the s-process during the evolution of massive stars.
7. Evidence of magnetic field decay in massive main-sequence stars
NASA Astrophysics Data System (ADS)
Fossati, L.; Schneider, F. R. N.; Castro, N.; Langer, N.; Simón-Díaz, S.; Müller, A.; de Koter, A.; Morel, T.; Petit, V.; Sana, H.; Wade, G. A.
2016-08-01
A significant fraction of massive main-sequence stars show strong, large-scale magnetic fields. The origin of these fields, their lifetimes, and their role in shaping the characteristics and evolution of massive stars are currently not well understood. We compile a catalogue of 389 massive main-sequence stars, 61 of which are magnetic, and derive their fundamental parameters and ages. The two samples contain stars brighter than magnitude 9 in the V-band and range in mass between 5 and 100 M⊙. We find that the fractional main-sequence age distribution of all considered stars follows what is expected for a magnitude limited sample, while that of magnetic stars shows a clear decrease towards the end of the main sequence. This dearth of old magnetic stars is independent of the choice of adopted stellar evolution tracks, and appears to become more prominent when considering only the most massive stars. We show that the decreasing trend in the distribution is significantly stronger than expected from magnetic flux conservation. We also find that binary rejuvenation and magnetic suppression of core convection are unlikely to be responsible for the observed lack of older magnetic massive stars, and conclude that its most probable cause is the decay of the magnetic field, over a time span longer than the stellar lifetime for the lowest considered masses, and shorter for the highest masses. We then investigate the spin-down ages of the slowly rotating magnetic massive stars and find them to exceed the stellar ages by far in many cases. The high fraction of very slowly rotating magnetic stars thus provides an independent argument for a decay of the magnetic fields.
8. Hot, Massive Stars in the Extremely Metal-Poor Galaxy, I Zw 18
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Malumuth, Eliot M.
2010-01-01
The extremely metal-poor galaxy I Zw 18, is the Rosetta Stone for understanding z=7-8 galaxies now being discovered by Hubb|e's Wide Field Camera 3 (HST/WFC3). Using HST/STIS images and recently obtained HST/COS ultraviolet spectra, we derive information about the hot, massive stars in this galaxy including stellar abundances, constraints on the stellar IMF and mass distribution of young clusters containing hot, massive stars.
9. The incidence of stellar mergers and mass gainers among massive stars
SciTech Connect
De Mink, S. E.; Sana, H.; Langer, N.; Izzard, R. G.; Schneider, F. R. N.
2014-02-10
Because the majority of massive stars are born as members of close binary systems, populations of massive main-sequence stars contain stellar mergers and products of binary mass transfer. We simulate populations of massive stars accounting for all major binary evolution effects based on the most recent binary parameter statistics and extensively evaluate the effect of model uncertainties. Assuming constant star formation, we find that 8{sub −4}{sup +9}% of a sample of early-type stars are the products of a merger resulting from a close binary system. In total we find that 30{sub −15}{sup +10}% of massive main-sequence stars are the products of binary interaction. We show that the commonly adopted approach to minimize the effects of binaries on an observed sample by excluding systems detected as binaries through radial velocity campaigns can be counterproductive. Systems with significant radial velocity variations are mostly pre-interaction systems. Excluding them substantially enhances the relative incidence of mergers and binary products in the non-radial velocity variable sample. This poses a challenge for testing single stellar evolutionary models. It also raises the question of whether certain peculiar classes of stars, such as magnetic O stars, are the result of binary interaction and it emphasizes the need to further study the effect of binarity on the diagnostics that are used to derive the fundamental properties (star-formation history, initial mass function, mass-to-light ratio) of stellar populations nearby and at high redshift.
10. Wolf-Rayet, Yellow and Red Supergiant in the single massive stars perspective
NASA Astrophysics Data System (ADS)
Georgy, Cyril; Hirschi, R.; Ekstrom, S.; Meynet, G.
2013-06-01
Rotation and mass loss are the key ingredients determining the fate of single massive stars. In recent years, a large effort has been made to compute whole grids of stellar models at different metallicities, including or not the effects of rotation, with the Geneva evolution code. In this talk, I will focus on the evolved stages of massive star evolution (red and yellow supergiants, Wolf-Rayet stars), in the framework of these new grids of models. I will highlight the effects of rotation and mass loss on the post-main sequence evolution of massive stars at solar and lower metallicity. In particular, I will discuss their impact on the maximum mass for a star to end its life as a RSG (leading to a type IIP supernova), on the possibility for a star to finish as a YSG, and on the initial mass ranges leading to various WR star subtypes. I will then compare the results predicted by our code with observed populations of evolved massive stars, bringing constraints on our computations, as well as some indications on the binary star fraction needed to reproduce them.
11. Low Mach Number Simulation of Core Convection in Massive Stars
NASA Astrophysics Data System (ADS)
Gilet, Candace Elise
This work presents three-dimensional simulations of core convection in a 15 solar mass star halfway through its main sequence lifetime. We examine the effects of two common modeling choices on the resulting convective flow: using a reduced domain size and using a monatomic, or single species, approximation. We compare a multi-species simulation on a full sphere (360 degree) domain with a multi-species simulation on an octant domain and also with a single species simulation on a full sphere domain. To perform the long-time calculations, we use the new low Mach number code MAESTRO. The first part of this work deals with numerical aspects of using MAESTRO for the core convection system, a new application for MAESTRO. We extend MAESTRO to include two new models, a single species model and a simplified two-dimensional planar model, to aid in the exploration of using MAESTRO for core convection in massive stars. We discuss using MAESTRO with a novel spherical geometry domain configuration, namely, with the outer boundary located in the interior of the star, and show how this can create spurious velocities that must be numerically damped using a sponging layer. We describe the preparation of the initial model for the simulation. We find that assuring neutral stratification in the convective core and reasonable resolution of the gravity waves in the stable layer are key factors in generating suitable initial conditions for the simulation. Further, we examine a numerical aspect of the velocity constraint that is part of the low Mach number formulation of the Euler equations. In particular, we investigate the numerical procedure for computing beta0, the density-like variable that captures background stratification in the velocity constraint, and find that the original method of computation remains a good choice. The three-dimensional simulation results show that using a single species model actually increases the computational cost of the simulation because the single
12. A RAPIDLY EVOLVING REGION IN THE GALACTIC CENTER: WHY S-STARS THERMALIZE AND MORE MASSIVE STARS ARE MISSING
SciTech Connect
Chen, Xian; Amaro-Seoane, Pau E-mail: [email protected]
2014-05-10
The existence of ''S-stars'' within a distance of 1'' from Sgr A* contradicts our understanding of star formation, due to Sgr A* 's forbiddingly violent environment. A suggested possibility is that they form far away and were brought in by some fast dynamical process, since they are young. Nonetheless, all conjectured mechanisms either fail to reproduce their eccentricities—without violating their young age—or cannot explain the problem of {sup i}nverse mass segregation{sup :} the fact that lighter stars (the S-stars) are closer to Sgr A* and more massive ones, Wolf-Rayet (WR) and O-stars, are farther out. In this Letter we propose that the mechanism responsible for both the distribution of the eccentricities and the paucity of massive stars is the Kozai-Lidov-like resonance induced by a sub-parsec disk recently discovered in the Galactic center. Considering that the disk probably extended to a smaller radius in the past, we show that in as short as (a few) 10{sup 6} yr, the stars populating the innermost 1'' region would redistribute in angular-momentum space and recover the observed ''super-thermal'' distribution. Meanwhile, WR and O-stars in the same region intermittently attain ample eccentricities that will lead to their tidal disruptions by the central massive black hole. Our results provide new evidences that Sgr A* was powered several millions years ago by an accretion disk as well as by tidal stellar disruptions.
13. Ultraviolet Imaging Telescope photometry of massive stars - The OB association NGC 206 in M31
NASA Technical Reports Server (NTRS)
Hill, Jesse K.; Pfarr, Barbara B.; Bohlin, Ralph C.; Isensee, Joan E.; O'Connell, Robert W.; Neff, Susan G.; Roberts, Morton S.; Smith, Andrew M.; Stecher, Theodore P.
1992-01-01
The Ultraviolet Imaging Telescope (UIT) obtained UV images of the giant M31 OB association NGC 206. Magnitudes in bands at 1520 and 2490 A were obtained for 30 massive stars, which demonstrate the effectiveness of UIT for photometry of moderately crowded hot stars to V about 21. The UV colors and magnitudes observed for stars in NGC 206 place them in the region of the color magnitude diagram occupied by evolutionary models for 30-60 solar mass stars, after correcting for extinction. The brighter stars are systematically redder than the fainter stars, indicating that they are supergiants of age about 4 Myr, while the fainter, bluer stars are nearer age zero. The relative numbers of probable supergiants measured by us and the number of probable main-sequence O stars measured from optical images are in agreement with the relative lifetimes. Calculated UIT colors are presented for a library of standard star spectra constructed from IUE and ground-based observations.
14. WIDE-FIELD INFRARED SURVEY EXPLORER OBSERVATIONS OF THE EVOLUTION OF MASSIVE STAR-FORMING REGIONS
SciTech Connect
Koenig, X. P.; Leisawitz, D. T.; Benford, D. J.; Padgett, D. L.; Rebull, L. M.
2012-01-10
We present the results of a mid-infrared survey of 11 outer Galaxy massive star-forming regions and 3 open clusters with data from the Wide-field Infrared Survey Explorer (WISE). Using a newly developed photometric scheme to identify young stellar objects and exclude extragalactic contamination, we have studied the distribution of young stars within each region. These data tend to support the hypothesis that latter generations may be triggered by the interaction of winds and radiation from the first burst of massive star formation with the molecular cloud material leftover from that earlier generation of stars. We dub this process the 'fireworks hypothesis' since star formation by this mechanism would proceed rapidly and resemble a burst of fireworks. We have also analyzed small cutout WISE images of the structures around the edges of these massive star-forming regions. We observe large (1-3 pc size) pillar and trunk-like structures of diffuse emission nebulosity tracing excited polycyclic aromatic hydrocarbon molecules and small dust grains at the perimeter of the massive star-forming regions. These structures contain small clusters of emerging Class I and Class II sources, but some are forming only a single to a few new stars.
15. Wide-Field Infrared Survey Explorer Observations of the Evolution of Massive Star-Forming Regions
NASA Technical Reports Server (NTRS)
Koenig, X. P.; Leisawitz, D. T.; Benford, D. J.; Rebull, L. M.; Padgett, D. L.; Assef, R. J.
2011-01-01
We present the results of a mid-infrared survey of 11 outer Galaxy massive star-forming regions and 3 open clusters with data from the Wide-field Infrared Survey Explorer (WISE). Using a newly developed photometric scheme to identify young stellar objects and exclude extragalactic contamination, we have studied the distribution of young stars within each region. These data tend to support the hypothesis that latter generations may be triggered by the interaction of winds and radiation from the first burst of massive star formation with the molecular cloud material leftover from that earlier generation of stars.We dub this process the "fireworks hypothesis" since star formation by this mechanism would proceed rapidly and resemble a burst of fireworks.We have also analyzed small cutout WISE images of the structures around the edges of these massive star-forming regions. We observe large (1-3 pc size) pillar and trunk-like structures of diffuse emission nebulosity tracing excited polycyclic aromatic hydrocarbon molecules and small dust grains at the perimeter of the massive star-forming regions. These structures contain small clusters of emerging Class I and Class II sources, but some are forming only a single to a few new stars.
16. Wide-Field Infrared Survey Explorer Observations of the Evolution of Massive Star-Forming Regions
NASA Technical Reports Server (NTRS)
Koenig, X. P.; Leisawitz, D. T.; Benford, D. J.; Rebull, L. M.; Padgett, D. L.; Asslef, R. J.
2012-01-01
We present the results of a mid-infrared survey of II outer Galaxy massive star-forming regions and 3 open clusters with data from the Wide-field Infrared Survey Explorer (WISE). Using a newly developed photometric scheme to identify young stellar objects and exclude extragalactic contamination, we have studied the distribution of young stars within each region. These data tend to support the hypothesis that latter generations may be triggered by the interaction of winds and radiation from the first burst of massive star formation with the molecular cloud material leftover from that earlier generation of stars. We dub this process the "fireworks hypothesis" since star formation by this mechanism would proceed rapidly and resemble a burst of fireworks. We have also analyzed small cutout WISE images of the structures around the edges of these massive star-forming regions. We observe large (1-3 pc size) pillar and trunk-like structures of diffuse emission nebulosity tracing excited polycyclic aromatic hydrocarbon molecules and small dust grains at the perimeter of the massive star-forming regions. These structures contain small clusters of emerging Class I and Class II sources, but some are forming only a single to a few new stars.
17. Influence of Entropy on Composition and Structure of Massive Protoneutron Stars
NASA Astrophysics Data System (ADS)
Hong, Bin; Jia, Huan-Yu; Mu, Xue-Ling; Zhou, Xia
2016-08-01
Adjusting the suitable coupling constants in relativistic mean Geld (RMF) theory and focusing on thermal effect of an entropy per baryon (S) from 0 to 3, we investigate the composition and structure of massive protoneutron stars corresponding PSR J1614-2230 and PSR J0348+0432. It is found that massive protoneutron stars (PNSs) have more hyperons than cold neutron stars. The entropy per baryon will stiffen the equation of state, and the influence on the pressure is more obvious at low density than high density, while the influence on the energy density is more obvious at high density than low density. It is found that higher entropy will give higher maximum mass, higher central temperature and lower central density. The entropy per baryon changes from 0 to 3, the radius of a PNS corresponding PSR J0348+0432 will increase from 12.86 km to 19.31 km and PSR J1612-2230 will increase from 13.03 km to 19.93 km. The entropy per baryon will raise the central temperature of massive PNSs in higher entropy per baryon, but the central temperature of massive PNSs maybe keep unchanged in lower entropy per baryon. The entropy per baryon will increase the moment of inertia of a massive protoneutron star, while decrease gravitational redshift of a massive neutron star. Supported by National Natural Science Foundation of China under Grant No. 11175147
18. THE MILKY WAY PROJECT: A STATISTICAL STUDY OF MASSIVE STAR FORMATION ASSOCIATED WITH INFRARED BUBBLES
SciTech Connect
Kendrew, S.; Robitaille, T. P.; Simpson, R.; Lintott, C. J.; Bressert, E.; Povich, M. S.; Sherman, R.; Schawinski, K.; Wolf-Chase, G.
2012-08-10
The Milky Way Project citizen science initiative recently increased the number of known infrared bubbles in the inner Galactic plane by an order of magnitude compared to previous studies. We present a detailed statistical analysis of this data set with the Red MSX Source (RMS) catalog of massive young stellar sources to investigate the association of these bubbles with massive star formation. We particularly address the question of massive triggered star formation near infrared bubbles. We find a strong positional correlation of massive young stellar objects (MYSOs) and H II regions with Milky Way Project bubbles at separations of <2 bubble radii. As bubble sizes increase, a statistically significant overdensity of massive young sources emerges in the region of the bubble rims, possibly indicating the occurrence of triggered star formation. Based on numbers of bubble-associated RMS sources, we find that 67% {+-} 3% of MYSOs and (ultra-)compact H II regions appear to be associated with a bubble. We estimate that approximately 22% {+-} 2% of massive young stars may have formed as a result of feedback from expanding H II regions. Using MYSO-bubble correlations, we serendipitously recovered the location of the recently discovered massive cluster Mercer 81, suggesting the potential of such analyses for discovery of heavily extincted distant clusters.
19. Angular Momentum Fluctuations in the Convective Helium Shell of Massive Stars
NASA Astrophysics Data System (ADS)
Gilkis, Avishai; Soker, Noam
2016-08-01
We find significant fluctuations of angular momentum within the convective helium shell of a pre-collapse massive star—a core-collapse supernova progenitor—that may facilitate the formation of accretion disks and jets that can explode the star. The convective flow in our model of an evolved {M}{ZAMS}=15{M}⊙ star, computed using the subsonic hydrodynamic solver MAESTRO, contains entire shells with net angular momentum in different directions. This phenomenon may have important implications for the late evolutionary stages of massive stars and for the dynamics of core collapse.
20. Angular Momentum Fluctuations in the Convective Helium Shell of Massive Stars
NASA Astrophysics Data System (ADS)
Gilkis, Avishai; Soker, Noam
2016-08-01
We find significant fluctuations of angular momentum within the convective helium shell of a pre-collapse massive star—a core-collapse supernova progenitor—that may facilitate the formation of accretion disks and jets that can explode the star. The convective flow in our model of an evolved {M}{ZAMS}=15{M}ȯ star, computed using the subsonic hydrodynamic solver MAESTRO, contains entire shells with net angular momentum in different directions. This phenomenon may have important implications for the late evolutionary stages of massive stars and for the dynamics of core collapse.
1. High molecular gas fractions in normal massive star-forming galaxies in the young Universe.
PubMed
Tacconi, L J; Genzel, R; Neri, R; Cox, P; Cooper, M C; Shapiro, K; Bolatto, A; Bouché, N; Bournaud, F; Burkert, A; Combes, F; Comerford, J; Davis, M; Schreiber, N M Förster; Garcia-Burillo, S; Gracia-Carpio, J; Lutz, D; Naab, T; Omont, A; Shapley, A; Sternberg, A; Weiner, B
2010-02-11
Stars form from cold molecular interstellar gas. As this is relatively rare in the local Universe, galaxies like the Milky Way form only a few new stars per year. Typical massive galaxies in the distant Universe formed stars an order of magnitude more rapidly. Unless star formation was significantly more efficient, this difference suggests that young galaxies were much more molecular-gas rich. Molecular gas observations in the distant Universe have so far largely been restricted to very luminous, rare objects, including mergers and quasars, and accordingly we do not yet have a clear idea about the gas content of more normal (albeit massive) galaxies. Here we report the results of a survey of molecular gas in samples of typical massive-star-forming galaxies at mean redshifts of about 1.2 and 2.3, when the Universe was respectively 40% and 24% of its current age. Our measurements reveal that distant star forming galaxies were indeed gas rich, and that the star formation efficiency is not strongly dependent on cosmic epoch. The average fraction of cold gas relative to total galaxy baryonic mass at z = 2.3 and z = 1.2 is respectively about 44% and 34%, three to ten times higher than in today's massive spiral galaxies. The slow decrease between z approximately 2 and z approximately 1 probably requires a mechanism of semi-continuous replenishment of fresh gas to the young galaxies.
2. Connecting the Dots: MUSE Unveils the Destructive Effect of Massive Stars
NASA Astrophysics Data System (ADS)
McLeod, A. F.; Ginsburg, A.; Klaassen, P.; Mottram, J.; Ramsay, S.; Testi, L.
2016-09-01
Throughout their entire lives, massive stars have a substantial impact on their surroundings, such as via protostellar outflows, stellar winds, ionising radiation and supernovae. Conceptually this is well understood, but the exact role of feedback mechanisms on the global star formation process and the stellar environment, as well as their dependence on the properties of the star-forming regions, are yet to be understood in detail. Observational quantification of the various feedback mechanisms is needed to precisely understand how high mass stars interact with and shape their environment, and which feedback mechanisms dominate under given conditions. We analysed the photo-evaporative effect of ionising radiation from massive stars on their surrounding molecular clouds using MUSE integral field data. This allowed us to determine the mass-loss rate of pillar-like structures (due to photo-evaporation) in different environments, and relate it to the ionising power of nearby massive stars. The resulting correlation is the first observational quantification of the destructive effect of ionising radiation from massive stars.
3. Kinematics of the inner thousand AU region around the young massive star AFGL 2591-VLA3: a massive disk candidate?
NASA Astrophysics Data System (ADS)
Wang, K.-S.; van der Tak, F. F. S.; Hogerheijde, M. R.
2012-07-01
Context. Recent detections of disks around young high-mass stars support the idea of massive star formation through accretion rather than coalescence, but the detailed kinematics in the equatorial region of the disk candidates is not well known, which limits our understanding of the accretion process. Aims: This paper explores the kinematics of the gas around a young massive star with millimeter-wave interferometry to improve our understanding of the formation of massive stars though accretion. Methods: We use Plateau de Bure interferometric images to probe the environment of the nearby (~1 kpc) and luminous (~20 000 L⊙) high-mass (10-16 M⊙) young star AFGL 2591-VLA3 in continuum and in lines of HDO, H_218O and SO2 in the 115 and 230 GHz bands. Radiative transfer calculations are employed to investigate the kinematics of the source. Results: At ~0.5″ (500 AU) resolution, the line images clearly resolve the velocity field of the central compact source (diameter of ~800 AU) and show linear velocity gradients in the northeast-southwest direction. Judging from the disk-outflow geometry, the observed velocity gradient results from rotation and radial expansion in the equatorial region of VLA3. Radiative transfer calculations suggest that the velocity field is consistent with sub-Keplerian rotation plus Hubble-law like expansion. The line profiles of the observed molecules suggest a layered structure, with HDO emission arising from the disk mid-plane, H_218O from the warm mid-layer, and SO2 from the upper disk. Conclusions: We propose AFGL 2591-VLA3 as a new massive disk candidate, with peculiar kinematics. The rotation of this disk is sub-Keplerian, probably due to magnetic braking, while the stellar wind may be responsible for the expansion of the disk. The expansion motion may also be an indirect evidence of disk accretion in the very inner region because of the conservation of angular momentum. The sub-Keplerian rotation discovered in our work suggests that
4. Massive runaway stars in the Small Magellanic Cloud
NASA Astrophysics Data System (ADS)
Gvaramadze, V. V.; Pflamm-Altenburg, J.; Kroupa, P.
2011-01-01
Using archival Spitzer Space Telescope data, we identified for the first time a dozen runaway OB stars in the Small Magellanic Cloud (SMC) through the detection of their bow shocks. The geometry of detected bow shocks allows us to infer the direction of motion of the associated stars and to determine their possible parent clusters and associations. One of the identified runaway stars, AzV 471, was already known as a high-velocity star on the basis of its high peculiar radial velocity, which is offset by ≃ 40 km s-1 from the local systemic velocity. We discuss implications of our findings for the problem of the origin of field OB stars. Several of the bow shock-producing stars are found in the confines of associations, suggesting that these may be “alien” stars contributing to the age spread observed for some young stellar systems. We also report the discovery of a kidney-shaped nebula attached to the early WN-type star SMC-WR3 (AzV 60a). We interpreted this nebula as an interstellar structure created owing to the interaction between the stellar wind and the ambient interstellar medium.
5. Slowly rotating neutron stars in scalar-tensor theories with a massive scalar field
NASA Astrophysics Data System (ADS)
Yazadjiev, Stoytcho S.; Doneva, Daniela D.; Popchev, Dimitar
2016-04-01
In the scalar-tensor theories with a massive scalar field, the coupling constants, and the coupling functions in general, which are observationally allowed, can differ significantly from those in the massless case. This fact naturally implies that the scalar-tensor neutron stars with a massive scalar field can have rather different structure and properties in comparison with their counterparts in the massless case and in general relativity. In the present paper, we study slowly rotating neutron stars in scalar-tensor theories with a massive gravitational scalar. Two examples of scalar-tensor theories are examined—the first example is the massive Brans-Dicke theory and the second one is a massive scalar-tensor theory indistinguishable from general relativity in the weak-field limit. In the latter case, we study the effect of the scalar field mass on the spontaneous scalarization of neutron stars. Our numerical results show that the inclusion of a mass term for the scalar field indeed changes the picture drastically compared to the massless case. It turns out that mass, radius, and moment of inertia for neutron stars in massive scalar-tensor theories can differ drastically from the pure general relativistic solutions if sufficiently large masses of the scalar field are considered.
6. VLTI and KI Interferometric Observations of Massive Evolved Stars and Their Dusty Circumstellar Environments
NASA Astrophysics Data System (ADS)
Wallace, Debra J.; Danchi, W. C.; Rajagopal, J.; Chesneau, O.; Lopez, B.; Menut, J.; Monnier, J.; Tuthill, P.; Ireland, M.; Barry, R.; Richardson, L. J.
2007-12-01
Recent aperture-masking and interferometric observations of late-type WC Wolf-Rayet stars strongly support the theory that dust formation in these objects is a result of colliding winds in binary systems. To explore and quantify this possible explanation, we have conducted a high-resolution interferometric survey of late-type massive stars utilizing the VLTI, KI, IOTA, and FGS1r interferometers. We present here the motivation for this study. We also present the first results from the MIDI instrument on the VLTI, and the KI and IOTA observations. Our VLTI study is aimed primarily at resolving and characterizing the dust around the WC9 star WR 85a and the LBV WR 122, both dust-producing but at different phases of massive star evolution. Our IOTA and KI interferometric observations resolve the WR star WR 137 into a dust-producing binary system.
7. s-Process Nucleosynthesis in Advanced Burning Phases of Massive Stars
NASA Astrophysics Data System (ADS)
The, Lih-Sin; El Eid, Mounib F.; Meyer, Bradley S.
2007-02-01
We present a detailed study of s-process nucleosynthesis in massive stars of solar-like initial composition and masses 15, 20, 25, and 30 Msolar. We update our previous results of s-process nucleosynthesis during the core He burning of these stars and then focus on an analysis of the s-process under the physical conditions encountered during the shell carbon burning. We show that the recent compilation of the 22Ne(α,n)25Mg rate leads to a remarkable reduction of the efficiency of the s-process during core He burning. In particular, this rate leads to the lowest overproduction factor of 80Kr found to date during core He burning in massive stars. The s-process yields resulting from shell carbon burning turn out to be very sensitive to the structural evolution of the carbon shell. This structure is influenced by the mass fraction of 12C attained at the end of core helium burning, which in turn is mainly determined by the 12C(α,γ)16O reaction. The still-present uncertainty in the rate for this reaction implies that the s-process in massive stars is also subject to this uncertainty. We identify some isotopes like 70Zn and 87Rb as the signatures of the s-process during shell carbon burning in massive stars. In determining the relative contribution of our s-only stellar yields to the solar abundances, we find it is important to take into account the neutron exposure of shell carbon burning. When we analyze our yields with a Salpeter initial mass function, we find that massive stars contribute at least 40% to s-only nuclei with mass A<=87. For s-only nuclei with mass A>90, massive stars contribute on average ~7%, except for 152Gd, 187Os, and 198Hg, which contribute ~14%, ~13%, and ~11%, respectively.
8. Massive Stars and their Siblings: the Extreme End of the Companion Mass Function
NASA Astrophysics Data System (ADS)
de Mink, Selma
2013-10-01
The gold-rush for detecting exoplanets has lead to an exponential improvement of optimization algorithms for high-contrast imaging optimized for HST. We propose to exploit these to probe the virtually unexplored population of low mass stars in the very close vicinity of young massive stars in order to I. progress our understanding of how low-mass stars form and survive under the influence of the ionizing radiation of their massive host and II. provide urgently needed constraints on competing theories of massive star formation by measuring their multiplicity. The high spatial and temporal stability of HST's point spread function is essential for the detection of very faint companions down to sub-arcsecond separations even in crowded regions at contrast up to delta-mag ~ 10, i.e. flux ratios up to 10,000. Furthermore the characterization of the low mass companions calls for wavelength bands largely affected by absorption by H2O in the earth's atmosphere. To achieve this goal we propose to use WFC3/IR to observe two adjacent fields in the center of the very young, nearby star cluster Trumpler 14, which harbors a rich population of massive stars.
9. Massive Stars and their Siblings: the Extreme End of the Companion Mass Function
NASA Astrophysics Data System (ADS)
de Mink, Selma
2014-10-01
The gold-rush for detecting exoplanets has lead to an exponential improvement of optimization algorithms for high-contrast imaging optimized for HST. We propose to exploit these to probe the virtually unexplored population of low mass stars in the very close vicinity of young massive stars in order to I. progress our understanding of how low-mass stars form and survive under the influence of the ionizing radiation of their massive host and II. provide urgently needed constraints on competing theories of massive star formation by measuring their multiplicity. The high spatial and temporal stability of HST's point spread function is essential for the detection of very faint companions down to sub-arcsecond separations even in crowded regions at contrast up to delta-mag ~ 10, i.e. flux ratios up to 10,000. Furthermore the characterization of the low mass companions calls for wavelength bands largely affected by absorption by H2O in the earth's atmosphere. To achieve this goal we propose to use WFC3/IR to observe two adjacent fields in the center of the very young, nearby star cluster Trumpler 14, which harbors a rich population of massive stars.
10. Theoretical Near-IR Spectra for Surface Abundance Studies of Massive Stars
NASA Technical Reports Server (NTRS)
Sonneborn, George; Bouret, J.
2011-01-01
We present initial results of a study of abundance and mass loss properties of O-type stars based on theoretical near-IR spectra computed with state-of-the-art stellar atmosphere models. The James Webb Space Telescope (JWST) will be a powerful tool to obtain high signal-to-noise ratio near-IR (1-5 micron) spectra of massive stars in different environments of local galaxies. Our goal is to analyze model near-IR spectra corresponding to those expected from NIRspec on JWST in order to map the wind properties and surface composition across the parameter range of 0 stars and to determine projected rotational velocities. As a massive star evolves, internal coupling, related mixing, and mass loss impact its intrinsic rotation rate. These three parameters form an intricate loop, where enhanced rotation leads to more mixing which in turn changes the mass loss rate, the latter thus affecting the rotation rate. Since the effects of rotation are expected to be much more pronounced at low metallicity, we pay special attention to models for massive stars in the the Small Magellanic Cloud. This galaxy provides a unique opportunity to probe stellar evolution, and the feedback of massive stars on galactic evol.ution in conditions similar to the epoch of maximal star formation. Plain-Language Abstract: We present initial results of a study of abundance and mass loss properties of massive stars based on theoretical near-infrared (1-5 micron) spectra computed with state-of-the-art stellar atmosphere models. This study is to prepare for observations by the James Webb Space Telescope.
11. Evolved massive stars in W33 and in GMC 23.3-0.3
NASA Astrophysics Data System (ADS)
Messineo, Maria; Clark, J. Simon; Figer, Donald F.; Menten, Karl M.; Kudritzki, Rolf-Peter; Najarro, Francisco; Rich, Michael; Ivanov, Valentin D.; Valenti, Elena; Trombley, Christine; Chen, Rosie; Davies, Ben; MacKenty, John W.
2015-08-01
We have conducted an infrared spectroscopic survey for massive evolved stars and/or clusters in the Galactic giant molecular clouds G23.3-0.3 and W33. A large number of extraordinary sub-clumps/clusters of massive stars were detected. The spatial and temporal distribution of these massive stars yields information on the star formation history of the clouds.In G23.3-0.3, we discovered a dozen massive O-type stars, one candidate luminous blue variable, and several red supergiants. The O-type stars have masses from 25 to 50 Msun and ages of 5-8 Myr, while the RSGs belong to a burst that occurred 20-30 Myr ago. Therefore, GMC G23.3-0.3 has had one of the longest known histories of star formation (20-30 Myr). GMC G23.3-0.3 is rich in HII regions and supernova remnants; we detected massive stars in the cores of SNR W41 and of SNR G22.7-0.2.In W33, we detected a few evolved O-type stars and one Wolf-Rayet star, but none of the late-type objects has the luminosity of a red supergiant. W33 is characterized by discrete sources and has had at least 3-5 Myr of star formation history, which is now propagating from west to east. While our detections of massive evolved stars in W33 are made on the west side of the cloud, several dense molecular cores that may harbor proto clusters have recently been detected on the east side of the cloud by Immer et al. (2014).Messineo, Maria; Menten, Karl M.; Figer, Donald F.; Davies, Ben; Clark, J. Simon; Ivanov, Valentin D.Kudritzki, Rolf-Peter; Rich, R. Michael; MacKenty, John W.; Trombley, Christine 2014A&A...569A..20MMessineo, Maria; Clark, J. Simon; Figer, Donald F.; Kudritzki, Rolf-Peter; Francisco, Najarro; Rich, R. Michael; Menten, Karl M.; Ivanov, Valentin D.; Valenti, Elena; Trombley, Christine; Chen, C.H. Rosie; Davies, Ben; submitted to ApJ.
12. Dense molecular cocoons in the massive protocluster W3 IRS5: a test case for models of massive star formation
NASA Astrophysics Data System (ADS)
Wang, K.-S.; Bourke, T. L.; Hogerheijde, M. R.; van der Tak, F. F. S.; Benz, A. O.; Megeath, S. T.; Wilson, T. L.
2013-10-01
Context. Two competing models describe the formation of massive stars in objects like the Orion Trapezium. In the turbulent core accretion model, the resulting stellar masses are directly related to the mass distribution of the cloud condensations. In the competitive accretion model, the gravitational potential of the protocluster captures gas from the surrounding cloud for which the individual cluster members compete. Aims: With high resolution submillimeter observations of the structure, kinematics, and chemistry of the proto-Trapezium cluster W3 IRS5, we aim to determine which mode of star formation dominates. Methods: We present 354 GHz Submillimeter Array observations at resolutions of 1″-3″ (1800-5400 AU) of W3 IRS5. The dust continuum traces the compact source structure and masses of the individual cores, while molecular lines of CS, SO, SO2, HCN, H2CS, HNCO, and CH3OH (and isotopologues) reveal the gas kinematics, density, and temperature. Results: The observations show five emission peaks (SMM1-5). SMM1 and SMM2 contain massive embedded stars (~20 M⊙); SMM3-5 are starless or contain low-mass stars (<8 M⊙). The inferred densities are high, ≥107 cm-3, but the core masses are small, 0.2-0.6 M⊙. The detected molecular emission reveals four different chemical zones. Abundant (X ~ few 10-7 to 10-6) SO and SO2 are associated with SMM1 and SMM2, indicating active sulfur chemistry. A low abundance (5 × 10-8) of CH3OH concentrated on SMM3/4 suggest the presence of a hot core that is only just turning on, possibly by external feedback from SMM1/2. The gas kinematics are complex with contributions from a near pole-on outflow traced by CS, SO, and HCN; rotation in SO2, and a jet in vibrationally excited HCN. Conclusions: The proto-Trapezium cluster W3 IRS5 is an ideal test case to discriminate between models of massive star formation. Either the massive stars accrete locally from their local cores; in this case the small core masses imply that W3 IRS5 is
13. On the onset of secondary stellar generations in giant star-forming regions and massive star clusters
SciTech Connect
Palouš, J.; Wünsch, R.; Tenorio-Tagle, G.
2014-09-10
Here we consider the strong evolution experienced by the matter reinserted by massive stars, both in giant star-forming regions driven by a constant star formation rate and in massive and coeval superstar clusters. In both cases we take into consideration the changes induced by stellar evolution on the number of massive stars, the number of ionizing photons, and the integrated mechanical luminosity of the star-forming regions. The latter is at all times compared with the critical luminosity that defines, for a given size, the lower mechanical luminosity limit above which the matter reinserted via strong winds and supernova explosions suffers frequent and recurrent thermal instabilities that reduce its temperature and pressure and inhibit its exit as part of a global wind. Instead, the unstable reinserted matter is compressed by the pervasive hot gas, and photoionization maintains its temperature at T ∼ 10{sup 4} K. As the evolution proceeds, more unstable matter accumulates and the unstable clumps grow in size. Here we evaluate the possible self-shielding of thermally unstable clumps against the UV radiation field. Self-shielding allows for a further compression of the reinserted matter, which rapidly develops a high-density neutral core able to absorb in its outer skin the incoming UV radiation. Under such conditions the cold (T ∼ 10 K) neutral cores soon surpass the Jeans limit and become gravitationally unstable, creating a new stellar generation with the matter reinserted by former massive stars. We present the results of several calculations of this positive star formation feedback scenario promoted by strong radiative cooling and mass loading.
14. On the Onset of Secondary Stellar Generations in Giant Star-forming Regions and Massive Star Clusters
NASA Astrophysics Data System (ADS)
Palouš, J.; Wünsch, R.; Tenorio-Tagle, G.
2014-09-01
Here we consider the strong evolution experienced by the matter reinserted by massive stars, both in giant star-forming regions driven by a constant star formation rate and in massive and coeval superstar clusters. In both cases we take into consideration the changes induced by stellar evolution on the number of massive stars, the number of ionizing photons, and the integrated mechanical luminosity of the star-forming regions. The latter is at all times compared with the critical luminosity that defines, for a given size, the lower mechanical luminosity limit above which the matter reinserted via strong winds and supernova explosions suffers frequent and recurrent thermal instabilities that reduce its temperature and pressure and inhibit its exit as part of a global wind. Instead, the unstable reinserted matter is compressed by the pervasive hot gas, and photoionization maintains its temperature at T ~ 104 K. As the evolution proceeds, more unstable matter accumulates and the unstable clumps grow in size. Here we evaluate the possible self-shielding of thermally unstable clumps against the UV radiation field. Self-shielding allows for a further compression of the reinserted matter, which rapidly develops a high-density neutral core able to absorb in its outer skin the incoming UV radiation. Under such conditions the cold (T ~ 10 K) neutral cores soon surpass the Jeans limit and become gravitationally unstable, creating a new stellar generation with the matter reinserted by former massive stars. We present the results of several calculations of this positive star formation feedback scenario promoted by strong radiative cooling and mass loading.
15. SOAR Optical and Near-infrared Spectroscopic Survey of Newly Discovered Massive Stars in the Periphery of Galactic Massive Star Clusters I-NGC 3603
NASA Astrophysics Data System (ADS)
Roman-Lopes, A.; Franco, G. A. P.; Sanmartim, D.
2016-06-01
In this work, we present the results of a spectroscopic study of very massive stars (VMSs) found outside the center of the massive stellar cluster NGC 3603. From the analysis of the associated Southern Astrophysical Research (SOAR) Telescope spectroscopic data and related optical-near-IR (NIR) photometry, we confirm the existence of several VMSs in the periphery of NGC 3603. The first group of objects (MTT58, WR42e, and RF7) is composed of three new Galactic exemplars of the OIf*/WN type, all of them with probable initial masses well above 100 {M}⊙ and estimated ages of about 1 Myr. Based on our Goodman blue-optical spectrum of another source in our sample (MTT68), we can confirm the previous finding in the NIR of the only other Galactic exemplar (besides HD 93129A) of the O2If* type known to date. Based on its position relative to a set of theoretical isochrones in a Hertzprung-Russel (H-R) diagram, we concluded that the new O2If* star could be one of the most massive (150 {M}⊙ ) and luminous (M V = -7.3) O-stars in the Galaxy. Also, another remarkable result is the discovery of a new O2v star (MTT31), which is the first exemplar of that class so far identified in the Milk Way. From its position in the H-R diagram it is found that this new star probably had an initial mass of 80 {M}⊙ , as well as an absolute magnitude of M V = -6.0, corresponding to a luminosity similar to other known O2v stars in the Large Magellanic Cloud. Finally, we also communicate the discovery of a new Galactic O3.5If* star (RFS8) that is quite an intriguing case. Indeed, it is located far to the south of the NGC 3603 center, in apparent isolation at a large radial projected linear distance of ˜62 pc. Its derived luminosity is similar to that of the other O3.5If* (Sh18) found in NGC 3603's innermost region, and the fact that a such high mass star is observed so isolated in the field led us to speculate that perhaps it could have been expelled from the innermost parts of the complex
16. SOAR Optical and Near-infrared Spectroscopic Survey of Newly Discovered Massive Stars in the Periphery of Galactic Massive Star Clusters I-NGC 3603
NASA Astrophysics Data System (ADS)
Roman-Lopes, A.; Franco, G. A. P.; Sanmartim, D.
2016-06-01
In this work, we present the results of a spectroscopic study of very massive stars (VMSs) found outside the center of the massive stellar cluster NGC 3603. From the analysis of the associated Southern Astrophysical Research (SOAR) Telescope spectroscopic data and related optical–near-IR (NIR) photometry, we confirm the existence of several VMSs in the periphery of NGC 3603. The first group of objects (MTT58, WR42e, and RF7) is composed of three new Galactic exemplars of the OIf*/WN type, all of them with probable initial masses well above 100 {M}ȯ and estimated ages of about 1 Myr. Based on our Goodman blue-optical spectrum of another source in our sample (MTT68), we can confirm the previous finding in the NIR of the only other Galactic exemplar (besides HD 93129A) of the O2If* type known to date. Based on its position relative to a set of theoretical isochrones in a Hertzprung–Russel (H–R) diagram, we concluded that the new O2If* star could be one of the most massive (150 {M}ȯ ) and luminous (M V = ‑7.3) O-stars in the Galaxy. Also, another remarkable result is the discovery of a new O2v star (MTT31), which is the first exemplar of that class so far identified in the Milk Way. From its position in the H–R diagram it is found that this new star probably had an initial mass of 80 {M}ȯ , as well as an absolute magnitude of M V = ‑6.0, corresponding to a luminosity similar to other known O2v stars in the Large Magellanic Cloud. Finally, we also communicate the discovery of a new Galactic O3.5If* star (RFS8) that is quite an intriguing case. Indeed, it is located far to the south of the NGC 3603 center, in apparent isolation at a large radial projected linear distance of ˜62 pc. Its derived luminosity is similar to that of the other O3.5If* (Sh18) found in NGC 3603's innermost region, and the fact that a such high mass star is observed so isolated in the field led us to speculate that perhaps it could have been expelled from the innermost parts of the
17. ON THE DIFFERENTIAL ROTATION OF MASSIVE MAIN-SEQUENCE STARS
SciTech Connect
Rogers, T. M.
2015-12-20
To date, asteroseismology has provided core-to-surface differential rotation measurements in eight main-sequence stars. These stars, ranging in mass from ∼1.5–9 M{sub ⊙}, show rotation profiles ranging from uniform to counter-rotation. Although they have a variety of masses, these stars all have convective cores and overlying radiative regions, conducive to angular momentum transport by internal gravity waves (IGWs). Using two-dimensional numerical simulations, we show that angular momentum transport by IGWs can explain all of these rotation profiles. We further predict that, should high mass, faster rotating stars be observed, the core-to-envelope differential rotation will be positive, but less than one.
18. A young massive planet in a star-disk system.
PubMed
Setiawan, J; Henning, Th; Launhardt, R; Müller, A; Weise, P; Kürster, M
2008-01-01
There is a general consensus that planets form within disks of dust and gas around newly born stars. Details of their formation process, however, are still a matter of ongoing debate. The timescale of planet formation remains unclear, so the detection of planets around young stars with protoplanetary disks is potentially of great interest. Hitherto, no such planet has been found. Here we report the detection of a planet of mass (9.8+/-3.3)M(Jupiter) around TW Hydrae (TW Hya), a nearby young star with an age of only 8-10 Myr that is surrounded by a well-studied circumstellar disk. It orbits the star with a period of 3.56 days at 0.04 au, inside the inner rim of the disk. This demonstrates that planets can form within 10 Myr, before the disk has been dissipated by stellar winds and radiation.
19. Statistics of magnetic fields and fluxes of massive OB stars and the origin of neutron star magnetic fields
NASA Astrophysics Data System (ADS)
Igoshev, A. P.; Kholtygin, A. F.
2011-12-01
Based on the newest measurements, statistical properties of rms mean magnetic fields of OB and neutron stars (NSs) were investigated. The magnetic field distribution function f(B) for OB stars was determined and a sharp decrease of f(B) for weak magnetic fields was found. The mean magnetic fluxes F for all massive stars and NSs with measured magnetic fields was estimated, and it was found that log F = 27.7 for OB stars and log F = 24.5 for NSs. To explain the large differences of the fluxes from normal and neutron stars we studied the birth and evolution of isolated neutron stars in the whole volume of our Galaxy with our new code of population synthesis. We started modeling %with our code from the birth of massive OB stars and followed their motion within the spiral arms to the point of supernova explosion. Next we considered the evolution of NS up to the death line with considering the magnetic field decay. We found that a significant magnetic field decay occurs during the first million years of a NS's life. We have estimated the mean time of the Ohmic decay for NS. We modeled the distributions of pulsar periods P, of period derivatives \\dot P, and of pulsar magnetic fields B, and found that they are in a good agreement with those taken from \\cite{ATNF}.
20. The multiplicity of massive stars: A high angular resolution survey with the HST fine guidance sensor
SciTech Connect
Aldoretta, E. J.; Gies, D. R.; Henry, T. J.; Jao, W.-C.; Norris, R. P. E-mail: [email protected] E-mail: [email protected]; and others
2015-01-01
We present the results of an all-sky survey made with the Fine Guidance Sensor on the Hubble Space Telescope to search for angularly resolved binary systems among massive stars. The sample of 224 stars is comprised mainly of Galactic O- and B-type stars and luminous blue variables, plus a few luminous stars in the Large Magellanic Cloud. The FGS TRANS mode observations are sensitive to the detection of companions with an angular separation between 0.″01 and 1.″0 and brighter than △m=5. The FGS observations resolved 52 binary and 6 triple star systems and detected partially resolved binaries in 7 additional targets (43 of these are new detections). These numbers yield a companion detection frequency of 29% for the FGS survey. We also gathered literature results on the numbers of close spectroscopic binaries and wider astrometric binaries among the sample, and we present estimates of the frequency of multiple systems and the companion frequency for subsets of stars residing in clusters and associations, field stars, and runaway stars. These results confirm the high multiplicity fraction, especially among massive stars in clusters and associations. We show that the period distribution is approximately flat in increments of logP. We identify a number of systems of potential interest for long-term orbital determinations, and we note the importance of some of these companions for the interpretation of the radial velocities and light curves of close binaries that have third companions.
1. VLT/X-shooter spectroscopy of massive pre-main-sequence stars in M17
NASA Astrophysics Data System (ADS)
Ramirez-Tannus, Maria Claudia; Kaper, Lex
2015-08-01
The formation process of massive stars is still poorly understood. Formation timescales are short, the corresponding accretion rates very high, and the forming stars are hidden from view due to vast amounts of interstellar extinction. On top of that, massive stars are rare, are located at relatively large distances, and play a major role in shaping the interstellar medium due to their strong UV radiation fields and stellar winds. Although massive stars show most spectral features in the UV and optical range, so far only for a handful of massive Young Stellar Objects (mYSOs) optical and near-infrared spectra have been obtained. For some of these their pre-main-sequence (PMS) nature has now been firmly established (e.g. Ochsendorf et al. 2011, Ellerbroek et al. 2013). The objective of our project is to determine the physical properties of mYSOs, to search for signatures remnant of their formation process and to better understand the feedback on their environment.To this aim the optical to near-infrared (300-2500 nm) spectra of six candidate mYSOs (Hanson et al. 1997), deeply embedded in the massive star forming region M17, have been obtained with X-Shooter on the ESO Very Large Telescope. These mYSO candidates have been identified based on their infrared excess and spectral features (double-peaked emission lines, CO band-head emission) indicating the presence of a disk. In most cases, we detect a photospheric spectrum allowing us to measure the physical properties of the mYSO and to confirm its PMS nature. We also uncover many emission features, including forbidden lines, providing information on the (active) formation process of these young (massive) stars.
2. The Formation of Massive Stars by Collisional Mergers: Theoretical Constraints and Observational Predictions
NASA Astrophysics Data System (ADS)
Zinnecker, Hans; Bally, John
2004-08-01
While accretional growth can lead to the formation of massive stars in isolation or in loose OB associations, collisional growth and mergers can only occur in high-density cluster environments. We will discuss the conditions in a very dense young star cluster under which the merger scenario of massive star formation may work, and whether these conditions are likely to occur somewhere in the our Galaxy (Orion BN/KL, NGC 3603, W3-IRS5), the Local Group (30 Dor, NGC 604), or other galaxies (NGC 5253, Henize 2-10, The Antennae clusters). We explore the observational consequences of the merger scenario. Protostellar mergers may produce high luminosity infrared flares. Mergers may be surrounded by thick tori of expanding debris, impulsive wide-angle outflows, shock-induced maser and radio continuum emission. The collision products are expected to have fast stellar rotation and a large multiplicity fraction. Massive stars growing by a series of mergers may produce eruptive bursts of wide-angle outflow activity with random orientations; the walls of the resulting outflow cavities may be observable as filaments of dense gas and dust pointing away from the massive star. The extremely rare merger of two stars close to the upper mass limit of the IMF may be a possible pathway to hypernova-generated gamma-ray bursters. We also speculate that the outflow "fingers" from the OMC1 core in the Orion molecular cloud were produced by a merger less than a thousand years ago (Bally and Zinnecker 2004, AJ submitted). Mergers may not occur in every dense young cluster, but certainly in some of them, especially those where dynamical mass segregation of massive stars has taken place (Freitag and Benz 2004, astro-ph 0403621).
3. The massive binary companion star to the progenitor of supernova 1993J.
PubMed
Maund, Justyn R; Smartt, Stephen J; Kudritzki, Rolf P; Podsiadlowski, Philipp; Gilmore, Gerard F
2004-01-01
The massive star that underwent a collapse of its core to produce supernova (SN)1993J was subsequently identified as a non-variable red supergiant star in images of the galaxy M81 taken before explosion. It showed an excess in ultraviolet and B-band colours, suggesting either the presence of a hot, massive companion star or that it was embedded in an unresolved young stellar association. The spectra of SN1993J underwent a remarkable transformation from the signature of a hydrogen-rich type II supernova to one of a helium-rich (hydrogen-deficient) type Ib. The spectral and photometric peculiarities were best explained by models in which the 13-20 solar mass supergiant had lost almost its entire hydrogen envelope to a close binary companion, producing a 'type IIb' supernova, but the hypothetical massive companion stars for this class of supernovae have so far eluded discovery. Here we report photometric and spectroscopic observations of SN1993J ten years after the explosion. At the position of the fading supernova we detect the unambiguous signature of a massive star: the binary companion to the progenitor. PMID:14712269
4. On the possibility that the most massive stars result from binary mergers
NASA Astrophysics Data System (ADS)
de Koter, A.; Bestenlehner, J. M.; de Mink, S. E.; Evans, C. J.; Gräfener, G.; Izzard, R. G.; Langer, N.; Ramírez-Agudelo, O. H.; Sana, H.; Schneider, F. R. N.; Simón-Díaz, S.; Vink, J. S.
2013-02-01
The VLT-FLAMES Tarantula Survey is an ESO Large Program from which we have obtained multi-epoch optical spectroscopy of over 800 massive stars in the 30 Doradus region of the Large Magellanic Cloud. This unprecedented dataset is being used to address outstanding questions in how massive stars evolve from the early main sequence to their deaths as core collapse supernovae. Here we focus on the rotation properties of the population of presumably single O stars and use binary population synthesis predictions to show that the rapid rotators among this population likely are post-interaction binaries. The same type of population synthesis can be used to study the mass function of massive young clusters. We argue - on the basis of predictions for the Arches and Quintuplet clusters - that a sizable fraction of the very massive WNh stars in 30 Doradus may also have such a binary interaction history. We single out the WNh star discovered in the VFTS, VFTS 682, and discuss its properties.
5. The massive binary companion star to the progenitor of supernova 1993J.
PubMed
Maund, Justyn R; Smartt, Stephen J; Kudritzki, Rolf P; Podsiadlowski, Philipp; Gilmore, Gerard F
2004-01-01
The massive star that underwent a collapse of its core to produce supernova (SN)1993J was subsequently identified as a non-variable red supergiant star in images of the galaxy M81 taken before explosion. It showed an excess in ultraviolet and B-band colours, suggesting either the presence of a hot, massive companion star or that it was embedded in an unresolved young stellar association. The spectra of SN1993J underwent a remarkable transformation from the signature of a hydrogen-rich type II supernova to one of a helium-rich (hydrogen-deficient) type Ib. The spectral and photometric peculiarities were best explained by models in which the 13-20 solar mass supergiant had lost almost its entire hydrogen envelope to a close binary companion, producing a 'type IIb' supernova, but the hypothetical massive companion stars for this class of supernovae have so far eluded discovery. Here we report photometric and spectroscopic observations of SN1993J ten years after the explosion. At the position of the fading supernova we detect the unambiguous signature of a massive star: the binary companion to the progenitor.
6. Molecular Clouds and Massive Star Formation in the Norma Spiral Arm
NASA Astrophysics Data System (ADS)
García, P.; Bronfman, L.; May, J.
2006-06-01
The Norma spiral arm in the Southern Galaxy contains the most massive molecular clouds as well as the most FIR luminous regions of massive star formation in the Galactic disk. The tangent region of this arm, at a well defined distance of ≈ 4.5 kpc from the Sun, is ideal to study in detail the process of massive star formation in GMCs (Bronfman et al. 1988, ApJ, 324, 248). We present maps of the major GMCs in ^{12}CO and C^{18}O obtained with the Nanten 4-m telescope, at a resolution of 2.5 arcmin. We have obtained also CS (2-1) and CS(5-4) maps of several OB star formation regions embedded in these GMCs (Bronfman et al. 1996, A&AS, 115, 81). What is the contribution from embedded OB stars to the total FIR emission from these GMCs? What is the fraction of cloud molecular gas involved in massive star formation?
7. The s-PROCESS Nucleosynthesis in Massive Metal-Poor Stars
NASA Astrophysics Data System (ADS)
2005-12-01
We present the s-process nucleosynthesis in massive stars with a wide range of metallicity, using the recent sets of reaction rates and stellar input physics. The decreasing metallicity makes poisoning effects of primary 16O larger at the late phase of core He burning, at which the s-process occurs actively in solar metallicity stars, and prevents the synthesis of heavy elements from being efficient. However, we find that the s-process proceeds very efficiently via neutron source reaction of 13C(α,n)16O at the end of core H burning phase when the metallicity decreases below Z ~ 10-8. These massive, extremely low metallicity stars may have an important contribution of light s-elements to observed extremely metal-poor stars.
8. Constraining the axion-photon coupling with massive stars.
PubMed
Friedland, Alexander; Giannotti, Maurizio; Wise, Michael
2013-02-01
We point out that stars in the mass window ~8-12M([circumpunct]) can serve as sensitive probes of the axion-photon interaction, g(Aγγ). Specifically, for these stars axion energy losses from the helium-burning core would shorten and eventually eliminate the blue loop phase of the evolution. This would contradict observational data, since the blue loops are required, e.g., to account for the existence of Cepheid stars. Using the MESA stellar evolution code, modified to include the extra cooling, we conservatively find g(Aγγ)
9. Stellar feedback from a massive Super Star Cluster in the Antennae merger
NASA Astrophysics Data System (ADS)
Herrera, Cinthya N.; Boulanger, Francois
2015-08-01
Super star clusters (SSCs), likely the progenitors of globular clusters, are one of the most extreme forms of star formation. Stellar feedback from such massive clusters is vital to galaxy evolution and star formation history in the Universe, as the intense radiation and stellar winds produced by massive stars are important in unbinding and dispersing large molecular clouds and affecting star formation efficiency and sequential star formation. Nearby galaxy mergers are ideal sites to investigate massive star feedback, and to form local analogous in high-redshift galaxies. Based on ALMA and VLT observations, we study this process in a SSC in the Antennae galaxies (NGC 4038/39, 22 Mpc), a spectacular example of a burst of star formation triggered by the encounter of two galaxies. We analyze a massive (~107 M⊙) and young (3.4 Myr) SSC, B1, which is associated with compact molecular and ionized emission, suggesting that it is still embedded in its parent molecular cloud. However, we found that the observed CO linewidth yields a conservative velocity expansion, the radiation pressure does not accelerate today the gas and the matter surrounding the cluster is clumpy, indicating that SSC B1 is not embedded in its parent cloud after all. We propose that radiation pressure was highly enhanced at the early stages of the SSC formation, early disrupting the parent cloud (< 3 Myr). The gas observed today surrounding the cluster did not participate on the cluster formation but are nearby clouds and/or gas accreted from the SGMC. We present evidences that outflowing gas from the parent cloud may be still observed in the broader, high velocity component of the CO gas, which has a bubble-like shape structure distributed around SSC B1. Higher angular resolution observations are needed to validate this interpretation and to understand the origin and fate of the component seen to be associated with SSC B1.
10. Colliding Winds in Massive Binaries Involving Wolf-Rayet Stars
NASA Astrophysics Data System (ADS)
Moffat, Anthony F. J.; Marchenko, Sergey V.; Bartzakos, Peter
1996-12-01
Wolf-Rayet stars are notorious for their very strong, hot winds. Their presence in binary systems can therefore lead to strong wind collisions, that manifest themselves as well-defined, phase-dependent distortions of the spectral lines. Turning this around, profile variations can be used to determine properties of the wind collision, as well as the winds and even the orbit itself. We review the present situation regarding colliding winds for WR stars in WR + O, WR + WR, and WR + c systems.
11. Formation of new stellar populations from gas accreted by massive young star clusters.
PubMed
Li, Chengyuan; de Grijs, Richard; Deng, Licai; Geller, Aaron M; Xin, Yu; Hu, Yi; Faucher-Giguère, Claude-André
2016-01-28
Stars in clusters are thought to form in a single burst from a common progenitor cloud of molecular gas. However, massive, old 'globular' clusters--those with ages greater than ten billion years and masses several hundred thousand times that of the Sun--often harbour multiple stellar populations, indicating that more than one star-forming event occurred during their lifetimes. Colliding stellar winds from late-stage, asymptotic-giant-branch stars are often suggested to be triggers of second-generation star formation. For this to occur, the initial cluster masses need to be greater than a few million solar masses. Here we report observations of three massive relatively young star clusters (1-2 billion years old) in the Magellanic Clouds that show clear evidence of burst-like star formation that occurred a few hundred million years after their initial formation era. We show that such clusters could have accreted sufficient gas to form new stars if they had orbited in their host galaxies' gaseous disks throughout the period between their initial formation and the more recent bursts of star formation. This process may eventually give rise to the ubiquitous multiple stellar populations in globular clusters. PMID:26819043
12. Formation of new stellar populations from gas accreted by massive young star clusters.
PubMed
Li, Chengyuan; de Grijs, Richard; Deng, Licai; Geller, Aaron M; Xin, Yu; Hu, Yi; Faucher-Giguère, Claude-André
2016-01-28
Stars in clusters are thought to form in a single burst from a common progenitor cloud of molecular gas. However, massive, old 'globular' clusters--those with ages greater than ten billion years and masses several hundred thousand times that of the Sun--often harbour multiple stellar populations, indicating that more than one star-forming event occurred during their lifetimes. Colliding stellar winds from late-stage, asymptotic-giant-branch stars are often suggested to be triggers of second-generation star formation. For this to occur, the initial cluster masses need to be greater than a few million solar masses. Here we report observations of three massive relatively young star clusters (1-2 billion years old) in the Magellanic Clouds that show clear evidence of burst-like star formation that occurred a few hundred million years after their initial formation era. We show that such clusters could have accreted sufficient gas to form new stars if they had orbited in their host galaxies' gaseous disks throughout the period between their initial formation and the more recent bursts of star formation. This process may eventually give rise to the ubiquitous multiple stellar populations in globular clusters.
13. Linking 1D evolutionary to 3D hydrodynamical simulations of massive stars
NASA Astrophysics Data System (ADS)
Cristini, A.; Meakin, C.; Hirschi, R.; Arnett, D.; Georgy, C.; Viallet, M.
2016-03-01
Stellar evolution models of massive stars are important for many areas of astrophysics, for example nucleosynthesis yields, supernova progenitor models and understanding physics under extreme conditions. Turbulence occurs in stars primarily due to nuclear burning at different mass coordinates within the star. The understanding and correct treatment of turbulence and turbulent mixing at convective boundaries in stellar models has been studied for decades but still lacks a definitive solution. This paper presents initial results of a study on convective boundary mixing (CBM) in massive stars. The ‘stiffness’ of a convective boundary can be quantified using the bulk Richardson number ({{Ri}}{{B}}), the ratio of the potential energy for restoration of the boundary to the kinetic energy of turbulent eddies. A ‘stiff’ boundary ({{Ri}}{{B}}˜ {10}4) will suppress CBM, whereas in the opposite case a ‘soft’ boundary ({{Ri}}{{B}}˜ 10) will be more susceptible to CBM. One of the key results obtained so far is that lower convective boundaries (closer to the centre) of nuclear burning shells are ‘stiffer’ than the corresponding upper boundaries, implying limited CBM at lower shell boundaries. This is in agreement with 3D hydrodynamic simulations carried out by Meakin and Arnett (2007 Astrophys. J. 667 448-75). This result also has implications for new CBM prescriptions in massive stars as well as for nuclear burning flame front propagation in super-asymptotic giant branch stars and also the onset of novae.
14. Feedback by massive stars and the emergence of superbubbles. I. Energy efficiency and Vishniac instabilities
NASA Astrophysics Data System (ADS)
Krause, M.; Fierlinger, K.; Diehl, R.; Burkert, A.; Voss, R.; Ziegler, U.
2013-02-01
Context. Massive stars influence their environment through stellar winds, ionising radiation, and supernova explosions. This is signified by observed interstellar bubbles. Such feedback is an important factor for galaxy evolution theory and galactic wind models. The efficiency of the energy injection into the interstellar medium (ISM) via bubbles and superbubbles is uncertain, and is usually treated as a free parameter for galaxy scale effects. In particular, since many stars are born in groups, it is interesting to study the dependence of the effective energy injection on the concentration of the stars. Aims: We aim to reproduce observations of superbubbles, their relation to the energy injection of the parent stars, and to understand their effective energy input into the ISM, as a function of the spatial configuration of the group of parent stars. Methods: We study the evolution of isolated and merging interstellar bubbles of three stars (25, 32, and 60 M⊙) in a homogeneous background medium with a density of 10mp cm-3 via 3D-hydrodynamic simulations with standard ISM thermodynamics (optically thin radiative cooling and photo-electric heating) and time-dependent energy and mass input according to stellar evolutionary tracks. We vary the position of the three stars relative to each other to compare the energy response for cases of isolated, merging and initially cospatial bubbles. Results: Mainly due to the Vishniac instability, our simulated bubbles develop thick shells and filamentary internal structures in column density. The shell widths reach tens of per cent of the outer bubble radius, which compares favourably to observations. More energy is retained in the ISM for more closely packed groups, by up to a factor of three and typically a factor of two for intermediate times after the first supernova. Once the superbubble is established, different positions of the contained stars make only a minor difference to the energy tracks. For our case of three massive
15. Very massive neutron stars in Ni's theory of gravity
NASA Technical Reports Server (NTRS)
Mikkelsen, D. R.
1977-01-01
It is shown that in Ni's theory of gravity, which is identical to general relativity in the post-Newtonian limit, neutron stars of arbitrarily large mass are possible. This result is independent, within reasonable bounds, of the equation of state of matter at supernuclear densities.
16. The delay time distribution of massive double compact star mergers
NASA Astrophysics Data System (ADS)
Mennekens, N.; Vanbeveren, D.
2016-05-01
To investigate the temporal evolution of binary populations, in general, and double compact-star binaries and mergers, in particular, within a galactic evolution context, a very straightforward method is obviously to implement a detailed binary evolutionary model in a galactic chemical evolution code. To our knowledge, the Brussels galactic chemical evolution code is the only one that fully and consistently accounts for the important effects of interacting binaries on the predictions of chemical evolution. With a galactic code that does not explicitly include binaries, the temporal evolution of the population of double compact star binaries and mergers can be estimated with reasonable accuracy if the delay time distribution (DTD) for these mergers is available. The DTD for supernovae type Ia has been studied extensively in the past decade. In the present paper we present the DTD for merging double neutron-star binaries and mixed systems consisting of a neutron star and a black hole. The latter mergers are very promising sites for producing r-process elements, and the DTDs can be used to study the galactic evolution of these elements with a code that does not explicitly account for binaries.
17. Southern Massive Stars at High Angular Resolution: Observational Campaign and Companion Detection
NASA Astrophysics Data System (ADS)
Sana, H.; Le Bouquin, J.-B.; Lacour, S.; Berger, J.-P.; Duvert, G.; Gauchet, L.; Norris, B.; Olofsson, J.; Pickel, D.; Zins, G.; Absil, O.; de Koter, A.; Kratter, K.; Schnurr, O.; Zinnecker, H.
2014-11-01
Multiplicity is one of the most fundamental observable properties of massive O-type stars and offers a promising way to discriminate between massive star formation theories. Nevertheless, companions at separations between 1 and 100 milliarcsec (mas) remain mostly unknown due to intrinsic observational limitations. At a typical distance of 2 kpc, this corresponds to projected physical separations of 2-200 AU. The Southern MAssive Stars at High angular resolution survey (SMaSH+) was designed to fill this gap by providing the first systematic interferometric survey of Galactic massive stars. We observed 117 O-type stars with VLTI/PIONIER and 162 O-type stars with NACO/Sparse Aperture Masking (SAM), probing the separation ranges 1-45 and 30-250 mas and brightness contrasts of ΔH < 4 and ΔH < 5, respectively. Taking advantage of NACO's field of view, we further uniformly searched for visual companions in an 8'' radius down to ΔH = 8. This paper describes observations and data analysis, reports the discovery of almost 200 new companions in the separation range from 1 mas to 8'' and presents a catalog of detections, including the first resolved measurements of over a dozen known long-period spectroscopic binaries. Excluding known runaway stars for which no companions are detected, 96 objects in our main sample (δ < 0° H < 7.5) were observed both with PIONIER and NACO/SAM. The fraction of these stars with at least one resolved companion within 200 mas is 0.53. Accounting for known but unresolved spectroscopic or eclipsing companions, the multiplicity fraction at separation ρ < 8'' increases to f m = 0.91 ± 0.03. The fraction of luminosity class V stars that have a bound companion reaches 100% at 30 mas while their average number of physically connected companions within 8'' is f c = 2.2 ± 0.3. This demonstrates that massive stars form nearly exclusively in multiple systems. The nine non-thermal radio emitters observed by SMaSH+ are all resolved, including the newly
18. SOUTHERN MASSIVE STARS AT HIGH ANGULAR RESOLUTION: OBSERVATIONAL CAMPAIGN AND COMPANION DETECTION
SciTech Connect
Sana, H.; Lacour, S.; Gauchet, L.; Pickel, D.; Berger, J.-P.; Norris, B.; Olofsson, J.; Absil, O.; De Koter, A.; Kratter, K.; Schnurr, O.; Zinnecker, H.
2014-11-01
Multiplicity is one of the most fundamental observable properties of massive O-type stars and offers a promising way to discriminate between massive star formation theories. Nevertheless, companions at separations between 1 and 100 milliarcsec (mas) remain mostly unknown due to intrinsic observational limitations. At a typical distance of 2 kpc, this corresponds to projected physical separations of 2-200 AU. The Southern MAssive Stars at High angular resolution survey (SMaSH+) was designed to fill this gap by providing the first systematic interferometric survey of Galactic massive stars. We observed 117 O-type stars with VLTI/PIONIER and 162 O-type stars with NACO/Sparse Aperture Masking (SAM), probing the separation ranges 1-45 and 30-250 mas and brightness contrasts of ΔH < 4 and ΔH < 5, respectively. Taking advantage of NACO's field of view, we further uniformly searched for visual companions in an 8'' radius down to ΔH = 8. This paper describes observations and data analysis, reports the discovery of almost 200 new companions in the separation range from 1 mas to 8'' and presents a catalog of detections, including the first resolved measurements of over a dozen known long-period spectroscopic binaries. Excluding known runaway stars for which no companions are detected, 96 objects in our main sample (δ < 0°; H < 7.5) were observed both with PIONIER and NACO/SAM. The fraction of these stars with at least one resolved companion within 200 mas is 0.53. Accounting for known but unresolved spectroscopic or eclipsing companions, the multiplicity fraction at separation ρ < 8'' increases to f {sub m} = 0.91 ± 0.03. The fraction of luminosity class V stars that have a bound companion reaches 100% at 30 mas while their average number of physically connected companions within 8'' is f {sub c} = 2.2 ± 0.3. This demonstrates that massive stars form nearly exclusively in multiple systems. The nine non-thermal radio emitters observed by SMaSH+ are all resolved
19. PRESUPERNOVA EVOLUTION AND EXPLOSIVE NUCLEOSYNTHESIS OF ZERO METAL MASSIVE STARS
SciTech Connect
Limongi, M.; Chieffi, A. E-mail: [email protected]
2012-04-01
We present a new set of zero metallicity models in the range 13-80 M{sub Sun} together to the associated explosive nucleosynthesis. These models are fully homogeneous with the solar metallicity set we published in Limongi and Chieffi and will be freely available at the Online Repository for the FRANEC Evolutionary Output Web site. A comparison between these yields and an average star that represents the average behavior of most of the very metal-poor stars in the range -5.0 < [Fe/H] < -2.5 confirms previous findings that only a fraction of the elemental [X/Fe] may be fitted by the ejecta of standard core collapse supernovae.
20. The Evolution of Massive Stars and the Concomitant Non-explosive and Explosive Nucleosynthesis
NASA Astrophysics Data System (ADS)
Arnould, Marcel
These lectures are concerned with some aspects of the evolution of massive stars and of the concomitant nucleosynthesis. They complement other lectures in this volume. Special emphasis is put on the production of the nuclides heavier than iron by the r- and p-processes.
1. Star formation in the massive cluster merger Abell 2744
NASA Astrophysics Data System (ADS)
Rawle, T. D.; Altieri, B.; Egami, E.; Pérez-González, P. G.; Richard, J.; Santos, J. S.; Valtchanov, I.; Walth, G.; Bouy, H.; Haines, C. P.; Okabe, N.
2014-07-01
We present a comprehensive study of star-forming (SF) galaxies in the Hubble Space Telescope (HST) Frontier Field recent cluster merger A2744 (z = 0.308). Wide-field, ultraviolet-infrared (UV-IR) imaging enables a direct constraint of the total star formation rate (SFR) for 53 cluster galaxies, with SFRUV+IR = 343 ± 10 M⊙ yr-1. Within the central 4 arcmin (1.1 Mpc) radius, the integrated SFR is complete, yielding a total SFRUV+IR = 201 ± 9 M⊙ yr-1. Focusing on obscured star formation, this core region exhibits a total SFRIR = 138 ± 8 M⊙ yr-1, a mass-normalized SFRIR of ΣSFR = 11.2 ± 0.7 M⊙ yr-1 per 1014 M⊙ and a fraction of IR-detected SF galaxies f_SF = 0.080^{+0.010}_{-0.037}. Overall, the cluster population at z ˜ 0.3 exhibits significant intrinsic scatter in IR properties (total SFRIR, Tdust distribution) apparently unrelated to the dynamical state: A2744 is noticeably different to the merging Bullet cluster, but similar to several relaxed clusters. However, in A2744 we identify a trail of SF sources including jellyfish galaxies with substantial unobscured SF due to extreme stripping (SFRUV/SFRIR up to 3.3). The orientation of the trail, and of material stripped from constituent galaxies, indicates that the passing shock front of the cluster merger was the trigger. Constraints on star formation from both IR and UV are crucial for understanding galaxy evolution within the densest environments.
2. Chemical abundances of massive stars in Local Group galaxies
NASA Astrophysics Data System (ADS)
Venn, Kim A.; Kaufer, Andreas; Tolstoy, Eline; Kudritzki, Rolf-Peter; Przybilla, Norbert; Smartt, Stephen J.; Lennon, Daniel J.
The relative abundances of elements in galaxies can provide valuable information on the stellar and chemical evolution of a galaxy. While nebulae can provide abundances for a variety of light elements, stars are the only way to directly determine the abundances of iron-group and s-process and r-process elements in a galaxy. The new 8m and 10m class telescopes and high-efficiency spectrographs now make high-quality spectral observations of bright supergiants possible in dwarf galaxies in the Local Group. We have been concentrating on elemental abundances in the metal-poor dwarf irregular galaxies, NGC 6822, WLM, Sextants A, and GR 8. Comparing abundance ratios to those predicted from their star formation histories, determined from color-magnitude diagrams, and comparing those ratios between these galaxies can give us new insights into the evolution of these dwarf irregular galaxies. Iron-group abundances also allow us to examine the metallicities of the stars in these galaxies directly, which affects their inferred mass loss rates and predicted stellar evolution properties.
3. The mass-radius relationship of massive compact stars
SciTech Connect
Chowdhury, Partha Roy
2015-02-24
The properties of pure hadronic and hybrid compact stars are reviewed using nuclear equation of state (EoS) for β-equilibrated neutron star (NS) matter obtained using a density-dependent M3Y (DDM3Y) effective nucleon-nucleon interaction. Depending on the model, the energy density of quark matter can be lower than that of this nuclear EoS at higher densities, implying the possibility of transition to quark matter inside the core and the transition density depends on the particular quark matter model used. The recent observations of the binary millisecond pulsar J1614–2230 by P.B. Demorest et al. [1] and PSR J0348+0432 by J. Antoniadis et al. [2] suggest that the masses lie within 1.97 ± 0.04 M{sub ⊙} and 2.01 ± 0.04 M{sub ⊙}, respectively, where M{sub ⊙} is the solar mass. In conformity with recent observations, a pure nucleonic EoS determines that the maximum mass of NS rotating with frequency ν∼ 667 Hz below r-mode instability is ∼ 1.95 M{sub ⊙} with radius ∼ 10 km. Compact stars with quark cores rotating with same frequency have the maximum mass of ∼ 1.72 M{sub ⊙} turns out to be lower than the observed masses.
4. A giant outburst two years before the core-collapse of a massive star.
PubMed
Pastorello, A; Smartt, S J; Mattila, S; Eldridge, J J; Young, D; Itagaki, K; Yamaoka, H; Navasardyan, H; Valenti, S; Patat, F; Agnoletto, I; Augusteijn, T; Benetti, S; Cappellaro, E; Boles, T; Bonnet-Bidaud, J-M; Botticella, M T; Bufano, F; Cao, C; Deng, J; Dennefeld, M; Elias-Rosa, N; Harutyunyan, A; Keenan, F P; Iijima, T; Lorenzi, V; Mazzali, P A; Meng, X; Nakano, S; Nielsen, T B; Smoker, J V; Stanishev, V; Turatto, M; Xu, D; Zampieri, L
2007-06-14
The death of massive stars produces a variety of supernovae, which are linked to the structure of the exploding stars. The detection of several precursor stars of type II supernovae has been reported (see, for example, ref. 3), but we do not yet have direct information on the progenitors of the hydrogen-deficient type Ib and Ic supernovae. Here we report that the peculiar type Ib supernova SN 2006jc is spatially coincident with a bright optical transient that occurred in 2004. Spectroscopic and photometric monitoring of the supernova leads us to suggest that the progenitor was a carbon-oxygen Wolf-Rayet star embedded within a helium-rich circumstellar medium. There are different possible explanations for this pre-explosion transient. It appears similar to the giant outbursts of luminous blue variable stars (LBVs) of 60-100 solar masses, but the progenitor of SN 2006jc was helium- and hydrogen-deficient (unlike LBVs). An LBV-like outburst of a Wolf-Rayet star could be invoked, but this would be the first observational evidence of such a phenomenon. Alternatively, a massive binary system composed of an LBV that erupted in 2004, and a Wolf-Rayet star exploding as SN 2006jc, could explain the observations. PMID:17568740
5. Asteroseismological study of massive ZZ Ceti stars with fully evolutionary models
SciTech Connect
Romero, A. D.; Kepler, S. O.; Córsico, A. H.; Althaus, L. G.
2013-12-10
We present the first asteroseismological study for 42 massive ZZ Ceti stars based on a large set of fully evolutionary carbon-oxygen core DA white dwarf models characterized by a detailed and consistent chemical inner profile for the core and the envelope. Our sample comprises all of the ZZ Ceti stars with spectroscopic stellar masses between 0.72 and 1.05 M {sub ☉} known to date. The asteroseismological analysis of a set of 42 stars enables study of the ensemble properties of the massive, pulsating white dwarf stars with carbon-oxygen cores, in particular the thickness of the hydrogen envelope and the stellar mass. A significant fraction of stars in our sample have stellar mass that is high enough to crystallize at the effective temperatures of the ZZ Ceti instability strip, which enables us to study the effects of crystallization on the pulsation properties of these stars. Our results show that the phase diagram presented in Horowitz et al. seems to be a good representation of the crystallization process inside white dwarf stars, in agreement with the results from white dwarf luminosity function in globular clusters.
6. A giant outburst two years before the core-collapse of a massive star.
PubMed
Pastorello, A; Smartt, S J; Mattila, S; Eldridge, J J; Young, D; Itagaki, K; Yamaoka, H; Navasardyan, H; Valenti, S; Patat, F; Agnoletto, I; Augusteijn, T; Benetti, S; Cappellaro, E; Boles, T; Bonnet-Bidaud, J-M; Botticella, M T; Bufano, F; Cao, C; Deng, J; Dennefeld, M; Elias-Rosa, N; Harutyunyan, A; Keenan, F P; Iijima, T; Lorenzi, V; Mazzali, P A; Meng, X; Nakano, S; Nielsen, T B; Smoker, J V; Stanishev, V; Turatto, M; Xu, D; Zampieri, L
2007-06-14
The death of massive stars produces a variety of supernovae, which are linked to the structure of the exploding stars. The detection of several precursor stars of type II supernovae has been reported (see, for example, ref. 3), but we do not yet have direct information on the progenitors of the hydrogen-deficient type Ib and Ic supernovae. Here we report that the peculiar type Ib supernova SN 2006jc is spatially coincident with a bright optical transient that occurred in 2004. Spectroscopic and photometric monitoring of the supernova leads us to suggest that the progenitor was a carbon-oxygen Wolf-Rayet star embedded within a helium-rich circumstellar medium. There are different possible explanations for this pre-explosion transient. It appears similar to the giant outbursts of luminous blue variable stars (LBVs) of 60-100 solar masses, but the progenitor of SN 2006jc was helium- and hydrogen-deficient (unlike LBVs). An LBV-like outburst of a Wolf-Rayet star could be invoked, but this would be the first observational evidence of such a phenomenon. Alternatively, a massive binary system composed of an LBV that erupted in 2004, and a Wolf-Rayet star exploding as SN 2006jc, could explain the observations.
7. SPITZER SAGE-SMC INFRARED PHOTOMETRY OF MASSIVE STARS IN THE SMALL MAGELLANIC CLOUD
SciTech Connect
Bonanos, A. Z.; Lennon, D. J.; Massa, D. L. E-mail: [email protected]
2010-08-15
We present a catalog of 5324 massive stars in the Small Magellanic Cloud (SMC), with accurate spectral types compiled from the literature, and a photometric catalog for a subset of 3654 of these stars, with the goal of exploring their infrared properties. The photometric catalog consists of stars with infrared counterparts in the Spitzer SAGE-SMC survey database, for which we present uniform photometry from 0.3to24 {mu}m in the UBVIJHK{sub s} +IRAC+MIPS24 bands. We compare the color-magnitude diagrams and color-color diagrams to those of stars in the Large Magellanic Cloud (LMC), finding that the brightest infrared sources in the SMC are also the red supergiants, supergiant B[e] (sgB[e]) stars, luminous blue variables, and Wolf-Rayet stars, with the latter exhibiting less infrared excess, the red supergiants being less dusty and the sgB[e] stars being on average less luminous. Among the objects detected at 24 {mu}m in the SMC are a few very luminous hypergiants, four B-type stars with peculiar, flat spectral energy distributions, and all three known luminous blue variables. We detect a distinct Be star sequence, displaced to the red, and suggest a novel method of confirming Be star candidates photometrically. We find a higher fraction of Oe and Be stars among O and early-B stars in our SMC catalog, respectively, when compared to the LMC catalog, and that the SMC Be stars occur at higher luminosities. We estimate mass-loss rates for the red supergiants, confirming the correlation with luminosity even at the metallicity of the SMC. Finally, we confirm the new class of stars displaying composite A and F type spectra, the sgB[e] nature of 2dFS1804 and find the F0 supergiant 2dFS3528 to be a candidate luminous blue variable with cold dust.
8. Are Young Massive Star Clusters in the Local Universe Analogous to Globular Clusters Progenitors?
NASA Astrophysics Data System (ADS)
Charbonnel, Corinne
2015-08-01
Several models do compete to reproduce the present-day characteristics of globular clusters (GC) and to explain the origin of the multiple stellar populations these systems are hosting.In parallel, independent clues on GC early evolution may be derived from observations of young massive clusters (YMC) in the Local Group.But are these two populations of clusters related? In this talk, we discuss how and if GC and YMC data can be reconciled.We revisit in particular the impact of massive stars on the early evolution of massive star clusters, as well as the question of early gas expulsion.We propose several tests to probe whether the YMC we are observing today can be considered as the analogues of GC progenitors.
9. EARLY-STAGE MASSIVE STAR FORMATION NEAR THE GALACTIC CENTER: Sgr C
SciTech Connect
Kendrew, S.; Johnston, K.; Beuther, H.; Ginsburg, A.; Bally, J.; Battersby, C.; Cyganowski, C. J.
2013-10-01
We present near-infrared spectroscopy and 1 mm line and continuum observations of a recently identified site of high mass star formation likely to be located in the Central Molecular Zone (CMZ) near Sgr C. Located on the outskirts of the massive evolved H II region associated with Sgr C, the area is characterized by an Extended Green Object (EGO) measuring ∼10'' in size (0.4 pc), whose observational characteristics suggest the presence of an embedded massive protostar driving an outflow. Our data confirm that early-stage star formation is taking place on the periphery of the Sgr C H II region, with detections of two protostellar cores and several knots of H{sub 2} and Brackett γ emission alongside a previously detected compact radio source. We calculate the cores' joint mass to be ∼10{sup 3} M {sub ☉}, with column densities of 1-2 × 10{sup 24} cm{sup –2}. We show the host molecular cloud to hold ∼10{sup 5} M {sub ☉} of gas and dust with temperatures and column densities favorable for massive star formation to occur, however, there is no evidence of star formation outside of the EGO, indicating that the cloud is predominantly quiescent. Given its mass, density, and temperature, the cloud is comparable to other remarkable non-star-forming clouds such as G0.253 in the eastern CMZ.
10. Can Very Massive Population III Stars Produce a Super-Collapsar?
NASA Astrophysics Data System (ADS)
Yoon, Sung-Chul; Kang, Jisu; Kozyreva, Alexandra
2015-03-01
A fraction of the first generation of stars in the early universe may be very massive (≳ 300 {{M}⊙ }) as they form in metal-free environments. Formation of black holes from these stars can be accompanied by supermassive collapsars to produce long gamma-ray bursts of a unique type having a very high total energy (˜ {{10}54} erg) as recently suggested by several authors. We present new stellar evolution models of very massive Population III stars including the effect of rotation to provide theoretical constraints on super-collapsar progenitors. We find that the angular momentum condition for a super-collapsar can be fulfilled if magnetic torques are ignored, in which case Eddington-Sweet circulations play the dominant role for the transport of angular momentum. We further find that the initial mass range for super-collapsar progenitors would be limited to 300 {{M}⊙ }≲ M≲ 700 {{M}⊙ }. However, all of our very massive star models of this mass range end their lives as red supergiants rather than blue supergiants, in good agreement with most of the previous studies. The predicted final fate of these stars is either a jet-powered type IIP supernova or an ultra-long, relatively faint gamma-ray transient, depending on the initial amount of angular momentum.
11. Gravitational waves from the collision of tidally disrupted stars with massive black holes
SciTech Connect
East, William E.
2014-11-10
We use simulations of hydrodynamics coupled with full general relativity to investigate the gravitational waves produced by a star colliding with a massive black hole when the star's tidal disruption radius lies far outside of the black hole horizon. We consider both main-sequence and white-dwarf compaction stars, and nonspinning black holes, as well as those with near-extremal spin. We study the regime in between where the star can be accurately modeled by a point particle, and where tidal effects completely suppress the gravitational wave signal. We find that nonnegligible gravitational waves can be produced even when the star is strongly affected by tidal forces, as well as when it collides with large angular momentum. We discuss the implications that these results have for the potential observation of gravitational waves from these sources with future detectors.
12. Curtain-Lifting Winds Allow Rare Glimpse into Massive Star Factory
NASA Astrophysics Data System (ADS)
2003-06-01
Formation of Exceedingly Luminous and Hot Stars in Young Stellar Cluster Observed Directly Summary Based on a vast observational effort with different telescopes and instruments, ESO-astronomer Dieter Nürnberger has obtained a first glimpse of the very first stages in the formation of heavy stars. These critical phases of stellar evolution are normally hidden from the view, because massive protostars are deeply embedded in their native clouds of dust and gas, impenetrable barriers to observations at all but the longest wavelengths. In particular, no visual or infrared observations have yet "caught" nascent heavy stars in the act and little is therefore known so far about the related processes. Profiting from the cloud-ripping effect of strong stellar winds from adjacent, hot stars in a young stellar cluster at the center of the NGC 3603 complex, several objects located near a giant molecular cloud were found to be bona-fide massive protostars, only about 100,000 years old and still growing. Three of these objects, designated IRS 9A-C, could be studied in more detail. They are very luminous (IRS 9A is about 100,000 times intrinsically brighter than the Sun), massive (more than 10 times the mass of the Sun) and hot (about 20,000 degrees). They are surrounded by relative cold dust (about 0°C), probably partly arranged in disks around these very young objects. Two possible scenarios for the formation of massive stars are currently proposed, by accretion of large amounts of circumstellar material or by collision (coalescence) of protostars of intermediate masses. The new observations favour accretion, i.e. the same process that is active during the formation of stars of smaller masses. PR Photo 16a/03: Stellar cluster and star-forming region NGC 3603. PR Photo 16b/03: Region near very young, massive stars IRS 9A-C in NGC 3603 (8 bands from J to Q). How do massive stars form? This question is easy to pose, but so far very difficult to answer. In fact, the processes
13. A THIRD MASSIVE STAR COMPONENT IN THE {sigma} ORIONIS AB SYSTEM
SciTech Connect
Simon-Diaz, S.; Caballero, J. A.; Lorenzo, J.
2011-11-20
We report on the detection of a third massive star component in the {sigma} Orionis AB system, traditionally considered as a binary system. The system has been monitored by the IACOB Spectroscopic Survey of Northern Massive Stars program, obtaining 23 high-resolution FIES-NOT spectra with a time span of {approx}2.5 years. The analysis of the radial velocity curves of the two spectroscopic components observed in the spectra has allowed us to obtain the orbital parameters of the system, resulting in a high eccentric orbit (e {approx} 0.78) with an orbital period of 143.5 {+-} 0.5 days. This result implies the actual presence of three stars in the {sigma} Orionis AB system when combined with previous results obtained from the study of the astrometric orbit (with an estimated period of {approx}157 years).
14. Rejuvenation of stellar mergers and the origin of magnetic fields in massive stars
NASA Astrophysics Data System (ADS)
Schneider, F. R. N.; Podsiadlowski, Ph.; Langer, N.; Castro, N.; Fossati, L.
2016-04-01
Approximately 10 per cent of massive OBA main-sequence (MS) and pre-MS stars harbour strong, large-scale magnetic fields. At the same time, there is a dearth of magnetic stars in close binaries. A process generating strong magnetic fields only in some stars must be responsible with the merging of pre-MS and MS stars being suggested as one such channel. Stars emerging from the coalescence of two MS stars are rejuvenated, appearing younger than they are. They can therefore be identified by comparison with reference clocks. Here, we predict the rejuvenation of MS merger products over a wide range of masses and binary configurations calibrated to smoothed-particle-hydrodynamical merger models. We find that the rejuvenation is of the order of the nuclear time-scale and is strongest in the lowest mass mergers and the most evolved binary progenitors with the largest mass ratios. These predictions allow us to put constraints on the binary progenitors of merger products. We show that the magnetic stars HR 2949 and τ Sco are younger than the potential binary companion HR 2948 and the Upper-Sco association, respectively, making them promising merger candidates. We find that the age discrepancies and the potential binary progenitors of both are consistent with them being rejuvenated merger products, implying that their magnetic fields may originate from this channel. Searching for age discrepancies in magnetic stars is therefore a powerful way to explore which fraction of magnetic stars may have obtained their strong magnetic field in MS mergers and to improve our understanding of magnetism in massive stars and their remnants.
15. Surveying Massive Star Formation in the Inner Galaxy
NASA Astrophysics Data System (ADS)
Dorda, R.; Negueruela, I.; González-Fernández, C.; Marco, A.
2016-10-01
The base of the Scutum arm is a Galactic region with a high density of red supergiant (RSG) stars, grouped in a few clusters which have similar ages, positions and radial velocities. We have performed an extensive survey using the multi-object spectrograph AAOmega, looking for new RSGs along the galactic plane from l˜24° to 30°. We have observed >1600 candidates, and identified them through an extensive study of the statistical behavior of RSG spectra, finding ˜200 new RSGs.
16. Iron-group opacities in the envelopes of massive stars
NASA Astrophysics Data System (ADS)
Le Pennec, Maëlle; Turck-Chièze, Sylvaine
2014-02-01
β Cephei and SPB stars are pulsating stars for which the excitation of modes by the κ mechanism, due to the iron-group opacity peak, seems puzzling. We have first investigated the origins of the differences noticed between OP and OPAL iron and nickel opacity calculations (up to a factor 2), a fact which complicates the interpretation. To accomplish this task, new well-qualified calculations (SCO-RCG, HULLAC and ATOMIC) have been performed and compared to values of these tables, and most of the differences are now well understood. Next, we have exploited a dedicated experiment on chromium, iron and nickel, conducted at the LULI 2000 facilities. We found that, in the case of iron, detailed calculations (OP, ATOMIC and HULLAC) show good agreement, contrary to all of the non-detailed calculations. However, in the case of nickel, OP calculations show large discrepancies with the experiments but also with other codes. Thus, the opacity tables need to be revised in the thermodynamical conditions corresponding to the peak of the iron group. Consequently we study the evolution of this iron peak with changes in stellar mass, age, and metallicity to determine the relevant region where these tables should be revised.
17. First detections of FS Canis Majoris stars in clusters. Evolutionary state as constrained by coeval massive stars
NASA Astrophysics Data System (ADS)
de la Fuente, D.; Najarro, F.; Trombley, C.; Davies, B.; Figer, D. F.
2015-03-01
Context. FS CMa stars are low-luminosity objects showing the B[e] phenomenon whose evolutionary state remains a puzzle. These stars are surrounded by compact disks of warm dust of unknown origin. Hitherto, membership of FS CMa stars to coeval populations has never been confirmed. Aims: The discovery of low-luminosity line emitters in the young massive clusters Mercer 20 and Mercer 70 prompts us to investigate the nature of such objects. We intend to confirm membership to coeval populations in order to characterize these emission-line stars through the cluster properties. Methods: Based on ISAAC/VLT medium-resolution spectroscopy and NICMOS/HST photometry of massive cluster members, new characterizations of Mercer 20 and Mercer 70 are performed. Coevality of each cluster and membership of the newly-discovered B[e] objects are investigated using our observations as well as literature data of the surroundings. Infrared excess and narrow-band photometric properties of the B[e] stars are also studied. Results: We confirm and classify 22 new cluster members, including Wolf-Rayet stars and blue hypergiants. Spectral types (O9-B1.5 V) and radial velocities of B[e] objects are compatible with the remaining cluster members, while emission features of Mg ii, Fe ii], and [Fe ii] are identified in their spectra. The ages of these stars are 4.5 and 6 Myr, and they show mild infrared excesses. Conclusions: We confirm the presence of FS CMa stars in the coeval populations of Mercer 20 and Mercer 70. We discuss the nature and evolutionary state of FS CMa stars, discarding a post-AGB nature and introducing a new hypothesis about mergers. A new search method for FS CMa candidates in young massive clusters based on narrow-band Paschen-α photometry is proposed and tested in photometric data of other clusters, yielding three new candidates. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, under program IDs 083.D
18. A magnetic confinement versus rotation classification of massive-star magnetospheres
NASA Astrophysics Data System (ADS)
Petit, V.; Owocki, S. P.; Wade, G. A.; Cohen, D. H.; Sundqvist, J. O.; Gagné, M.; Maíz Apellániz, J.; Oksala, M. E.; Bohlender, D. A.; Rivinius, T.; Henrichs, H. F.; Alecian, E.; Townsend, R. H. D.; ud-Doula, A.; MiMeS Collaboration
2013-02-01
Building on results from the Magnetism in Massive Stars (MiMeS) project, this paper shows how a two-parameter classification of massive-star magnetospheres in terms of the magnetic wind confinement (which sets the Alfvén radius RA) and stellar rotation (which sets the Kepler co-rotation radius RK) provides a useful organization of both observational signatures and theoretical predictions. We compile the first comprehensive study of inferred and observed values for relevant stellar and magnetic parameters of 64 confirmed magnetic OB stars with Teff ≳ 16 kK. Using these parameters, we locate the stars in the magnetic confinement-rotation diagram, a log-log plot of RK versus RA. This diagram can be subdivided into regimes of centrifugal magnetospheres (CM), with RA > RK, versus dynamical magnetospheres (DM), with RK > RA. We show how key observational diagnostics, like the presence and characteristics of Hα emission, depend on a star's position within the diagram, as well as other parameters, especially the expected wind mass-loss rates. In particular, we identify two distinct populations of magnetic stars with Hα emission: namely, slowly rotating O-type stars with narrow emission consistent with a DM, and more rapidly rotating B-type stars with broader emission associated with a CM. For O-type stars, the high mass-loss rates are sufficient to accumulate enough material for line emission even within the relatively short free-fall time-scale associated with a DM: this high mass-loss rate also leads to a rapid magnetic spindown of the stellar rotation. For the B-type stars, the longer confinement of a CM is required to accumulate sufficient emitting material from their relatively weak winds, which also lead to much longer spindown time-scales. Finally, we discuss how other observational diagnostics, e.g. variability of UV wind lines or X-ray emission, relate to the inferred magnetic properties of these stars, and summarize prospects for future developments in our
19. THE COSMIC CORE-COLLAPSE SUPERNOVA RATE DOES NOT MATCH THE MASSIVE-STAR FORMATION RATE
SciTech Connect
Horiuchi, Shunsaku; Beacom, John F.; Kochanek, Christopher S.; Stanek, K. Z.; Thompson, Todd A.; Prieto, Jose L.
2011-09-10
We identify a 'supernova rate problem': the measured cosmic core-collapse supernova rate is a factor of {approx}2 smaller (with significance {approx}2{sigma}) than that predicted from the measured cosmic massive-star formation rate. The comparison is critical for topics from galaxy evolution and enrichment to the abundance of neutron stars and black holes. We systematically explore possible resolutions. The accuracy and precision of the star formation rate data and conversion to the supernova rate are well supported, and proposed changes would have far-reaching consequences. The dominant effect is likely that many supernovae are missed because they are either optically dim (low-luminosity) or dark, whether intrinsically or due to obscuration. We investigate supernovae too dim to have been discovered in cosmic surveys by a detailed study of all supernova discoveries in the local volume. If possible supernova impostors are included, then dim supernovae are common enough by fraction to solve the supernova rate problem. If they are not included, then the rate of dark core collapses is likely substantial. Other alternatives are that there are surprising changes in our understanding of star formation or supernova rates, including that supernovae form differently in small galaxies than in normal galaxies. These possibilities can be distinguished by upcoming supernova surveys, star formation measurements, searches for disappearing massive stars, and measurements of supernova neutrinos.
20. Super massive black hole in galactic nuclei with tidal disruption of stars
SciTech Connect
Zhong, Shiyan; Berczik, Peter; Spurzem, Rainer
2014-09-10
Tidal disruption of stars by super massive central black holes from dense star clusters is modeled by high-accuracy direct N-body simulation. The time evolution of the stellar tidal disruption rate, the effect of tidal disruption on the stellar density profile, and, for the first time, the detailed origin of tidally disrupted stars are carefully examined and compared with classic papers in the field. Up to 128k particles are used in simulation to model the star cluster around a super massive black hole, and we use the particle number and the tidal radius of the black hole as free parameters for a scaling analysis. The transition from full to empty loss-cone is analyzed in our data, and the tidal disruption rate scales with the particle number, N, in the expected way for both cases. For the first time in numerical simulations (under certain conditions) we can support the concept of a critical radius of Frank and Rees, which claims that most stars are tidally accreted on highly eccentric orbits originating from regions far outside the tidal radius. Due to the consumption of stars moving on radial orbits, a velocity anisotropy is found inside the cluster. Finally we estimate the real galactic center based on our simulation results and the scaling analysis.
1. A massive hypergiant star as the progenitor of the supernova SN 2005gl.
PubMed
Gal-Yam, A; Leonard, D C
2009-04-16
Our understanding of the evolution of massive stars before their final explosions as supernovae is incomplete, from both an observational and a theoretical standpoint. A key missing piece in the supernova puzzle is the difficulty of identifying and studying progenitor stars. In only a single case-that of supernova SN 1987A in the Large Magellanic Cloud-has a star been detected at the supernova location before the explosion, and been subsequently shown to have vanished after the supernova event. The progenitor of SN 1987A was a blue supergiant, which required a rethink of stellar evolution models. The progenitor of supernova SN 2005gl was proposed to be an extremely luminous object, but the association was not robustly established (it was not even clear that the putative progenitor was a single luminous star). Here we report that the previously proposed object was indeed the progenitor star of SN 2005gl. This very massive star was likely a luminous blue variable that standard stellar evolution predicts should not have exploded in that state. PMID:19305392
2. A Spectroscopic Survey of Massive Stars in M31 and M33
NASA Astrophysics Data System (ADS)
Massey, Philip; Neugent, Kathryn F.; Smart, Brianna M.
2016-09-01
We describe our spectroscopic follow-up to the Local Group Galaxy Survey (LGGS) photometry of M31 and M33. We have obtained new spectroscopy of 1895 stars, allowing us to classify 1496 of them for the first time. Our study has identified many foreground stars, and established membership for hundreds of early- and mid-type supergiants. We have also found nine new candidate luminous blue variables and a previously unrecognized Wolf-Rayet star. We republish the LGGS M31 and M33 catalogs with improved coordinates, and including spectroscopy from the literature and our new results. The spectroscopy in this paper is responsible for the vast majority of the stellar classifications in these two nearby spiral neighbors. The most luminous (and hence massive) of the stars in our sample are early-type B supergiants, as expected; the more massive O stars are more rare and fainter visually, and thus mostly remain unobserved so far. The majority of the unevolved stars in our sample are in the 20-40 M ⊙ range. The spectroscopic observations reported here were obtained at the MMT Observatory, a joint facility of the University of Arizona and the Smithsonian Institution. MMT telescope time was granted by NOAO, through the Telescope System Instrumentation Program (TSIP). TSIP is funded by the National Science Foundation. This paper uses data products produced by the OIR Telescope Data Center, supported by the Smithsonian Astrophysical Observatory.
3. A Spectroscopic Survey of Massive Stars in M31 and M33
NASA Astrophysics Data System (ADS)
Massey, Philip; Neugent, Kathryn F.; Smart, Brianna M.
2016-09-01
We describe our spectroscopic follow-up to the Local Group Galaxy Survey (LGGS) photometry of M31 and M33. We have obtained new spectroscopy of 1895 stars, allowing us to classify 1496 of them for the first time. Our study has identified many foreground stars, and established membership for hundreds of early- and mid-type supergiants. We have also found nine new candidate luminous blue variables and a previously unrecognized Wolf–Rayet star. We republish the LGGS M31 and M33 catalogs with improved coordinates, and including spectroscopy from the literature and our new results. The spectroscopy in this paper is responsible for the vast majority of the stellar classifications in these two nearby spiral neighbors. The most luminous (and hence massive) of the stars in our sample are early-type B supergiants, as expected; the more massive O stars are more rare and fainter visually, and thus mostly remain unobserved so far. The majority of the unevolved stars in our sample are in the 20–40 M ⊙ range. The spectroscopic observations reported here were obtained at the MMT Observatory, a joint facility of the University of Arizona and the Smithsonian Institution. MMT telescope time was granted by NOAO, through the Telescope System Instrumentation Program (TSIP). TSIP is funded by the National Science Foundation. This paper uses data products produced by the OIR Telescope Data Center, supported by the Smithsonian Astrophysical Observatory.
4. Super Massive Black Hole in Galactic Nuclei with Tidal Disruption of Stars
NASA Astrophysics Data System (ADS)
Zhong, Shiyan; Berczik, Peter; Spurzem, Rainer
2014-09-01
Tidal disruption of stars by super massive central black holes from dense star clusters is modeled by high-accuracy direct N-body simulation. The time evolution of the stellar tidal disruption rate, the effect of tidal disruption on the stellar density profile, and, for the first time, the detailed origin of tidally disrupted stars are carefully examined and compared with classic papers in the field. Up to 128k particles are used in simulation to model the star cluster around a super massive black hole, and we use the particle number and the tidal radius of the black hole as free parameters for a scaling analysis. The transition from full to empty loss-cone is analyzed in our data, and the tidal disruption rate scales with the particle number, N, in the expected way for both cases. For the first time in numerical simulations (under certain conditions) we can support the concept of a critical radius of Frank & Rees, which claims that most stars are tidally accreted on highly eccentric orbits originating from regions far outside the tidal radius. Due to the consumption of stars moving on radial orbits, a velocity anisotropy is found inside the cluster. Finally we estimate the real galactic center based on our simulation results and the scaling analysis.
5. YOUNG STELLAR OBJECTS IN THE MASSIVE STAR-FORMING REGION W49
SciTech Connect
Saral, G.; Hora, J. L.; Willis, S. E.; Koenig, X. P.; Gutermuth, R. A.; Saygac, A. T.
2015-11-01
We present the initial results of our investigation of the star-forming complex W49, one of the youngest and most luminous massive star-forming regions in our Galaxy. We used Spitzer/Infrared Array Camera (IRAC) data to investigate massive star formation with the primary objective of locating a representative set of protostars and the clusters of young stars that are forming around them. We present our source catalog with the mosaics from the IRAC data. In this study we used a combination of IRAC, MIPS, Two Micron All Sky Survey, and UKIRT Deep Infrared Sky Survey (UKIDSS) data to identify and classify the young stellar objects (YSOs). We identified 232 Class 0/I YSOs, 907 Class II YSOs, and 74 transition disk candidate objects using color–color and color–magnitude diagrams. In addition, to understand the evolution of star formation in W49, we analyzed the distribution of YSOs in the region to identify clusters using a minimal spanning tree method. The fraction of YSOs that belong to clusters with ≥7 members is found to be 52% for a cutoff distance of 96″, and the ratio of Class II/I objects is 2.1. We compared the W49 region to the G305 and G333 star-forming regions and concluded that W49 has the richest population, with seven subclusters of YSOs.
6. A massive hypergiant star as the progenitor of the supernova SN 2005gl.
PubMed
Gal-Yam, A; Leonard, D C
2009-04-16
Our understanding of the evolution of massive stars before their final explosions as supernovae is incomplete, from both an observational and a theoretical standpoint. A key missing piece in the supernova puzzle is the difficulty of identifying and studying progenitor stars. In only a single case-that of supernova SN 1987A in the Large Magellanic Cloud-has a star been detected at the supernova location before the explosion, and been subsequently shown to have vanished after the supernova event. The progenitor of SN 1987A was a blue supergiant, which required a rethink of stellar evolution models. The progenitor of supernova SN 2005gl was proposed to be an extremely luminous object, but the association was not robustly established (it was not even clear that the putative progenitor was a single luminous star). Here we report that the previously proposed object was indeed the progenitor star of SN 2005gl. This very massive star was likely a luminous blue variable that standard stellar evolution predicts should not have exploded in that state.
7. Massive stars dying alone: Extremely remote environments of SN2009ip and SN2010jp
NASA Astrophysics Data System (ADS)
Smith, Nathan
2014-10-01
We propose an imaging study of the astonishingly remote environments of two recent supernovae (SNe): SN2009ip and SN2010jp. Both were unusual Type IIn explosions that crashed into dense circumstellar material (CSM) ejected by the star shortly before explosion. The favored progenitors of these SNe are very massive luminous blue variable (LBV) stars. In fact, SN2009ip presents an extraordinay case where the LBV-like progenitor was actually detected directly in archival HST data, and where we obtained spectra and photometry for numerous pre-SN eruptions. No other SN has this treasure trove of detailed information about the progenitor (not even SN1987A). SN2010jp represents a possible collapsar-powered event, since it showed evidence of a fast bipolar jet in spectra and a low 56Ni mass; this would be an analog of the black-hole forming explosions that cause gamma ray bursts, but where the relativistic jet is damped by a residual H envelope on the star. In both cases, the only viable models for these SNe involve extremely massive (initial masses of 40-100 Msun) progenitor stars. This seems at odds with their extremely remote environments in the far outskirts of their host galaxies, with no detected evidence for an underlying massive star population in ground-based data (nor in the single shallow WFPC2/F606W image of SN2009ip). Here we propose deep UV HST images to search for any mid/late O-type stars nearby, deep red images to detect any red supergiants, and an H-alpha image to search for any evidence of ongoing star formation in the vicinity. These observations will place important and demanding constraints on the initial masses and ages of these progenitors.
8. A detailed study of feedback from a massive star
NASA Astrophysics Data System (ADS)
Geen, Sam; Rosdahl, Joakim; Blaizot, Jeremy; Devriendt, Julien; Slyz, Adrianne
2015-04-01
We present numerical simulations of a 15 M⊙ star in a suite of idealized environments in order to quantify the amount of energy transmitted to the interstellar medium (ISM). We include models of stellar winds, UV photoionization and the subsequent supernova based on theoretical models and observations of stellar evolution. The system is simulated in 3D using RAMSES-RT, an Adaptive Mesh Refinement Radiation Hydrodynamics code. We find that stellar winds have a negligible impact on the system owing to their relatively low luminosity compared to the other processes. The main impact of photoionization is to reduce the density of the medium into which the supernova explodes, reducing the rate of radiative cooling of the subsequent supernova. Finally, we present a grid of models quantifying the energy and momentum of the system that can be used to motivate simulations of feedback in the ISM unable to fully resolve the processes discussed in this work.
9. Main sequence models for massive zero-metal stars
NASA Technical Reports Server (NTRS)
Cary, N.
1974-01-01
Zero-age main-sequence models for stars of 20, 10, 5, and 2 solar masses with no heavy elements are constructed for three different possible primordial helium abundances: Y=0.00, Y=0.23, and Y=0.30. The latter two values of Y bracket the range of primordial helium abundances cited by Wagoner. With the exceptions of the two 20 solar mass models that contain helium, these models are found to be self-consistent in the sense that the formation of carbon through the triple-alpha process during premain sequence contraction is not sufficient to bring the CN cycle into competition with the proton-proton chain on the ZAMS. The zero-metal models of the present study have higher surface and central temperatures, higher central densities, smaller radii, and smaller convective cores than do the population I models with the same masses.
10. The metal and dust yields of the first massive stars
NASA Astrophysics Data System (ADS)
Marassi, Stefania; Schneider, Raffaella; Limongi, Marco; Chieffi, Alessandro; Bocchio, Marco; Bianchi, Simone
2015-12-01
We quantify the role of Population (Pop) III core-collapse supernovae (SNe) as the first cosmic dust polluters. Starting from a homogeneous set of stellar progenitors with masses in the range [13-80] M⊙, we find that the mass and composition of newly formed dust depend on the mixing efficiency of the ejecta and the degree of fallback experienced during the explosion. For standard Pop III SNe, whose explosions are calibrated to reproduce the average elemental abundances of Galactic halo stars with [Fe/H] < -2.5, between 0.18 and 3.1 M⊙ (0.39-1.76 M⊙) of dust can form in uniformly mixed (unmixed) ejecta, and the dominant grain species are silicates. We also investigate dust formation in the ejecta of faint Pop III SN, where the ejecta experience a strong fallback. By examining a set of models, tailored to minimize the scatter with the abundances of carbon-enhanced Galactic halo stars with [Fe/H] < -4, we find that amorphous carbon is the only grain species that forms, with masses in the range 2.7 × 10^{-3}-0.27 M_{⊙} (7.5 × 10^{-4} -0.11 M_{⊙}) for uniformly mixed (unmixed) ejecta models. Finally, for all the models we estimate the amount and composition of dust that survives the passage of the reverse shock, and find that, depending on circumstellar medium densities, between 3 and 50 per cent (10-80 per cent) of dust produced by standard (faint) Pop III SNe can contribute to early dust enrichment.
11. Probing dust-obscured star formation in the most massive gamma-ray burst host galaxies
NASA Astrophysics Data System (ADS)
Greiner, Jochen; Michałowski, Michał J.; Klose, Sylvio; Hunt, Leslie K.; Gentile, Gianfranco; Kamphuis, Peter; Herrero-Illana, Rubén; Wieringa, Mark; Krühler, Thomas; Schady, Patricia; Elliott, Jonathan; Graham, John F.; Ibar, Eduardo; Knust, Fabian; Nicuesa Guelbenzu, Ana; Palazzi, Eliana; Rossi, Andrea; Savaglio, Sandra
2016-08-01
Context. As a result of their relation to massive stars, long-duration gamma-ray bursts (GRBs) allow the pinpointing of star formation in galaxies independent of redshift, dust obscuration, or galaxy mass/size, thus providing a unique tool to investigate star formation history over cosmic time. Aims: About half of the optical afterglows of long-duration GRBs are missed owing to dust extinction and are primarily located in the most massive GRB hosts. It is important to investigate the amount of obscured star formation in these GRB host galaxies to understand this bias. Methods: Radio emission of galaxies correlates with star formation, but does not suffer extinction as do the optical star formation estimators. We selected 11 GRB host galaxies with either large stellar mass or large UV-based and optical-based star formation rates (SFRs) and obtained radio observations of these with the Australia Telescope Compact Array and the Karl Jansky Very Large Array. Results: Despite intentionally selecting GRB hosts with expected high SFRs, we do not find any radio emission related to star formation in any of our targets. Our upper limit for GRB 100621A implies that the earlier reported radio detection was due to afterglow emission. We detect radio emission from the position of GRB 020819B, but argue that it is in large part, if not completely, due to afterglow contamination. Conclusions: Half of our sample has radio-derived SFR limits, which are only a factor 2-3 above the optically measured SFRs. This supports other recent studies that the majority of star formation in GRB hosts is not obscured by dust. Based on observations collected with ATCA under ID C2718, and at VLA under ID 13B-017.
12. Models of the circumstellar medium of evolving, massive runaway stars moving through the Galactic plane
NASA Astrophysics Data System (ADS)
Meyer, D. M.-A.; Mackey, J.; Langer, N.; Gvaramadze, V. V.; Mignone, A.; Izzard, R. G.; Kaper, L.
2014-11-01
At least 5 per cent of the massive stars are moving supersonically through the interstellar medium (ISM) and are expected to produce a stellar wind bow shock. We explore how the mass-loss and space velocity of massive runaway stars affect the morphology of their bow shocks. We run two-dimensional axisymmetric hydrodynamical simulations following the evolution of the circumstellar medium of these stars in the Galactic plane from the main sequence to the red supergiant phase. We find that thermal conduction is an important process governing the shape, size and structure of the bow shocks around hot stars, and that they have an optical luminosity mainly produced by forbidden lines, e.g. [O III]. The Hα emission of the bow shocks around hot stars originates from near their contact discontinuity. The Hα emission of bow shocks around cool stars originates from their forward shock, and is too faint to be observed for the bow shocks that we simulate. The emission of optically thin radiation mainly comes from the shocked ISM material. All bow shock models are brighter in the infrared, i.e. the infrared is the most appropriate waveband to search for bow shocks. Our study suggests that the infrared emission comes from near the contact discontinuity for bow shocks of hot stars and from the inner region of shocked wind for bow shocks around cool stars. We predict that, in the Galactic plane, the brightest, i.e. the most easily detectable bow shocks are produced by high-mass stars moving with small space velocities.
13. The life of massive stars seen through optical/infrared interferometry
NASA Astrophysics Data System (ADS)
Sanchez-Bermudez, J.; Alberdi, A.; Schödel, R.
2015-05-01
During the last decade, optical/infrared interferometry has become an essential tool to contribute to the understanding of stellar astrophysics. We present our results in the study of different aspects in the life of massive stars using optical interferometry. Particularly, we focused the discussion in our findings about multiplicity, interactions of the massive stars with the interstellar medium, and the early stages of high-mass stars. Our near-infrared observations comprise both: (i) long-baseline interferometry making use of AMBER/VLTI, and (ii) sparse aperture masking with VLT/NACO/SAM. These data have been obtained by our research group in the previous years, and the results have been published in several peer-reviewed papers. The principles of the optical/near-infrared interferometry are briefly presented. Particularly, we describe how to get the calibrated Interferometric observables. Henceforth, we present our results of two massive systems (HD150136 and Herschel 36) for which we discovered their triple nature using AMBER/VLTI. Finally, we will present the recently found evidence of a disk and a binary system in a very massive young stellar object known as IRS 9A in the NGC 3603 region.
14. Massive stars at low metallicity. Evolution and surface abundances of O dwarfs in the SMC
NASA Astrophysics Data System (ADS)
Bouret, J.-C.; Lanz, T.; Martins, F.; Marcolino, W. L. F.; Hillier, D. J.; Depagne, E.; Hubeny, I.
2013-07-01
Aims: We aim to study the properties of massive stars at low metallicity, with an emphasis on their evolution, rotation, and surface abundances. We focus on O-type dwarfs in the Small Magellanic Cloud. These stars are expected to have weak winds that do not remove significant amounts of their initial angular momentum. Methods: We analyzed the UV and optical spectra of twenty-three objects using the NLTE stellar atmosphere code cmfgen and derived photospheric and wind properties. Results: The observed binary fraction of the sample is ≈26%, which is consistent with more systematic studies if one considers that the actual binary fraction is potentially larger owing to low-luminosity companions and that the sample was biased because it excluded obvious spectroscopic binaries. The location of the fastest rotators in the Hertzsprung-Russell (H-R) diagram built with fast-rotating evolutionary models and isochrones indicates that these could be several Myr old. The offset in the position of these fast rotators compared with the other stars confirms the predictions of evolutionary models that fast-rotating stars tend to evolve more vertically in the H-R diagram. Only one star of luminosity class Vz, expected to best characterize extreme youth, is located on the zero-age main sequence, the other two stars are more evolved. We found that the distribution of O and B stars in the ɛ(N) - vsin i diagram is the same, which suggests that the mechanisms responsible for the chemical enrichment of slowly rotating massive stars depend only weakly on the star's mass. We furthermore confirm that the group of slowly rotating N-rich stars is not reproduced by the evolutionary tracks. Even for more massive stars and faster rotators, our results call for stronger mixing in the models to explain the range of observed N abundances. All stars have an N/C ratio as a function of stellar luminosity that match the predictions of the stellar evolution models well. More massive stars have a higher
15. Stars and (furry) black holes in Lorentz breaking massive gravity
NASA Astrophysics Data System (ADS)
Comelli, D.; Nesti, F.; Pilo, L.
2011-04-01
We study the exact spherically symmetric solutions in a class of Lorentz-breaking massive gravity theories, using the effective-theory approach where the graviton mass is generated by the interaction with a suitable set of Stückelberg fields. We find explicitly the exact black-hole solutions which generalizes the familiar Schwarzschild one, which shows a nonanalytic hair in the form of a powerlike term rγ. For realistic self-gravitating bodies, we find interesting features, linked to the effective violation of the Gauss law: (i) the total gravitational mass appearing in the standard 1/r term gets a multiplicative renormalization proportional to the area of the body itself; (ii) the magnitude of the powerlike hairy correction is also linked to size of the body. The novel features can be ascribed to the presence of the Goldstones fluid turned on by matter inside the body; its equation of state approaching that of dark energy near the center. The Goldstones fluid also changes the matter equilibrium pressure, leading to an upper limit for the graviton mass, m≲10-28÷29eV, derived from the largest stable gravitational bound states in the Universe.
16. Super- and massive AGB stars - IV. Final fates - initial-to-final mass relation
NASA Astrophysics Data System (ADS)
Doherty, Carolyn L.; Gil-Pons, Pilar; Siess, Lionel; Lattanzio, John C.; Lau, Herbert H. B.
2015-01-01
We explore the final fates of massive intermediate-mass stars by computing detailed stellar models from the zero-age main sequence until near the end of the thermally pulsing phase. These super-asymptotic giant branch (super-AGB) and massive AGB star models are in the mass range between 5.0 and 10.0 M⊙ for metallicities spanning the range Z = 0.02-0.0001. We probe the mass limits Mup, Mn and Mmass, the minimum masses for the onset of carbon burning, the formation of a neutron star and the iron core-collapse supernovae, respectively, to constrain the white dwarf/electron-capture supernova (EC-SN) boundary. We provide a theoretical initial-to-final mass relation for the massive and ultra-massive white dwarfs and specify the mass range for the occurrence of hybrid CO(Ne) white dwarfs. We predict EC-SN rates for lower metallicities which are significantly lower than existing values from parametric studies in the literature. We conclude that the EC-SN channel (for single stars and with the critical assumption being the choice of mass-loss rate) is very narrow in initial mass, at most ≈0.2 M⊙. This implies that between ˜2 and 5 per cent of all gravitational collapse supernova are EC-SNe in the metallicity range Z = 0.02-0.0001. With our choice for mass-loss prescription and computed core growth rates, we find, within our metallicity range, that CO cores cannot grow sufficiently massive to undergo a Type 1.5 SN explosion.
17. The Inflow Signature toward Different Evolutionary Phases of Massive Star Formation
NASA Astrophysics Data System (ADS)
Jin, Mihwa; Lee, Jeong-Eun; Kim, Kee-Tae; Evans, Neal J., II
2016-08-01
We analyze both HCN J = 1-0 and HNC J = 1-0 line profiles to study the inflow motions in different evolutionary stages of massive star formation: 54 infrared dark clouds (IRDCs), 69 high-mass protostellar objects (HMPOs), and 54 ultra-compact H ii regions (UCHIIs). Inflow asymmetry in the HCN spectra seems to be prevalent throughout all the three evolutionary phases, with IRDCs showing the largest excess in the blue profile. In the case of the HNC spectra, the prevalence of blue sources does not appear, apart from for IRDCs. We suggest that this line is not appropriate to trace the inflow motion in the evolved stages of massive star formation, because the abundance of HNC decreases at high temperatures. This result highlights the importance of considering chemistry in dynamics studies of massive star-forming regions. The fact that the IRDCs show the highest blue excess in both transitions indicates that the most active inflow occurs in the early phase of star formation, i.e., in the IRDC phase rather than in the later phases. However, mass is still inflowing onto some UCHIIs. We also find that the absorption dips of the HNC spectra in six out of seven blue sources are redshifted relative to their systemic velocities. These redshifted absorption dips may indicate global collapse candidates, although mapping observations with better resolution are needed to examine this feature in more detail.
18. The Inflow Signature toward Different Evolutionary Phases of Massive Star Formation
NASA Astrophysics Data System (ADS)
Jin, Mihwa; Lee, Jeong-Eun; Kim, Kee-Tae; Evans, Neal J., II
2016-08-01
We analyze both HCN J = 1–0 and HNC J = 1–0 line profiles to study the inflow motions in different evolutionary stages of massive star formation: 54 infrared dark clouds (IRDCs), 69 high-mass protostellar objects (HMPOs), and 54 ultra-compact H ii regions (UCHIIs). Inflow asymmetry in the HCN spectra seems to be prevalent throughout all the three evolutionary phases, with IRDCs showing the largest excess in the blue profile. In the case of the HNC spectra, the prevalence of blue sources does not appear, apart from for IRDCs. We suggest that this line is not appropriate to trace the inflow motion in the evolved stages of massive star formation, because the abundance of HNC decreases at high temperatures. This result highlights the importance of considering chemistry in dynamics studies of massive star-forming regions. The fact that the IRDCs show the highest blue excess in both transitions indicates that the most active inflow occurs in the early phase of star formation, i.e., in the IRDC phase rather than in the later phases. However, mass is still inflowing onto some UCHIIs. We also find that the absorption dips of the HNC spectra in six out of seven blue sources are redshifted relative to their systemic velocities. These redshifted absorption dips may indicate global collapse candidates, although mapping observations with better resolution are needed to examine this feature in more detail.
19. Narrow-band Imaging of Massive Star-Forming Regions: Tracing Outflows and the Rate of Star-Formation
NASA Astrophysics Data System (ADS)
Hall, Kendall; Willis, Sarah; Hora, Joseph L.
2016-01-01
Narrowband images targeting ionized hydrogen (Brackett gamma, 2.17 microns) and molecular hydrogen (2.12 microns) were obtained for six massive star-forming regions within the Milky Way, NGC 6334, G305, G3333, G3264, G3266, and G351. These regions are within 1-4 kpc from our solar system. The narrowband flux in Brackett gamma was used as a star-formation tracer to calculate a star-formation rate for each region. This is compared with other star-formation rates found using other methods such as the count of young stars and YSOs, and rates calculated from using other tracers (e.g. 70 micron monochromatic luminosity). The molecular hydrogen narrowband images were manually searched to locate outflows from young stars. Once these outflows are identified, it may help to get a better survey of the young stellar population. A better understanding of the stellar population distribution can lead to more accurate star-formation rates to compare to those calculated from star-formation tracers. We found the regions NGC 6334 and G3266 to have the highest levels of ongoing star formation activity as indicated by the number of molecular hydrogen objects (MHOs) detected. There are a total of 279 cataloged MHOs in 181 categorized systems for the six regions. There are a total of 150 identified potential driving sources.This work was supported in part by the NSF REU and DoD ASSURE programs under NSF grant no. 1262851 and by the Smithsonian Institution.
20. A Link Between Massive Binary Stars and Non-thermal Radio Emission
NASA Astrophysics Data System (ADS)
Wallace, Debra
1999-07-01
Non-thermal radio emission in Wolf-Rayet {WR} stars is explained in terms of synchrotron emission from shocks in the wind. For single star models, the shocks arise from instabilities in the wind itself, whereas in binary models, the shocks form at the wind-wind interaction zone. In Niemela et al. 1998 {from WFPC2 data}, we support the binary theory, for two WR stars, linking the non-thermal emission with the colliding wind region. Before we can conclusively link non- thermal emission to binarity, we must demonstrate that all non-thermal emitters are binary, and that all thermal emitters are either single stars or binary systems with separations that are either too wide or too close to result in a wind-wind interaction that produces shocks. We cannot yet conclusively state this because WFPC2 does not resolve binaries with separations less than about 0.100''. We propose to use the FGS to observe 9 non-thermal and 9 thermal WR stars to search for binary companions. The FGS ca n resolve angular separations as s mall as .007''. If the non-thermal stars are resolved as binaries and the thermal emitters are determined to be single, the single star theory of non-thermal emission can be disavowed. Co-latterally, we will have demonstrated a new method of detecting massive binaries, and, for all WR stars, we will establish a more accurate binary incidence rate.
1. Massive Stars and the Energy Balance of the Interstellar Medium. 1; The Impact of an Isolated 60 M. Star
NASA Technical Reports Server (NTRS)
Freyer, Tim; Hensler, Gerhard; Yorke, Harold W.
2003-01-01
We present results of numerical simulations carried out with a two-dimensional radiation hydrodynamics code in order to study the impact of massive stars on their surrounding interstellar medium. This first paper deals with the evolution of the circumstellar gas around an isolated 60 M. star. The interaction of the photo- ionized H II region with the stellar wind bubble forms a variety of interesting structures like shells, clouds, fingers, and spokes. These results demonstrate that complex structures found in H II regions are not necessarily relics from the time before the gas became ionized but may result from dynamical processes during the course of the H II region evolution. We have also analyzed the transfer and deposit of the stellar wind and radiation energy into the circumstellar medium until the star explodes as a supernova. Although the total mechanical wind energy supplied by the star is negligible compared to the accumulated energy of the Lyman continuum photons, the kinetic energy imparted to the circumstellar gas over the star s lifetime is 4 times higher than for a comparable windless simulation. Furthermore, the thermal energy of warm photoionized gas is lower by some 55%). Our results document the necessity to consider both ionizing radiation and stellar winds for an appropriate description of the interaction of OB stars with their circumstellar environment.
2. Small-scale hero: Massive-star enrichment in the Hercules dwarf spheroidal
NASA Astrophysics Data System (ADS)
Koch, Andreas; Matteucci, Francesca; Feltzing, Sofia
2012-09-01
Dwarf spheroidal galaxies are often conjectured to be the sites of the first stars. The best current contenders for finding the chemical imprints from the enrichment by those massive objects are the ultrafaint dwarfs'' (UFDs). Here we present evidence for remarkably low heavy element abundances in the metal poor Hercules UFD. Combined with other peculiar abundance patterns this indicates that Hercules was likely only influenced by very few, massive explosive events - thus bearing the traces of an early, localized chemical enrichment with only very little other contributions from other sources at later times.
3. Properties of massive star-forming clumps with infall motions
NASA Astrophysics Data System (ADS)
He, Yu-Xin; Zhou, Jian-Jun; Esimbek, Jarken; Ji, Wei-Guang; Wu, Gang; Tang, Xin-Di; Komesh, Toktarkhan; Yuan, Ye; Li, Da-Lei; Baan, W. A.
2016-09-01
In this work, we aim to characterize high-mass clumps with infall motions. We selected 327 clumps from the Millimetre Astronomy Legacy Team 90-GHz survey, and identified 100 infall candidates. Combined with the results of He et al., we obtained a sample of 732 high-mass clumps, including 231 massive infall candidates and 501 clumps where infall is not detected. Objects in our sample were classified as pre-stellar, proto-stellar, H II or photodissociation region (PDR). The detection rates of the infall candidates in the pre-stellar, proto-stellar, H II and PDR stages are 41.2 per cent, 36.6 per cent, 30.6 per cent and 12.7 per cent, respectively. The infall candidates have a higher H2 column density and volume density compared with the clumps where infall is not detected at every stage. For the infall candidates, the median values of the infall rates at the pre-stellar, proto-stellar, H II and PDR stages are 2.6 × 10-3, 7.0 × 10-3, 6.5 × 10-3 and 5.5 × 10-3 M⊙ yr-1, respectively. These values indicate that infall candidates at later evolutionary stages are still accumulating material efficiently. It is interesting to find that both infall candidates and clumps where infall is not detected show a clear trend of increasing mass from the pre-stellar to proto-stellar, and to the H II stages. The power indices of the clump mass function are 2.04 ± 0.16 and 2.17 ± 0.31 for the infall candidates and clumps where infall is not detected, respectively, which agree well with the power index of the stellar initial mass function (2.35) and the cold Planck cores (2.0).
4. Chemical differentiation in regions of massive star formation
NASA Technical Reports Server (NTRS)
Rodgers, S. D.; Charnley, S. B.
2001-01-01
We have reexamined the origin of the apparent differentiation between nitrogen-bearing molecules and complex oxygen-bearing molecules that is observed in hot molecular cores associated with massive protostars. Observations show that methanol is an ubiquitous and abundant component of protostellar ices. Recent observations suggest that ammonia may constitute an appreciable fraction of the ices toward some sources. In contrast to previous theories that suggested that N/O differentiation was caused by an anticorrelation between methanol and ammonia in the precursor grain mantles, we show that the presence of ammonia in mantles and the core temperature are key quantities in determining N/O differentiation. Calculations are presented which show that when large amounts of ammonia are evaporated alkyl cation transfer reactions are suppressed and the abundances of complex O-bearing organic molecules greatly reduced. Cooler cores (100 K) eventually evolve to an oxygen-rich chemical state similar to that attained when no ammonia was injected, but on a timescale that is an order of magnitude longer (10(5) yr). Hotter cores (300 K) never evolve an O-rich chemistry unless ammonia is almost absent from the mantles. In this latter case, a complex O-rich chemistry develops on a timescale of 10(4) yr, as in previous models, but disappears in about 2 x 10(5) yr, after which time the core is rich in NH3, HCN, and other N-bearing molecules. There are thus two ways in which N-rich cores can occur. We briefly discuss the implications for the determination of hot-core ages and for explaining N/O differentiation in several well-studied sources.
5. Massive open star clusters using the VVV survey. II. Discovery of six clusters with Wolf-Rayet stars
NASA Astrophysics Data System (ADS)
Chené, A.-N.; Borissova, J.; Bonatto, C.; Majaess, D. J.; Baume, G.; Clarke, J. R. A.; Kurtev, R.; Schnurr, O.; Bouret, J.-C.; Catelan, M.; Emerson, J. P.; Feinstein, C.; Geisler, D.; de Grijs, R.; Hervé, A.; Ivanov, V. D.; Kumar, M. S. N.; Lucas, P.; Mahy, L.; Martins, F.; Mauro, F.; Minniti, D.; Moni Bidin, C.
2013-01-01
Context. The ESO Public Survey "VISTA Variables in the Vía Láctea" (VVV) provides deep multi-epoch infrared observations for an unprecedented 562 sq. degrees of the Galactic bulge, and adjacent regions of the disk. Nearly 150 new open clusters and cluster candidates have been discovered in this survey. Aims: This is the second in a series of papers about young, massive open clusters observed using the VVV survey. We present the first study of six recently discovered clusters. These clusters contain at least one newly discovered Wolf-Rayet (WR) star. Methods: Following the methodology presented in the first paper of the series, wide-field, deep JHKs VVV observations, combined with new infrared spectroscopy, are employed to constrain fundamental parameters for a subset of clusters. Results: We find that the six studied stellar groups are real young (2-7 Myr) and massive (between 0.8 and 2.2 × 103 M⊙) clusters. They are highly obscured (AV ~ 5-24 mag) and compact (1-2 pc). In addition to WR stars, two of the six clusters also contain at least one red supergiant star, and one of these two clusters also contains a blue supergiant. We claim the discovery of 8 new WR stars, and 3 stars showing WR-like emission lines which could be classified WR or OIf. Preliminary analysis provides initial masses of ~30-50 M⊙ for the WR stars. Finally, we discuss the spiral structure of the Galaxy using the six new clusters as tracers, together with the previously studied VVV clusters. Based on observations with ISAAC, VLT, ESO (programme 087.D-0341A), New Technology Telescope at ESO's La Silla Observatory (programme 087.D-0490A) and with the Clay telescope at the Las Campanas Observatory (programme CN2011A-086). Also based on data from the VVV survey (programme 172.B-2002).
6. Low Mach Number Modeling of Core Convection in Massive Stars
NASA Astrophysics Data System (ADS)
Gilet, C.; Almgren, A. S.; Bell, J. B.; Nonaka, A.; Woosley, S. E.; Zingale, M.
2013-08-01
This work presents three-dimensional simulations of core convection in a 15 M ⊙ star halfway through its main sequence lifetime. To perform the necessary long-time calculations, we use the low Mach number code MAESTRO, with initial conditions taken from a one-dimensional stellar model. We first identify several key factors that the one-dimensional initial model must satisfy to ensure efficient simulation of the convection process. We then use the three-dimensional simulations to examine the effects of two common modeling choices on the resulting convective flow: using a fixed composition approximation and using a reduced domain size. We find that using a fixed composition model actually increases the computational cost relative to using the full multi-species model because the fixed composition system takes longer to reach convection that is in a quasi-static state. Using a reduced (octant rather than full sphere) simulation domain yields flow with statistical properties that are within a factor of two of the full sphere simulation values. Both the octant and full sphere simulations show similar mixing across the convection zone boundary that is consistent with the turbulent entrainment model. However, the global character of the flow is distinctly different in the octant simulation, showing more rapid changes in the large-scale structure of the flow and thus a more isotropic flow on average.
7. HADRON-QUARK CROSSOVER AND MASSIVE HYBRID STARS WITH STRANGENESS
SciTech Connect
Masuda, Kota; Hatsuda, Tetsuo; Takatsuka, Tatsuyuki
2013-02-10
Using the idea of smooth crossover from hadronic matter with hyperons to quark matter with strangeness, we show that the maximum mass (M {sub max}) of neutron stars with quark matter cores can be larger than those without quark matter cores. This is in contrast to the conventional softening of the equation of state due to exotic components at high density. The essential conditions for reaching our conclusion are that (1) the crossover takes place at relatively low densities, around three times the normal nuclear density and (2) the quark matter is strongly interacting in the crossover region. From these, the pressure of the system can be greater than that of purely hadronic matter at a given baryon density in the crossover density region and leads to M {sub max} greater than 2 solar mass. This conclusion is insensitive to the different choice of the hadronic equation of state with hyperons. We remark upon several implications of this result to the nuclear incompressibility, the hyperon mixing, and the neutrino cooling.
8. LOW MACH NUMBER MODELING OF CORE CONVECTION IN MASSIVE STARS
SciTech Connect
Gilet, C.; Almgren, A. S.; Bell, J. B.; Nonaka, A.; Woosley, S. E.; Zingale, M.
2013-08-20
This work presents three-dimensional simulations of core convection in a 15 M{sub Sun} star halfway through its main sequence lifetime. To perform the necessary long-time calculations, we use the low Mach number code MAESTRO, with initial conditions taken from a one-dimensional stellar model. We first identify several key factors that the one-dimensional initial model must satisfy to ensure efficient simulation of the convection process. We then use the three-dimensional simulations to examine the effects of two common modeling choices on the resulting convective flow: using a fixed composition approximation and using a reduced domain size. We find that using a fixed composition model actually increases the computational cost relative to using the full multi-species model because the fixed composition system takes longer to reach convection that is in a quasi-static state. Using a reduced (octant rather than full sphere) simulation domain yields flow with statistical properties that are within a factor of two of the full sphere simulation values. Both the octant and full sphere simulations show similar mixing across the convection zone boundary that is consistent with the turbulent entrainment model. However, the global character of the flow is distinctly different in the octant simulation, showing more rapid changes in the large-scale structure of the flow and thus a more isotropic flow on average.
9. Formation of Massive Primordial Stars: Intermittent UV Feedback with Episodic Mass Accretion
NASA Astrophysics Data System (ADS)
Hosokawa, Takashi; Hirano, Shingo; Kuiper, Rolf; Yorke, Harold W.; Omukai, Kazuyuki; Yoshida, Naoki
2016-06-01
We present coupled stellar evolution (SE) and 3D radiation-hydrodynamic (RHD) simulations of the evolution of primordial protostars, their immediate environment, and the dynamic accretion history under the influence of stellar ionizing and dissociating UV feedback. Our coupled SE RHD calculations result in a wide diversity of final stellar masses covering 10 {M}ȯ ≲ M * ≲ 103 {M}ȯ . The formation of very massive (≳250 {M}ȯ ) stars is possible under weak UV feedback, whereas ordinary massive (a few ×10 {M}ȯ ) stars form when UV feedback can efficiently halt the accretion. This may explain the peculiar abundance pattern of a Galactic metal-poor star recently reported by Aoki et al., possibly the observational signature of very massive precursor primordial stars. Weak UV feedback occurs in cases of variable accretion, in particular when repeated short accretion bursts temporarily exceed 0.01 {M}ȯ {{{yr}}}-1, causing the protostar to inflate. In the bloated state, the protostar has low surface temperature and UV feedback is suppressed until the star eventually contracts, on a thermal adjustment timescale, to create an H ii region. If the delay time between successive accretion bursts is sufficiently short, the protostar remains bloated for extended periods, initiating at most only short periods of UV feedback. Disk fragmentation does not necessarily reduce the final stellar mass. Quite the contrary, we find that disk fragmentation enhances episodic accretion as many fragments migrate inward and are accreted onto the star, thus allowing continued stellar mass growth under conditions of intermittent UV feedback. This trend becomes more prominent as we improve the resolution of our simulations. We argue that simulations with significantly higher resolution than reported previously are needed to derive accurate gas mass accretion rates onto primordial protostars.
10. Formation of Massive Primordial Stars: Intermittent UV Feedback with Episodic Mass Accretion
NASA Astrophysics Data System (ADS)
Hosokawa, Takashi; Hirano, Shingo; Kuiper, Rolf; Yorke, Harold W.; Omukai, Kazuyuki; Yoshida, Naoki
2016-06-01
We present coupled stellar evolution (SE) and 3D radiation-hydrodynamic (RHD) simulations of the evolution of primordial protostars, their immediate environment, and the dynamic accretion history under the influence of stellar ionizing and dissociating UV feedback. Our coupled SE RHD calculations result in a wide diversity of final stellar masses covering 10 {M}⊙ ≲ M * ≲ 103 {M}⊙ . The formation of very massive (≳250 {M}⊙ ) stars is possible under weak UV feedback, whereas ordinary massive (a few ×10 {M}⊙ ) stars form when UV feedback can efficiently halt the accretion. This may explain the peculiar abundance pattern of a Galactic metal-poor star recently reported by Aoki et al., possibly the observational signature of very massive precursor primordial stars. Weak UV feedback occurs in cases of variable accretion, in particular when repeated short accretion bursts temporarily exceed 0.01 {M}⊙ {{{yr}}}-1, causing the protostar to inflate. In the bloated state, the protostar has low surface temperature and UV feedback is suppressed until the star eventually contracts, on a thermal adjustment timescale, to create an H ii region. If the delay time between successive accretion bursts is sufficiently short, the protostar remains bloated for extended periods, initiating at most only short periods of UV feedback. Disk fragmentation does not necessarily reduce the final stellar mass. Quite the contrary, we find that disk fragmentation enhances episodic accretion as many fragments migrate inward and are accreted onto the star, thus allowing continued stellar mass growth under conditions of intermittent UV feedback. This trend becomes more prominent as we improve the resolution of our simulations. We argue that simulations with significantly higher resolution than reported previously are needed to derive accurate gas mass accretion rates onto primordial protostars.
11. Hyperon puzzle, hadron-quark crossover and massive neutron stars
NASA Astrophysics Data System (ADS)
Masuda, Kota; Hatsuda, Tetsuo; Takatsuka, Tatsuyuki
2016-03-01
Bulk properties of cold and hot neutron stars are studied on the basis of the hadron-quark crossover picture where a smooth transition from the hadronic phase to the quark phase takes place at finite baryon density. By using a phenomenological equation of state (EOS) "CRover", which interpolates the two phases at around 3 times the nuclear matter density (ρ0, it is found that the cold NSs with the gravitational mass larger than 2M_{odot} can be sustained. This is in sharp contrast to the case of the first-order hadron-quark transition. The radii of the cold NSs with the CRover EOS are in the narrow range (12.5 ± 0.5) km which is insensitive to the NS masses. Due to the stiffening of the EOS induced by the hadron-quark crossover, the central density of the NSs is at most 4 ρ0 and the hyperon-mixing barely occurs inside the NS core. This constitutes a solution of the long-standing hyperon puzzle. The effect of color superconductivity (CSC) on the NS structures is also examined with the hadron-quark crossover. For the typical strength of the diquark attraction, a slight softening of the EOS due to two-flavor CSC (2SC) takes place and the maximum mass is reduced by about 0.2M_{odot}. The CRover EOS is generalized to the supernova matter at finite temperature to describe the hot NSs at birth. The hadron-quark crossover is found to decrease the central temperature of the hot NSs under isentropic condition. The gravitational energy release and the spin-up rate during the contraction from the hot NS to the cold NS are also estimated.
12. Formation of massive black holes through runaway collisions in dense young star clusters.
PubMed
Zwart, Simon F Portegies; Baumgardt, Holger; Hut, Piet; Makino, Junichiro; McMillan, Stephen L W
2004-04-15
A luminous X-ray source is associated with MGG 11--a cluster of young stars approximately 200 pc from the centre of the starburst galaxy M 82 (refs 1, 2). The properties of this source are best explained by invoking a black hole with a mass of at least 350 solar masses (350 M(o)), which is intermediate between stellar-mass and supermassive black holes. A nearby but somewhat more massive cluster (MGG 9) shows no evidence of such an intermediate-mass black hole, raising the issue of just what physical characteristics of the clusters can account for this difference. Here we report numerical simulations of the evolution and motion of stars within the clusters, where stars are allowed to merge with each other. We find that for MGG 11 dynamical friction leads to the massive stars sinking rapidly to the centre of the cluster, where they participate in a runaway collision. This produces a star of 800-3,000 M(o) which ultimately collapses to a black hole of intermediate mass. No such runaway occurs in the cluster MGG 9, because the larger cluster radius leads to a mass segregation timescale a factor of five longer than for MGG 11.
13. The MiMeS survey of magnetism in massive stars: introduction and overview
NASA Astrophysics Data System (ADS)
Wade, G. A.; Neiner, C.; Alecian, E.; Grunhut, J. H.; Petit, V.; Batz, B. de; Bohlender, D. A.; Cohen, D. H.; Henrichs, H. F.; Kochukhov, O.; Landstreet, J. D.; Manset, N.; Martins, F.; Mathis, S.; Oksala, M. E.; Owocki, S. P.; Rivinius, Th.; Shultz, M. E.; Sundqvist, J. O.; Townsend, R. H. D.; ud-Doula, A.; Bouret, J.-C.; Braithwaite, J.; Briquet, M.; Carciofi, A. C.; David-Uraz, A.; Folsom, C. P.; Fullerton, A. W.; Leroy, B.; Marcolino, W. L. F.; Moffat, A. F. J.; Nazé, Y.; Louis, N. St; Aurière, M.; Bagnulo, S.; Bailey, J. D.; Barbá, R. H.; Blazère, A.; Böhm, T.; Catala, C.; Donati, J.-F.; Ferrario, L.; Harrington, D.; Howarth, I. D.; Ignace, R.; Kaper, L.; Lüftinger, T.; Prinja, R.; Vink, J. S.; Weiss, W. W.; Yakunin, I.
2016-02-01
The MiMeS (Magnetism in Massive Stars) project is a large-scale, high-resolution, sensitive spectropolarimetric investigation of the magnetic properties of O- and early B-type stars. Initiated in 2008 and completed in 2013, the project was supported by three Large Program allocations, as well as various programmes initiated by independent principal investigators, and archival resources. Ultimately, over 4800 circularly polarized spectra of 560 O and B stars were collected with the instruments ESPaDOnS (Echelle SpectroPolarimetric Device for the Observation of Stars) at the Canada-France-Hawaii Telescope, Narval at the Télescope Bernard Lyot and HARPSpol at the European Southern Observatory La Silla 3.6 m telescope, making MiMeS by far the largest systematic investigation of massive star magnetism ever undertaken. In this paper, the first in a series reporting the general results of the survey, we introduce the scientific motivation and goals, describe the sample of targets, review the instrumentation and observational techniques used, explain the exposure time calculation designed to provide sensitivity to surface dipole fields above approximately 100 G, discuss the polarimetric performance, stability and uncertainty of the instrumentation, and summarize the previous and forthcoming publications.
14. Hot, Massive Stars in the Extremely Metal-Poor Galaxy, I Zw 18
NASA Technical Reports Server (NTRS)
Heap, Sara R.; Malumuth, Eliot M.
2010-01-01
The carbon-enhanced metal-poor galaxy, I Zw 18, is the Rosetta Stone for understanding galaxies in the early universe by providing constraints on the IMF of massive stars, the role of galaxies in reionization of the universe, mixing of newly synthesized material in the ISM, and gamma-ray bursts at low metallicity, and on the earliest generations of stars producing the observed abundance pattern. We describe these constraints as derived from analyses of HST/COS spectra of I Zw 18 including stellar atmosphere analysis and photo-ionization modeling of both the emission and absorption spectra of the nebular material and interstellar medium.
15. Modeling X-ray emission line profiles from massive star winds - A review
NASA Astrophysics Data System (ADS)
Ignace, Richard
2016-09-01
The Chandra and XMM-Newton X-ray telescopes have led to numerous advances in the study and understanding of astrophysical X-ray sources. Particularly important has been the much increased spectral resolution of modern X-ray instrumentation. Wind-broadened emission lines have been spectroscopically resolved for many massive stars. This contribution reviews approaches to the modeling of X-ray emission line profile shapes from single stars, including smooth winds, winds with clumping, optically thin versus thick lines, and the effect of a radius-dependent photoabsorption coefficient.
16. Massive open star clusters using the VVV survey. IV. WR 62-2, a new very massive star in the core of the VVV CL041 cluster
NASA Astrophysics Data System (ADS)
Chené, A.-N.; Ramírez Alegría, S.; Borissova, J.; O'Leary, E.; Martins, F.; Hervé, A.; Kuhn, M.; Kurtev, R.; Consuelo Amigo Fuentes, P.; Bonatto, C.; Minniti, D.
2015-12-01
Context. The ESO Public Survey VISTA Variables in the Vía Láctea (VVV) provides deep multi-epoch infrared observations for an unprecedented 562 sq. deg of the Galactic bulge and adjacent regions of the disk. Nearly 150 new open clusters and cluster candidates have been discovered in this survey. Aims: We present the fourth article in a series of papers focussed on young and massive clusters discovered in the VVV survey. This article is dedicated to the cluster VVV CL041, which contains a new very massive star candidate, WR 62-2. Methods: Following the methodology presented in the first paper of the series, wide-field, deep JHKs VVV observations, combined with new infrared spectroscopy, are employed to constrain fundamental parameters (distance, reddening, mass, age) of VVV CL041. Results: We confirm that the cluster VVV CL041 is a young (less than 4 Myr) and massive (3 ± 2 × 103 M⊙) cluster, and not a simple asterism. It is located at a distance of 4.2 ± 0.9 kpc, and its reddening is AV = 8.0 ± 0.2 mag, which is slightly lower than the average for the young clusters towards the centre of the Galaxy. Spectral analysis shows that the most luminous star of the cluster, of the WN8h spectral type, is a candidate to have an initial mass larger than 100 M⊙. Based on observations taken within the ESO VISTA Public Survey VVV, Programme ID 179.B-2002, and on observations with VLT/ISAAC at ESO (programme 087.D.0341A) and Flamingos-2 at Gemini (programme GS-2014A-Q-72).The photometric catalogue is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/584/A31
17. NGC346: Looking in the Cradle of a Massive Star Cluster
NASA Astrophysics Data System (ADS)
Gouliermis, Dimitrios; Hony, Sacha; Dib, Sami; Galliano, Frederic; Cormier, Diane; Ralf, Klessen
2015-08-01
How a star cluster of more than few 10,000 solar masses forms? We present the case of the cluster NGC 346 in the Small Magellanic Cloud, still embedded in its natal star-forming region N66, and we propose a scenario for its formation, based on the rich resolved stellar populations found in the region. Young massive clusters (YMCs) host a significant amount of early-type stars, indicating an extremely high star formation efficiency. The Milky Way galaxy hosts several YMCs that fill the gap between young low-mass open clusters and old massive globular clusters. Only a handful, though, is relatively close to their formation, and the investigation of their gaseous natal environments suffers from contamination by the Galactic disk. YMCs are very abundant in distant starburst and interacting galaxies, but the distance of their hosting galaxies do not also allow a detailed analysis of their formation. The Magellanic Clouds, on the other hand, host YMCs in a wide range of ages with the youngest being still embedded in their giant HII regions. Hubble Space Telescope (HST) imaging of such star-forming complexes provide a complete stellar sampling with a high dynamic range in stellar masses, allowing the detailed study of star formation at scales typical for molecular clouds. The distribution of newly-born stars in N66 shows that star formation in the region proceeds in a clumpy hierarchical fashion, leading to the formation of both a dominant YMC, hosting about half of the observed pre--main-sequence (PMS) stars, and a self-similar dispersed distribution of the remaining PMS population. We investigate the correlation between stellar surface density (and star formation rate derived from star-counts) and molecular gas surface density (derived from dust column density) in an attempt to disentangle the physical conditions that gave birth to NGC 346. We discuss our findings in terms of stellar clustering, its relation to the turbulent interstellar medium, and the observed
18. Explosive nucleosynthesis in massive stars - Comparison with the Cassiopeia A fast-moving knots
NASA Technical Reports Server (NTRS)
Johnston, M. D.; Yahil, A.
1984-01-01
If the ejecta of a Type II supernova do not undergo extensive mixing, then, based on the explosion of current presupernova models, only a small fraction approximately equal to or less than 0.1 solar mass of the mantle of a massive star can yield abundances similar to those observed in the fast-moving knots of Cas A. This is shown to be independent of the detailed structure of the mantle and the supernova energy. Lack of mixing in Cas A is indicated by strong upper limits on the abundance ratios Ne/O, and Fe/O. If this is confirmed by further observations, then either Cas A is not the result of a standard progenitor of approximately equal to or less than 25 solar masses disrupted by a Type II supernova, or the picture of the last stages of stellar evolution in massive stars needs to be modified substantially.
19. VizieR Online Data Catalog: Spectra of W49 massives young stars (Wu+, 2016)
NASA Astrophysics Data System (ADS)
Wu, S.-W.; Bik, A.; Bestenlehner, J. M.; Henning, T.; Pasquali, A.; Brandner, W.; Stolte, A.
2016-02-01
Near-infrared observations have been carried out with LUCI mounted on the Large Binocular Telescope (LBT), Mount Graham, Arizona. LUCI is a near-infrared multi-mode instrument capable of Multi-Object Spectroscopy (MOS), long-slit spectroscopy and imaging. The spectra of the massive stars in W49 have been taken in MOS mode based on K-band pre-image also obtained with LUCI. Additional archival data were used to complement the LUCI data. Medium-resolution (R=10,000) K-band spectra of five massive stars in W49 obtained with ISAAC mounted on Antu (UT1) of ESO's Very Large Telescope (VLT), Paranal, Chile, and J- and H -band images obtained with SOFI at the New Technology Telescope (NTT), La Silla, Chile, were downloaded from the ESO archive. (2 data files).
20. HST/STIS ULTRAVIOLET SPECTROSCOPY OF THE COMPONENTS OF THE MASSIVE TRIPLE STAR δ ORI A
SciTech Connect
Richardson, Noel D.; Moffat, Anthony F. J.; Gull, Theodore R.; Lindler, Don J.; Gies, Douglas R.; Corcoran, Michael F.
2015-07-20
The multiple star system of δ Orionis is one of the closest examples of a system containing a luminous O-type, bright giant star (component Aa1). It is often used as a spectral-type standard and has the highest observed X-ray flux of any hot-star binary. The main component Aa1 is orbited by two lower mass stars, faint Aa2 in a 5.7 day eclipsing binary, and Ab, an astrometric companion with an estimated period of 346 years. Generally the flux from all three stars is recorded in ground-based spectroscopy, and the spectral decomposition of the components has proved difficult. Here we present Hubble Space Telescope/Space Telescope Imaging Spectrograph ultraviolet spectroscopy of δ Ori A that provides us with spatially separated spectra of Aa and Ab for the first time. We measured radial velocities for Aa1 and Ab in two observations made near the velocity extrema of Aa1. We show tentative evidence for the detection of the Aa2 component in cross-correlation functions of the observed and model spectra. We discuss the appearance of the UV spectra of Aa1 and Ab with reference to model spectra. Both stars have similar effective temperatures, but Ab is fainter and is a rapid rotator. The results will help in the interpretation of ground-based spectroscopy and in understanding the physical and evolutionary parameters of these massive stars.
1. Massive Stars: Some Open Questions and the Role of Multi-Object Spectroscopy
NASA Astrophysics Data System (ADS)
Herrero, A.
2016-10-01
Massive stars are key ingredients in the evolution of the Universe. Yet, important uncertainties and limits persist in our understanding of these objects, even in their early phases, limiting their application as tools to interpret the Universe. Here we review some of these open questions and argue that large samples are needed to answer them, both in the Milky Way and nearby galaxies. Multi-object spectroscopy plays a crucial role in this process.
2. SPITZER SAGE INFRARED PHOTOMETRY OF MASSIVE STARS IN THE LARGE MAGELLANIC CLOUD
SciTech Connect
Bonanos, A. Z.; Massa, D. L.; Sewilo, M. E-mail: [email protected]
2009-10-15
We present a catalog of 1750 massive stars in the Large Magellanic Cloud (LMC), with accurate spectral types compiled from the literature, and a photometric catalog for a subset of 1268 of these stars, with the goal of exploring their infrared properties. The photometric catalog consists of stars with infrared counterparts in the Spitzer SAGE survey database, for which we present uniform photometry from 0.3 to 24 {mu}m in the UBVIJHK{sub s} +IRAC+MIPS24 bands. The resulting infrared color-magnitude diagrams illustrate that the supergiant B[e], red supergiant, and luminous blue variable (LBV) stars are among the brightest infrared point sources in the LMC, due to their intrinsic brightness, and at longer wavelengths, due to dust. We detect infrared excesses due to free-free emission among {approx}900 OB stars, which correlate with luminosity class. We confirm the presence of dust around 10 supergiant B[e] stars, finding the shape of their spectral energy distributions (SEDs) to be very similar, in contrast to the variety of SED shapes among the spectrally variable LBVs. The similar luminosities of B[e] supergiants (log L/L {sub sun} {>=} 4) and the rare, dusty progenitors of the new class of optical transients (e.g., SN 2008S and NGC 300 OT), plus the fact that dust is present in both types of objects, suggests a common origin for them. We find the infrared colors for Wolf-Rayet stars to be independent of spectral type and their SEDs to be flatter than what models predict. The results of this study provide the first comprehensive roadmap for interpreting luminous, massive, resolved stellar populations in nearby galaxies at infrared wavelengths.
3. Radio Emission Toward Regions of Massive Star Formation in the Large Magellanic Cloud
NASA Astrophysics Data System (ADS)
2015-01-01
Four regions of massive star formation in the Large Magellanic Cloud (LMC) were observed for water and methanol maser emission and radio continuum emission. A total of 42 radio detections were made including 27 new radio sources, four water masers, and eight compact H II regions. The lobes of a radio galaxy were resolved for the first time, and the host galaxy identified. Seven sources were associated with known massive young stellar objects (YSOs). A multi-wavelength analysis using both the infrared and radio spectrum was used to characterize the sources. Mid-infrared color-magnitude selection criteria for ultracompact H II (UCHII) regions in the LMC are presented, yielding 136 UCHII region candidates throughout that galaxy. New maser detections identified two previously unknown massive YSOs. No methanol masers were detected, consistent with previous studies and supporting the hypothesis that the LMC may be deficient in these molecules. These discoveries contribute to the history of star formation in the LMC, which will lead to a better understanding of star formation in the Milky Way and throughout the universe.
4. THE MASSIVE STAR-FORMING REGIONS OMNIBUS X-RAY CATALOG
SciTech Connect
Townsley, Leisa K.; Broos, Patrick S.; Feigelson, Eric D.; Getman, Konstantin V.; Kuhn, Michael A.; Garmire, Gordon P.; Bouwman, Jeroen; Povich, Matthew S.
2014-07-01
We present the Massive Star-forming Regions (MSFRs) Omnibus X-ray Catalog (MOXC), a compendium of X-ray point sources from Chandra/ACIS observations of a selection of MSFRs across the Galaxy, plus 30 Doradus in the Large Magellanic Cloud. MOXC consists of 20,623 X-ray point sources from 12 MSFRs with distances ranging from 1.7 kpc to 50 kpc. Additionally, we show the morphology of the unresolved X-ray emission that remains after the cataloged X-ray point sources are excised from the ACIS data, in the context of Spitzer and WISE observations that trace the bubbles, ionization fronts, and photon-dominated regions that characterize MSFRs. In previous work, we have found that this unresolved X-ray emission is dominated by hot plasma from massive star wind shocks. This diffuse X-ray emission is found in every MOXC MSFR, clearly demonstrating that massive star feedback (and the several-million-degree plasmas that it generates) is an integral component of MSFR physics.
5. CIRCUMVENTING THE RADIATION PRESSURE BARRIER IN THE FORMATION OF MASSIVE STARS VIA DISK ACCRETION
SciTech Connect
Kuiper, Rolf; Klahr, Hubert; Beuther, Henrik; Henning, Thomas
2010-10-20
We present radiation hydrodynamic simulations of the collapse of massive pre-stellar cores. We treat frequency-dependent radiative feedback from stellar evolution and accretion luminosity at a numerical resolution down to 1.27 AU. In the 2D approximation of axially symmetric simulations, for the first time it is possible to simulate the whole accretion phase (up to the end of the accretion disk epoch) for a forming massive star and to perform a broad scan of the parameter space. Our simulation series evidently shows the necessity to incorporate the dust sublimation front to preserve the high shielding property of massive accretion disks. While confirming the upper mass limit of spherically symmetric accretion, our disk accretion models show a persistent high anisotropy of the corresponding thermal radiation field. This yields the growth of the highest-mass stars ever formed in multi-dimensional radiation hydrodynamic simulations, far beyond the upper mass limit of spherical accretion. Non-axially symmetric effects are not necessary to sustain accretion. The radiation pressure launches a stable bipolar outflow, which grows in angle with time, as presumed from observations. For an initial mass of the pre-stellar host core of 60, 120, 240, and 480 M{sub sun} the masses of the final stars formed in our simulations add up to 28.2, 56.5, 92.6, and at least 137.2 M{sub sun}, respectively.
6. Distribution of HNCO 505-404 in massive star-forming regions
NASA Astrophysics Data System (ADS)
Li, J.; Wang, J. Z.; Gu, Q. S.; Zheng, X. W.
2013-07-01
Aims: The goal of this paper is to study the spatial distribution of HNCO in massive star-forming region and to investigate both its spatial association with infrared sources and physical conditions in region of HNCO emission. Methods: We mapped nine massive star-forming regions in HNCO 505-404 with the Purple Mountain Observatory 13.7 m telescope. The C18O maps of these sources were obtained simultaneously. Results: The HNCO emission shows compact distribution, with the emission peak centered on water masers. Nearly all the HNCO clumps show signs of embedded mid-infrared or far-infrared sources. The FWHM sizes of HNCO clumps are significantly smaller than C18O clumps but rather similar to HC3N clumps. We find a good correlation between the integrated intensities, linewidths, and LSR velocities of HNCO and HC3N emission, implying similar excitation mechanisms for these two species. As such, collisional excitation is likely to be the dominant excitation mechanism for HNCO 505 - 404 emission in galactic massive star-forming regions.
7. A distance-limited sample of massive star-forming cores from the RMS
NASA Astrophysics Data System (ADS)
Maud, L. T.; Lumsden, S. L.; Moore, T. J. T.; Mottram, J. C.; Urquhart, J. S.; Cicchini, A.
2015-09-01
We analyse C18O (J = 3-2) data from a sample of 99 infrared (IR)-bright massive young stellar objects (MYSOs) and compact H II regions that were identified as potential molecular-outflow sources in the Red MSX Source survey. We extract a distance-limited (D < 6 kpc) sample shown to be representative of star formation covering the transition between the source types. At the spatial resolution probed, Larson-like relationships are found for these cores, though the alternative explanation, that Larson's relations arise where surface-density-limited samples are considered, is also consistent with our data. There are no significant differences found between source properties for the MYSOs and H II regions, suggesting that the core properties are established prior to the formation of massive stars, which subsequently have little impact at the later evolutionary stages investigated. There is a strong correlation between dust-continuum and C18O-gas masses, supporting the interpretation that both trace the same material in these IR-bright sources. A clear linear relationship is seen between the independently established core masses and luminosities. The position of MYSOs and compact H II regions in the mass-luminosity plane is consistent with the luminosity expected from the most massive protostar in the cluster when using an ˜40 per cent star formation efficiency and indicates that they are at a similar evolutionary stage, near the end of the accretion phase.
8. A BUTTERFLY-SHAPED 'PAPILLON' NEBULA YIELDS SECRETS OF MASSIVE STAR BIRTH
NASA Technical Reports Server (NTRS)
2002-01-01
A NASA Hubble Space Telescope view of a turbulent cauldron of starbirth, called N159, taking place 170,000 light-years away in our satellite galaxy, the Large Magellanic Cloud (LMC). Torrential stellar winds from hot newborn massive stars within the nebula sculpt ridges, arcs, and filaments in the vast cloud, which is over 150 light-years across. A rare type of compact ionized 'blob' is resolved for the first time to be a butterfly-shaped or 'Papillon' (French for 'butterfly') nebula, buried in the center of the maelstrom of glowing gases and dark dust. The unprecedented details of the structure of the Papillon, itself less than 2 light-years in size (about 2 arcseconds in the sky), are seen in the inset. A possible explanation of this bipolar shape is the outflow of gas from massive stars (over 10 times the mass of our sun) hidden in the central absorption zone. Such stars are so hot that their radiation pressure halts the infall of gas and directs it away from the stars in two opposite directions. Presumably, a dense equatorial disk formed by matter still trying to fall in onto the stars focuses the outstreaming matter into the bipolar directions. This observation is part of a search for young massive stars in the LMC. Rare are the cases where we can see massive stars so early after their birth. The red in this true-color image is from the emission of hydrogen and the yellow from high excitation ionized oxygen. The picture was taken on September 5, 1998 with the Wide Field Planetary Camera 2. The Hubble observations of the Papillon nebula were conducted by the European astronomers Mohammad Heydari-Malayeri (Paris Observatory, France) and co-investigators Michael Rosa (Space Telescope-European Coordinating Facility, European Southern Observatory, Germany), Vassilis Charmandaris (Paris Observatory), Lise Deharveng (Marseille Observatory, France), and Hans Zinnecker (Astrophysical Institute, Potsdam, Germany). Their work is submitted for publication in the European
9. Double core evolution. 7: The infall of a neutron star through the envelope of its massive star companion
NASA Technical Reports Server (NTRS)
Terman, James L.; Taam, Ronald E.; Hernquist, Lars
1995-01-01
Binary systems with properties similar to those of high-mass X-ray binaries are evolved through the common envelope phase. Three-dimensional simulations show that the timescale of the infall phase of the neutron star depends upon the evolutionary state of its massive companion. We find that tidal torques more effectively accelerate common envelope evolution for companions in their late core helium-burning stage and that the infall phase is rapid (approximately several initial orbital periods). For less evolved companions the decay of the orbit is longer; however, once the neutron star is deeply embedded within the companion's envelope the timescale for orbital decay decreases rapidly. As the neutron star encounters the high-density region surrounding the helium core of its massive companion, the rate of energy loss from the orbit increases dramatically leading to either partial or nearly total envelope ejection. The outcome of the common envelope phase depends upon the structure of the evolved companion. In particular, it is found that the entire common envelope can be ejected by the interaction of the neutron star with a red supergiant companion in binaries with orbital periods similar to those of long-period Be X-ray binaries. For orbital periods greater than or approximately equal to 0.8-2 yr (for companions of mass 12-24 solar mass) it is likely that a binary will survive the common envelope phase. For these systems, the structure of the progenitor star is characterized by a steep density gradient above the helium core, and the common envelope phase ends with a spin up of the envelope to within 50%-60% of corotation and with a slow mass outflow. The efficiency of mass ejection is found to be approximately 30%-40%. For less evolved companions, there is insufficient energy in the orbit to unbind the common envelope and only a fraction of it is ejected. Since the timescale for orbital decay is always shorter than the mass-loss timescale from the common envelope
10. Type Ic core-collapse supernova explosions evolved from very massive stars
NASA Astrophysics Data System (ADS)
Yoshida, Takashi; Okita, Shinpei; Umeda, Hideyuki
2014-03-01
We investigate the possibility of a superluminous Type Ic core-collapse supernovae (SNe) producing a large amount of 56Ni. Very massive stars with a main-sequence mass larger than 100 M⊙ and a metallicity 0.001 < Z ≲ 0.004 are expected to explode as superluminous Type Ic SNe. Stars with ˜110-150 M⊙ and Z ≲ 0.001 would explode as Type Ic pulsational pair-instability SNe if the whole H and He layer has been lost by the mass-loss during pulsational pair instability. We evaluate the total ejecta mass and the yields of 56Ni, O and Si in core-collapse SNe evolved from very massive stars. We adopt 43.1 and 61.1 M⊙ WO stars with Z = 0.004 as SN progenitors expected to explode as Type Ic core-collapse SNe. These progenitors have masses of 110 and 250 M⊙ at the zero-age main sequence. Spherical explosions with an explosion energy larger than 2 × 1052 erg produce more than 3.5 M⊙56Ni, enough to reproduce the light curve of SN 2007bi. Asphericity of the explosion affects the total ejecta mass as well as the yields of 56Ni, O and Si. Aspherical explosions of the 110 and 250 M⊙ models reproduce the 56Ni yield of SN 2007bi. These explosions will also show large velocity dispersion. An aspherical core-collapse SN evolved from a very massive star is a possibility of the explosion of SN 2007bi.
11. THE ROTATION RATES OF MASSIVE STARS: THE ROLE OF BINARY INTERACTION THROUGH TIDES, MASS TRANSFER, AND MERGERS
SciTech Connect
De Mink, S. E.; Langer, N.; Izzard, R. G.; Sana, H.; De Koter, A.
2013-02-20
Rotation is thought to be a major factor in the evolution of massive stars-especially at low metallicity-with consequences for their chemical yields, ionizing flux, and final fate. Deriving the birth spin distribution is of high priority given its importance as a constraint on theories of massive star formation and as input for models of stellar populations in the local universe and at high redshift. Recently, it has become clear that the majority of massive stars interact with a binary companion before they die. We investigate how this affects the distribution of rotation rates, through stellar winds, expansion, tides, mass transfer, and mergers. For this purpose, we simulate a massive binary-star population typical for our Galaxy assuming continuous star formation. We find that, because of binary interaction, 20{sup +5} {sub -10}% of all massive main-sequence stars have projected rotational velocities in excess of 200 km s{sup -1}. We evaluate the effect of uncertain input distributions and physical processes and conclude that the main uncertainties are the mass transfer efficiency and the possible effect of magnetic braking, especially if magnetic fields are generated or amplified during mass accretion and stellar mergers. The fraction of rapid rotators we derive is similar to that observed. If indeed mass transfer and mergers are the main cause for rapid rotation in massive stars, little room remains for rapidly rotating stars that are born single. This implies that spin-down during star formation is even more efficient than previously thought. In addition, this raises questions about the interpretation of the surface abundances of rapidly rotating stars as evidence for rotational mixing. Furthermore, our results allow for the possibility that all early-type Be stars result from binary interactions and suggest that evidence for rotation in explosions, such as long gamma-ray bursts, points to a binary origin.
12. Mass ejection by pulsational pair instability in very massive stars and implications for luminous supernovae
NASA Astrophysics Data System (ADS)
Yoshida, Takashi; Umeda, Hideyuki; Maeda, Keiichi; Ishii, Tatsuo
2016-03-01
Massive stars having a CO core of ˜40-60 M⊙ experience pulsational pair-instability (PPI) after carbon-burning. This instability induces strong pulsations of the whole star and a part of outer envelope is ejected. We investigate the evolution and mass ejection of metal-poor very massive stars which experience PPI. We use stellar models with initial masses of 140, 200, and 250 M⊙ and the metallicity Z = 0.004. Their masses decrease to 54.09, 58.65, and 61.03 M⊙ before the neon-burning owing to mass-loss and He mass fraction at the surface becomes about 20 per cent. During the PPI period of ˜1-2000 yr, they experience six, four, and three pulsations, respectively. The larger CO-core model has the longer PPI period and ejects the larger amount of mass. Since almost all surface He has been lost by the pulsations, these stars become Type Ic supernovae if they explode. Light curves during the PPI stage and supernovae are investigated and are implicated in luminous supernovae. The luminosity created by the interaction of different PPI ejecta becomes Mbol ˜ -16 to -20. The interaction between the circumstellar shell ejected by PPI and the supernova ejecta can be more luminous. These luminous transients could be an origin of Type I superluminous supernovae and supernovae with precursor.
13. The Fragmentation of Magnetized, Massive Star-forming Cores with Radiative Feedback
NASA Astrophysics Data System (ADS)
Myers, Andrew T.; McKee, Christopher F.; Cunningham, Andrew J.; Klein, Richard I.; Krumholz, Mark R.
2013-04-01
We present a set of three-dimensional, radiation-magnetohydrodynamic calculations of the gravitational collapse of massive (300 M ⊙), star-forming molecular cloud cores. We show that the combined effects of magnetic fields and radiative feedback strongly suppress core fragmentation, leading to the production of single-star systems rather than small clusters. We find that the two processes are efficient at suppressing fragmentation in different regimes, with the feedback most effective in the dense, central region and the magnetic field most effective in more diffuse, outer regions. Thus, the combination of the two is much more effective at suppressing fragmentation than either one considered in isolation. Our work suggests that typical massive cores, which have mass-to-flux ratios of about 2 relative to critical, likely form a single-star system, but that cores with weaker fields may form a small star cluster. This result helps us understand why the observed relationship between the core mass function and the stellar initial mass function holds even for ~100 M ⊙ cores with many thermal Jeans masses of material. We also demonstrate that a ~40 AU Keplerian disk is able to form in our simulations, despite the braking effect caused by the strong magnetic field.
14. Signatures of multiple stellar populations in unresolved extragalactic globular/young massive star clusters
SciTech Connect
Peacock, Mark B.; Zepf, Stephen E.; Finzell, Thomas
2013-06-01
We present an investigation of potential signatures of the formation of multiple stellar populations in recently formed extragalactic star clusters. All of the Galactic globular clusters for which good samples of individual stellar abundances are available show evidence for multiple populations. This appears to require that multiple episodes of star formation and light element enrichment are the norm in the history of a globular cluster. We show that there are detectable observational signatures of multiple formation events in the unresolved spectra of massive, young extragalactic star clusters. We present the results of a pilot program to search for one of the cleanest signatures that we identify—the combined presence of emission lines from a very recently formed population and absorption lines from a somewhat older population. A possible example of such a system is identified in the Antennae galaxies. This source's spectrum shows evidence of two stellar populations with ages of 8 Myr and 80 Myr. Further investigation shows that these populations are in fact physically separated, but only by a projected distance of 59 pc. We show that the clusters are consistent with being bound and discuss the possibility that their coalescence could result in a single globular cluster hosting multiple stellar populations. While not the prototypical system proposed by most theories of the formation of multiple populations in clusters, the detection of this system in a small sample is both encouraging and interesting. Our investigation suggests that expanded surveys of massive young star clusters should detect more clusters with such signatures.
15. A Rare Encounter with Very Massive Stars in NGC~3125-A1
NASA Astrophysics Data System (ADS)
Wofford, A.; Leitherer, C.; Chandar, R.; Bouret, J. C.
2014-09-01
Super star cluster A1 in the nearby starburst galaxy NGC~3125 shows broad He II λ1640 emission (FWHM ~ 1200 km/s) of unprecedented strength (equivalent width, EW = 7.1+/-0.4 angstroms). Previous attempts to characterize A1's massive star content were hampered by the low resolution of the UV spectrum and the lack of co-spatial panchromatic data. We obtained far-UV to near-IR spectroscopy of the two principal emitting regions in the galaxy with the Space Telescope Imaging Spectrograph and the Cosmic Origins Spectrograph on board the Hubble Space Telescope. We use these data to derive the ages, reddenings, masses, and Wolf-Rayet (WR) to O star ratios of three compact clusters in the galaxy. We rule out that the extraordinary HeII lambda 1640 emission and OV lambda 1371 absorption in A1 are due to an extremely flat upper Initial Mass Function (IMF), and suggest that they originate in the winds of Very Massive Stars ( > 120 Msun, VMS). In order to reproduce the properties of peculiar clusters such as A1, the stellar evolution tracks implemented in Starburst99 need to be extended to masses >120 Msun.
16. Gamma-ray bursts from massive Population-III stars: clues from the radio band
NASA Astrophysics Data System (ADS)
Burlon, D.; Murphy, T.; Ghirlanda, G.; Hancock, P. J.; Parry, R.; Salvaterra, R.
2016-07-01
Current models suggest gamma-ray bursts could be used as a way of probing Population-III stars - the first stars in the early Universe. In this paper, we use numerical simulations to demonstrate that late-time radio observations of gamma-ray burst afterglows could provide a means of identifying bursts that originate from Population-III stars, if these were highly massive, independently from their redshift. We then present the results from a pilot study using the Australia Telescope Compact Array at 17 GHz, designed to test the hypothesis that there may be Population-III gamma-ray bursts amongst the current sample of known events. We observed three candidates plus a control gamma-ray burst, and make no detections with upper limits of 20-40 μJy at 500-1300 d post-explosion.
17. [A new automated method to identify emission line star from massive spectra].
PubMed
Pan, Jing-Chang; Zhang, Cai-Ming; Wei, Peng; Luo, A-Li; Zhao, Yong-Heng
2012-06-01
Stellar spectra are characterized by obvious absorption lines or absorption bands, while those with emission lines are usually special stars such as cataclysmic variable stars (CVs), HerbigAe/Be etc. The further study of this kind of spectra is meaningful. The present paper proposed a new method to identify emission line stars (ELS) spectra automatically. After the continuum normalization is done for the original spectral flux, line detection is made by comparing the normalized flux with the mean and standard deviation of the flux in its neighbor region The results of the experiment on massive spectra from SDSS DR8 indicate that the method can identify ELS spectra completely and accurately. Since no complex transformation and computation are involved in this method, the identifying process is fast and it is ideal for the ELS detection in large sky survey projects like LAMOST and SDSS. PMID:22870668
18. [A new automated method to identify emission line star from massive spectra].
PubMed
Pan, Jing-Chang; Zhang, Cai-Ming; Wei, Peng; Luo, A-Li; Zhao, Yong-Heng
2012-06-01
Stellar spectra are characterized by obvious absorption lines or absorption bands, while those with emission lines are usually special stars such as cataclysmic variable stars (CVs), HerbigAe/Be etc. The further study of this kind of spectra is meaningful. The present paper proposed a new method to identify emission line stars (ELS) spectra automatically. After the continuum normalization is done for the original spectral flux, line detection is made by comparing the normalized flux with the mean and standard deviation of the flux in its neighbor region The results of the experiment on massive spectra from SDSS DR8 indicate that the method can identify ELS spectra completely and accurately. Since no complex transformation and computation are involved in this method, the identifying process is fast and it is ideal for the ELS detection in large sky survey projects like LAMOST and SDSS.
19. Sizes and Shapes of Young, Massive Star Clusters in M83
NASA Astrophysics Data System (ADS)
Ryon, Jenna E.; Bastian, Nate; Adamo, Angela; Silva-Villa, Esteban; Gallagher, John S.
2015-01-01
Using HST imaging, the surface brightness profiles of individual star clusters in nearby galaxies can be resolved, in that clusters are clearly more extended than the stellar PSF. Previous studies of the sizes and shapes of star clusters find little variation with cluster age, mass, or galaxy environment. We use observations from seven pointings on M83 from HST/WFC3 programs GO/DD-11360 (PI O'Connell) and GO-12513 (PI Blair) to obtain a large sample of young, massive star clusters. We measure the half-light radii and power-law indices of the EFF light profile (Elson, Fall, & Freeman 1987) of these clusters using the galfit software package (Peng et al. 2002). We present our results on the relationships between cluster size, shape, age, mass, and environment in the disk of M83.
20. Toward Detecting Fast Moving Massive Stars around the 30 Doradus Region
NASA Astrophysics Data System (ADS)
Platais, Imants; Sabbi, E.; Anderson, J.; Lennon, D. J.; van der Marel, R. P.; Bellini, A. J.; de Mink, S. E.; Sohn, S. T.; Bedin, L. R.
2012-05-01
We have started an HST proper motion survey in the 30 Dor region of the Large Magellanic Cloud with the goal to find directions of tangential velocities of massive runaway stars and, hence, test the suggested production mechanisms (point of origin) of such stars. While the thrust of this survey is based on yet-to-be-completed two-epoch observations with the HST wide field cameras, there appears to be a considerable potential in achieving complementary aims by combining archival data from the HST WFPC2 with the latest extant observations. We report the first results of this approach and explore the level of proper-motion precision achievable with these data sets in the regime of sub-optimal images for the brighter main target stars. SdM acknowledges the NASA Hubble Fellowship grant HST-HF-51270.01-A awarded by STScI, operated by AURA for NASA, contract NAS 5-26555.
1. Detailed abundance analysis of the brightest star in Segue 2, the least massive galaxy
NASA Astrophysics Data System (ADS)
Roederer, Ian U.; Kirby, Evan N.
2014-05-01
We present the first high-resolution spectroscopic observations of one red giant star in the ultra-faint dwarf galaxy Segue 2, which has the lowest total mass (including dark matter) estimated for any known galaxy. These observations were made using the Magellan Inamori Kyocera Echelle (MIKE) spectrograph on the Magellan II Telescope at Las Campanas Observatory. We perform a standard abundance analysis of this star, SDSS J021933.13+200830.2, and present abundances of 21 species of 18 elements as well as upper limits for 25 additional species. We derive [Fe/H] = -2.9, in excellent agreement with previous estimates from medium-resolution spectroscopy. Our main result is that this star bears the chemical signatures commonly found in field stars of similar metallicity. The heavy elements produced by neutron-capture reactions are present, but they are deficient at levels characteristic of stars in other ultra-faint dwarf galaxies and a few luminous dwarf galaxies. The otherwise normal abundance patterns suggest that the gas from which this star formed was enriched by metals from multiple Type II supernovae reflecting a relatively well-sampled IMF. This adds to the growing body of evidence indicating that Segue 2 may have been substantially more massive in the past.
2. Non-standard s-process in low metallicity massive rotating stars
NASA Astrophysics Data System (ADS)
Frischknecht, U.; Hirschi, R.; Thielemann, F.-K.
2012-02-01
Context. Rotation is known to have a strong impact on the nucleosynthesis of light elements in massive stars, mainly by inducing mixing in radiative zones. In particular, rotation boosts the primary nitrogen production, and models of rotating stars are able to reproduce the nitrogen observed in low-metallicity halo stars. Aims: Here we present the first grid of stellar models for rotating massive stars at low metallicity, where a full s-process network is used to study the impact of rotation-induced mixing on the neutron capture nucleosynthesis of heavy elements. Methods: We used the Geneva stellar evolution code that includes an enlarged reaction network with nuclear species up to bismuth to calculate 25 M⊙ models at three different metallicities (Z = 10-3,10-5, and 10-7) and with different initial rotation rates. Results: First, we confirm that rotation-induced mixing (shear) between the convective H-shell and He-core leads to a large production of primary 22Ne (0.1 to 1% in mass fraction), which is the main neutron source for the s-process in massive stars. Therefore rotation boosts the s-process in massive stars at all metallicities. Second, the neutron-to-seed ratio increases with decreasing Z in models including rotation, which leads to the complete consumption of all iron seeds at metallicities below Z = 10-3 by the end of core He-burning. Thus at low Z, the iron seeds are the main limitation for this boosted s-process. Third, as the metallicity decreases, the production of elements up to the Ba peak increases at the expense of the elements of the Sr peak. We studied the impact of the initial rotation rate and of the highly uncertain 17O(α,γ) rate (which strongly affects the strength of 16O as a neutron poison) on our results. This study shows that rotating models can produce significant amounts of elements up to Ba over a wide range of Z, which has important consequences for our understanding of the formation of these elements in low
3. Lithium and zirconium abundances in massive Galactic O-rich AGB stars
NASA Astrophysics Data System (ADS)
García-Hernández, D. A.; García-Lario, P.; Plez, B.; Manchado, A.; D'Antona, F.; Lub, J.; Habing, H.
2007-02-01
Lithium and zirconium abundances (the latter taken as representative of s-process enrichment) are determined for a large sample of massive Galactic O-rich AGB stars, for which high-resolution optical spectroscopy has been obtained (R˜ 40 000{-}50 000). This was done by computing synthetic spectra based on classical hydrostatic model atmospheres for cool stars and using extensive line lists. The results are discussed in the framework of "hot bottom burning" (HBB) and nucleosynthesis models. The complete sample is studied for various observational properties such as the position of the stars in the IRAS two-colour diagram ([ 12] - [25] vs. [ 25] - [60] ), Galactic distribution, expansion velocity (derived from the OH maser emission), and period of variability (when available). We conclude that a considerable fraction of these sources are actually massive AGB stars (M>3{-}4 M⊙) experiencing HBB, as deduced from the strong Li overabundances we found. A comparison of our results with similar studies carried out in the past for the Magellanic Clouds (MCs) reveals that, in contrast to MC AGB stars, our Galactic sample does not show any indication of s-process element enrichment. The differences observed are explained as a consequence of metallicity effects. Finally, we discuss the results obtained in the framework of stellar evolution by comparing our results with the data available in the literature for Galactic post-AGB stars and PNe. Based on observations at the 4.2 m William Herschel Telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofisica de Canarias. Also based on observations with the ESO 3.6 m telescope at La Silla Observatory (Chile). Tables [see full text]-[see full text] are only available in electronic form at http://www.aanda.org
4. MOLECULAR LINE EMISSION FROM A PROTOPLANETARY DISK IRRADIATED EXTERNALLY BY A NEARBY MASSIVE STAR
SciTech Connect
Walsh, Catherine; Millar, T. J.; Nomura, Hideko
2013-04-01
Star formation often occurs within or nearby stellar clusters. Irradiation by nearby massive stars can photoevaporate protoplanetary disks around young stars (so-called proplyds) which raises questions regarding the ability of planet formation to take place in these environments. We investigate the two-dimensional physical and chemical structure of a protoplanetary disk surrounding a low-mass (T Tauri) star which is irradiated by a nearby massive O-type star to determine the survivability and observability of molecules in proplyds. Compared with an isolated star-disk system, the gas temperature ranges from a factor of a few (in the disk midplane) to around two orders of magnitude (in the disk surface) higher in the irradiated disk. Although the UV flux in the outer disk, in particular, is several orders of magnitude higher, the surface density of the disk is sufficient for effective shielding of the disk midplane so that the disk remains predominantly molecular in nature. We also find that non-volatile molecules, such as HCN and H{sub 2}O, are able to freeze out onto dust grains in the disk midplane so that the formation of icy planetesimals, e.g., comets, may also be possible in proplyds. We have calculated the molecular line emission from the disk assuming LTE and determined that multiple transitions of atomic carbon, CO (and isotopologues, {sup 13}CO and C{sup 18}O), HCO{sup +}, CN, and HCN may be observable with ALMA, allowing characterization of the gas column density, temperature, and optical depth in proplyds at the distance of Orion ( Almost-Equal-To 400 pc).
5. They Might Be Giants: Confirming Candidate OB Stars While Netting a Large Sample of Massive Star Spectra in the Great Nebula in Carina
NASA Astrophysics Data System (ADS)
Povich, Matthew S.; McSwain, M. Virginia
2013-02-01
We propose one night of observations with the AAOmega instrument on the Anglo-Australian Telescope to obtain spectra of a large sample of massive stars in the Great Nebula in Carina, the nearest analog of extragalactic starburst regions. Our targets include >100 spectroscopically classified OB stars plus 55 candidate OB stars that we recently identified via X-ray emission and infrared (IR) spectral energy distributions (SEDs). These observations will confirm or reject individual candidate OB stars as massive members of the Carina Nebula stellar population, a vital test for our methodology that will pave the way to discovering new massive stars in other regions. Determining the nature of the candidate OB stars is critical to any census of the massive stellar population in Carina, impacting our understanding of the energetics and stellar initial mass function in this well-studied region. We will employ spectral modeling and broadband optical-IR SED fitting to derive physical properties (e.g. temperature, bolometric luminosity, surface gravity, and mass) of the known OB stars and those newly-confirmed candidate OB stars with high (ga100) signal-to- noise spectra.
6. A rare encounter with very massive stars in NGC 3125-A1
SciTech Connect
Wofford, Aida; Leitherer, Claus; Chandar, Rupali; Bouret, Jean-Claude
2014-02-01
Super star cluster A1 in the nearby starburst galaxy NGC 3125 is characterized by broad He II λ1640 emission (FWHM ∼ 1200 km s{sup –1}) of unprecedented strength (equivalent width, EW = 7.1 ± 0.4 Å). Previous attempts to characterize the massive star content in NGC 3125-A1 were hampered by the low resolution of the UV spectrum and the lack of co-spatial panchromatic data. We obtained far-UV to near-IR spectroscopy of the two principal emitting regions in the galaxy with the Space Telescope Imaging Spectrograph and the Cosmic Origins Spectrograph on board the Hubble Space Telescope. We use these data to study three clusters in the galaxy, A1, B1, and B2. We derive cluster ages of 3-4 Myr, intrinsic reddenings of E(B – V) = 0.13, 0.15, and 0.13, and cluster masses of 1.7 × 10{sup 5}, 1.4 × 10{sup 5}, and 1.1 × 10{sup 5} M {sub ☉}, respectively. A1 and B2 show O V λ1371 absorption from massive stars, which is rarely seen in star-forming galaxies, and have Wolf-Rayet (WR) to O star ratios of N(WN5-6)/N(O) = 0.23 and 0.10, respectively. The high N(WN5-6)/N(O) ratio of A1 cannot be reproduced by models that use a normal initial mass function (IMF) and generic WR star line luminosities. We rule out that the extraordinary He II λ1640 emission and O V λ1371 absorption of A1 are due to an extremely flat upper IMF exponent, and suggest that they originate in the winds of very massive (>120 M {sub ☉}) stars. In order to reproduce the properties of peculiar clusters such as A1, the present grid of stellar evolution tracks implemented in Starburst99 needs to be extended to masses >120 M {sub ☉}.
7. A Rare Encounter with Very Massive Stars in NGC 3125-A1
NASA Astrophysics Data System (ADS)
Wofford, Aida; Leitherer, Claus; Chandar, Rupali; Bouret, Jean-Claude
2014-02-01
Super star cluster A1 in the nearby starburst galaxy NGC 3125 is characterized by broad He II λ1640 emission (FWHM ~ 1200 km s-1) of unprecedented strength (equivalent width, EW = 7.1 ± 0.4 Å). Previous attempts to characterize the massive star content in NGC 3125-A1 were hampered by the low resolution of the UV spectrum and the lack of co-spatial panchromatic data. We obtained far-UV to near-IR spectroscopy of the two principal emitting regions in the galaxy with the Space Telescope Imaging Spectrograph and the Cosmic Origins Spectrograph on board the Hubble Space Telescope. We use these data to study three clusters in the galaxy, A1, B1, and B2. We derive cluster ages of 3-4 Myr, intrinsic reddenings of E(B - V) = 0.13, 0.15, and 0.13, and cluster masses of 1.7 × 105, 1.4 × 105, and 1.1 × 105 M ⊙, respectively. A1 and B2 show O V λ1371 absorption from massive stars, which is rarely seen in star-forming galaxies, and have Wolf-Rayet (WR) to O star ratios of N(WN5-6)/N(O) = 0.23 and 0.10, respectively. The high N(WN5-6)/N(O) ratio of A1 cannot be reproduced by models that use a normal initial mass function (IMF) and generic WR star line luminosities. We rule out that the extraordinary He II λ1640 emission and O V λ1371 absorption of A1 are due to an extremely flat upper IMF exponent, and suggest that they originate in the winds of very massive (>120 M ⊙) stars. In order to reproduce the properties of peculiar clusters such as A1, the present grid of stellar evolution tracks implemented in Starburst99 needs to be extended to masses >120 M ⊙.
8. Ages of young star clusters, massive blue stragglers, and the upper mass limit of stars: Analyzing age-dependent stellar mass functions
SciTech Connect
Schneider, F. R. N.; Izzard, R. G.; Langer, N.; Stolte, A.; Hußmann, B.; De Mink, S. E.; De Koter, A.; Sana, H.; Gvaramadze, V. V.; Liermann, A.
2014-01-10
Massive stars rapidly change their masses through strong stellar winds and mass transfer in binary systems. The latter aspect is important for populations of massive stars as more than 70% of all O stars are expected to interact with a binary companion during their lifetime. We show that such mass changes leave characteristic signatures in stellar mass functions of young star clusters that can be used to infer their ages and to identify products of binary evolution. We model the observed present-day mass functions of the young Galactic Arches and Quintuplet star clusters using our rapid binary evolution code. We find that the shaping of the mass function by stellar wind mass loss allows us to determine the cluster ages as 3.5 ± 0.7 Myr and 4.8 ± 1.1 Myr, respectively. Exploiting the effects of binary mass exchange on the cluster mass function, we find that the most massive stars in both clusters are rejuvenated products of binary mass transfer, i.e., the massive counterpart of classical blue straggler stars. This resolves the problem of an apparent age spread among the most luminous stars exceeding the expected duration of star formation in these clusters. We perform Monte Carlo simulations to probe stochastic sampling, which support the idea of the most massive stars being rejuvenated binary products. We find that the most massive star is expected to be a binary product after 1.0 ± 0.7 Myr in Arches and after 1.7 ± 1.0 Myr in Quintuplet. Today, the most massive 9 ± 3 stars in Arches and 8 ± 3 in Quintuplet are expected to be such objects. Our findings have strong implications for the stellar upper mass limit and solve the discrepancy between the claimed 150 M {sub ☉} limit and observations of four stars with initial masses of 165-320 M {sub ☉} in R136 and of supernova 2007bi, which is thought to be a pair-instability supernova from an initial 250 M {sub ☉} star. Using the stellar population of R136, we revise the upper mass limit to values in the range
9. The Massive Stellar Content in the Starburst NGC 3049: A Test for Hot-Star Models
NASA Astrophysics Data System (ADS)
González Delgado, Rosa M.; Leitherer, Claus; Stasińska, Grażyna; Heckman, Timothy M.
2002-12-01
The objective of this work is twofold. First, we seek evidence for or against the depletion of massive stars in metal-rich starbursts. A second, equally important goal is to perform a consistency test of the latest generation of starburst models in such a high-metallicity environment. We have obtained high spatial resolution ultraviolet and optical STIS spectroscopy and imaging of the metal-rich nuclear starburst in NGC 3049. The stellar continuum and the absorption-line spectrum in the ultraviolet are used to constrain the massive stellar population. The strong, blueshifted stellar lines of C IV and Si IV detected in the UV spectra indicate a metal-rich, compact, massive (~106 Msolar) cluster of age 3-4 Myr emitting the UV-optical continuum. We find strong evidence against a depletion of massive stars in this metal-rich cluster. The derived age and the upper mass-limit cutoff of the initial mass function are also consistent with the detection of Wolf-Rayet (W-R) features at optical wavelengths. As a second, independent constraint on the massive stellar content, the nebular emission-line spectrum is modeled with photoionization codes using stellar spectra from evolutionary synthesis models. The morphology of the nuclear starburst of NGC 3049 from the STIS images indicates a simple geometry for the nebular emission-line region. However, the nebular lines are badly reproduced by 3-4 Myr instantaneous bursts, as required by the UV line spectrum, when unblanketed W-R and/or Kurucz stellar atmospheres are used. The corresponding number of photons above 24 and 54 eV in the synthetic models is too high in comparison with values suggested by the observed line ratios. Since the ionizing spectrum in this regime is dominated by emission from W-R stars, this discrepancy between observations and models is most likely the result of incorrect assumptions about the W-R stars. Thus, we conclude that the nebular spectrum of high-metallicity starbursts is poorly reproduced by models
10. Global collapse of molecular clouds as a formation mechanism for the most massive stars
NASA Astrophysics Data System (ADS)
Peretto, N.; Fuller, G. A.; Duarte-Cabral, A.; Avison, A.; Hennebelle, P.; Pineda, J. E.; André, Ph.; Bontemps, S.; Motte, F.; Schneider, N.; Molinari, S.
2013-07-01
The relative importance of primordial molecular cloud fragmentation versus large-scale accretion still remains to be assessed in the context of massive core/star formation. Studying the kinematics of the dense gas surrounding massive-star progenitors can tell us the extent to which large-scale flow of material impacts the growth in mass of star-forming cores. Here we present a comprehensive dataset of the 5500(±800) M⊙ infrared dark cloud SDC335.579-0.272 (hereafter SDC335), which exhibits a network of cold, dense, parsec-long filaments. Atacama Large Millimeter Array (ALMA) Cycle 0 observations reveal two massive star-forming cores, MM1 and MM2, sitting at the centre of SDC335 where the filaments intersect. With a gas mass of 545(-385+770) M⊙ contained within a source diameter of 0.05 pc, MM1 is one of the most massive, compact protostellar cores ever observed in the Galaxy. As a whole, SDC335 could potentially form an OB cluster similar to the Trapezium cluster in Orion. ALMA and Mopra single-dish observations of the SDC335 dense gas furthermore reveal that the kinematics of this hub-filament system are consistent with a global collapse of the cloud. These molecular-line data point towards an infall velocity Vinf = 0.7( ± 0.2) km s-1, and a total mass infall rate Ṁinf ≃ 2.5(±1.0) × 10-3 M⊙ yr-1 towards the central pc-size region of SDC335. This infall rate brings 750(±300) M⊙ of gas to the centre of the cloud per free-fall time (tff = 3 × 105 yr). This is enough to double the mass already present in the central pc-size region in 3.5-1.0+2.2 × tff. These values suggest that the global collapse of SDC335 over the past million year resulted in the formation of an early O-type star progenitor at the centre of the cloud's gravitational potential well.
11. The Coevolution of Nuclear Star Clusters, Massive Black Holes, and Their Host Galaxies
NASA Astrophysics Data System (ADS)
Antonini, Fabio; Barausse, Enrico; Silk, Joseph
2015-10-01
Studying how nuclear star clusters (NSCs) form and how they are related to the growth of the central massive black holes (MBHs) and their host galaxies is fundamental for our understanding of the evolution of galaxies and the processes that have shaped their central structures. We present the results of a semi-analytical galaxy formation model that follows the evolution of dark matter halos along merger trees, as well as that of the baryonic components. This model allows us to study the evolution of NSCs in a cosmological context, by taking into account the growth of NSCs due to both dynamical-friction-driven migration of stellar clusters and star formation triggered by infalling gas, while also accounting for dynamical heating from (binary) MBHs. We find that in situ star formation contributes a significant fraction (up to ∼80%) of the total mass of NSCs in our model. Both NSC growth through in situ star formation and that through star cluster migration are found to generate NSC—host galaxy scaling correlations that are shallower than the same correlations for MBHs. We explore the role of galaxy mergers on the evolution of NSCs and show that observational data on NSC—host galaxy scaling relations provide evidence of partial erosion of NSCs by MBH binaries in luminous galaxies. We show that this observational feature is reproduced by our models, and we make predictions about the NSC and MBH occupation fraction in galaxies. We conclude by discussing several implications for theories of NSC formation.
12. Bow shock nebulae of hot massive stars in a magnetized medium
NASA Astrophysics Data System (ADS)
Meyer, D. M.-A.; Mignone, A.; Kuiper, R.; Raga, A.; Kley, W.
2016-10-01
A significant fraction of OB-type, main-sequence massive stars are classified as runaway and move supersonically through the interstellar medium (ISM). Their strong stellar winds interact with their surroundings where the typical strength of the local ISM magnetic field is about 3.5-7 μ G, which can result in the formation of bow shock nebulae. We investigate the effects of such magnetic fields, aligned with the motion of the flow, on the formation and emission properties of these circumstellar structures. Our axisymmetric, magneto-hydrodynamical simulations with optically-thin radiative cooling, heating and anisotropic thermal conduction show that the presence of the background ISM magnetic field affects the projected optical emission our bow shocks at Hα and [OIII] λ 5007 which become fainter by about 1-2 orders of magnitude, respectively. Radiative transfer calculations against dust opacity indicate that the magnetic field slightly diminishes their projected infrared emission and that our bow shocks emit brightly at 60 μ m. This may explain why the bow shocks generated by ionizing runaway massive stars are often difficult to identify. Finally, we discuss our results in the context of the bow shock of ζ Ophiuchi and we support the interpretation of its imperfect morphology as an evidence of the presence of an ISM magnetic field not aligned with the motion of its driving star.
13. Big Fish in Small Ponds: massive stars in the low-mass clusters of M83
SciTech Connect
Andrews, J. E.; Calzetti, D.; McElwee, Sean; Chandar, R.; Elmegreen, B. G.; Kennicutt, R. C.; Kim, Hwihyun; Krumholz, Mark R.; Lee, J. C.; Whitmore, B.; O'Connell, R. W. E-mail: [email protected]
2014-09-20
We have used multi-wavelength Hubble Space Telescope WFC3 data of the starbursting spiral galaxy M83 in order to measure variations in the upper end of the stellar initial mass function (uIMF) using the production rate of ionizing photons in unresolved clusters with ages ≤ 8 Myr. As in earlier papers on M51 and NGC 4214, the uIMF in M83 is consistent with a universal IMF, and stochastic sampling of the stellar populations in the ∼<10{sup 3} M {sub ☉} clusters are responsible for any deviations in this universality. The ensemble cluster population, as well as individual clusters, also imply that the most massive star in a cluster does not depend on the cluster mass. In fact, we have found that these small clusters seem to have an over-abundance of ionizing photons when compared to an expected universal or truncated IMF. This also suggests that the presence of massive stars in these clusters does not affect the star formation in a destructive way.
14. Enhanced Star Formation of Less Massive Galaxies in a Protocluster at z = 2.5
NASA Astrophysics Data System (ADS)
Hayashi, Masao; Kodama, Tadayuki; Tanaka, Ichi; Shimakawa, Rhythm; Koyama, Yusei; Tadaki, Ken-ichi; Suzuki, Tomoko L.; Yamamoto, Moegi
2016-08-01
We investigate a correlation between star formation rate (SFR) and stellar mass for Hα emission-line galaxies (HAEs) in one of the richest protoclusters ever known at z ˜ 2.5, the USS 1558-003 protocluster. This study is based on a 9.7 hr narrowband imaging data with MOIRCS on the Subaru telescope. We are able to construct a sample in combination with additional H-band data taken with WFC3 on the Hubble Space Telescope, of 100 HAEs reaching the dust-corrected SFRs down to 3 M ⊙ yr-1 and the stellar masses down to 108.0 M ⊙. We find that while the star-forming galaxies with ≳109.3 M ⊙ are located on the universal SFR-mass main sequence (MS) irrespective of the environment, less massive star-forming galaxies with ≲109.3 M ⊙ show a significant upward scatter from the MS in this protocluster. This suggests that some less massive galaxies are in a starburst phase, although we do not know yet if this is due to environmental effects.
15. A NEW MECHANISM FOR MASS ACCRETION UNDER RADIATION PRESSURE IN MASSIVE STAR FORMATION
SciTech Connect
Tanaka, Kei E. I.; Nakamoto, Taishi
2010-05-01
During the formation of a massive star, strong radiation pressure from the central star acts on the dust sublimation front and tends to halt the accretion flow. To overcome this strong radiation pressure, it has been considered that a strong ram pressure produced by a high-mass accretion rate of 10{sup -3} M{sub sun} yr{sup -1} or more is needed. We reinvestigated the necessary condition to overcome the radiation pressure and found a new mechanism for overcoming it. Accumulated mass in a stagnant flow near the dust sublimation front helps the mass accretion by its weight. This mechanism relaxes the condition for the massive star formation. We call this mechanism the 'OMOSHI effect', where OMOSHI is an acronym for 'One Mechanism for Overcoming Stellar High radiation pressure by weIght'. Additionally, in Japanese, OMOSHI is a noun meaning a weight that is put on something to prevent it from moving. We investigate the generation of the OMOSHI effect using local one-dimensional radiation hydrodynamics simulations. The radiation pressure and the gravitational force are connected through the gas pressure, and to sum up, the radiation pressure is balanced or overcome by the gravitational force. We also discuss the global structure and temporal variation of the accretion flow.
16. Weak-interaction-mediated rates on iron isotopes for presupernova evolution of massive stars
NASA Astrophysics Data System (ADS)
Nabi, J.-Un
2009-05-01
During the presupernova evolution of massive stars, the isotopes of iron, 54, 55, 56Fe , are advocated to play a key role inside the cores primarily decreasing the electron-to-baryon ratio (Ye) mainly via electron capture processes thereby reducing the pressure support. Electron decay and positron capture on 55Fe , on the other hand, also have a consequential role in increasing the lepton ratio during the silicon burning phases of massive stars. The neutrinos and antineutrinos produced, as a result of these weak-interaction reactions, are transparent to the stellar matter and assist in cooling the core thereby reducing the entropy. The structure of the presupernova star is altered both by the changes in Ye and the entropy of the core material. The aim of this paper is to report the improved microscopic calculation of Gamow-Teller (GT±) strength distributions of these key isotopes of iron using the pn-QRPA theory. The main improvement comes from the incorporation of experimental deformation values for these nuclei. Additionally six different weak-interaction rates, namely electron and positron capture, electron and positron decay, and, neutrino and antineutrino cooling rates, were also calculated in presupernova matter. The calculated electron capture and neutrino cooling rates due to isotopes of iron are in good agreement with the large-scale shell model (LSSM) results. The calculated beta decay rates, however, are suppressed by three to five orders of magnitude.
17. Spitzer view of massive star formation in the tidally stripped Magellanic Bridge
SciTech Connect
Chen, C.-H. Rosie; Indebetouw, Remy; Muller, Erik; Kawamura, Akiko; Gordon, Karl D.; Meixner, Margaret; Seale, Jonathan P.; Shiao, Bernie; Sewiło, Marta; Whitney, Barbara A.; Meade, Marilyn R.; Fukui, Yasuo; Madden, Suzanne C.; Robitaille, Thomas P.
2014-04-20
The Magellanic Bridge is the nearest low-metallicity, tidally stripped environment, offering a unique high-resolution view of physical conditions in merging and forming galaxies. In this paper, we present an analysis of candidate massive young stellar objects (YSOs), i.e., in situ, current massive star formation (MSF) in the Bridge using Spitzer mid-IR and complementary optical and near-IR photometry. While we definitely find YSOs in the Bridge, the most massive are ∼10 M {sub ☉}, <<45 M {sub ☉} found in the LMC. The intensity of MSF in the Bridge also appears to be decreasing, as the most massive YSOs are less massive than those formed in the past. To investigate environmental effects on MSF, we have compared properties of massive YSOs in the Bridge to those in the LMC. First, YSOs in the Bridge are apparently less embedded than in the LMC: 81% of Bridge YSOs show optical counterparts, compared to only 56% of LMC sources with the same range of mass, circumstellar dust mass, and line-of-sight extinction. Circumstellar envelopes are evidently more porous or clumpy in the Bridge's low-metallicity environment. Second, we have used whole samples of YSOs in the LMC and the Bridge to estimate the probability of finding YSOs at a given H I column density, N(H I). We found that the LMC has ∼3 × higher probability than the Bridge for N(H I) >12 × 10{sup 20} cm{sup –2}, but the trend reverses at lower N(H I). Investigating whether this lower efficiency relative to H I is due to less efficient molecular cloud formation or to less efficient cloud collapse, or to both, will require sensitive molecular gas observations.
18. Spitzer View of Massive Star Formation in the Tidally Stripped Magellanic Bridge
NASA Astrophysics Data System (ADS)
Chen, C.-H. Rosie; Indebetouw, Remy; Muller, Erik; Kawamura, Akiko; Gordon, Karl D.; Sewiło, Marta; Whitney, Barbara A.; Fukui, Yasuo; Madden, Suzanne C.; Meade, Marilyn R.; Meixner, Margaret; Oliveira, Joana M.; Robitaille, Thomas P.; Seale, Jonathan P.; Shiao, Bernie; van Loon, Jacco Th.
2014-04-01
The Magellanic Bridge is the nearest low-metallicity, tidally stripped environment, offering a unique high-resolution view of physical conditions in merging and forming galaxies. In this paper, we present an analysis of candidate massive young stellar objects (YSOs), i.e., in situ, current massive star formation (MSF) in the Bridge using Spitzer mid-IR and complementary optical and near-IR photometry. While we definitely find YSOs in the Bridge, the most massive are ~10 M ⊙, Lt45 M ⊙ found in the LMC. The intensity of MSF in the Bridge also appears to be decreasing, as the most massive YSOs are less massive than those formed in the past. To investigate environmental effects on MSF, we have compared properties of massive YSOs in the Bridge to those in the LMC. First, YSOs in the Bridge are apparently less embedded than in the LMC: 81% of Bridge YSOs show optical counterparts, compared to only 56% of LMC sources with the same range of mass, circumstellar dust mass, and line-of-sight extinction. Circumstellar envelopes are evidently more porous or clumpy in the Bridge's low-metallicity environment. Second, we have used whole samples of YSOs in the LMC and the Bridge to estimate the probability of finding YSOs at a given H I column density, N(H I). We found that the LMC has ~3 × higher probability than the Bridge for N(H I) >12 × 1020 cm-2, but the trend reverses at lower N(H I). Investigating whether this lower efficiency relative to H I is due to less efficient molecular cloud formation or to less efficient cloud collapse, or to both, will require sensitive molecular gas observations.
19. SINGLE-STAR H II REGIONS AS A PROBE OF MASSIVE STAR SPECTRAL ENERGY DISTRIBUTIONS
SciTech Connect
Zastrow, Jordan; Oey, M. S.; Pellegrini, E. W.
2013-06-01
The shape of OB-star ionizing spectral energy distributions (SEDs) is a critical component in many diagnostics of galaxy and interstellar medium properties. To quantitatively examine the shape of the OB-star SED, we compare long slit observations of single-star, Large Magellanic Cloud H II regions to the predictions from CLOUDY photoionization simulations that use CoStar, TLUSTY, and WM-basic stellar atmosphere models as the ionizing source. For each atmosphere model, we run grids of H II region simulations with the effective temperature (T{sub eff}) of the star as a free parameter. The best SEDs from each atmosphere code are found by matching the predicted emission-line spectra with those observed from the nebulae. By assuming a clumpy gas distribution, all atmosphere codes are able to reproduce the observed emission lines, except at the highest energy transitions {approx}> 40 eV. Taking into account both low and high energy transitions, we find that simulations using WM-basic produce the best agreement with the observed line ratios. The rates of ionizing photons from different atmosphere models vary systematically with the relative hardness of the SEDs. However, in general the rates produced by the model SEDs, for standard log(g) = 4.0 models, are consistent with the rates derived from the H{alpha} luminosities. We find that our effective temperatures inferred from the nebular ionization balance are consistent with those predicted by conventional photospheric-based calibrations from the literature. We suggest that future spectral type to T{sub eff} calibrations can be constructed from nebular data.
20. Mass loss from evolved massive stars: self-consistent modeling of the wind and photosphere
NASA Astrophysics Data System (ADS)
Groh, J. H.
2007-03-01
This work analyzes the mass loss phenomenon in evolved massive stars through self-consistent modeling of the wind and photosphere of such stars, using the radiative transfer code CMFGEN. In the first part, fundamental physical parameters of Wolf-Rayet stars of spectral types WN3-w (WR 46 e WR 152) and WN6-s (WR 136) were obtained. The results clearly indicate that hydrogen is present on the surface of those stars in a considerable fraction, defying current evolutionary models. For both WN subtypes, significant difference between the physical parameters obtained here and in previous works were noticed. The 20-year evolution of the luminous blue variable (LBV) AG Carinae was analyzed in detail in the second part of this work. The results indicate unexpected changes in the current paradigm of massive star evolution during the S Dor cycle. In this work, the high rotational velocity obtained during the hot phases, and the transition between the bistability regimes of line-driven winds were detected for the first time in LBVs. Those results need to be considered in future analysis of such massive stars. This Thesis also presents a pioneering study about the impact of the time variability effects on the analysis of the winds of LBVs. The results achieved here are valid for the whole LBV class, and show that the mass-loss rates derived from Hα and radio free-free emission are affected by time-dependent effects. The mass-loss rate evolution during the S Dor cycle, derived using time-dependent models, implies that LBV eruptions begin well before the maximum in the visual lightcurve during this phase. The analysis of the full S Dor cycle of AG Car rule out that the S Dor variability is caused exclusively by an expanding pseudo-photosphere. The AG Car hydrostatic radius was found to vary by a factor of six between cool and hot phases, while the bolometric luminosity is 50% higher during the hot phase. Both results provide observational contraints for the physical mechanism
1. The Schmidt Law in Six Galactic Massive Star-forming Regions
NASA Astrophysics Data System (ADS)
Willis, S.; Guzman, A.; Marengo, M.; Smith, H. A.; Martínez-Galarza, J. R.; Allen, L.
2015-08-01
We present a census of young stars in five massive star-forming regions in the 4th Galactic quadrant, G305, G326-4, G326-6, G333 (RCW 106), and G351, and combine this census with an earlier census of young stars in NGC 6334. Each region was observed at J, H, and Ks with the NOAO Extremely Wide-Field Infrared Imager and combined with deep observations taken with the Infrared Array Camera (IRAC) on board the Spitzer Space Telescope at the wavelengths 3.6 and 4.5 μm. We derived a five band point-source catalog containing >200,000 infrared sources in each region. We have identified a total of 2871 YSO candidates, 363 Class I YSOs, and 2508 Class II YSOs. We mapped the column density of each cloud using observations from Herschel between 160 and 500 μm and near-infrared extinction maps in order to determine the average gas surface density above AV > 2. We study the surface density of the YSOs and the star-formation rate as a function of the column density within each cloud and compare them to the results for nearby star-forming regions. We find a range in power-law indices across the clouds, with the dispersion in the local relations in an individual cloud much lower than the average over the six clouds. We find the average over the six clouds to be {{{Σ }}}{SFR}∼ {{{Σ }}}{gas}2.15+/- 0.41 and power-law exponents ranging from 1.77 to 2.86, similar to the values derived within nearby star-forming regions, including Taurus and Orion. The large dispersion in the power-law relations between individual Milky Way molecular clouds reinforces the idea that there is not a direct universal connection between Σgas and a cloud's observed star-formation rate.
2. HST/STIS Ultraviolet Spectroscopy of the Components of the Massive Triple Star δ Ori A
NASA Astrophysics Data System (ADS)
Richardson, Noel D.; Moffat, Anthony F. J.; Gull, Theodore R.; Lindler, Don J.; Gies, Douglas R.; Corcoran, Michael F.; Chené, André-Nicolas
2015-07-01
The multiple star system of δ Orionis is one of the closest examples of a system containing a luminous O-type, bright giant star (component Aa1). It is often used as a spectral-type standard and has the highest observed X-ray flux of any hot-star binary. The main component Aa1 is orbited by two lower mass stars, faint Aa2 in a 5.7 day eclipsing binary, and Ab, an astrometric companion with an estimated period of 346 years. Generally the flux from all three stars is recorded in ground-based spectroscopy, and the spectral decomposition of the components has proved difficult. Here we present Hubble Space Telescope/Space Telescope Imaging Spectrograph ultraviolet spectroscopy of δ Ori A that provides us with spatially separated spectra of Aa and Ab for the first time. We measured radial velocities for Aa1 and Ab in two observations made near the velocity extrema of Aa1. We show tentative evidence for the detection of the Aa2 component in cross-correlation functions of the observed and model spectra. We discuss the appearance of the UV spectra of Aa1 and Ab with reference to model spectra. Both stars have similar effective temperatures, but Ab is fainter and is a rapid rotator. The results will help in the interpretation of ground-based spectroscopy and in understanding the physical and evolutionary parameters of these massive stars. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program #13450.
3. Hydrodynamical simulations of the tidal stripping of binary stars by massive black holes
NASA Astrophysics Data System (ADS)
Mainetti, Deborah; Lupi, Alessandro; Campana, Sergio; Colpi, Monica
2016-04-01
In a galactic nucleus, a star on a low angular momentum orbit around the central massive black hole can be fully or partially disrupted by the black hole tidal field, lighting up the compact object via gas accretion. This phenomenon can repeat if the star, not fully disrupted, is on a closed orbit. Because of the multiplicity of stars in binary systems, also binary stars may experience in pairs such a fate, immediately after being tidally separated. The consumption of both the binary components by the black hole is expected to power a double-peaked flare. In this paper, we perform for the first time, with GADGET2, a suite of smoothed particle hydrodynamics simulations of binary stars around a galactic central black hole in the Newtonian regime. We show that accretion luminosity light curves from double tidal disruptions reveal a more prominent knee, rather than a double peak, when decreasing the impact parameter of the encounter and when elevating the difference between the mass of the star which leaves the system after binary separation and the mass of the companion. The detection of a knee can anticipate the onset of periodic accretion luminosity flares if one of the stars, only partially disrupted, remains bound to the black hole after binary separation. Thus knees could be precursors of periodic flares, which can then be predicted, followed up and better modelled. Analytical estimates in the black hole mass range 105-108 M⊙ show that the knee signature is enhanced in the case of black holes of mass 106-107 M⊙.
4. The exclusion of a significant range of ages in a massive star cluster.
PubMed
Li, Chengyuan; de Grijs, Richard; Deng, Licai
2014-12-18
Stars spend most of their lifetimes on the main sequence in the Hertzsprung-Russell diagram. The extended main-sequence turn-off regions--containing stars leaving the main sequence after having spent all of the hydrogen in their cores--found in massive (more than a few tens of thousands of solar masses), intermediate-age (about one to three billion years old) star clusters are usually interpreted as evidence of internal age spreads of more than 300 million years, although young clusters are thought to quickly lose any remaining star-forming fuel following a period of rapid gas expulsion on timescales of order 10(7) years. Here we report, on the basis of a combination of high-resolution imaging observations and theoretical modelling, that the stars beyond the main sequence in the two-billion-year-old cluster NGC 1651, characterized by a mass of about 1.7 × 10(5) solar masses, can be explained only by a single-age stellar population, even though the cluster has a clearly extended main-sequence turn-off region. The most plausible explanation for the existence of such extended regions invokes a population of rapidly rotating stars, although the secondary effects of the prolonged stellar lifetimes associated with such a stellar population mixture are as yet poorly understood. From preliminary analysis of previously obtained data, we find that similar morphologies are apparent in the Hertzsprung-Russell diagrams of at least five additional intermediate-age star clusters, suggesting that an extended main-sequence turn-off region does not necessarily imply the presence of a significant internal age dispersion. PMID:25519133
5. The exclusion of a significant range of ages in a massive star cluster.
PubMed
Li, Chengyuan; de Grijs, Richard; Deng, Licai
2014-12-18
Stars spend most of their lifetimes on the main sequence in the Hertzsprung-Russell diagram. The extended main-sequence turn-off regions--containing stars leaving the main sequence after having spent all of the hydrogen in their cores--found in massive (more than a few tens of thousands of solar masses), intermediate-age (about one to three billion years old) star clusters are usually interpreted as evidence of internal age spreads of more than 300 million years, although young clusters are thought to quickly lose any remaining star-forming fuel following a period of rapid gas expulsion on timescales of order 10(7) years. Here we report, on the basis of a combination of high-resolution imaging observations and theoretical modelling, that the stars beyond the main sequence in the two-billion-year-old cluster NGC 1651, characterized by a mass of about 1.7 × 10(5) solar masses, can be explained only by a single-age stellar population, even though the cluster has a clearly extended main-sequence turn-off region. The most plausible explanation for the existence of such extended regions invokes a population of rapidly rotating stars, although the secondary effects of the prolonged stellar lifetimes associated with such a stellar population mixture are as yet poorly understood. From preliminary analysis of previously obtained data, we find that similar morphologies are apparent in the Hertzsprung-Russell diagrams of at least five additional intermediate-age star clusters, suggesting that an extended main-sequence turn-off region does not necessarily imply the presence of a significant internal age dispersion.
6. Dynamics of dusty radiation-pressure-driven shells and clouds: fast outflows from galaxies, star clusters, massive stars, and AGN
NASA Astrophysics Data System (ADS)
Thompson, Todd A.; Fabian, Andrew C.; Quataert, Eliot; Murray, Norman
2015-05-01
It is typically assumed that radiation-pressure-driven winds are accelerated to an asymptotic velocity of v∞ ≃ vesc, where vesc is the escape velocity from the central source. We note that this is not the case for dusty shells and clouds. Instead, if the shell or cloud is initially optically thick to the UV emission from the source of luminosity L, then there is a significant boost in v∞ that reflects the integral of the momentum absorbed as it is accelerated. For shells reaching a generalized Eddington limit, we show that v∞ ≃ (4RUVL/Mshc)1/2, in both point-mass and isothermal-sphere potentials, where RUV is the radius where the shell becomes optically thin to UV photons, and Msh is the mass of the shell. The asymptotic velocity significantly exceeds vesc for typical parameters, and can explain the ˜1000-2000 km s-1 outflows observed from rapidly star-forming galaxies and active galactic nuclei (AGN) if the surrounding halo has low gas density. Similarly fast outflows from massive stars can be accelerated on ˜few-103 yr time-scales. These results carry over to clouds that subtend only a small fraction of the solid angle from the source of radiation and that expand as a consequence of their internal sound speed. We further consider the dynamics of shells that sweep up a dense circumstellar or circumgalactic medium. We calculate the momentum ratio' dot{M} v/(L/c) in the shell limit and show that it can only significantly exceed ˜2 if the effective optical depth of the shell to re-radiated far-infrared photons is much larger than unity. We discuss simple prescriptions for the properties of galactic outflows for use in large-scale cosmological simulations. We also briefly discuss applications to the dusty ejection episodes of massive stars, the disruption of giant molecular clouds, and AGN.
7. A search for kinematic proof of the triggering of massive star formation
NASA Astrophysics Data System (ADS)
Urquhart, James; Hoare, Melvin; Moore, Toby; van Loo, Sven; Morgan, Larry
2009-04-01
We propose to search for kinematic signatures that will prove that a shockwave has passed across regions that appear to be have been triggered into forming massive stars. Our sample selection starts from the massive young stellar objects in our Red MSX Source (RMS) survey that appear to be triggered from their morphology in the Spitzer GLIMPSE survey. These have a bright point source lying in a dark cloud that has strong PAH emission indicating interaction along one side. Recent modeling of magnetically supported clouds hit by an external shock show that a slow-mode wave continues on into the gas behind the dense collapsing clumps and should be observable as a 'smoking gun? of triggering. We propose single-dish observations to narrow down the best targets for future interferometric searches for this kinematic signature.
8. The Blob, the Very Rare Massive Star and the Two Populations
NASA Astrophysics Data System (ADS)
2005-04-01
The nebula N214 [1] is a large region of gas and dust located in a remote part of our neighbouring galaxy, the Large Magellanic Cloud. N214 is a quite remarkable site where massive stars are forming. In particular, its main component, N214C (also named NGC 2103 or DEM 293), is of special interest since it hosts a very rare massive star, known as Sk-71 51 [2] and belonging to a peculiar class with only a dozen known members in the whole sky. N214C thus provides an excellent opportunity for studying the formation site of such stars. Using ESO's 3.5-m New Technology telescope (NTT) located at La Silla (Chile) and the SuSI2 and EMMI instruments, astronomers from France and the USA [3] studied in great depth this unusual region by taking the highest resolution images so far as well as a series of spectra of the most prominent objects present. N214C is a complex of ionised hot gas, a so-called H II region [4], spreading over 170 by 125 light-years (see ESO PR Photo 12b/05). At the centre of the nebula lies Sk-71 51, the region's brightest and hottest star. At a distance of ~12 light-years north of Sk-71 51 runs a long arc of highly compressed gas created by the strong stellar wind of the star. There are a dozen less bright stars scattered across the nebula and mainly around Sk-71 51. Moreover, several fine, filamentary structures and fine pillars are visible. The green colour in the composite image, which covers the bulk of the N214C region, comes from doubly ionised oxygen atoms [5] and indicates that the nebula must be extremely hot over a very large extent. The Star Sk-71 51 decomposed ESO PR Photo 12c/05 ESO PR Photo 12c/05 The Cluster Around Sk-71 51 [Preview - JPEG: 400 x 620 pix - 189k] [Normal - JPEG: 800 x 1239 pix - 528k] Caption: ESO PR Photo 12c/05 shows a small field around the hot star Sk-71 51 as seen through the V filter. The left image shows a single frame after subtraction of the nebular background. The image quality - or seeing - is roughly 8.5 pixels
9. An x-ray study of massive star forming regions with CHANDRA
NASA Astrophysics Data System (ADS)
Wang, Junfeng
2007-08-01
Massive stars are characterized by powerful stellar winds, strong ultraviolet (UV) radiation, and consequently devastating supernovae explosions, which have a profound influence on their natal clouds and galaxy evolution. However, the formation and evolution of massive stars themselves and how their low-mass siblings are affected in the wind-swept and UV-radiation-dominated environment are not well understood. Much of the stellar populations inside of the massive star forming regions (MSFRs) are poorly studied in the optical and IR wavelengths because of observational challenges caused by large distance, high extinction, and heavy contamination from unrelated sources. Although it has long been recognized that X-rays open a new window to sample the young stellar populations residing in the MSFRs, the low angular resolution of previous generation X-ray telescopes has limited the outcome from such studies. The sensitive high spatial resolution X-ray observations enabled by the Chandra X- ray Observatory and the Advanced CCD Imaging Spectrometer (ACIS) have significantly improved our ability to study the X-ray-emitting populations in the MSFRs in the last few years. In this thesis, I analyzed seven high spatial resolution Chandra /ACIS images of two massive star forming complexes, namely the NGC 6357 region hosting the 1 Myr old Pismis 24 cluster (Chapter 3) and the Rosette Complex including the 2 Myr old NGC 2244 cluster immersed in the Rosette Nebula (Chapter 4), embedded clusters in the Rosette Molecular Cloud (RMC; Chapter 5), and a triggered cluster NGC 2237 (Chapter 6). The X-ray sampled stars were studied in great details. The unique power of X-ray selection of young stellar cluster members yielded new knowledge in the stellar populations, the cluster structures, and the star formation histories. The census of cluster members is greatly improved in each region. A large fraction of the X-ray detections have optical or near-infrared (NIR) stellar counterparts
10. ON THE EFFECTS OF OPTICALLY THICK GAS (DISKS) AROUND MASSIVE STARS
SciTech Connect
Kuiper, Rolf; Yorke, Harold W. E-mail: [email protected]
2013-02-15
Numerical simulations have shown that the often cited radiation pressure barrier to accretion onto massive stars can be circumvented, when the radiation field is highly anisotropic in the presence of a circumstellar accretion disk with high optical depth. Here, these studies of the so-called flashlight effect are expanded by including the opacity of the innermost dust-free but potentially optically thick gas regions around forming massive stars. In addition to frequency-dependent opacities for the dust grains, we use temperature- and density-dependent Planck and Rosseland mean opacities for the gas. The simulations show that the innermost dust-free parts of the accretion disks are optically thick to the stellar radiation over a substantial fraction of the solid angle above and below the disk's midplane. The temperature in the shielded disk region decreases faster with radius than in a comparison simulation with a lower constant gas opacity, and the dust sublimation front is shifted to smaller radii. The shielding by the dust-free gas in the inner disk thus contributes to an enhanced flashlight effect, which ultimately results in a smaller opening angle of the radiation pressure driven outflow and in a much longer timescale of sustained feeding of the circumstellar disk by the molecular cloud core. We conclude that it is necessary to properly account for the opacity of the inner dust-free disk regions around forming massive stars in order to correctly assess the effectiveness of the flashlight effect, the opening angle of radiation pressure driven outflows, and the lifetime and morphological evolution of the accretion disk.
11. Quasithermal neutrinos from rotating protoneutron stars born during core collapse of massive stars
NASA Astrophysics Data System (ADS)
Murase, Kohta; Dasgupta, Basudeb; Thompson, Todd A.
2014-02-01
Rotating and magnetized protoneutron stars may drive relativistic magnetocentrifugally accelerated winds as they cool immediately after core collapse. The wind fluid near the star is composed of neutrons and protons, and the neutrons become relativistic while collisionally coupled with the ions. Here, we argue that the neutrons in the flow eventually undergo inelastic collisions around the termination shock inside the stellar material, producing ˜0.1-1 GeV neutrinos, without relying on cosmic-ray acceleration mechanisms. Even higher-energy neutrinos may be produced via particle acceleration mechanisms. We show that Precision IceCube Next Generation Upgrade and Hyper-Kamiokande can detect such neutrinos from nearby core-collapse supernovae, by reducing the atmospheric neutrino background via coincident detection of MeV neutrinos or gravitational waves and optical observations. Detection of these GeV and/or higher-energy neutrinos would provide important clues to the physics of magnetic acceleration, nucleosynthesis, the relation between supernovae and gamma-ray bursts, and the properties of newly born neutron stars.
12. Evolution and explosion of the most massive asymptotic giant branch star
SciTech Connect
Takahashi, Koh; Umeda, Hideyuki; Yoshida, Takashi
2014-05-02
The most massive asymptotic giant branch (AGB) stars can form a critical mass of ONe core at its center. The collapse of such a critical ONe core may end up as an Electron Capture Supernova (ECSN). We have accomplished a progenitor calculation for ECSN for the first time in more than two decades and have updated a pre-explosion structure for this model. Some details for ONe core formation and important mechanisms for the core contraction are shown. We discuss how the envelope mass loss affects the predicted existence of ECSN, and what physics is needed to model for a plausible structure of ECSN progenitor.
13. Stability boundaries for massive stars in the sHR diagram
NASA Astrophysics Data System (ADS)
Saio, Hideyuki; Georgy, Cyril; Meynet, Georges
2015-01-01
Stability boundaries of radial pulsations in massive stars are compared with positions of variable and non-variable blue-supergiants in the spectroscopic HR (sHR) diagram (Langer & Kudritzki 2014), whose vertical axis is 4 log T eff - log g(= log L/M). Observational data indicate that variables tend to have higher L/M than non-variables in agreement with the theoretical prediction. However, many variable blue-supergiants are found to have values of L/M below the theoretical stability boundary; i.e., surface gravities seem to be too high by around 0.2-0.3 dex.
14. A circumstellar molecular gas structure associated with the massive young star Cepheus A-HW 2
NASA Technical Reports Server (NTRS)
Torrelles, Jose M.; Rodriguez, Luis F.; Canto, Jorge; Ho, Paul T. P.
1993-01-01
We report the detection via VLA-D observations of ammonia of a circumstellar high-density molecular gas structure toward the massive young star related to the object Cepheus A-HW 2, a firm candidate for the powering source of the high-velocity molecular outflow in the region. We suggest that the circumstellar molecular gas structure could be related to the circumstellar disk previously suggested from infrared, H2O, and OH maser observations. We consider as a plausible scenario that the double radio continuum source of HW 2 could represent the ionized inner part of the circumstellar disk, in the same way as proposed to explain the double radio source in L1551. The observed motions in the circumstellar molecular gas can be produced by bound motions (e.g., infall or rotation) around a central mass of about 10-20 solar masses (B0.5 V star or earlier).
15. Gas expulsion in massive star clusters?. Constraints from observations of young and gas-free objects
NASA Astrophysics Data System (ADS)
Krause, Martin G. H.; Charbonnel, Corinne; Bastian, Nate; Diehl, Roland
2016-03-01
Context. Gas expulsion is a central concept in some of the models for multiple populations and the light-element anti-correlations in globular clusters. If the star formation efficiency was around 30 per cent and the gas expulsion happened on the crossing timescale, this process could preferentially expel stars born with the chemical composition of the proto-cluster gas, while stars with special composition born in the centre would remain bound. Recently, a sample of extragalactic, gas-free, young massive clusters has been identified that has the potential to test the conditions for gas expulsion. Aims: We investigate the conditions required for residual gas expulsion on the crossing timescale. We consider a standard initial mass function and different models for the energy production in the cluster: metallicity-dependent stellar winds, radiation, supernovae and more energetic events, such as hypernovae, which are related to gamma ray bursts. The latter may be more energetic than supernovae by up to two orders of magnitude. Methods: We computed a large number of thin-shell models for the gas dynamics, and calculated whether the Rayleigh-Taylor instability is able to disrupt the shell before it reaches the escape speed. Results: We show that the success of gas expulsion depends on the compactness index of a star cluster C5 ≡ (M∗/ 105 M⊙)/(rh/ pc), with initial stellar mass M∗ and half-mass radius rh. For given C5, a certain critical, local star formation efficiency is required to remove the rest of the gas. Common stellar feedback processes may not lead to gas expulsion with significant loss of stars above C5 ≈ 1. Considering pulsar winds and hypernovae, the limit increases to C5 ≈ 30. If successful, gas expulsion generally takes place on the crossing timescale. Some observed young massive clusters have 1
16. Spectacular Spitzer images of the Trifid Nebula: Protostars in a young, massive-star-forming region
NASA Astrophysics Data System (ADS)
Rho, Jeonghee; Reach, W. T.; Lefloch, B.; Fazio, G.
2005-07-01
Spitzer IRAC and MIPS images of the Trifid Nebula (M20) reveal its spectacular appearance in infrared light, demonstrating its special evolutionary stage: recently-formed massive protostars and numerous young stars, including a single O star that illuminates the surrounding molecular cloud from which it formed and unveiling large-scale, filamentary dark clouds. The hot dust grains show contrasting infrared colors in shells, arcs, bow-shocks and dark cores. Multiple protostars, previously defined as Class 0 from dust continuum and molecular outflow observations, are revealed in the infrared within the cold dust continuum peaks TC3 and TC4. The cold dust continuum cores of TC1 and TC2 contain only one protostar each; the newly-discovered infrared protostar in TC2 is the driving source of the HH399 jet. The Spitzer color-color diagram allowed us to identify ~150 young stellar objects (YSO) and classify them into different evolutionary stages, and also revealed a new class of YSO which are bright at 24μm but with spectral energy distribution peaking at 5-8μm; we name these sources Hot excess'' YSO. Despite of expectation that Class 0 sources would be starless'' cores, the Spitzer images, with unprecedented sensitivity, uncover mid-infrared emission from these Class 0 protostars. The mid-infrared detections of Class 0 protostars show that the emission escapes the dense, cold envelope of young protostars; the mid-infrared emission cannot arise from the same location as the mm-wave emission, and instead must arise from a much smaller region with less intervening extinction to the central accretion. The presence of multiple protostars within the cold cores of Class 0 objects implies that clustering occurs at this early stage of star formation. The most massive stars are located at the center of the cluster and are formed simultaneously with low-mass stars. The angular and mass distributions of protostars within the dust cores imply that these early protostars are
17. Massive stars dying alone: The extremely remote environment of SN 2009ip
NASA Astrophysics Data System (ADS)
Smith, Nathan; Andrews, Jennifer E.; Mauerhan, Jon C.
2016-09-01
We present late-time Hubble Space Telescope (HST) images of the site of supernova (SN) 2009ip taken almost 3 yr after its bright 2012 luminosity peak. SN 2009ip is now slightly fainter in broad filters than the progenitor candidate detected by HST in 1999. The current source continues to be dominated by ongoing late-time CSM interaction that produces strong Hα emission and a weak pseudo-continuum, as found previously for 1-2 yr after explosion. The intent of these observations was to search for evidence of recent star formation in the local (˜1kpc; 10″) environment around SN 2009ip, in the remote outskirts of its host spiral galaxy NGC 7259. We can rule out the presence of any massive star-forming complexes like 30 Dor or the Carina Nebula at the SN site or within a few kpc. If the progenitor of SN 2009ip was really a 50-80 M⊙ star as archival HST images suggested, then it is strange that there is no sign of this type of massive star formation anywhere in the vicinity. A possible explanation is that the progenitor was the product of a merger or binary mass transfer, rejuvenated after a lifetime that was much longer than 4-5 Myr, allowing its natal H II region to have faded. A smaller region like the Orion Nebula would be an unresolved but easily detected point source. This is ruled out within ˜1.5 kpc around SN 2009ip, but a small H II region could be hiding in the glare of SN 2009ip itself. Later images after a few more years have passed are needed to confirm that the progenitor candidate is truly gone and to test for the possibility of a small H II region or cluster at the SN position.
18. Submillimeter Array Observations Toward the Massive Star-forming Core MM1 of W75N
NASA Astrophysics Data System (ADS)
Minh, Y. C.; Su, Y.-N.; Chen, H.-R.; Liu, S.-Y.; Yan, C.-H.; Kim, S.-J.
2010-11-01
The massive star-forming core MM1 of W75N was observed using the Submillimeter Array with ~1'' and 2'' spatial resolutions at 217 and 347 GHz, respectively. From the 217 GHz continuum we found that the MM1 core consists of two sources, separated by about 1'': MM1a (~0.6 M sun) and MM1b (~1.4 M sun), located near the radio continuum sources VLA 2/VLA 3 and VLA 1, respectively. Within MM1b, two gas clumps were found to be expanding away from VLA 1 at about ±3 km s-1, as a result of the most recent star formation activity in the region. Observed molecular lines show emission peaks at two positions, MM1a and MM1b: sulfur-bearing species have emission peaks toward MM1a, but methanol and saturated species at MM1b. We identified high-temperature (~200 K) gas toward MM1a and the hot core in MM1b. This segregation may result from the evolution of the massive star-forming core. In the very early phase of star formation, the hot core is seen through the evaporation of dust ice-mantle species. As the mantle species are consumed via evaporation the high-temperature gas species (such as the sulfur-bearing molecules) become bright. The SiO molecule is unique in having an emission peak exactly at the VLA 2 position, probably tracing a shock powered by VLA 2. The observed sulfur-bearing species show similar abundances both in MM1a and MM1b, whereas the methanol and saturated species show significant abundance enhancement toward MM1b, by about an order of magnitude, compared to MM1a.
19. s-process production in rotating massive stars at solar and low metallicities
NASA Astrophysics Data System (ADS)
Frischknecht, Urs; Hirschi, Raphael; Pignatari, Marco; Maeder, André; Meynet, George; Chiappini, Cristina; Thielemann, Friedrich-Karl; Rauscher, Thomas; Georgy, Cyril; Ekström, Sylvia
2016-02-01
Rotation was shown to have a strong impact on the structure and light element nucleosynthesis in massive stars. In particular, models including rotation can reproduce the primary nitrogen observed in halo extremely metal poor (EMP) stars. Additional exploratory models showed that rotation may enhance s-process production at low metallicity. Here we present a large grid of massive star models including rotation and a full s-process network to study the impact of rotation on the weak s-process. We explore the possibility of producing significant amounts of elements beyond the strontium peak, which is where the weak s-process usually stops. We used the Geneva stellar evolution code coupled to an enlarged reaction network with 737 nuclear species up to bismuth to calculate 15-40 M⊙ models at four metallicities (Z = 0.014, 10-3, 10-5 and 10-7) from the main sequence up to the end of oxygen burning. We confirm that rotation-induced mixing between the convective H-shell and He-core enables an important production of primary 14N and 22Ne and s-process at low metallicity. At low metallicity, even though the production is still limited by the initial number of iron seeds, rotation enhances the s-process production, even for isotopes heavier than strontium, by increasing the neutron-to-seed ratio. The increase in this ratio is a direct consequence of the primary production of 22Ne. Despite nuclear uncertainties affecting the s-process production and stellar uncertainties affecting the rotation-induced mixing, our results show a robust production of s-process at low metallicity when rotation is taken into account. Considering models with a distribution of initial rotation rates enables us to reproduce the observed large range of the [Sr/Ba] ratios in (carbon-enhanced and normal) EMP stars.
20. Massive pulsating stars observed by BRITE-Constellation. I. The triple system β Centauri (Agena)
NASA Astrophysics Data System (ADS)
Pigulski, A.; Cugier, H.; Popowicz, A.; Kuschnig, R.; Moffat, A. F. J.; Rucinski, S. M.; Schwarzenberg-Czerny, A.; Weiss, W. W.; Handler, G.; Wade, G. A.; Koudelka, O.; Matthews, J. M.; Mochnacki, St.; Orleański, P.; Pablo, H.; Ramiaramanantsoa, T.; Whittaker, G.; Zocłońska, E.; Zwintz, K.
2016-04-01
Context. Asteroseismology of massive pulsating stars of β Cep and SPB types can help us to uncover the internal structure of massive stars and understand certain physical phenomena that are taking place in their interiors. We study β Centauri (Agena), a triple system with two massive fast-rotating early B-type components which show p- and g-mode pulsations; the system's secondary is also known to have a measurable magnetic field. Aims: This paper aims to precisely determine the masses and detect pulsation modes in the two massive components of β Cen with BRITE-Constellation photometry. In addition, seismic models for the components are considered and the effects of fast rotation are discussed. This is done to test the limitations of seismic modeling for this very difficult case. Methods: A simultaneous fit of visual and spectroscopic orbits is used to self-consistently derive the orbital parameters, and subsequently the masses, of the components. Time-series analysis of BRITE-Constellation data is used to detect pulsation modes and derive their frequencies, amplitudes, phases, and rates of frequency change. Theoretically-predicted frequencies are calculated for the appropriate evolutionary models and their stability is checked. The effects of rotational splitting and coupling are also presented. Results: The derived masses of the two massive components are equal to 12.02 ± 0.13 and 10.58 ± 0.18 M⊙. The parameters of the wider, A-B system, presently approaching periastron passage, are constrained. Analysis of the combined blue- and red-filter BRITE-Constellation photometric data of the system revealed the presence of 19 periodic terms, of which eight are likely g modes, nine are p modes, and the remaining two are combination terms. It cannot be excluded that one or two low-frequency terms are rotational frequencies. It is possible that both components of β Cen are β Cep/SPB hybrids. An attempt to use the apparent changes of frequency to distinguish which
1. Rapid growth of black holes in massive star-forming galaxies.
PubMed
Alexander, D M; Smail, I; Bauer, F E; Chapman, S C; Blain, A W; Brandt, W N; Ivison, R J
2005-04-01
The tight relationship between the masses of black holes and galaxy spheroids in nearby galaxies implies a causal connection between the growth of these two components. Optically luminous quasars host the most prodigious accreting black holes in the Universe, and can account for greater than or approximately equal to 30 per cent of the total cosmological black-hole growth. As typical quasars are not, however, undergoing intense star formation and already host massive black holes (> 10(8)M(o), where M(o) is the solar mass), there must have been an earlier pre-quasar phase when these black holes grew (mass range approximately (10(6)-10(8))M(o)). The likely signature of this earlier stage is simultaneous black-hole growth and star formation in distant (redshift z > 1; >8 billion light years away) luminous galaxies. Here we report ultra-deep X-ray observations of distant star-forming galaxies that are bright at submillimetre wavelengths. We find that the black holes in these galaxies are growing almost continuously throughout periods of intense star formation. This activity appears to be more tightly associated with these galaxies than any other coeval galaxy populations. We show that the black-hole growth from these galaxies is consistent with that expected for the pre-quasar phase.
2. THE HCN/HNC ABUNDANCE RATIO TOWARD DIFFERENT EVOLUTIONARY PHASES OF MASSIVE STAR FORMATION
SciTech Connect
Jin, Mihwa; Lee, Jeong-Eun; Kim, Kee-Tae E-mail: [email protected]
2015-07-20
Using the H{sup 13}CN and HN{sup 13}C J = 1–0 line observations, the abundance ratio of HCN/HNC has been estimated for different evolutionary stages of massive star formation: infrared dark clouds (IRDCs), high-mass protostellar objects (HMPOs), and ultracompact H ii regions (UCH iis). IRDCs were divided into “quiescent IRDC cores (qIRDCc)” and “active IRDC cores (aIRDCc),” depending on star formation activity. The HCN/HNC ratio is known to be higher at active and high temperature regions related to ongoing star formation, compared to cold and quiescent regions. Our observations toward 8 qIRDCc, 16 aIRDCc, 23 HMPOs, and 31 UCH iis show consistent results; the ratio is 0.97 (±0.10), 2.65 (±0.88), 4.17 (±1.03), and 8.96 (±3.32) in these respective evolutionary stages, increasing from qIRDCc to UCH iis. The change of the HCN/HNC abundance ratio, therefore, seems directly associated with the evolutionary stages of star formation, which have different temperatures. One suggested explanation for this trend is the conversion of HNC to HCN, which occurs effectively at higher temperatures. To test the explanation, we performed a simple chemical model calculation. In order to fit the observed results, the energy barrier of the conversion must be much lower than the value provided by theoretical calculations.
3. Deviations from a uniform period spacing of gravity modes in a massive star.
PubMed
Degroote, Pieter; Aerts, Conny; Baglin, Annie; Miglio, Andrea; Briquet, Maryline; Noels, Arlette; Niemczura, Ewa; Montalban, Josefina; Bloemen, Steven; Oreiro, Raquel; Vucković, Maja; Smolders, Kristof; Auvergne, Michel; Baudin, Frederic; Catala, Claude; Michel, Eric
2010-03-11
The life of a star is dominantly determined by the physical processes in the stellar interior. Unfortunately, we still have a poor understanding of how the stellar gas mixes near the stellar core, preventing precise predictions of stellar evolution. The unknown nature of the mixing processes as well as the extent of the central mixed region is particularly problematic for massive stars. Oscillations in stars with masses a few times that of the Sun offer a unique opportunity to disentangle the nature of various mixing processes, through the distinct signature they leave on period spacings in the gravity mode spectrum. Here we report the detection of numerous gravity modes in a young star with a mass of about seven solar masses. The mean period spacing allows us to estimate the extent of the convective core, and the clear periodic deviation from the mean constrains the location of the chemical transition zone to be at about 10 per cent of the radius and rules out a clear-cut profile.
4. COLLAPSE OF MOLECULAR CLOUD CORES WITH RADIATION TRANSFER: FORMATION OF MASSIVE STARS BY ACCRETION
SciTech Connect
Sigalotti, Leonardo Di G.; Daza-Montero, Judith; De Felice, Fernando
2009-12-20
Most early radiative transfer calculations of protostellar collapse have suggested an upper limit of approx40 M{sub sun} for the final stellar mass before radiation pressure can exceed the star's gravitational pull and halt the accretion. Here we perform further collapse calculations, using frequency-dependent radiation transfer coupled to a frequency-dependent dust model that includes amorphous carbon particles, silicates, and ice-coated silicates. The models start from pressure-bounded, logatropic spheres of mass between 5 M{sub sun} and 150 M{sub sun} with an initial nonsingular density profile. We find that in a logatrope the infall is never reversed by the radiative forces on the dust and that stars with masses approx>100 M{sub sun} may form by continued accretion. Compared to previous models that start the collapse with a rho propor to r{sup -2} density configuration, our calculations result in higher accretion times and lower average accretion rates with peak values of approx5.8 x 10{sup -5} M{sub sun} yr{sup -1}. The radii and bolometric luminosities of the produced massive stars (approx>90 M{sub sun}) are in good agreement with the figures reported for detected stars with initial masses in excess of 100 M{sub sun}. The spectral energy distribution from the stellar photosphere reproduces the observed fluxes for hot molecular cores with peaks of emission from mid- to near-infrared.
5. A LIBRARY OF THEORETICAL ULTRAVIOLET SPECTRA OF MASSIVE, HOT STARS FOR EVOLUTIONARY SYNTHESIS
SciTech Connect
Leitherer, Claus; Ortiz Otalvaro, Paula A.; Bresolin, Fabio; Kudritzki, Rolf-Peter; Lo Faro, Barbara; Pauldrach, Adalbert W. A.; Pettini, Max; Rix, Samantha A. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
2010-08-15
We computed a comprehensive set of theoretical ultraviolet spectra of hot, massive stars with the radiation-hydrodynamics code WM-Basic. This model atmosphere and spectral synthesis code is optimized for computing the strong P Cygni type lines originating in the winds of hot stars, which are the strongest features in the ultraviolet spectral region. The computed set is suitable as a spectral library for inclusion in evolutionary synthesis models of star clusters and star-forming galaxies. The chosen stellar parameters cover the upper left Hertzsprung-Russell diagram at L {approx}> 10{sup 2.75} L {sub sun} and T {sub eff} {approx}> 20,000 K. The adopted elemental abundances are 0.05 Z {sub sun}, 0.2 Z {sub sun}, 0.4 Z {sub sun}, Z {sub sun}, and 2 Z {sub sun}. The spectra cover the wavelength range from 900 to 3000 A and have a resolution of 0.4 A. We compared the theoretical spectra to data of individual hot stars in the Galaxy and the Magellanic Clouds obtained with the International Ultraviolet Explorer and Far Ultraviolet Spectroscopic Explorer satellites and found very good agreement. We built a library with the set of spectra and implemented it into the evolutionary synthesis code Starburst99 where it complements and extends the existing empirical library toward lower chemical abundances. Comparison of population synthesis models at solar and near-solar composition demonstrates consistency between synthetic spectra generated with either library. We discuss the potential of the new library for the interpretation of the rest-frame ultraviolet spectra of star-forming galaxies. Properties that can be addressed with the models include ages, initial mass function, and heavy-element abundance. The library can be obtained both individually or as part of the Starburst99 package.
6. Arp 65 interaction debris: massive H I displacement and star formation
NASA Astrophysics Data System (ADS)
Sengupta, C.; Scott, T. C.; Paudel, S.; Saikia, D. J.; Dwarakanath, K. S.; Sohn, B. W.
2015-12-01
Context. Pre-merger tidal interactions between pairs of galaxies are known to induce significant changes in the morphologies and kinematics of the stellar and interstellar medium components. Large amounts of gas and stars are often found to be disturbed or displaced as tidal debris. This debris then evolves, sometimes forming stars and occasionally forming tidal dwarf galaxies. Here we present results from our H i study of Arp 65, an interacting pair hosting extended H i tidal debris. Aims: In an effort to understand the evolution of tidal debris produced by interacting pairs of galaxies, including in situ star and tidal dwarf galaxy formation, we are mapping H i in a sample of interacting galaxy pairs. The Arp 65 pair is the latest member of this sample to be mapped. Methods: Our resolved H i 21 cm line survey is being carried out using the Giant Metrewave Radio Telescope. We used our H i survey data as well as available SDSS optical, Spitzer infra-red and GALEX UV data to study the evolution of the tidal debris and the correlation of H i with the star-forming regions within it. Results: In Arp 65 we see a high impact pre-merger tidal interaction involving a pair of massive galaxies (NGC 90 and NGC 93) that have a stellar mass ratio of ~1:3. The interaction, which probably occurred ~1.0-2.5 × 108 yr ago, appears to have displaced a large fraction of the H i in NGC 90 (including the highest column density H i) beyond its optical disk. We also find extended on-going star formation in the outer disk of NGC 90. In the major star-forming regions, we find the H i column densities to be ~4.7 × 1020 cm-2 or lower. But no signature of star formation was found in the highest column density H i debris SE of NGC 90. This indicates conditions within the highest density H i debris remain hostile to star formation and it reaffirms that high H i column densities may be a necessary but not sufficient criterion for star formation.
7. Mass-loss predictions for evolved very metal-poor massive stars
NASA Astrophysics Data System (ADS)
Muijres, L.; Vink, J. S.; de Koter, A.; Hirschi, R.; Langer, N.; Yoon, S.-C.
2012-10-01
Context. The first couple of stellar generations may have been massive, of order 100 M⊙, and to have played a dominant role in galaxy formation and the chemical enrichment of the early Universe. Some fraction of these objects may have died as pair-instability supernovae or gamma-ray bursts. The winds of these stars may have played an important role in determining these outcomes. As the winds are driven by radiation pressure on spectral lines, their strengths are expected to vary with metallicity. Until now, most mass-loss predictions for metal-poor O-type stars have assumed a scaled-down solar-abundance pattern. However, Population III evolutionary tracks show significant surface enrichment through rotational mixing of CNO-processed material, because even metal-poor stars switch to CNO-burning early on. Aims: We address the question of whether the CNO surface enhanced self-enrichment in the first few generations of stars could impact their mass-loss properties. Methods: We employ Monte Carlo simulations to establish the local line-force and solve for the momentum equation of the stellar outflow, testing whether an outflow can actually be established by assessing the net acceleration at the sonic point of the flow. Stellar evolution models of rotating metal-poor stars are used to specify the surface chemical composition, focussing on the phases of early enrichment. Results: We find that the mass-loss rates of CNO enhanced metal-poor stars are higher than those of non-enriched stars, but they are much lower than those rates where the CNO abundance is included in the total abundance Z. Metal-poor stars hotter than ~50 000 K, in the metallicity range investigated here (with an initial metallicity Z ≲ 10-4) are found to have no wind, as the high-ionization species of the CNO elements have too few strong lines to drive an outflow. We present a heuristic formula that provides mass-loss estimates for CNO-dominated winds in relation to scaled-down solar abundances
8. An X-ray and radio study of the massive star-forming cluster IRAS 20126+4104
NASA Astrophysics Data System (ADS)
Montes, Virginie; Hofner, Peter; Anderson, Crystal; Rosero, Viviana
2015-08-01
Two main competitive theories intent to explain massive star formation: the turbulent core model, which is an extension of the low-mass star formation model (McKee & Tan 2003), and models involving competitive accretion or stellar collisions (Bonnell & Bate 2006). The characterization of the cluster in which massive stars remain can help discriminate between the two main scenarios of their formation.Until recently it was believed that massive stars were only formed in dense molecular clouds leading to a substantial cluster. However, a previous study of the massive star forming region IRAS 20126+4104 using Spitzer observations by Qiu et al. (2008), suggested that the massive protostar was isolated, and the region was showing no obvious cluster.Here we adopt a multiwavelength technique to characterize the stellar environment of the IRAS 20126+4104 region combining Chandra X-ray ACIS-I and VLA 6cm continuum observations, and near-infrared (2MASS) data of the region. We detected 150 X-ray sources in the ACIS-I field and 13 radio sources within the 9’.2 VLA primary beam. Associating X-ray sources with their near-infrared counterparts from the 2MASS catalog and a color study of those counterparts, allow us to determine the galactic foreground/background contamination, and we conclude that 90 X-ray sources are associated with the region.This study shows an increasing surface density of X-ray sources toward the massive protostar and a number of at least 42 YSOs within 1.2 pc distance from the massive protostar. This number is consistent with typical B-type stars clusters (Lada & Lada 2003).
9. The Blob, the Very Rare Massive Star and the Two Populations
NASA Astrophysics Data System (ADS)
2005-04-01
The nebula N214 [1] is a large region of gas and dust located in a remote part of our neighbouring galaxy, the Large Magellanic Cloud. N214 is a quite remarkable site where massive stars are forming. In particular, its main component, N214C (also named NGC 2103 or DEM 293), is of special interest since it hosts a very rare massive star, known as Sk-71 51 [2] and belonging to a peculiar class with only a dozen known members in the whole sky. N214C thus provides an excellent opportunity for studying the formation site of such stars. Using ESO's 3.5-m New Technology telescope (NTT) located at La Silla (Chile) and the SuSI2 and EMMI instruments, astronomers from France and the USA [3] studied in great depth this unusual region by taking the highest resolution images so far as well as a series of spectra of the most prominent objects present. N214C is a complex of ionised hot gas, a so-called H II region [4], spreading over 170 by 125 light-years (see ESO PR Photo 12b/05). At the centre of the nebula lies Sk-71 51, the region's brightest and hottest star. At a distance of ~12 light-years north of Sk-71 51 runs a long arc of highly compressed gas created by the strong stellar wind of the star. There are a dozen less bright stars scattered across the nebula and mainly around Sk-71 51. Moreover, several fine, filamentary structures and fine pillars are visible. The green colour in the composite image, which covers the bulk of the N214C region, comes from doubly ionised oxygen atoms [5] and indicates that the nebula must be extremely hot over a very large extent. The Star Sk-71 51 decomposed ESO PR Photo 12c/05 ESO PR Photo 12c/05 The Cluster Around Sk-71 51 [Preview - JPEG: 400 x 620 pix - 189k] [Normal - JPEG: 800 x 1239 pix - 528k] Caption: ESO PR Photo 12c/05 shows a small field around the hot star Sk-71 51 as seen through the V filter. The left image shows a single frame after subtraction of the nebular background. The image quality - or seeing - is roughly 8.5 pixels
10. An outburst from a massive star 40 days before a supernova explosion.
PubMed
Ofek, E O; Sullivan, M; Cenko, S B; Kasliwal, M M; Gal-Yam, A; Kulkarni, S R; Arcavi, I; Bildsten, L; Bloom, J S; Horesh, A; Howell, D A; Filippenko, A V; Laher, R; Murray, D; Nakar, E; Nugent, P E; Silverman, J M; Shaviv, N J; Surace, J; Yaron, O
2013-02-01
Some observations suggest that very massive stars experience extreme mass-loss episodes shortly before they explode as supernovae, as do several models. Establishing a causal connection between these mass-loss episodes and the final explosion would provide a novel way to study pre-supernova massive-star evolution. Here we report observations of a mass-loss event detected 40 days before the explosion of the type IIn supernova SN 2010mc (also known as PTF 10tel). Our photometric and spectroscopic data suggest that this event is a result of an energetic outburst, radiating at least 6 × 10(47) erg of energy and releasing about 10(-2) solar masses of material at typical velocities of 2,000 km s(-1). The temporal proximity of the mass-loss outburst and the supernova explosion implies a causal connection between them. Moreover, we find that the outburst luminosity and velocity are consistent with the predictions of the wave-driven pulsation model, and disfavour alternative suggestions.
11. Clumping Effects on Non-Thermal Particle Spectra in Massive Star Systems
SciTech Connect
Reimer, A.; /Stanford U., HEPL /KIPAC, Menlo Park
2007-11-09
Observational evidence exists that winds of massive stars are clumped. Many massive star systems are known as non-thermal particle production sites, as indicated by their synchrotron emission in the radio band. As a consequence they are also considered as candidate sites for non-thermal high-energy photon production up to gamma-ray energies. The present work considers the effects of wind clumpiness expected on the emitting relativistic particle spectrum in colliding wind systems, built up from the pool of thermal wind particles through diffusive particle acceleration, and taking into account inverse Compton and synchrotron losses. In comparison to a homogeneous wind, a clumpy wind causes flux variations of the emitting particle spectrum when the clump enters the wind collision region. It is found that the spectral features associated with this variability moves temporally from low to high energy bands with the time shift between any two spectral bands being dependent on clump size, filling factor, and the energy-dependence of particle energy gains and losses.
12. FEEDBACK FROM MASSIVE STARS AND GAS EXPULSION FROM PROTO-GLOBULAR CLUSTERS
SciTech Connect
Calura, F.; Romano, D.; D’Ercole, A.; Few, C. G.
2015-11-20
Globular clusters (GCs) are considerably more complex structures than previously thought, harboring at least two stellar generations that present clearly distinct chemical abundances. Scenarios explaining the abundance patterns in GCs mostly assume that originally the clusters had to be much more massive than today, and that the second generation of stars originates from the gas shed by stars of the first generation (FG). The lack of metallicity spread in most GCs further requires that the supernova-enriched gas ejected by the FG is completely lost within ∼30 Myr, a hypothesis never tested by means of three-dimensional hydrodynamic simulations. In this paper, we use 3D hydrodynamic simulations including stellar feedback from winds and supernovae, radiative cooling and self-gravity to study whether a realistic distribution of OB associations in a massive proto-GC of initial mass M{sub tot} ∼ 10{sup 7} M{sub ⊙} is sufficient to expel its entire gas content. Our numerical experiment shows that the coherence of different associations plays a fundamental role: as the bubbles interact, distort, and merge, they carve narrow tunnels that reach deeper and deeper toward the innermost cluster regions, and through which the gas is able to escape. Our results indicate that after 3 Myr, the feedback from stellar winds is responsible for the removal of ∼40% of the pristine gas, and that after 14 Myr, 99% of the initial gas mass has been removed.
13. THE THREE-DIMENSIONAL EVOLUTION TO CORE COLLAPSE OF A MASSIVE STAR
SciTech Connect
Couch, Sean M.; Chatzopoulos, Emmanouil; Arnett, W. David; Timmes, F. X.
2015-07-20
We present the first three-dimensional (3D) simulation of the final minutes of iron core growth in a massive star, up to and including the point of core gravitational instability and collapse. We capture the development of strong convection driven by violent Si burning in the shell surrounding the iron core. This convective burning builds the iron core to its critical mass and collapse ensues, driven by electron capture and photodisintegration. The non-spherical structure and motion generated by 3D convection is substantial at the point of collapse, with convective speeds of several hundreds of km s{sup −1}. We examine the impact of such physically realistic 3D initial conditions on the core-collapse supernova mechanism using 3D simulations including multispecies neutrino leakage and find that the enhanced post-shock turbulence resulting from 3D progenitor structure aids successful explosions. We conclude that non-spherical progenitor structure should not be ignored, and should have a significant and favorable impact on the likelihood for neutrino-driven explosions. In order to make simulating the 3D collapse of an iron core feasible, we were forced to make approximations to the nuclear network making this effort only a first step toward accurate, self-consistent 3D stellar evolution models of the end states of massive stars.
14. Evolution of Massive Stars Up to the End of Central Oxygen Burning
NASA Astrophysics Data System (ADS)
El Eid, M. F.; Meyer, B. S.; The, L.-S.
2004-08-01
We present a detailed study of the evolution of massive stars of masses 15, 20, 25, and 30 Msolar assuming solar-like initial chemical composition. The stellar sequences were evolved through the advanced burning phases up to the end of core oxygen burning. We present a careful analysis of the physical characteristics of the stellar models. In particular, we investigate the effect of the still-unsettled reaction 12C(α,γ)16O on the advanced evolution by using recent compilations of this rate. We find that this rate has a significant impact on the evolution not only during the core helium burning phase but also during the late burning phases, especially the shell carbon burning. We have also considered the effect of different treatments of convective instability based on the Ledoux criterion in regions of varying molecular weight gradient during the hydrogen- and helium-burning phases. We compare our results with other investigations whenever available. Finally, our present study constitutes the basis of analyzing the nucleosynthesis processes in massive stars. In particular, we will present a detailed analysis of the s-process in a forthcoming paper.
15. Explosive nucleosynthesis of N15 in a massive-star model
NASA Astrophysics Data System (ADS)
Bojazi, Michael J.; Meyer, Bradley S.
2014-02-01
Background: Presolar meteoritic graphite grains from supernovas show spatially correlated excesses in N15 and O18. These excesses signal the helium-rich layers of supernova ejecta as important source material for the grains. Purpose: Elucidate the explosive nucleosynthesis of N15 in massive stars, especially during shock passage through the helium-rich stellar layers. Method: A simple but realistic model of shock passage through the outer layers of exploding massive stars is used to follow the important N15 nucleosynthesis production pathways and their sensitivity to explosion energy and governing reaction rates in a particular stellar model. All calculations are performed with open-source, freely available codes. Results: Recent reaction rate updates tend to decrease by ˜4× the explosive helium-burning yield of N15 relative to some commonly used stellar model outputs. Conclusions: Neutron-capture reactions on F18 play an important role in the explosive production of N15 in helium-rich stellar layers. This neutron-induced nucleosynthesis is likely connected to that of other isotopic signatures in presolar supernova grains. The Supplemental Material provides instructions that interested readers can follow for their own calculations of explosive nucleosynthesis and nuclear reaction rate sensitivities.
16. Probing the Final Stages of Massive Star Evolution With Type IIn Supernovae
NASA Astrophysics Data System (ADS)
Fox, Ori
2013-06-01
Type IIn supernovae (SNe IIn), defined by their dense circumstellar medium (CSM), have gained considerable attention over the past decade given their association with massive star progenitors. Due to the nature of the dense CSM, many SNe IIn have been linked to the eruptive state of Luminous Blue Variables (LBVs), but the identification of a single progenitor class remains ambiguous. The pre-SN mass-loss history of SNe IIn must be better understood since it is the progenitors smoking gun, serving as a direct probe of the late stages of massive star evolution. Differences in wind speeds, densities, compositions, and asymmetries result in distinguishable observational behaviors. Many SNe IIn observations have been obtained at relatively early epochs (<100 days), which provides only an instantaneous snapshot of the CSM characteristics at small radii. Shock interaction and dust formation in the dense CSM, however, often result in significant emission ranging from X-ray to radio for many years post-explosion. Here I will present recent observations of the diverse late-time (>100 days) multi-wavelength evolution of SNe IIn, tracing the complete mass-loss history of the progenitors out to larger radii.
17. Can 55Co Give us the Desired Prompt Explosion of Massive STARS?
NASA Astrophysics Data System (ADS)
Jameel-Un-Nabi
2007-04-01
Core collapse simulators are striving hard to achieve prompt explosion of a collapsing core of a massive star. Various parameters need to be fed into the simulation code before results of such a complex problem emerge. The most important nuclear physics input parameters to such codes include weak interaction rates (electron capture and beta decay rates) of key nuclides. So far, the weak rates fed into the code resulted in an undesired delayed explosion. Simulators attribute this result partly to somewhat suppressed electron capture rates of these key nuclides. Recently I calculated electron capture rates of these key nuclides using the proton-neutron quasi-particle random phase approximation (pn-QRPA) theory in a microscopic fashion. The microscopic results of QRPA certainly yield more enhanced rates for these nuclides. 55Co is not only present in abundance in presupernova phase but is also advocated to play a decisive role in the core collapse of massive stars. The important question to ask is "can QRPA rates contribute to triggering a prompt explosion?"
18. OUTFLOWS AND MASSIVE STARS IN THE PROTOCLUSTER IRAS 05358+3543
SciTech Connect
Ginsburg, Adam G.; Bally, John; Yan Chihung; Williams, Jonathan P. E-mail: [email protected]
2009-12-10
We present new near-IR H{sub 2}, CO J = 2-1, and CO J = 3-2 observations to study outflows in the massive star-forming region IRAS 05358+3543. The Canada-France-Hawaii Telescope H{sub 2} images and James Clerk Maxwell Telescope CO data cubes of the IRAS 05358 region reveal several new outflows, most of which emerge from the dense cluster of submillimeter cores associated with the Sh 2-233IR NE cluster to the northeast of IRAS 05358. We used Apache Point Observatory JHK spectra to determine line-of-sight velocities of the outflowing material. Analysis of archival Very Large Array cm continuum data and previously published very long baseline interferometry observations reveal a massive star binary as a probable source of one or two of the outflows. We have identified probable sources for six outflows and candidate counterflows for seven out of a total of 11 seen to be originating from the IRAS 05358 clusters. We classify the clumps within Sh 2-233IR NE as an early protocluster and Sh 2-233IR SW as a young cluster, and conclude that the outflow energy injection rate approximately matches the turbulent decay rate in Sh 2-233IR NE.
19. Core-Halo Structure of a Chemically Homogeneous Massive Star and Bending of the Zero-Age Main Sequence
NASA Astrophysics Data System (ADS)
Ishii, Mie; Ueno, Munetaka; Kato, Mariko
1999-08-01
We have recalculated the interior structure of very massive stars of uniform chemical composition with the OPAL opacity. Very massive stars are found to develop a core-halo structure with an extended radiative-envelope. With the core-halo structure, because a more massive star has a more extended envelope, the track of the upper zero-age main-sequence (ZAMS) curves redward in the H-R diagram at > 100 MO (Z=0.02), >70 MO (Z=0.05), and > 15 MO for helium ZAMS (X=0, Z=0.02). Therefore, the effective temperatures of very massive ZAMS stars are rather low: e.g., for a 200 MO star, log T_eff=4.75 (Z=0.004), 4.60 (Z=0.02), 4.46 (Z=0.05), and 4.32 (Z=0.10). The effective temperatures of very luminous stars (> 120 MO ) found in the LMC, the SMC, and the Galaxy are discussed in relation to this metal dependence of a curving upper main-sequence.
20. Down-regulation of T-STAR, a growth inhibitory protein, after SV40-mediated immortalization.
PubMed
Kool, J; van Zaane, W; van der Eb, A J; Terleth, C
2001-11-01
Normal human cells can undergo a limited number of divisions, whereas transformed cells may have an extended life span and can give rise to immortal cells. To isolate genes involved in the immortalization process, gene expression in SV40-transformed preimmortal human fibroblasts was compared with expression in SV40-transformed immortalized fibroblasts using an mRNA differential display. We found that the growth-inhibitory protein testis-signal transduction and activation of RNA (T-STAR) a homologue of cell-cycle regulator Sam68, is strongly down-regulated in immortalized cells. Overexpression of T-STAR in the SV40-transformed immortalized cells resulted in a strong reduction of colony formation, whereas deletion of the RNA-binding domain of T-STAR abrogated this effect. Down-regulation of testis-signal transduction and activation of RNA (T-STAR) expression is found only in immortal cells isolated after a proliferative crisis accompanied with massive cell death. The strict correlation of down-regulation of T-STAR expression only in those immortal cells that arose after a clear proliferative crisis suggests that the loss of T-STAR might be necessary to bypass crisis. PMID:11714634
1. Influence of the weakly interacting light U boson on the properties of massive protoneutron stars
NASA Astrophysics Data System (ADS)
Hong, Bin; Jia, Huan-Yu; Mu, Xue-Ling; Zhou, Xia
2016-06-01
Considering the octet baryons in relativistic mean field theory and selecting entropy per baryon S=1, we calculate and discuss the influence of U bosons on the equation of state, mass-radius, moment of inertia and gravitational redshift of massive protoneutron stars (PNSs). The effective coupling constant g U of U bosons and nucleons is selected from 0 to 70 GeV-2. The results indicate that U bosons will stiffen the equation of state (EOS). The influence of U bosons on the pressure is more obvious at low density than high density, while the influence of U bosons on the energy density is more obvious at high density than low density. The U bosons play a significant role in increasing the maximum mass and radius of PNS. When the value of g U changes from 0 to 70 GeV-2, the maximum mass of a massive PNS increases from 2.11M ⊙ to 2.58M ⊙, and the radius of a PNS corresponding to PSR J0348+0432 increases from 13.71 km to 24.35 km. The U bosons will increase the moment of inertia and decrease the gravitational redshift of a PNS. For the PNS of the massive PSR J0348+0432, the radius and moment of inertia vary directly with g U, and the gravitational redshift varies approximately inversely with g U. Supported by National Natural Science Foundation of China (11175147)
2. Massive star formation in Wolf-Rayet galaxies. I. Optical and NIR photometric results
NASA Astrophysics Data System (ADS)
López-Sánchez, Á. R.; Esteban, C.
2008-11-01
Aims: We have performed a comprehensive multiwavelength analysis of a sample of 20 starburst galaxies that show the presence of a substantial population of massive stars. The main aims are the study of the massive star formation and stellar populations in these galaxies, and the role that interactions with or between dwarf galaxies and/or low surface companion objects have in triggering the bursts. In this series of papers, we present our new optical and near-infrared photometric and spectroscopic observations, and complete with data at other wavelengths (X-ray, far-infrared, and radio) available in the literature. In this paper, the first in the series, we analyze the morphology, stellar population age, and star-formation rate of each system. Methods: We completed new deep optical and NIR broad-band images, as well as the new continuum-subtracted Hα maps, of our sample of Wolf-Rayet galaxies. We analyze the morphology of each system and its surroundings and quantify the photometric properties of all important objects. All data were corrected for both extinction and nebular emission using our spectroscopic data. The age of the most recent star-formation burst is estimated and compared with the age of the underlying older low-luminosity population. The Hα-based star-formation rate, number of O7V equivalent stars, mass of ionized gas, and mass of the ionizing star cluster are also derived. Results: We found interaction features in many (15 up to 20) of the analyzed objects, which were extremely evident in the majority. We checked that the correction for nebular emission to the broad-band filter fluxes is important in compact objects and/or with intense nebular emission to obtain realistic colors and compare with the predictions of evolutionary synthesis models. The estimate of the age of the most recent star-formation burst is derived consistently. In general, the Hα-based star formation rate agrees with the estimates given by independent multiwavelength methods
3. THE COEVOLUTION OF NUCLEAR STAR CLUSTERS, MASSIVE BLACK HOLES, AND THEIR HOST GALAXIES
SciTech Connect
Antonini, Fabio; Barausse, Enrico; Silk, Joseph
2015-10-10
Studying how nuclear star clusters (NSCs) form and how they are related to the growth of the central massive black holes (MBHs) and their host galaxies is fundamental for our understanding of the evolution of galaxies and the processes that have shaped their central structures. We present the results of a semi-analytical galaxy formation model that follows the evolution of dark matter halos along merger trees, as well as that of the baryonic components. This model allows us to study the evolution of NSCs in a cosmological context, by taking into account the growth of NSCs due to both dynamical-friction-driven migration of stellar clusters and star formation triggered by infalling gas, while also accounting for dynamical heating from (binary) MBHs. We find that in situ star formation contributes a significant fraction (up to ∼80%) of the total mass of NSCs in our model. Both NSC growth through in situ star formation and that through star cluster migration are found to generate NSC—host galaxy scaling correlations that are shallower than the same correlations for MBHs. We explore the role of galaxy mergers on the evolution of NSCs and show that observational data on NSC—host galaxy scaling relations provide evidence of partial erosion of NSCs by MBH binaries in luminous galaxies. We show that this observational feature is reproduced by our models, and we make predictions about the NSC and MBH occupation fraction in galaxies. We conclude by discussing several implications for theories of NSC formation.
4. Chemical Pollution and Evolution of Massive Starbursts: Cleaning up the Environment in Star-Forming Galaxies
NASA Astrophysics Data System (ADS)
Kobulnicky, C.
1996-12-01
I present the results of a research program seeking to characterize the impact of massive star-clusters on the chemical and dynamical evolution of metal-poor, irregular and blue compact galaxies. The evolution of high mass stars is thought to contribute the bulk of heavy element enrichment in the interstellar medium, especially alpha -process elements like O, Si, etc. Yet, in actively star-forming galaxies, localized chemical inhomogeneities are seldom observed. Spatially-resolved optical and ultraviolet spectroscopy from the Hubble Space Telescope and ground-based observatories is used to search for chemical enrichment in the vicinity of young star clusters in nearby galaxies. VLA aperture synthesis maps are used to examine the neutral hydrogen content, dynamics, and local environment of the sample galaxies. Despite the spread in evolutionary state of the starbursts determined by the EW of Balmer emission lines and the radio continuum spectral index, few instances of localized enrichment are found. In light of these data, the instantaneous enrichment'' scenario for extragalactic HII regions appears less probable than one which operates on long timescales and global spatial scales. The results are consistent with the idea that starburst driven winds expel freshly synthesized metals in a hot 10(6) K phase into the halos of galaxies where they cool, condense into globules, and mix homogeneously with the rest of the galaxy on long (dynamical) timescales. The C/O and N/O ratios of the galaxies are used as new tools for measuring the recent star formation history. Implications for chemical evolution of galaxies both locally and cosmologically are developed.
5. Dust Heating By Low-mass Stars in Massive Galaxies at z< 1
NASA Astrophysics Data System (ADS)
Kajisawa, M.; Morishita, T.; Taniguchi, Y.; Kobayashi, M. A. R.; Ichikawa, T.; Fukui, Y.
2015-03-01
Using the Hubble Space Telescope/Wide Field Camera 3 imaging data and multi-wavelength photometric catalog, we investigated the dust temperature of passively evolving and star-forming galaxies at 0.2\\lt z\\lt 1.0 in the CANDELS fields. We estimated the stellar radiation field by low-mass stars from the stellar mass and surface brightness profile of these galaxies and then calculated their steady-state dust temperature. At first, we tested our method using nearby early-type galaxies with the deep far-IR data by the Herschel Virgo cluster survey and confirmed that the estimated dust temperatures are consistent with the observed temperatures within the uncertainty. We then applied the method to galaxies at 0.2\\lt z\\lt 1.0, and found that most passively evolving galaxies with {{M}star}\\gt {{10}10} {{M}⊙ } have relatively high dust temperatures of {{T}dust}\\gt 20 K, for which the formation efficiency of molecular hydrogen on the surface of dust grains in the diffuse ISM is expected to be very low from the laboratory experiments. The fraction of passively evolving galaxies strongly depends on the expected dust temperature at all redshifts and increases rapidly increasing temperature around {{T}dust}˜ 20 K. These results suggest that the dust heating by low-mass stars in massive galaxies plays an important role in the continuation of their passive evolution because the lack of the shielding effect of the molecular hydrogen on the UV radiation can prevent the gas cooling and formation of new stars.
6. Linking star formation and galaxy kinematics in the massive cluster Abell 2163
NASA Astrophysics Data System (ADS)
Menacho, Veronica; Verdugo, Miguel
2015-02-01
The origin of the morphology-density relation is still an open question in galaxy evolution. It is most likely driven by the combination of the efficient star formation in the highest peaks of the mass distribution at high-z and the transformation by environmental processes at later times as galaxies fall into more massive halos. To gain additional insights about these processes we study the kinematics, star formation and structural properties of galaxies in Abell 2163 a very massive (~4×1015 M⊙, Holz & Perlmutter 2012) merging cluster at z = 0.2. We use high resolution spectroscopy with VLT/VIMOS to derive rotation curves and dynamical masses for galaxies that show regular kinematics. Galaxies that show irregular rotation are also analysed to study the origin of their distortion. This information is combined with stellar masses and structural parameters obtained from high quality CFHT imaging. From narrow band photometry (2.2m/WFI), centered on the redshifted Hα line, we obtain star formation rates. Although our sample is still small, field and cluster galaxies lie in a similar Tully-Fisher relation as local galaxies. Controlling by additional parameters like SFRs or bulge-to-disk ratio do not affect this result. We find however that ~50% of the cluster galaxies display irregular kinematics in contrast to what is found in the field at similar redshifts (~30%, Böhm et al. 2004) and in agreement with other studies in clusters (e.g. Bösch et al. 2013, Kutdemir et al. 2010) which points out to additional processes operating in clusters that distort the galaxy kinematics.
7. The formation of massive primordial stars in the presence of moderate UV backgrounds
SciTech Connect
Latif, M. A.; Schleicher, D. R. G.; Bovino, S.; Grassi, T.; Spaans, M.
2014-09-01
Radiative feedback produced by stellar populations played a vital role in early structure formation. In particular, photons below the Lyman limit can escape the star-forming regions and produce a background ultraviolet (UV) flux, which consequently may influence the pristine halos far away from the radiation sources. These photons can quench the formation of molecular hydrogen by photodetachment of H{sup –}. In this study, we explore the impact of such UV radiation on fragmentation in massive primordial halos of a few times 10{sup 7} M {sub ☉}. To accomplish this goal, we perform high resolution cosmological simulations for two distinct halos and vary the strength of the impinging background UV field in units of J {sub 21} assuming a blackbody radiation spectrum with a characteristic temperature of T {sub rad} = 10{sup 4} K. We further make use of sink particles to follow the evolution for 10,000 yr after reaching the maximum refinement level. No vigorous fragmentation is observed in UV-illuminated halos while the accretion rate changes according to the thermal properties. Our findings show that a few 10{sup 2}-10{sup 4} solar mass protostars are formed when halos are irradiated by J {sub 21} = 10-500 at z > 10 and suggest a strong relation between the strength of the UV flux and mass of a protostar. This mode of star formation is quite different from minihalos, as higher accretion rates of about 0.01-0.1 M {sub ☉} yr{sup –1} are observed by the end of our simulations. The resulting massive stars are potential cradles for the formation of intermediate-mass black holes at earlier cosmic times and contribute to the formation of a global X-ray background.
8. Radiation-hydrodynamic Simulations of Massive Star Formation with Protostellar Outflows
NASA Astrophysics Data System (ADS)
Cunningham, Andrew J.; Klein, Richard I.; Krumholz, Mark R.; McKee, Christopher F.
2011-10-01
9. Radiation-Hydrodynamic Simulations of Massive Star Formation with Protostellar Outflows
SciTech Connect
Cunningham, A J; Klein, R I; Krumholz, M R; McKee, C F
2011-03-02
10. Spectacular Spitzer Images of the Trifid Nebula: Protostars in a Young, Massive-Star-forming Region
NASA Astrophysics Data System (ADS)
Rho, Jeonghee; Reach, William T.; Lefloch, Bertrand; Fazio, Giovanni G.
2006-06-01
Spitzer IRAC and MIPS images of the Trifid Nebula (M20) reveal its spectacular appearance in infrared light, highlighting the nebula's special evolutionary stage. The images feature recently formed massive protostars and numerous young stellar objects, and a single O star that illuminates the surrounding molecular cloud from which it formed, and unveil large-scale, filamentary dark clouds. Multiple protostars are detected in the infrared, within the cold dust cores of TC3 and TC4, which were previously defined as Class 0. The cold dust continuum cores of TC1 and TC2 contain only one protostar each. The Spitzer color-color diagram allowed us to identify ~160 young stellar objects (YSOs) and classify them into different evolutionary stages. The diagram also revealed a unique group of YSOs that are bright at 24 μm but have the spectral energy distribution peaking at 5-8 μm. Despite expectation that Class 0 sources would be starless'' cores, the Spitzer images, with unprecedented sensitivity, uncover mid-infrared emission from these Class 0 protostars. The mid-infrared detections of Class 0 protostars show that the emission escapes the dense, cold envelope of young protostars. The mid-infrared emission of the protostars can be fit by two temperatures of 150 and 400 K; the hot core region is probably optically thin in the mid-infrared regime, and the size of hot core is much smaller than that of the cold envelope. The presence of multiple protostars within the cold cores of Class 0 objects implies that clustering occurs at this early stage of star formation. The most massive star in the TC3 cluster is located at the center of the cluster and at the bottom of the gravitational potential well.
11. THE IMPACT OF INTERACTIONS, BARS, BULGES, AND ACTIVE GALACTIC NUCLEI ON STAR FORMATION EFFICIENCY IN LOCAL MASSIVE GALAXIES
SciTech Connect
Saintonge, Amelie; Fabello, Silvia; Wang Jing; Catinella, Barbara; Tacconi, Linda J.; Genzel, Reinhard; Gracia-Carpio, Javier; Wuyts, Stijn; Kramer, Carsten; Moran, Sean; Heckman, Timothy M.; Schiminovich, David; Schuster, Karl
2012-10-20
Using atomic and molecular gas observations from the GASS and COLD GASS surveys and complementary optical/UV data from the Sloan Digital Sky Survey and the Galaxy Evolution Explorer, we investigate the nature of the variations in the molecular gas depletion time observed across the local massive galaxy population. The large and unbiased COLD GASS sample allows us for the first time to statistically assess the relative importance of galaxy interactions, bar instabilities, morphologies, and the presence of active galactic nuclei (AGNs) in regulating star formation efficiency. We find that both the H{sub 2} mass fraction and depletion time vary as a function of the distance of a galaxy from the main sequence traced by star-forming galaxies in the SFR-M {sub *} plane. The longest gas depletion times are found in below-main-sequence bulge-dominated galaxies ({mu}{sub *} >5 Multiplication-Sign 10{sup 8} M {sub Sun} kpc{sup -2}, C > 2.6) that are either gas-poor (M{sub H{sub 2}}/M {sub *} <1.5%) or else on average less efficient by a factor of {approx}2 than disk-dominated galaxies at converting into stars any cold gas they may have. We find no link between the presence of AGNs and these long depletion times. In the regime where galaxies are disk-dominated and gas-rich, the galaxies undergoing mergers or showing signs of morphological disruptions have the shortest molecular gas depletion times, while those hosting strong stellar bars have only marginally higher global star formation efficiencies as compared to matched control samples. Our interpretation is that the molecular gas depletion time variations are caused by changes in the ratio between the gas mass traced by the CO(1-0) observations and the gas mass in high-density star-forming cores (as traced by observations of, e.g., HCN(1-0)). While interactions, mergers, and bar instabilities can locally increase pressure and raise the ratio of efficiently star-forming gas to CO-detected gas (therefore lowering the CO
12. A new massive double-lined spectroscopic binary system: The Wolf-Rayet star WR 68a
NASA Astrophysics Data System (ADS)
Collado, A.; Gamen, R.; Barbá, R. H.; Morrell, N.
2015-09-01
Double-lined spectroscopic binary systems, containing a Wolf-Rayet and a massive O-type star, are key objects for the study of massive star evolution because these kinds of systems allow the determination of fundamental astrophysical parameters of their components. We have performed spectroscopic observations of the star WR 68a as part of a dedicated monitoring program of WR stars to discover new binary systems. We identified spectral lines of the two components of the system and disentangled the spectra. We measured the radial velocities in the separated spectra and determined the orbital solution. We discovered that WR 68a is a double-lined spectroscopic binary with an orbital period of 5.2207 days, very small or null eccentricity, and inclination ranging between 75 and 85 deg. We classified the binary components as WN6 and O5.5-6. The WN star is less massive than the O-type star with minimum masses of 15 ± 5 M⊙ and 30 ± 4 M⊙, respectively. The equivalent width of the He ii λ4686 emission line shows variations with the orbital phase, presenting a minimum when the WN star is in front of the system. The light curve constructed from available photometric data presents minima in both conjunctions of the system. Table 2 is available in electronic form at http://www.aanda.org
13. Peering to the Heart of Massive Star Birth - IV. Surveying Across Evolution, Environment and the IMF
NASA Astrophysics Data System (ADS)
Tan, Jonathan
2014-10-01
We propose to continue our Cycle 2 survey of MIR/FIR (10-40 micron) emission from massive protostars, utilizing the unique capabilities of SOFIA-FORCAST. We have demonstrated theoretically and observationally that 10-40 micron observations are crucial for defining the spectral energy distribution of massive protostars and thus the bolometric flux directed towards us. The 40 micron peak brightness is typically very close to the actual protostar's position, while at shorter wavelengths this is often not the case due to re-radiation via outflow cavities. SOFIA's relatively high angular resolution at 30-40 microns, i.e. ~3" compared to ~6" of Herschel at 70 microns, is thus important for disentangling massive star formation activity, especially that occurring in crowded regions. With source G35.2 we have also demonstrated SOFIA's ability to deliver high contrast imaging revealing fainter extended emission from the protostellar envelope that is impractical to observe from the ground at 10-20 microns. Combined with sophisticated radiative transfer modeling, analysis of this emission constrains the geometry of the outflow cavities, allowing more reliable measurement of the true bolometric luminosity and thus protostellar mass. Our goal is to continue to apply these techniques to a much larger sample of protostars, spanning a range of evolutionary and environmental states, from relatively isolated sources in Infrared Dark Clouds, to less extincted sources with compact (often jet-like) radio emission, to ultra-compact HII regions (where radio emission extends beyond MIR emission), to sources in crowded regions. We also include 10 intermediate-mass protostars to allow comparison with their more massive cousins. A typical observation will take ~60 minutes and the ~50 targeted fields of view will yield ~60 protostars: enough to begin to provide statistically significant samples in these different evolutionary and environmental states.
14. Multi-dimensional models of circumstellar shells around evolved massive stars
NASA Astrophysics Data System (ADS)
van Marle, A. J.; Keppens, R.
2012-11-01
Context. Massive stars shape their surrounding medium through the force of their stellar winds, which collide with the circumstellar medium. Because the characteristics of these stellar winds vary over the course of the evolution of the star, the circumstellar matter becomes a reflection of the stellar evolution and can be used to determine the characteristics of the progenitor star. In particular, whenever a fast wind phase follows a slow wind phase, the fast wind sweeps up its predecessor in a shell, which is observed as a circumstellar nebula. Aims: We make 2D and 3D numerical simulations of fast stellar winds sweeping up their slow predecessors to investigate whether numerical models of these shells have to be 3D, or whether 2D models are sufficient to reproduce the shells correctly. Methods: We use the MPI-AMRVAC code, using hydrodynamics with optically thin radiative losses included, to make numerical models of circumstellar shells around massive stars in 2D and 3D and compare the results. We focus on those situations where a fast Wolf-Rayet star wind sweeps up the slower wind emitted by its predecessor, being either a red supergiant or a luminous blue variable. Results: As the fast Wolf-Rayet wind expands, it creates a dense shell of swept up material that expands outward, driven by the high pressure of the shocked Wolf-Rayet wind. These shells are subject to a fair variety of hydrodynamic-radiative instabilities. If the Wolf-Rayet wind is expanding into the wind of a luminous blue variable phase, the instabilities will tend to form a fairly small-scale, regular filamentary lattice with thin filaments connecting knotty features. If the Wolf-Rayet wind is sweeping up a red supergiant wind, the instabilities will form larger interconnected structures with less regularity. The numerical resolution must be high enough to resolve the compressed, swept-up shell and the evolving instabilities, which otherwise may not even form. Conclusions: Our results show that 3D
15. Spectacular Spitzer images of the Trifid Nebula: Protostars in a young, massive-star-forming region
NASA Astrophysics Data System (ADS)
Rho, J.; Reach, W. T.; Lefloch, B.; Fazio, G.
Spitzer IRAC and MIPS images of the Trifid Nebula (M20; see Figure 1) reveal its spectacular appearance in infrared light, demonstrating its special evolutionary stage: recently-formed massive protostars and numerous young stars, including a single O star that illuminates the surrounding molecular cloud from which it formed and unveiling large-scale, filamentary dark clouds. The hot dust grains show contrasting infrared colors in shells, arcs, bow-shocks and dark cores. Multiple protostars, previously defined as Class 0 from dust continuum and molecular outflow observations, are revealed in the infrared within the cold dust continuum peaks TC3 and TC4. The cold dust continuum cores of TC1 and TC2 contain only one protostar each; the newly-discovered infrared protostar in TC2 is the driving source of the HH399 jet. The Spitzer color-color diagram allowed us to identify ~150 young stellar objects (YSO) and classify them into different evolutionary stages, and also revealed a new class of YSO which are bright at 24μm but with spectral energy distribution peaking at 5-8μm; we name these sources "Hot excess" YSO. Despite of expectation that Class 0 sources would be "starless" cores, the Spitzer images, with unprecedented sensitivity, uncover mid-infrared emission from these Class 0 protostars. The mid-infrared detections of Class 0 protostars show that the emission escapes the dense, cold envelope of young protostars; the mid-infrared emission cannot arise from the same location as the mm-wave emission, and instead must arise from a much smaller region with less intervening extinction to the central accretion. The presence of multiple protostars within the cold cores of Class 0 objects implies that clustering occurs at this early stage of star formation. The most massive stars are located at the center of the cluster and are formed simultaneously with low-mass stars. The angular and mass distributions of protostars within the dust cores imply that these early
16. Uncertainties in the production of p nuclei in massive stars obtained from Monte Carlo variations
NASA Astrophysics Data System (ADS)
Rauscher, T.; Nishimura, N.; Hirschi, R.; Cescutti, G.; Murphy, A. St. J.; Heger, A.
2016-09-01
Nuclear data uncertainties in the production of p nuclei in massive stars have been quantified in a Monte Carlo procedure. Bespoke temperature-dependent uncertainties were assigned to different types of reactions involving nuclei from Fe to Bi. Their simultaneous impact was studied in postprocessing explosive trajectories for three different stellar models. It was found that the grid of mass zones in the model of a 25 M⊙ star, which is widely used for investigations of p nucleosynthesis, is too crude to properly resolve the detailed temperature changes required for describing the production of p nuclei. Using models with finer grids for 15 M⊙ and 25 M⊙ stars with initial solar metallicity, it was found that most of the production uncertainties introduced by nuclear reaction uncertainties are smaller than a factor of two. Since a large number of rates were varied at the same time in the Monte Carlo procedure, possible cancellation effects of several uncertainties could be taken into account. Key rates were identified for each p nucleus, which provide the dominant contribution to the production uncertainty. These key rates were found by examining correlations between rate variations and resulting abundance changes. This method is superior to studying flow patterns, especially when the flows are complex, and to individual, sequential variation of a few rates.
17. A massive star origin for an unusual helium-rich supernova in an elliptical galaxy.
PubMed
Kawabata, K S; Maeda, K; Nomoto, K; Taubenberger, S; Tanaka, M; Deng, J; Pian, E; Hattori, T; Itagaki, K
2010-05-20
The unusual helium-rich (type Ib) supernova SN 2005E is distinguished from all supernovae hitherto observed by its faint and rapidly fading light curve, prominent calcium lines in late-phase spectra and lack of any mark of recent star formation near the supernova location. These properties are claimed to be explained by a helium detonation in a thin surface layer of an accreting white dwarf. Here we report that the observed properties of SN 2005cz, which appeared in an elliptical galaxy, resemble those of SN 2005E. We argue that these properties are best explained by a core-collapse supernova at the low-mass end (8-12 solar masses) of the range of massive stars that explode. Such a low-mass progenitor lost its hydrogen-rich envelope through binary interaction, had very thin oxygen-rich and silicon-rich layers above the collapsing core, and accordingly ejected a very small amount of radioactive (56)Ni and oxygen. Although the host galaxy NGC 4589 is an elliptical, some studies have revealed evidence of recent star-formation activity, consistent with the core-collapse model. PMID:20485430
18. The massive star binary fraction in young open clusters - II. NGC6611 (Eagle Nebula)
NASA Astrophysics Data System (ADS)
Sana, H.; Gosset, E.; Evans, C. J.
2009-12-01
Based on a set of over 100 medium- to high-resolution optical spectra collected from 2003 to 2009, we investigate the properties of the O-type star population in NGC6611 in the core of the Eagle Nebula (M16). Using a much more extended data set than previously available, we revise the spectral classification and multiplicity status of the nine O-type stars in our sample. We confirm two suspected binaries and derive the first SB2 orbital solutions for two systems. We further report that two other objects are displaying a composite spectrum, suggesting possible long-period binaries. Our analysis is supported by a set of Monte Carlo simulations, allowing us to estimate the detection biases of our campaign and showing that the latter do not affect our conclusions. The absolute minimal binary fraction in our sample is fmin = 0.44 but could be as high as 0.67 if all the binary candidates are confirmed. As in NGC6231 (see Paper I), up to 75 per cent of the O star population in NGC6611 are found in an O+OB system, thus implicitly excluding random pairing from a classical IMF as a process to describe the companion association in massive binaries. No statistical difference could be further identified in the binary fraction, mass-ratio and period distributions between NGC6231 and NGC 6611, despite the difference in age and environment of the two clusters.
19. Investigating star formation properties of galaxies in massive clusters with Herschel and ALMA
NASA Astrophysics Data System (ADS)
Wu, John F.; Baker, Andrew J.; Aguirre, Paula; Barkats, D.; Halpern, Mark; Hilton, Matt; Hughes, John Patrick; Infante, Leopoldo; Lindner, Robert; Marriage, Tobias; Menanteau, Felipe; Sifon, Cristobal; Weiss, Axel; ACT Collaboration
2016-01-01
I will present results from an investigation of star formation properties of galaxies residing in two massive z ~ 1 clusters (including the 'El Gordo' merger) that were initially selected via their Sunyaev-Zeldovich decrements by the Atacama Cosmology Telescope (ACT) southern survey. This study uses new Herschel Space Observatory and Atacama Large Millimeter/submillimeter Array (ALMA) Cycle 2 observations, which provide information about the dust and cold gas content of galaxies in our targeted clusters. We have detected CO (4-3) and [CI] in individual star-forming cluster galaxies, and also measured stacked continuum and spectral line fluxes at long (e.g., far-infrared, submillimeter, and radio) wavelengths. We use these results to explore the relations between star formation and local environment and cluster dynamical state.This work has been supported by (i) an award issued by JPL/Caltech in association with Herschel, which is a European Space Agency Cornerstone Mission with significant participation by NASA, and (ii) the National Science Foundation through award GSSP SOSPA2-018 from the National Radio Astronomy Observatory, which is operated under cooperative agreement by Associated Universities, Inc.
20. Different Evolutionary Stages in the Massive Star-forming Region W3 Main Complex
NASA Astrophysics Data System (ADS)
Wang, Yuan; Beuther, Henrik; Zhang, Qizhou; Bik, Arjan; Rodón, Javier A.; Jiang, Zhibo; Fallscheer, Cassandra
2012-08-01
We observed three high-mass star-forming regions in the W3 high-mass star formation complex with the Submillimeter Array and IRAM 30 m telescope. These regions, i.e., W3 SMS1 (W3 IRS5), SMS2 (W3 IRS4) and SMS3, are in different evolutionary stages and are located within the same large-scale environment, which allows us to study rotation and outflows as well as chemical properties in an evolutionary sense. While we find multiple millimeter continuum sources toward all regions, these three subregions exhibit different dynamical and chemical properties, which indicate that they are in different evolutionary stages. Even within each subregion, massive cores of different ages are found, e.g., in SMS2, sub-sources from the most evolved ultracompact H II region to potential starless cores exist within 30,000 AU of each other. Outflows and rotational structures are found in SMS1 and SMS2. Evidence for interactions between the molecular cloud and the H II regions is found in the 13CO channel maps, which may indicate triggered star formation.
1. Different Evolutionary Stages in the Massive Star-forming Complex W3 Main
NASA Astrophysics Data System (ADS)
Wang, Yuan; Beuther, Henrik; Zhang, Qizhou; Bik, Arjan; Rodón, Javier A.; Jiang, Zhibo; Fallscheer, Cassandra
2013-03-01
We observed with the Submillimeter Array and IRAM 30 m telescope three high-mass star-forming regions in different evolutionary stages in the W3 high-mass star formation complex. These regions, i.e. W3 SMS1 (W3 IRS5), SMS2 (W3 IRS4) and SMS3, are located within the same large-scale environment, which allows us to study rotation and outflows as well as chemical properties in an evolutionary sense. While we find multiple mm continuum sources toward all regions, these three subregions exhibit different dynamical and chemical properties, which indicates that they are in different evolutionary stages. Even within each sub-region, massive cores of different ages are found, e.g. in SMS2, sub-sources from the most evolved UCHii region to potential starless cores exist within 30 000 AU (left panel, Fig. 1). Outflows and rotational structures are found in SMS1 and SMS2. Evidence for interactions between the molecular cloud and the HII regions is found in the 13CO channel maps (right panel, Fig. 1), which may indicate triggered star formation.
2. Clarifying Our View of Milky Way Massive Young Star Clusters with Adaptive Optics
NASA Astrophysics Data System (ADS)
Lu, Jessica R.; Ghez, A. M.; McCrady, N.; Yelda, S.
2011-01-01
We present Keck laser guide star adaptive optics (AO) observations of the massive young star clusters W51 G48.9-0.3 and W49A Cluster 1 in an effort to test the universality of the initial mass function (IMF) in extreme star forming environments. High-precision AO astrometry over a 1 year time baseline is successfully used to separate cluster members from contaminating field objects with differential proper motions as small as 0.5 mas/yr (15 km/s at 6 pc). We have developed improved AO photometric analysis techniques and use the near-infrared photometry of the proper motion selected cluster members to construct mass functions corrected for spatially varying extinction and incompleteness. Contrary to previous results for W51, we measure a mass function that has a high-mass end slope consistent with a Salpeter IMF and find that the observed cluster mass within 0.3 pc is <700 solar masses between 1 and 60 solar masses.
3. A massive star origin for an unusual helium-rich supernova in an elliptical galaxy.
PubMed
Kawabata, K S; Maeda, K; Nomoto, K; Taubenberger, S; Tanaka, M; Deng, J; Pian, E; Hattori, T; Itagaki, K
2010-05-20
The unusual helium-rich (type Ib) supernova SN 2005E is distinguished from all supernovae hitherto observed by its faint and rapidly fading light curve, prominent calcium lines in late-phase spectra and lack of any mark of recent star formation near the supernova location. These properties are claimed to be explained by a helium detonation in a thin surface layer of an accreting white dwarf. Here we report that the observed properties of SN 2005cz, which appeared in an elliptical galaxy, resemble those of SN 2005E. We argue that these properties are best explained by a core-collapse supernova at the low-mass end (8-12 solar masses) of the range of massive stars that explode. Such a low-mass progenitor lost its hydrogen-rich envelope through binary interaction, had very thin oxygen-rich and silicon-rich layers above the collapsing core, and accordingly ejected a very small amount of radioactive (56)Ni and oxygen. Although the host galaxy NGC 4589 is an elliptical, some studies have revealed evidence of recent star-formation activity, consistent with the core-collapse model.
4. Exploring Bias and Uncertainty in Gaussian Mixture Models of Young, Massive Star Clusters
NASA Astrophysics Data System (ADS)
Elrod, Aunna; Clarkson, William I.
2016-06-01
Mixture models are important for studies of star clusters observed against a foreground or background field population. By directly estimating both the distribution parameters of the components and the component fractions (and thus the formal membership probabilities), the populations of interest can be fit directly without recourse to binning. Gaussian Mixtures are a highly popular choice when modeling star clusters, and their determination using the Expectation Maximization algorithm, or its extension to cases with strongly varying measurement uncertainty (e.g. Bovy et al.’s Extreme Deconvolution) now appears in some statistics textbooks.Here we describe our Monte Carlo study to estimate the effect of the choice of instrumental setup, particularly different field of views, on parameter recovery for simulated star clusters under a variety of situations. We simulate observations of a Young, Massive Cluster like those near the Galactic Center, focusing mainly on scenarios where the same cluster is observed from ground and from space. We characterize the bias and uncertainty that might be introduced when using this fairly recent yet increasingly popular technique across heterogenous instrumental setups.
5. Massive Star Formation in a Gravitationally-Lensed H II Galaxy at z = 3.357
SciTech Connect
Villar-Martin, M; Stern, D; Hook, R N; Rosati, P; Lombardi, M; Humphrey, A; Fosbury, R; Stanford, S A; Holden, B P
2004-03-02
The Lynx arc, with a redshift of 3.357, was discovered during spectroscopic follow-up of the z = 0.570 cluster RX J0848+4456 from the ROSAT Deep Cluster Survey. The arc is characterized by a very red R - K color and strong, narrow emission lines. Analysis of HST WFPC 2 imaging and Keck optical and infrared spectroscopy shows that the arc is an H II galaxy magnified by a factor of {approx} 10 by a complex cluster environment. The high intrinsic luminosity, the emission line spectrum, the absorption components seen in Ly{alpha} and C IV, and the restframe ultraviolet continuum are all consistent with a simple H II region model containing {approx} 10{sup 6} hot O stars. The best fit parameters for this model imply a very hot ionizing continuum (T{sub BB} {approx} 80, 000 K), high ionization parameter (log U {approx} -1), and low nebular metallicity (Z/Z{sub {circle_dot}} {approx} 0.05). The narrowness of the emission lines requires a low mass-to-light ratio for the ionizing stars, suggestive of an extremely low metallicity stellar cluster. The apparent overabundance of silicon in the nebula could indicate enrichment by past pair instability supernovae, requiring stars more massive than {approx}140M{sub {circle_dot}}.
6. The massive star population in M101. II. Spatial variations in the recent star formation history
SciTech Connect
Grammer, Skyler; Humphreys, Roberta M. E-mail: [email protected]
2014-09-01
We investigate star formation history (SFH) as a function of radius in M101 using archival Hubble Space Telescope Advanced Camera for Surveys photometry. We derive the SFH from the resolved stellar populations in five 2' wide annuli. Binning the SFH into time frames corresponding to stellar populations traced by Hα, far-ultraviolet, and near-ultraviolet emission, we find that the fraction of stellar populations young enough to contribute in Hα is 15%-35% in the inner regions, compared to less than 5% in the outer regions. This provides a sufficient explanation for the lack of Hα emission at large radii. We also model the blue to red supergiant ratio in our five annuli, examine the effects that a metallicity gradient and variable SFH have on the predicted ratios, and compare to the observed values. We find that the radial behavior of our modeled blue to red supergiant ratios is highly sensitive to both spatial variations in the SFH and metallicity. Incorporating the derived SFH into modeled ratios, we find that we are able to reproduce the observed values at large radii (low metallicity), but at small radii (high metallicity) the modeled and observed ratios are discrepant.
7. NATURE OF W51e2: MASSIVE CORES AT DIFFERENT PHASES OF STAR FORMATION
SciTech Connect
Shi Hui; Han, J. L.; Zhao Junhui E-mail: [email protected]
2010-02-10
We present high-resolution continuum images of the W51e2 complex processed from archival data of the Submillimeter Array (SMA) at 0.85 and 1.3 mm and the Very Large Array at 7 and 13 mm. We also made line images and profiles of W51e2 for three hydrogen radio recombination lines (RRLs; H26alpha, H53alpha, and H66alpha) and absorption of two molecular lines of HCN(4-3) and CO(2-1). At least four distinct continuum components have been detected in the 3'' region of W51e2 from the SMA continuum images at 0.85 and 1.3 mm with resolutions of 0.''3 x 0.''2 and 1.''4 x 0.''7, respectively. The west component, W51e2-W, coincides with the ultracompact H II region reported from previous radio observations. The H26alpha line observation reveals an unresolved hyper-compact ionized core (<0.''06 or <310 AU) with a high electron temperature of 1.2 x 10{sup 4} K, with the corresponding emission measure EM>7 x 10{sup 10} pc cm{sup -6} and the electron density N{sub e} >7 x 10{sup 6} cm{sup -3}. The inferred Lyman continuum flux implies that the H II region W51e2-W requires a newly formed massive star, an O8 star or a cluster of B-type stars, to maintain the ionization. W51e2-E, the brightest component at 0.85 mm, is located 0.''9 east from the hyper-compact ionized core. It has a total mass of {approx}140 M{sub sun} according to our spectral energy distribution analysis and a large infall rate of >1.3 x 10{sup -3} M{sub sun} yr{sup -1} inferred from the absorption of HCN. W51e2-E appears to be the accretion center in W51e2. Given the fact that no free-free emission and no RRLs have been detected, the massive core of W51e2-E appears to host one or more growing massive proto-stars. Located 2'' northwest from W51e2-E, W51e2-NW is detected in the continuum emission at 0.85 and 1.3 mm. No continuum emission has been detected at lambda>= 7 mm. Along with the maser activities previously observed, our analysis suggests that W51e2-NW is at an earlier phase of star formation. W51e2-N is
8. Massive open star clusters using the VVV survey. III. A young massive cluster at the far edge of the Galactic bar
NASA Astrophysics Data System (ADS)
Ramírez Alegría, S.; Borissova, J.; Chené, A. N.; O'Leary, E.; Amigo, P.; Minniti, D.; Saito, R. K.; Geisler, D.; Kurtev, R.; Hempel, M.; Gromadzki, M.; Clarke, J. R. A.; Negueruela, I.; Marco, A.; Fierro, C.; Bonatto, C.; Catelan, M.
2014-04-01
Context. Young massive clusters are key to map the Milky Way's structure, and near-infrared large area sky surveys have contributed strongly to the discovery of new obscured massive stellar clusters. Aims: We present the third article in a series of papers focused on young and massive clusters discovered in the VVV survey. This article is dedicated to the physical characterization of VVV CL086, using part of its OB-stellar population. Methods: We physically characterized the cluster using JHKS near-infrared photometry from ESO public survey VVV images, using the VVV-SkZ pipeline, and near-infrared K-band spectroscopy, following the methodology presented in the first article of the series. Results: Individual distances for two observed stars indicate that the cluster is located at the far edge of the Galactic bar. These stars, which are probable cluster members from the statistically field-star decontaminated CMD, have spectral types between O9 and B0 V. According to our analysis, this young cluster (1.0 Myr < age < 5.0 Myr) is located at a distance of 11+5-6 kpc, and we estimate a lower limit for the cluster total mass of (2.8+1.6-1.4) · 103 M⊙. It is likely that the cluster contains even earlier and more massive stars. Based on observations taken within the ESO VISTA Public Survey VVV (programme ID 179.B-2002), and with ISAAC, VLT, ESO (programme 087.D-0341A).Near-IR photometry of the most probable cluster members is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/564/L9
9. IDENTIFICATION OF A POPULATION OF X-RAY-EMITTING MASSIVE STARS IN THE GALACTIC PLANE
SciTech Connect
Anderson, Gemma E.; Gaensler, B. M.; Kaplan, David L.; Posselt, Bettina; Slane, Patrick O.; Murray, Stephen S.; Drake, Jeremy J.; Grindlay, Jonathan E.; Hong, Jaesub; Lee, Julia C.; Mauerhan, Jon C.; Benjamin, Robert A.; Brogan, Crystal L.; Chakrabarty, Deepto; Drew, Janet E.; Lazio, T. Joseph W.; Steeghs, Danny T. H.; Van Kerkwijk, Marten H.
2011-02-01
We present X-ray, infrared, optical, and radio observations of four previously unidentified Galactic plane X-ray sources: AX J163252-4746, AX J184738-0156, AX J144701-5919, and AX J144547-5931. Detection of each source with the Chandra X-ray Observatory has provided sub-arcsecond localizations, which we use to identify bright infrared counterparts to all four objects. Infrared and optical spectroscopy of these counterparts demonstrate that all four X-ray sources are extremely massive stars, with spectral classifications: Ofpe/WN9 (AX J163252-4746), WN7 (AX J184738-0156 = WR121a), WN7-8h (AX J144701-5919), and OIf{sup +} (AX J144547-5931). AX J163252-4746 and AX J184738-0156 are both luminous, hard, X-ray emitters with strong Fe XXV emission lines in their X-ray spectra at {approx}6.7 keV. The multi-wavelength properties of AX J163252-4746 and AX J184738-0156 are not consistent with isolated massive stars or accretion onto a compact companion; we conclude that their X-ray emission is most likely generated in a colliding-wind binary (CWB) system. For both AX J144701-5919 and AX J144547-5931, the X-ray emission is an order of magnitude less luminous and with a softer spectrum. These properties are consistent with a CWB interpretation for these two sources also, but other mechanisms for the generation of X-rays cannot be excluded. There are many other as yet unidentified X-ray sources in the Galactic plane, with X-ray properties similar to those seen for AX J163252-4746, AX J184738-0156, AX J144701-5919, and AX J144547-5931. This may indicate a substantial population of X-ray-emitting massive stars and CWBs in the Milky Way.
10. AN UPDATED LOOK AT BINARY CHARACTERISTICS OF MASSIVE STARS IN THE CYGNUS OB2 ASSOCIATION
SciTech Connect
Kiminki, Daniel C.; Kobulnicky, Henry A.
2012-05-20
This work provides a statistical analysis of the massive star binary characteristics in the Cygnus OB2 association using radial velocity information of 114 B3-O5 primary stars and orbital properties for the 24 known binaries. We compare these data to a series of Monte Carlo simulations to infer the intrinsic binary fraction and distributions of mass ratios, periods, and eccentricities. We model the distribution of mass ratio, log-period, and eccentricity as power laws and find best-fitting indices of {alpha} = 0.1 {+-} 0.5, {beta} = 0.2 {+-} 0.4, and {gamma} = -0.6 {+-} 0.3, respectively. These distributions indicate a preference for massive companions, short periods, and low eccentricities. Our analysis indicates that the binary fraction of the cluster is 44% {+-} 8% if all binary systems are (artificially) assumed to have P < 1000 days; if the power-law period distribution is extrapolated to 10{sup 4} years, then a plausible upper limit for bound systems, the binary fraction is {approx}90% {+-} 10%. Of these binary (or higher order) systems, {approx}45% will have companions close enough to interact during pre- or post-main-sequence evolution (semi-major axis {approx}<4.7 AU). The period distribution for P < 26 days is not well reproduced by any single power law owing to an excess of systems with periods around 3-5 days (0.08-0.31 AU) and a relative shortage of systems with periods around 7-14 days (0.14-0.62 AU). We explore the idea that these longer-period systems evolved to produce the observed excess of short-period systems. The best-fitting binary parameters imply that secondaries generate, on average, {approx}16% of the V-band light in young massive populations. This means that photometrically based distance measurements for young massive clusters and associations will be systematically low by {approx}8% (0.16 mag in the distance modulus) if the luminous contributions of unresolved secondaries are not taken into account.
11. Massive star evolution in close binaries. Conditions for homogeneous chemical evolution
NASA Astrophysics Data System (ADS)
Song, H. F.; Meynet, G.; Maeder, A.; Ekström, S.; Eggenberger, P.
2016-01-01
compact. We also study the impact of different processes for the angular momentum transport on the surface abundances and velocities in single and close binaries. In models where strong internal coupling is assumed, strong surface enrichments are always associated with high surface velocities in binary or single star models. In contrast, models computed with mild coupling may produce strong surface enrichments associated with low surface velocities. This observable difference can be used to probe different models for the transport of the angular momentum in stars. Homogeneous evolution is more easily obtained in models (with or without tidal interactions) with solid body rotation. Conclusions: Close binary models help us to understand homogeneous massive stars, fast rotating Wolf-Rayet stars, and progenitors of long soft gamma-ray bursts, even at high metallicities.
12. Stellar evolution with rotation. V. Changes in all the outputs of massive star models
NASA Astrophysics Data System (ADS)
Meynet, G.; Maeder, A.
2000-09-01
Grids of models for rotating stars are constructed in the range of 9 to 120 Msun at solar metallicity. The following effects of rotation are included: shellular rotation, new structure equations for non-conservative case, surface distortions, increase of mass loss with rotation, meridional circulation and interaction with horizontal turbulence, shear instability and coupling with thermal effects, advection and diffusion of angular momentum treated in the non-stationary regime, transport and diffusion of the chemical elements. Globally we find that for massive stars the effects of rotation have an importance comparable to those of mass loss. Due to meridional circulation the internal rotation law Omega (r) rapidly converges, in 1-2% of the MS lifetime, towards a near equilibrium profile which then slowly evolves during the MS phase. The circulation shows two main cells. In the deep interior, circulation rises along the polar axis and goes down at the equator, while due to the Gratton-Öpik term it is the inverse in outer layers. This external inverse circulation grows in depth as evolution proceeds. We emphasize that a stationary approximation and a diffusive treatment of meridional circulation would be inappropriate. After the MS phase, the effects of core contraction and envelope expansion dominate the evolution of the angular momentum. The surface velocities decrease very much during the MS evolution of the most massive stars, due to their high mass loss, which also removes a lot of angular momentum. This produces some convergence of the velocities, but not necessarily towards the break-up velocities. However, stars with masses below ~ 12 Msun with initially high rotation may easily reach the break-up velocities near the end of the MS phase, which may explain the occurrence of Be-stars. Some other interesting properties of the rotational velocities are pointed out. For an average rotation, the tracks in the HR diagram are modified like a moderate overshoot would
13. A Submillimetre Study of Massive Star Formation Within the W51 Complex and Infrared Dark Clouds
NASA Astrophysics Data System (ADS)
Parsons, Harriet Alice Louise
Despite its importance the fundamental question of how massive stars form remains unanswered, with improvements to both models and observations having crucial roles to play. To quote Bate et al. (2003) computational models of star formation are limited because "conditions in molecular clouds are not sufficiently well understood to be able to select a representative sample of cloud cores for the initial conditions". It is this notion that motivates the study of the environments within Giant Molecular Clouds (GMCs) and Infrared Dark Clouds (IRDCs), known sites of massive star formation, at the clump and core level. By studying large populations of these objects, it is possible to make conclusions based on global properties. With this in mind I study the dense molecular clumps within one of the most massive GMCs in the Galaxy: the W51 GMC. New observations of the W51 GMC in the 12CO, 13CO and C18O (3-2) transitions using the HARP instrument on the JCMT are presented. With the help of the clump finding algorithm CLUMPFIND a total of 1575 dense clumps are identified of which 1130 are associated with the W51 GMC, yielding a dense mass reservoir of 1.5 × 10^5 M contained within these clumps. Of these clumps only 1% by number are found to be super-critical, yielding a super-critical clump formation efficiency of 0.5%, below current SFE estimates of the region. This indicates star formation within the W51 GMC will diminish over time although evidence from the first search for molecular outflows presents the W51 GMC in an active light with a lower limit of 14 outflows. The distribution of the outflows within the region searched found them concentrated towards the W51A region. Having much smaller sizes and masses, obtaining global properties of clumps and cores within IRDCs required studying a large sample of these objects. To do this pre-existing data from the SCUBA Legacy Catalogue was utilised to study IRDCs within a catalogues based on 8 μm data. This data identified
14. The onset of massive star formation: The evolution of temperature and density structure in an infrared dark cloud
SciTech Connect
Battersby, Cara; Ginsburg, Adam; Bally, John; Darling, Jeremy; Longmore, Steve; Dunham, Miranda
2014-06-01
We present new NH{sub 3} (1, 1), (2, 2), and (4, 4) observations from the Karl G. Jansky Very Large Array compiled with work in the literature to explore the range of conditions observed in young, massive star-forming regions. To sample the effects of evolution independent from those of distance/resolution, abundance, and large-scale environment, we compare clumps in different evolutionary stages within a single infrared dark cloud (IRDC), G32.02+0.06. We find that the early stages of clustered star formation are characterized by dense, parsec-scale filamentary structures interspersed with complexes of dense cores (<0.1 pc cores clustered in complexes separated by ∼1 pc) with masses from about 10 to 100 M {sub ☉}. The most quiescent core is the most extended while the star forming cores are denser and more compact, showing very similar column density structure before and shortly after the onset of massive star formation, with peak surface densities Σ ≳ 1 g cm{sup –2}. Quiescent cores and filaments show smoothly varying temperatures from 10 to 20 K, rising to over 40 K in star-forming cores. We calculate virial parameters for 16 cores and find that the level of support provided by turbulence is generally insufficient to support them against gravitational collapse ((α{sub vir}) ∼ 0.6). The star-forming filaments show smooth velocity fields, punctuated by discontinuities at the sites of active star formation. We discuss the massive molecular filament (M ∼ 10{sup 5} M {sub ☉}, length >60 pc) hosting the IRDC, hypothesizing that it may have been shaped by previous generations of massive stars.
15. HD 179821 (V1427 Aql, IRAS 19114+0002) - a massive post-red supergiant star?
NASA Astrophysics Data System (ADS)
Şahin, T.; Lambert, David L.; Klochkova, Valentina G.; Panchuk, Vladimir E.
2016-10-01
We have derived elemental abundances of a remarkable star, HD 179821, with unusual composition (e.g. [Na/Fe] = 1.0 ± 0.2 dex) and extra-ordinary spectral characteristics. Its metallicity at [Fe/H] = 0.4 dex places it among the most metal-rich stars yet analysed. The abundance analysis of this luminous star is based on high-resolution and high-quality (S/N ≈ 120-420) optical echelle spectra from McDonald Observatory and Special Astronomy Observatory. The data includes five years of observations over 21 epochs. Standard 1D local thermodynamic equilibrium analysis provides a fresh determination of the atmospheric parameters over all epochs: Teff = 7350 ± 200 K, log g= +0.6 ± 0.3, and a microturbulent velocity ξ = 6.6 ± 1.6 km s-1 and [Fe/H] = 0.4 ± 0.2, and a carbon abundance [C/Fe] = -0.19 ± 0.30. We find oxygen abundance [O/Fe] = -0.25 ± 0.28 and an enhancement of 0.9 dex in N. A supersonic macroturbulent velocity of 22.0 ± 2.0 km s-1 is determined from both strong and weak Fe I and Fe II lines. Elemental abundances are obtained for 22 elements. HD 179821 is not enriched in s-process products. Eu is overabundant relative to the anticipated [X/Fe] ≈ 0.0. Some peculiarities of its optical spectrum (e.g. variability in the spectral line shapes) is noticed. This includes the line profile variations for H α line. Based on its estimated luminosity, effective temperature and surface gravity, HD 179821 is a massive star evolving to become a red supergiant and finally a Type II supernova.
16. Massive star-forming host galaxies of quasars on Sloan digital sky survey stripe 82
SciTech Connect
Matsuoka, Yoshiki; Strauss, Michael A.; Price, Ted N. III; DiDonato, Matthew S.
2014-01-10
The stellar properties of about 800 galaxies hosting optically luminous, unobscured quasars at z < 0.6 are analyzed. Deep co-added Sloan Digital Sky Survey (SDSS) images of the quasars on Stripe 82 are decomposed into nucleus and host galaxy using point spread function and Sérsic models. The systematic errors in the measured galaxy absolute magnitudes and colors are estimated to be less than 0.5 mag and 0.1 mag, respectively, with simulated quasar images. The effect of quasar light scattered by the interstellar medium is also carefully addressed. The measured quasar-to-galaxy ratio in total flux decreases toward longer wavelengths, from ∼8 in the u band to ∼1 in the i and z bands. We find that the SDSS quasars are hosted exclusively by massive galaxies (stellar mass M {sub star} > 10{sup 10} M {sub ☉}), which is consistent with previous results for less luminous narrow-line (obscured) active galactic nuclei (AGNs). The quasar hosts are very blue and almost absent on the red sequence, showing stark contrast to the color-magnitude distribution of normal galaxies. The fact that more powerful AGNs reside in galaxies with higher star-formation efficiency may indicate that negative AGN feedback, if it exists, is not concurrent with the most luminous phase of AGNs. We also find positive correlation between the mass of supermassive black holes (SMBHs; M {sub BH}) and host stellar mass, but the M {sub BH}-M {sub star} relation is offset toward large M {sub BH} or small M {sub star} compared to the local relation. While this could indicate that SMBHs grow earlier than do their host galaxies, such an argument is not conclusive, as the effect may be dominated by observational biases.
17. A Rigid-Field Hydrodynamics approach to modelling the magnetospheres of massive stars
NASA Astrophysics Data System (ADS)
Townsend, R. H. D.; Owocki, S. P.; Ud-Doula, A.
2007-11-01
We introduce a new Rigid-Field Hydrodynamics approach to modelling the magnetospheres of massive stars in the limit of very strong magnetic fields. Treating the field lines as effectively rigid, we develop hydrodynamical equations describing the one-dimensional flow along each, subject to pressure, radiative, gravitational and centrifugal forces. We solve these equations numerically for a large ensemble of field lines to build up a three-dimensional time-dependent simulation of a model star with parameters similar to the archetypal Bp star σOriE. Since the flow along each field line can be solved independently of other field lines, the computational cost of this approach is a fraction of an equivalent magnetohydrodynamical treatment. The simulations confirm many of the predictions of previous analytical and numerical studies. Collisions between wind streams from opposing magnetic hemispheres lead to strong shock heating. The post-shock plasma cools initially via X-ray emission, and eventually accumulates into a warped, rigidly rotating disc defined by the locus of minima of the effective (gravitational plus centrifugal) potential. However, a number of novel results also emerge. For field lines extending far from the star, the rapid area divergence enhances the radiative acceleration of the wind, resulting in high shock velocities (up to ~3000kms-1) and hard X-rays. Moreover, the release of centrifugal potential energy continues to heat the wind plasma after the shocks, up to temperatures around twice those achieved at the shocks themselves. Finally, in some circumstances the cool plasma in the accumulating disc can oscillate about its equilibrium position, possibly due to radiative cooling instabilities in the adjacent post-shock regions.
18. The Physical Environment of the Massive Star-forming Region W42
NASA Astrophysics Data System (ADS)
Dewangan, L. K.; Luna, A.; Ojha, D. K.; Anandarao, B. G.; Mallick, K. K.; Mayya, Y. D.
2015-10-01
We present an analysis of multi-wavelength observations from various data sets and Galactic plane surveys to study the star-formation process in the W42 complex. A bipolar appearance of the W42 complex is evident due to the ionizing feedback from the O5-O6 type star in a medium that is highly inhomogeneous. The Very Large Telescope/NACO adaptive-optics K and L‧ images (resolutions ˜0.″2-0.″1) resolved this ionizing source into multiple point-like sources below ˜5000 AU scale. The position angle ˜15° of the W42 molecular cloud is consistent with the H-band starlight mean polarization angle, which in turn is close to the Galactic magnetic field, suggesting the influence of the Galactic field on the evolution of the W42 molecular cloud. Herschel sub-millimeter data analysis reveals three clumps located along the waist axis of the bipolar nebula, with the peak column densities of ˜(3-5) × 1022 cm-2 corresponding to visual extinctions of AV ˜ 32-53.5 mag. The Herschel temperature map traces a temperature gradient in W42, revealing regions of 20 K, 25 K, and 30-36 K. Herschel maps reveal embedded filaments (length ˜1-3 pc) that appear to be radially pointed to the denser clump associated with the O5-O6 star, forming a hub-filament system. A total of 512 candidate young stellar objects (YSOs) are identified in the complex, ˜40% of which are present in clusters distributed mainly within the molecular cloud, including the Herschel filaments. Our data sets suggest that the YSO clusters, including the massive stars, are located at the junction of the filaments, similar to those seen in the Rosette Molecular Cloud.
19. Initial phases of massive star formation in high infrared extinction clouds *. I. Physical parameters
NASA Astrophysics Data System (ADS)
Rygl, K. L. J.; Wyrowski, F.; Schuller, F.; Menten, K. M.
2010-06-01
Aims: The earliest phases of massive star formation are found in cold and dense infrared dark clouds (IRDCs). Since the detection method of IRDCs is very sensitive to the local properties of the background emission, we present here an alternative method to search for high column density in the Galactic plane by using infrared extinction maps. Using this method we find clouds between 1 and 5 kpc, of which many were missed by previous surveys. By studying the physical conditions of a subsample of these clouds, we aim at a better understanding of the initial conditions of massive star formation. Methods: We have made extinction maps of the Galactic plane based on the 3.6-4.5 μm color excess between the two shortest wavelength Spitzer IRAC bands, reaching to visual extinctions of ~100 mag and column densities of 9 × 1022 cm-2. From this we compiled a new sample of cold and compact high extinction clouds. We used the MAMBO array at the IRAM 30 m telescope to study the morphology, masses and densities of the clouds and the dense clumps within them. The latter were followed up by pointed ammonia observations with the 100 m Effelsberg telescope, to determine rotational temperatures and kinematic distances. Results: Extinction maps of the Galactic plane trace large scale structures such as the spiral arms. The extinction method probes lower column densities, NH2 ~ 4 × 1022 cm-2, than the 1.2 mm continuum, which reaches up to NH2 ~ 3 × 1023 cm-2 but is less sensitive to large scale structures. The 1.2 mm emission maps reveal that the high extinction clouds contain extended cold dust emission, from filamentary structures to still diffuse clouds. Most of the clouds are dark in 24 μm, but several show already signs of star formation via maser emission or bright infrared sources, suggesting that the high extinction clouds contain a variety of evolutionary stages. The observations suggest an evolutionary scheme from dark, cold and diffuse clouds, to clouds with a stronger 1
20. HATS-15b and HATS-16b: Two Massive Planets Transiting Old G Dwarf Stars
NASA Astrophysics Data System (ADS)
Ciceri, S.; Mancini, L.; Henning, T.; Bakos, G.; Penev, K.; Brahm, R.; Zhou, G.; Hartman, J. D.; Bayliss, D.; Jordán, A.; Csubry, Z.; de Val-Borro, M.; Bhatti, W.; Rabus, M.; Espinoza, N.; Suc, V.; Schmidt, B.; Noyes, R.; Howard, A. W.; Fulton, B. J.; Isaacson, H.; Marcy, G. W.; Butler, R. P.; Arriagada, P.; Crane, J. D.; Shectman, S.; Thompson, I.; Tan, T. G.; Lázár, J.; Papp, I.; Sari, P.
2016-07-01
1. Feedback from quasars in star-forming galaxies and the triggering of massive galactic winds
NASA Astrophysics Data System (ADS)
Monaco, Pierluigi; Fontanot, Fabio
2005-05-01
The shining of quasars is a likely trigger of massive galactic winds, able to remove most interstellar medium (ISM) from a star-forming spheroid. However, the mechanism responsible for the deposition of energy into the ISM is still unclear. Starting from a model for feedback in galaxy formation with a two-phase medium (Monaco), we propose that the perturbation induced by radiative heating from a quasar on the ISM triggers a critical change of feedback regime. In the feedback model, supernova remnants (SNRs) expanding in the hot and pressurized phase of a star-forming spheroid typically become pressure confined before the hot interior gas is able to cool. In the presence of runaway radiative heating by a quasar, a mass flow from the cold to the hot phase develops; whenever this evaporation flow is significant with respect to the star formation rate, owing to the increased density of the hot phase the SNRs reach the point where their interior gas cools before being confined, forming a thick cold shell. We show that in this case the consequent drop in pressure leads quickly to the percolation of all the shells and to the formation of a super shell of cold gas that sweeps the whole galaxy. Radiation pressure is then very effective in removing such a shell from the galaxy. This self-limiting mechanism leads to a correlation between black hole and bulge masses for more massive bulges than 1010 Msolar. The insertion of a motivated wind trigger criterion in a hierarchical galaxy formation model shows, however, that winds are not necessary to obtain a good black hole-bulge correlation. In the absence of winds, good results are obtained if the mechanism responsible for the creation of a reservoir of low-angular momentum gas (able to accrete on to the black hole) deposits mass at a rate proportional to the star formation rate. Using a novel galaxy formation model, we show under which conditions black hole masses are self-limited by the wind mechanism described above, and
2. The Most Complete View Yet of Massive Star formation in the Local Universe
NASA Astrophysics Data System (ADS)
We propose to take advantage of the nearly all-sky coverage of the Galaxy Evolution Explorer and Wide Field Infrared Surveyor missions to construct a combined atlas of ultraviolet and mid-infrared intensity images for almost all massive galaxies within 40 Mpc as well as several key local galaxy surveys beyond this volume. Following established methodology, we will use these to construct resolved estimates of the star formation rate surface density (the recent rate of star formation per unit area) across the whole local galaxy population. We will then use this atlas to measure basic facts about star formation in the local universe: where are most stars forming? Where are galaxies of different masses and morphologies most rapidly increasing their mass and where are they quenched? How common are extreme'' events like nuclear or off-nuclear starbursts? The limited resolution of infrared telescopes has made it difficult to address these questions in large samples before the latest generation of NASA missions. These local galaxies have been, and will remain, the subject of much focused study. The atlas will also serve as a reference point to place smaller samples studied in greater detail into the full context of the galaxy population; for example, we highlight the ability to place detailed studies of gas and dust in moderate-size galaxies into the broader context of galaxy evolution. The prospect to make homogenously constructed, extinction robust, resolved maps of a huge set of galaxies is only now available and offers a powerful chance to link these two fields (nearby galaxy studies and statistical studies of galaxy evolution and population). It will also serve as an invaluable resource to target future detailed studies with telescopes like the James Webb Space Telescope that are optimized for extraordinarily detailed study of comparatively small areas and so require the sort of “finding chart” that we propose to produce. This goal of mapping all star formation
3. Chandra and NTT Observations of Massive Young Stars in the Heavily Reddened Galactic Cluster Westerlund 1
NASA Astrophysics Data System (ADS)
Skinner, S. L.; Damineli, A.; Palla, F.; Zhekov, S. A.; Simmons, A. E.; Teodoro, M.
2005-12-01
The southern galactic starburst cluster Westerlund 1 (Wd1) contains a rich population of massive young stars that is spectacularly revealed in infrared images. Recent studies give a mean extinction in the range Av = 9.5 - 13.6 mag and age estimates of ˜3 - 5 Myr (Brandner et al. 2005, Clark et al. 2005). The cluster contains numerous supergiants, hypergiants, a LBV candidate, and at least 19 Wolf-Rayet (WR) stars. We present new results from Chandra X-ray and NTT near-IR observations of Wd1. Our immediate objectives are to obtain an X-ray census, identify optical or near-IR counterparts to the X-ray sources, and quantify the X-ray properties of the cluster members. Chandra detections include a newly-discovered 10.61 sec pulsar, the unusual B[e] supergiant W9, and half of the currently known WR stars in the cluster. The Chandra ACIS-S CCD spectrum of the Wd1 pulsar (CXO J164710.2-455217) can be acceptably reproduced by an absorbed soft blackbody emission model, but the model is not uniquely constrained by the existing data. A high-temperature component is clearly present in the X-ray spectrum of W9, suggesting that it is a close binary or unresolved multiple. Most of the Chandra WR detections are nitrogen-rich WN stars, but a few carbon-rich WC stars are surprisingly detected. At an assumed distance of 4 kpc, the X-ray luminosity of W87-239 (WC9) is two orders of magnitude greater than upper limits previously obtained for closer less-obscured single WC stars such as WR 135 (WC8, log Lx < 29.82 ergs/s; Skinner et al. 2005). The luminous X-ray emission and hot plasma in W87-239 point toward binarity. This study was supported by NASA/SAO grants GO5-6009X (PI: S.S.) and GO4-5003X (PI: S.Z.).
4. Analytical solutions for radiation-driven winds in massive stars. I. The fast regime
SciTech Connect
Araya, I.; Curé, M.; Cidale, L. S.
2014-11-01
Accurate mass-loss rate estimates are crucial keys in the study of wind properties of massive stars and for testing different evolutionary scenarios. From a theoretical point of view, this implies solving a complex set of differential equations in which the radiation field and the hydrodynamics are strongly coupled. The use of an analytical expression to represent the radiation force and the solution of the equation of motion has many advantages over numerical integrations. Therefore, in this work, we present an analytical expression as a solution of the equation of motion for radiation-driven winds in terms of the force multiplier parameters. This analytical expression is obtained by employing the line acceleration expression given by Villata and the methodology proposed by Müller and Vink. On the other hand, we find useful relationships to determine the parameters for the line acceleration given by Müller and Vink in terms of the force multiplier parameters.
5. SMA Observations of the Massive Star-forming Regions NGC 6334 I & I(N)
NASA Astrophysics Data System (ADS)
Hunter, Todd R.; Beuther, Henrik; Megeath, Tom; Menten, Karl; Thorwirth, Sven; Zhang, Qizhou
We present high-resolution observations of the massive star-formation regions NGC 6334 I and I(N) in the 230 GHz band. Data were obtained during Spring 2004 and 2005 in the compact and extended configurations of the Submillimeter Array (SMA), a joint venture of the Smithsonian Astrophysical Observatory and the Academica Sinica Institute of Astronomy and Astrophysics. Various pieces of evidence, including a molecular line survey by Thorwirth et al. (2003), have suggested that these two fields exist in different evolutionary stages, with field I(N) being younger. Our new observations will help to explore this hypothesis. We have detected and imaged a number of molecular lines that trace the outflow activity and dense gas in both fields. In field I, we have begun to resolve the strong dust continuum emission into multiple sources. In a separate work, these new sources were found to coincide with strong thermal centimeter lines of ammonia and methanol (Beuther et al. 2005).
6. A possible relativistic jetted outburst from a massive black hole fed by a tidally disrupted star.
PubMed
Bloom, Joshua S; Giannios, Dimitrios; Metzger, Brian D; Cenko, S Bradley; Perley, Daniel A; Butler, Nathaniel R; Tanvir, Nial R; Levan, Andrew J; O'Brien, Paul T; Strubbe, Linda E; De Colle, Fabio; Ramirez-Ruiz, Enrico; Lee, William H; Nayakshin, Sergei; Quataert, Eliot; King, Andrew R; Cucchiara, Antonino; Guillochon, James; Bower, Geoffrey C; Fruchter, Andrew S; Morgan, Adam N; van der Horst, Alexander J
2011-07-01
Gas accretion onto some massive black holes (MBHs) at the centers of galaxies actively powers luminous emission, but most MBHs are considered dormant. Occasionally, a star passing too near an MBH is torn apart by gravitational forces, leading to a bright tidal disruption flare (TDF). Although the high-energy transient Sw 1644+57 initially displayed none of the theoretically anticipated (nor previously observed) TDF characteristics, we show that observations suggest a sudden accretion event onto a central MBH of mass about 10(6) to 10(7) solar masses. There is evidence for a mildly relativistic outflow, jet collimation, and a spectrum characterized by synchrotron and inverse Compton processes; this leads to a natural analogy of Sw 1644+57 to a temporary smaller-scale blazar.
7. Nucleosynthesis in a massive star associated with magnetohydrodynamical jets from collapsars
SciTech Connect
Ono, M.; Hashimoto, M.; Fujimoto, S.; Kotake, K.; Yamada, S.
2012-11-12
We investigate the nucleosynthesis during the stellar evolution and the jet-like supernova explosion of a massive star of 70 M{sub Circled-Dot-Operator} having the solar metallicity in the main sequence stage. The nucleosynthesis calculations have been performed with large nuclear reaction networks, where the weak s-, p-, and r-processes are taken into account. As a result s-elements of 60 > A > 90 and r-elements of 90 > A > 160 are highly overproduced relative to the solar system abundances. We find that the Sr-Y-Zr isotopes are primarily synthesized in the explosive nucleosynthesis which could be one of the sites of the lighter element primary process (LEPP).
8. Xenon, osmium, and lead formed in O-shells and C-shells of massive stars
NASA Technical Reports Server (NTRS)
Heymann, D.; Dziczkaniec, M.
1980-01-01
In this paper it is shown that the explosive products from O-shells of massive stars which contain Xe-124 with large overproduction factors do not contain any of the naturally occurring isotopes of Os and Pb. Further, it is shown that the explosive products from C-shells (explosive carbon burning) do contain Os and Pb along with Xe which is strongly enriched in r-Xe of anomalous isotopic composition. The composition of Os in this matter is probably s-like rather than r-like. Pb in this matter is enriched in Pb-208. The results and arguments of this paper have implications for studies of isotopic compositions of Xe, Os, and Pb in residues of the Allende and other carbonaceous chondrites.
9. A Possible Relativistic Jetted Outburst from a Massive Black Hole Fed by a Tidally Disrupted Star
NASA Astrophysics Data System (ADS)
Bloom, Joshua S.; Giannios, Dimitrios; Metzger, Brian D.; Cenko, S. Bradley; Perley, Daniel A.; Butler, Nathaniel R.; Tanvir, Nial R.; Levan, Andrew J.; O'Brien, Paul T.; Strubbe, Linda E.; De Colle, Fabio; Ramirez-Ruiz, Enrico; Lee, William H.; Nayakshin, Sergei; Quataert, Eliot; King, Andrew R.; Cucchiara, Antonino; Guillochon, James; Bower, Geoffrey C.; Fruchter, Andrew S.; Morgan, Adam N.; van der Horst, Alexander J.
2011-07-01
Gas accretion onto some massive black holes (MBHs) at the centers of galaxies actively powers luminous emission, but most MBHs are considered dormant. Occasionally, a star passing too near an MBH is torn apart by gravitational forces, leading to a bright tidal disruption flare (TDF). Although the high-energy transient Sw 1644+57 initially displayed none of the theoretically anticipated (nor previously observed) TDF characteristics, we show that observations suggest a sudden accretion event onto a central MBH of mass about 106 to 107 solar masses. There is evidence for a mildly relativistic outflow, jet collimation, and a spectrum characterized by synchrotron and inverse Compton processes; this leads to a natural analogy of Sw 1644+57 to a temporary smaller-scale blazar.
10. SOAR Near-Infrared and Optical Survey of OIf* and OIf*/WN Stars in the Periphery of Galactic Massive Star Forming Regions
NASA Astrophysics Data System (ADS)
Roman-Lopes, A.; Franco, G. A. P.; Sanmartin, D.
In this contribution we present some preliminary results obtained from a SOAR-Goodman optical spectroscopic survey aimed to confirm the OIf* - OIf*/WN nature of a sample of Galactic candidates that were previously confirmed as massive stars based on near-infrared spectra taken with OSIRIS at SOAR. With only a few of such stars known in the Galaxy to date, our study significantly contributes to improve the number of known Galactic O2If* stars, as well as almost doubling the number of known members of the galactic sample of the rare type OIf*/WN.
11. X-RAY EMISSION LINE PROFILES FROM WIND CLUMP BOW SHOCKS IN MASSIVE STARS
SciTech Connect
Ignace, R.; Waldron, W. L.; Cassinelli, J. P.; Burke, A. E. E-mail: [email protected] E-mail: [email protected]
2012-05-01
The consequences of structured flows continue to be a pressing topic in relating spectral data to physical processes occurring in massive star winds. In a preceding paper, our group reported on hydrodynamic simulations of hypersonic flow past a rigid spherical clump to explore the structure of bow shocks that can form around wind clumps. Here we report on profiles of emission lines that arise from such bow shock morphologies. To compute emission line profiles, we adopt a two-component flow structure of wind and clumps using two 'beta' velocity laws. While individual bow shocks tend to generate double-horned emission line profiles, a group of bow shocks can lead to line profiles with a range of shapes with blueshifted peak emission that depends on the degree of X-ray photoabsorption by the interclump wind medium, the number of clump structures in the flow, and the radial distribution of the clumps. Using the two beta law prescription, the theoretical emission measure and temperature distribution throughout the wind can be derived. The emission measure tends to be power law, and the temperature distribution is broad in terms of wind velocity. Although restricted to the case of adiabatic cooling, our models highlight the influence of bow shock effects for hot plasma temperature and emission measure distributions in stellar winds and their impact on X-ray line profile shapes. Previous models have focused on geometrical considerations of the clumps and their distribution in the wind. Our results represent the first time that the temperature distribution of wind clump structures are explicitly and self-consistently accounted for in modeling X-ray line profile shapes for massive stars.
12. Local Radiation Hydrodynamic Simulations of Massive Star Envelopes at the Iron Opacity Peak
NASA Astrophysics Data System (ADS)
Jiang, Yan-Fei; Cantiello, Matteo; Bildsten, Lars; Quataert, Eliot; Blaes, Omer
2015-11-01
We perform three-dimensional radiation hydrodynamic simulations of the structure and dynamics of the radiation-dominated envelopes of massive stars at the location of the iron opacity peak. One-dimensional hydrostatic calculations predict an unstable density inversion at this location, whereas our simulations reveal a complex interplay of convective and radiative transport whose behavior depends on the ratio of the photon diffusion time to the dynamical time. The latter is set by the ratio of the optical depth per pressure scale height, {τ }0, to {τ }{{c}}=c/{c}{{g}}, where {c}{{g}}≈ 50 {km} {{{s}}}-1 is the isothermal sound speed in the gas alone. When {τ }0\\gg {τ }{{c}}, convection reduces the radiation acceleration and removes the density inversion. The turbulent energy transport in the simulations agrees with mixing length theory and provides its first numerical calibration in the radiation-dominated regime. When {τ }0\\ll {τ }{{c}}, convection becomes inefficient and the turbulent energy transport is negligible. The turbulent velocities exceed cg, driving shocks and large density fluctuations that allow photons to preferentially diffuse out through low-density regions. However, the effective radiation acceleration is still larger than the gravitational acceleration so that the time average density profile contains a modest density inversion. In addition, the simulated envelope undergoes large-scale oscillations with periods of a few hours. The turbulent velocity field may affect the broadening of spectral lines and therefore stellar rotation measurements in massive stars, while the time variable outer atmosphere could lead to variations in their mass loss and stellar radius.
13. Locating and Measuring the High Mass Ejecta from the Unstable Massive Star System eta Carinae
NASA Astrophysics Data System (ADS)
Morris, Patrick
2014-10-01
The luminous, massive binary system eta Carinae is both one of the nearest and most unstable objects in a class of evolved massive stars, near the end of its lifetime before expected destruction in a supernova. It experienced a major outburst in 1843, producing the well-known Homunculus nebula, containing some 15 to 40 Msun in warm (~170 K) and cool (90-110 K) dust and associated gas, according to mid-infrared ISO spectroscopy. The location of this material is very uncertain, due to large apertures of the spectroscopic observations, and lack of direct imaging beyond 25 microns. We propose to use the FORCAST imager with long wavelength filters to better locate and estimate the mass in thermal components of this material that may be resolved, constraining it to the interior regions or bipolar lobes of the Homunculus nebula, or in outer ejecta that would support the hypothesis of a major event prior to the 1843 eruption. This is crucial to understanding the mass-loss history of this object on the edge of a final supernova explosion, and provide constraints on the distribution and extinction properties of the dust in 3D hydrodynamical + radiative transfer numerical modeling of the Homunculus nebula.
14. Peering to the Heart of Massive Star Birth - III. Surveying Across Evolution and Environment
NASA Astrophysics Data System (ADS)
Tan, Jonathan
2013-10-01
We propose to utilize the unique capabilities of SOFIA-FORCAST to perform a 30-40 micron imaging survey of massive protostars, building upon our Basic Science results on G35.20-0.74 (hereafter G35.2) and our approved Cycle 1 observations of several more sources. We have demonstrated theoretically and observationally that 30-40 micron observations are crucial for defining the spectral energy distribution of massive protostars and thus the bolometric flux directed towards us. The 40 micron peak brightness is typically very close to the actual protostar's position, while at shorter wavelengths this is often not the case due to re-radiation via outflow cavities. SOFIA's relatively high angular resolution at 30-40 microns, i.e. ~3" compared to ~6" of Herschel at 70 microns, is thus important for disentangling massive star formation activity, especially that occurring in crowded regions. With G35.2 we have also demonstrated SOFIA's ability to deliver high contrast imaging revealing fainter extended emission from the protostellar envelope that is impractical to observe from the ground at 10-20 microns. Combined with sophisticated radiative transfer modeling, analysis of this emission constrains the geometry of the outflow cavities, allowing more reliable measurement of the true bolometric luminosity and thus protostellar mass. Our goal now is to apply these techniques to a much larger sample of protostars, spanning a wider range of evolutionary and environmental states, from relatively isolated sources in Infrared Dark Clouds, to less extincted sources with compact (often jet-like) radio emission, to ultra-compact HII regions (where radio emission extends beyond MIR emission), to sources in crowded regions. A typical observation will take ~60 minutes and the ~40 targeted fields of view will yield >~50 protostars: enough to begin to provide statistically significant samples in these different evolutionary and environmental states.
15. RADIATION TRANSFER OF MODELS OF MASSIVE STAR FORMATION. I. DEPENDENCE ON BASIC CORE PROPERTIES
SciTech Connect
Zhang Yichen; Tan, Jonathan C. E-mail: [email protected]
2011-05-20
Radiative transfer calculations of massive star formation are presented. These are based on the Turbulent Core Model of McKee and Tan and self-consistently included a hydrostatic core, an inside-out expansion wave, a zone of free-falling rotating collapse, wide-angle dust-free outflow cavities, an active accretion disk, and a massive protostar. For the first time for such models, an optically thick inner gas disk extends inside the dust destruction front. This is important to conserve the accretion energy naturally and for its shielding effect on the outer region of the disk and envelope. The simulation of radiation transfer is performed with the Monte Carlo code of Whitney, yielding spectral energy distributions (SEDs) for the model series, from the simplest spherical model to the fiducial one, with the above components each added step by step. Images are also presented in different wavebands of various telescope cameras, including Spitzer IRAC and MIPS, SOFIA FORCAST, and Herschel PACS and SPIRE. The existence of the optically thick inner disk produces higher optical wavelength fluxes but reduces near- and mid-IR emission. The presence of outflow cavities, the inclination angle to the line of sight, and the thickness of the disk all affect the SEDs and images significantly. For the high-mass surface density cores considered here, the mid-IR emission can be dominated by the outflow cavity walls, as has been suggested by De Buizer. The effect of varying the pressure of the environment bounding the surface of the massive core is also studied. With lower surface pressures, the core is larger, has lower extinction and accretion rates, and the observed mid-IR flux from the disk can then be relatively high even though the accretion luminosity is lower. In this case the silicate absorption feature becomes prominent, in contrast to higher density cores forming under higher pressures.
16. Irradiated interfaces in the Ara OB1, Carina, Eagle Nebula, and Cyg OB2 massive star formation regions
NASA Astrophysics Data System (ADS)
Hartigan, P.; Palmer, J.; Cleeves, L. I.
2012-12-01
Regions of massive star formation offer some of the best and most easily-observed examples of radiation hydrodynamics. Boundaries where fully-ionized H II regions transition to neutral/molecular photodissociation regions (PDRs) are of particular interest because marked temperature and density contrasts across the boundaries lead to evaporative flows and fluid dynamical instabilities that can evolve into spectacular pillar-like structures. When detached from their parent clouds, pillars become ionized globules that often harbor one or more young stars. H2 molecules at the interface between a PDR and an H II region absorb ultraviolet light from massive stars, and the resulting fluoresced infrared emission lines are an ideal way to trace this boundary independent of obscuring dust. This paper presents H2 images of four regions of massive star formation that illustrate different types of PDR boundaries. The Ara OB1 star formation region contains a striking long wall that has several wavy structures which are present in H2, but the emission is not particularly bright because the ambient UV fluxes are relatively low. In contrast, the Carina star formation region shows strong H2 fluorescence both along curved walls and at the edges of spectacular pillars that in some cases have become detached from their parent clouds. The less-spectacular but more well-known Eagle Nebula has two regions that have strong fluorescence in addition to its pillars. While somewhat older than the other regions, Cyg OB2 has the highest number of massive stars of the regions surveyed and contains many isolated, fluoresced globules that have head-tail morphologies which point towards the sources of ionizing radiation. These images provide a collection of potential astrophysical analogs that may relate to ablated interfaces observed in laser experiments of radiation hydrodynamics.
17. Sub-mm free-free emission from the winds of massive stars in the age of ALMA
NASA Astrophysics Data System (ADS)
Daley-Yates, S.; Stevens, I. R.; Crossland, T. D.
2016-09-01
The thermal radio and sub-mm emission from the winds of massive stars is investigated and the contribution to the emission due to the stellar wind acceleration region and clumping of the wind is quantified. Building upon established theory, a method for calculating the thermal radio and sub-mm emission using results for a line-driven stellar outflow according to Castor, Abbott & Klein (1975) is presented. The results show strong variation of the spectral index for 102 GHz <ν < 104 GHz. This corresponds both to the wind acceleration region and clumping of the wind, leading to a strong dependence on the wind velocity law and clumping parameters. The Atacama Large Millimeter/sub-mm Array (ALMA) is the first observatory to have both the spectral window and sensitivity to observe at the high frequencies required to probe the acceleration regions of massive stars. The deviations in the predicted flux levels as a result of the inclusion of the wind acceleration region and clumping are sufficient to be detected by ALMA, through deviations in the spectral index in different portions of the radio/sub-mm spectra of massive stars, for a range of reasonable mass-loss rates and distances. Consequently both mechanisms need to be included to fully understand the mass-loss rates of massive stars.
18. Evidence for the Non-destruction of the Most Massive Molecular Clouds even After they have Given Birth to Massive Star Clusters
NASA Astrophysics Data System (ADS)
Zaragoza-Cardiel, Javier; Beckman, John Etienne; Font, Joan; Camps-Fariña, Artemi
2015-08-01
We have observed the interacting galaxies system, the Antennae, using the Fabry-Perot interferometer GHαFaS on the 4.2m William Herschel Telescope at the Observatorio del Roque de los Muchachos, La Palma, deriving the Hα surface brightness, velocity and velocity dispersion maps, and extracting key physical parameters (mean electron density, mass, velocity dispersion, and effective radius) of 303 HII regions, using a technique for which 3D mapping, including velocity, is essential. We also derived the CO(3-2) surface brightness, velocity, and velocity dispersion maps, and extracted the relevant parameters (size, CO luminosity, velocity dispersion and mass) of ~142 GMC's, using observations from the ALMA archive.We compared the properties of HII regions with GMC's, finding that the two distinct populations of HII regions are related to two populations of GMC's, as both show bimodal mass functions with a break at 106.5 solar masses. The classical Larson scaling laws need modification for the more massive population of GMC's, as the surface gas density increases with mass, which leads to enhanced star formation efficiency.The analysis of the turbulent velocity dispersion of the regions suggests that the more massive regions are bound by their own gravity, while the less massive star forming regions are confined by external pressure. If the two population of HII regions are derived from the twopopulations of GMC's, our results show the GMC's do not dissolve after they have given birth to massive stars, at least for the regime of the population of high mass clouds.
19. A massive neutron star in the millisecond pulsar PSR J2215+5135
NASA Astrophysics Data System (ADS)
Shahbaz, Tariq
2016-07-01
Binary evolution may increase neutron masses via accretion. Hence the most massive neutron stars (NSs) are expected to be located amongst the binary millisecond pulsars (MSPs) spun-up within X-ray binaries. Most MSPs are found with brown dwarf lookalikes or ˜0.2 M stars in systems called "black widows" and "redbacks", respectively, because these companions are ablated by the pulsar wind. These systems offer some advantages over white dwarf-pulsar binaries: they are typically brighter, they present strongly irradiated hemispheres, and they fill significant fractions of their Roche lobes. As a result, their optical light curves exhibit variability due to a combination of their ellipsoidal shape and irradiation, which can be modelled in order to determine orbital parameters such as the mass ratio and inclination. Combining these with optical spectroscopy and/or pulsar timing enables one to determine a reliable NS masses. Here we present the results of our detailed modelling of the optical lightcurves and radial velocity curves of J2215+5135, which allows us to determine various ystem parameters, including the NS mass.
20. VizieR Online Data Catalog: Massive LMC stars AAOmega spectroscopy (Evans+, 2015)
NASA Astrophysics Data System (ADS)
Evans, C. J.; van Loon, J. T.; Hainich, R.; Bailey, M.
2015-08-01
This catalogue comprises ascii versions of the optical spectra of 263 massive stars in the Large Magellanic Cloud, obtained with the AAOmega spectrograph on the Anglo Australian Telescope. Spectra from the first night (2006 Feb 22) were obtained with a 1700B grating at two wavelength settings. The spectra published here were obtained by median combining the two exposures at both settings, and then median combining them in the overlap region (spanning ~4375-4400Å). Spectra from the second night (2006 Feb 23) were obtained with a 1500V grating at one central wavelength setting (4375Å). The spectra The published spectra have been normalised/rectified using an automated script, which uses pre-defined regions (selected to avoid known absorption lines in early-type stars) to create a polynomial fit to the notional continuum in each spectrum. The published spectra have been divided by those fits to rectify them to unity. As such, we caution the user that quantitative analysis of these data would benefit from tailored rectification of the spectra. In particular, at the ends of the spectral range, and across broad emission features (such as that around HeII 4686 in luminous O-type supergiants). Also note that there were a number of 'hot' columns in the AAOmega CCDs, leading to small breaks (at multiple wavelengths) in the large majority of the spectra. (5 data files).
1. RADIATION-HYDRODYNAMIC MODELS OF THE EVOLVING CIRCUMSTELLAR MEDIUM AROUND MASSIVE STARS
SciTech Connect
Toala, J. A.; Arthur, S. J.
2011-08-20
We study the evolution of the interstellar and circumstellar media around massive stars (M {>=} 40 M{sub sun}) from the main sequence (MS) through to the Wolf-Rayet (WR) stage by means of radiation-hydrodynamic simulations. We use publicly available stellar evolution models to investigate the different possible structures that can form in the stellar wind bubbles around WR stars. We find significant differences between models with and without stellar rotation, and between models from different authors. More specifically, we find that the main ingredients in the formation of structures in the WR wind bubbles are the duration of the red supergiant (or luminous blue variable) phase, the amount of mass lost, and the wind velocity during this phase, in agreement with previous authors. Thermal conduction is also included in our models. We find that MS bubbles with thermal conduction are slightly smaller, due to extra cooling which reduces the pressure in the hot, shocked bubble, but that thermal conduction does not appear to significantly influence the formation of structures in post-MS bubbles. Finally, we study the predicted X-ray emission from the models and compare our results with observations of the WR bubbles S 308, NGC 6888, and RCW 58. We find that bubbles composed primarily of clumps have reduced X-ray luminosity and very soft spectra, while bubbles with shells correspond more closely to observations.
2. The x-ray and spectropolarimetric view of mass loss and transfer in massive binary stars
NASA Astrophysics Data System (ADS)
Lomax, Jamie R.
2013-03-01
The majority of massive stars are members of binary systems. In order to have a better understanding of their evolutionary pathways, the mass and angular momentum loss from massive binaries needs to be well understood. Self consistent explanations for their behavior need to be valid across many wavelength regimes in order to illuminate key phases of mass loss to completely determine how it affects their evolution. In this dissertation I present the results of X-ray and specropolarimetric studies on one Roche-lobe overflow binary (beta Lyr) and two colliding wind binaries (V444 Cyg and WR 140). In beta Lyr a repeatable discrepancy between the secondary eclipse in total and polarized light indicates that an accretion hot spot has formed on the edge of the disk in the system. This hot spot may also be the source of the bipolar outflows within the system. The existence of a hot spot and its relationship to bipolar outflows is important in understanding the mass transfer dynamics of Roche-lobe overflow binaries. The absorption of the 2.0 keV spectral fit component in V444 Cyg suggests that the shock has a large opening angle while analysis of the X-ray light curves places the stagnation point farther away from the O star than theoretically expected. Combining this with evidence of polarimetric variability in V444 Cyg's optical emission lines shows that the effects of radiative inhibition or braking are significant for this close binary and may be important in other colliding wind systems. Long term X-ray monitoring of the shock formed by the winds in WR 140 shows conflicting evidence for unexpected intrinsic hard X-ray emission. Spectral analysis shows that the low energy thermal tail is causing the observed higher energy emission. On the other hand, light curve analysis of the absorption feature near periastron passage suggests that there may be intrinsic hard X-ray emission from the system. WR 140's polarimetric behavior is consistent with the formation of dust near
3. Molecular gas observations and enhanced massive star formation efficiencies in M 100.
NASA Astrophysics Data System (ADS)
Knapen, J. H.; Beckman, J. E.; Cepa, J.; Nakai, N.
1996-04-01
We present new J=1->0 ^12^CO observations along the northern spiral arm of the grand-design spiral galaxy M 100 (NGC 4321), and study the distribution of molecular hydrogen as derived from these observations, comparing the new data with a set of data points on the southern arm published previously. We compare these measurements on both spiral arms and on the interarm regions with observations of the atomic and ionized hydrogen components. We determine massive star formation efficiency parameters, defined as the ratio of Hα luminosity to total gas mass, along the arms and compare the values to those in the interarm regions adjacent to the arms. We find that these parameters in the spiral arms are on average a factor of 3 higher than outside the arms, a clear indication of triggering of the star formation in the spiral arms. We discuss possible mechanisms for this triggering, and conclude that a density wave system is probably responsible for it. We discuss several possible systematical effects in some detail, and infer that the conclusions on triggering are sound. We specifically discuss the possible effects of extinction in Hα, or a non-standard CO to H_2_ conversion factor (X), and find that our conclusions on the enhancement of the efficiencies in the arms are reinforced rather than weakened by these considerations. A simple star forming scheme involving threshold densities for gravitational collapse is discussed for NGC 4321, and comparison is made with M 51. We find that the gas between the arms is generally stable against gravitational collapse whereas the gas in the arms is not, possibly leading to the observed arm-interarm differences in efficiency, but also note that these results, unlike the others obtained in this paper, do depend critically on the assumed value for the conversion factor.
4. The star-formation rates of 1.5 < z < 2.5 massive galaxies
NASA Astrophysics Data System (ADS)
Nordon, R.; Lutz, D.; Shao, L.; Magnelli, B.; Berta, S.; Altieri, B.; Andreani, P.; Aussel, H.; Bongiovanni, A.; Cava, A.; Cepa, J.; Cimatti, A.; Daddi, E.; Dominguez, H.; Elbaz, D.; Förster Schreiber, N. M.; Genzel, R.; Grazian, A.; Magdis, G.; Maiolino, R.; Pérez García, A. M.; Poglitsch, A.; Popesso, P.; Pozzi, F.; Riguccini, L.; Rodighiero, G.; Saintonge, A.; Sanchez-Portal, M.; Santini, P.; Sturm, E.; Tacconi, L.; Valtchanov, I.; Wetzstein, M.; Wieprecht, E.
2010-07-01
The star formation rate (SFR) is a key parameter in the study of galaxy evolution. The accuracy of SFR measurements at z ~ 2 has been questioned following a disagreement between observations and theoretical models. The latter predict SFRs at this redshift that are typically a factor 4 or more lower than the measurements. We present star-formation rates based on calorimetric measurements of the far-infrared (FIR) luminosities for massive 1.5 < z < 2.5, normal star-fo | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8698338270187378, "perplexity": 3095.878719120483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463610374.3/warc/CC-MAIN-20170528181608-20170528201608-00461.warc.gz"} |
https://www.physicsforums.com/threads/why-some-materials-have-bigger-refractive-index-than-others.214155/ | # Why some materials have bigger refractive index than others?
1. Feb 9, 2008
### Physicsissuef
why some materials have bigger refractive index than others? Lets say why air have 1.003 refractive index, and quartz glass 1.46?
2. Feb 9, 2008
### John Creighto
The refractive index depends upon the dielectric constant and the magnetic permeability.
http://en.wikipedia.org/wiki/Refractive_index
3. Feb 9, 2008
### Physicsissuef
Ok, I understand something. But is it actually the atoms and molecules and their bonds?
4. Feb 10, 2008
### pam
The permittivity depends on how easy it is for an electric field to polarize the molecules.
5. Feb 10, 2008
### Claude Bile
Yes. In optical media, the refractive index depends almost exclusively on the electric permittivity of the medium, which in turn depends on the characteristics of the atomic and molecular bonds within the medium.
Claude.
Have something to add?
Similar Discussions: Why some materials have bigger refractive index than others? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8854179382324219, "perplexity": 1524.7959275793476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280872.69/warc/CC-MAIN-20170116095120-00034-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/electron-density-when-introducing-impurites.854458/ | # Electron Density when Introducing Impurites
Tags:
1. Jan 28, 2016
### PeoplesChamp
How do you go about calculating electron densities, especially when impurities are involved?
Last edited: Jan 28, 2016
2. Jan 28, 2016
### analogdesign
I assume you are referring to carrier density, since even in doped semiconductor the overall electron density is essentially unchanged since the dopant density is so low compared to the number of lattice sites.
Given that assumption, dopant impurities are almost entirely ionized at room temperature. So, in N-type material the carrier density is equal to Nd which is the donor impurity density. In P-type material you would use the equation ni^2 = pn where ni is the intrinsic carrier density (due to thermal effects), p is the hole density and n is the electron density, so in that case, n = (ni^2)/p.
This is all well explained on the following web page: | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9456024765968323, "perplexity": 1382.7330740582913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591140.45/warc/CC-MAIN-20180719144851-20180719164851-00351.warc.gz"} |
https://www.physicsforums.com/threads/electric-potential-and-energy.253864/ | # Electric potential and energy
1. Sep 5, 2008
### devanlevin
2 balls are held at a distance of 1m from each other and then released, what will each of their velocities be when they are 2m from one another
m1=0.05 (kg)
q1=6*10^-6 (c)
m2=0.1 (kg)
q2=5*10^-6 (c)
what i tried was looking at each ball seperately
ball1
Energy===> Ue(initial)=Ue(final)+Ek
K(q2q1)/r(initial)=K(Q2q1)/r(final)+(1/2)mv^2
K(q2q1)/1=K(Q2q1)/2)+(1/2)mv^2
v^2=(Kq2q1)/m1=5.4
v1=2.3237m/s
V^2=(Kq2q1)/m2=2.7
v2=1.643m/s
but the answers in my text book are
v1=1.9m/s
v2=-0.95m/s
where have i gone wrong here
2. Sep 5, 2008
### jackiefrost
The (1/2)mv^2 (highlighted red, above) is the final total kinetic energy. So, it is actually (1/2*m1*v1^2)+(1/2*m2*v2^2). You then have to ask yourself how that total is split between the two masses to arrive at the individual velocities.
jf
3. Sep 5, 2008
### devanlevin
so then, i say
(1/2)mv^2=(1/2*m1*v1^2)+(1/2*m2*v2^2)=27/200
v^2=9/5
v=1.34m/s
then from there i suppose i should split the velocity between the 2 as the ratio of their mass, but i still dont come to the correct answer, i see in the answer that the ratio of the velocity is the same as the ratio of the mass which is logical, but how do i get those numbers
4. Sep 5, 2008
### jackiefrost
The total final kinetic energy KE is (1/2)m1v1^2 + (1/2)m2v2^2 and equals the work done by the electric field w. That work is equal to the difference between the initial and final potential energy.
KE = -w = (kq1q2/r2 - kq1q2/r1) [r2=final dist 2m, r1=initial dist 1m]
The total kinetic energy is divided according to the proportion of the individual masses to the total mass m1+m2. Note however that the ke2 of the more massive m2 is affected proportionally less than that for m1. This makes sense since the electric force acts to accelerate the less massive m1 to a proportionally higher velocity than m2 and it's those new velocities that account for the new individual kinetic energies since the masses haven't changed. That's why the following equations don't look correct (the kinetic energy of m1 using the m2/m1+m2 proportion instead of m1/m1+m2, etc...)
ke1 = -w(m2/m1+m2) [ke1 is kinetic energy of m1, etc]
ke2 = -w(m1/m1+m2)
then you can solve for v1 using ke1 and m1, etc...
jf | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9652554988861084, "perplexity": 1066.4518027283898}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171066.47/warc/CC-MAIN-20170219104611-00532-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://alienscientist.com/trigonometry/ | Below is a rough guide to trigonometry mostly borrowed from Wikipedia. Probably the easiest way to learn trig is through some of the excellent online instructional videos out there, which take you through all the material in baby steps. Below is a video from YouTube user KhanAcademy If you’d like an excellent series of instructional videos in mathematics, I highly recommend checking out this guy’s videos and website.
For more educational resources be sure to visit http://www.khanacademy.org/
## Overview
In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. These ratios are given by the following trigonometric functions of the known angle A, where ab and c refer to the lengths of the sides in the accompanying figure:
• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$
• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.
$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$
• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,.$
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsinearccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.
With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.
### Extending the definitions
Graphs of the functions sin(x) and cos(x), where the angle x is measured in radians.
The above definitions apply to angles between 0 and 90 degrees (0 and p/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2p radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or p radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
ex + iy = ex(cosy + isiny).
See Euler’s and De Moivre’s formulas.
### Mnemonics
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sinecosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in *SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
• “TOA CAH SOH” — pronounced as ‘tua’ ‘ca’ (i.e. the word ‘car’ without the ‘r’) ‘so’ — means “Big Foot Lady” in Chinese dialect, which could help in remembering the ratios.
Also used is its reverse, OHSAHCOAT (pronounced oh-sah-coat). It is interpreted as OHS-AHC-OAT, as shown below.
Opposite ÷ Hypotenuse = Sine
One way to remember the letters is to sound them out phonetically (i.e. “SOHCAHTOA”).[13] Another method is to expand the letters into a phrase, such as “Some Old Horses Chew Apples Happily Throughout Old Age”.[14]
### Calculating trigonometric functions
Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.
## Applications of trigonometry
Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theoryacousticsoptics, analysis of financial markets, electronicsprobability theorystatisticsbiologymedical imaging (CAT scans and ultrasound), pharmacychemistrynumber theory (and hence cryptology), seismologymeteorologyoceanography, many physical sciences, land surveying and geodesyarchitecturephoneticseconomicselectrical engineeringmechanical engineeringcivil engineeringcomputer graphicscartographycrystallography and game development.
## Standard Identities
Identities are those equations that hold true for any value.
1.:sin2A + cos2A = 1
2.:sec2A – tan2A = 1
3.:cosec2A – cot2A = 1
## Angle Transformation Formulas
1.:sin(A + B) = sinA * cosB + cosA * sinB
2.:cos(A + B) = cosA * cosB – sinA * sinB
3.:sin(A – B) = sinA * cosB – cosA * sinB
4.:cos(A – B) = cosA * cosB + sinA * sinB
## Common formulas
Triangle with sides a,b,c and respectively opposite angles A,B,C
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
In the following identities, AB and C are the angles of a triangle and ab and c are the lengths of sides of the triangle opposite the respective angles.
### Law of sines
The law of sines (also known as the “sine rule”) for an arbitrary triangle states:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R,$
where R is the radius of the circumscribed circle of the triangle:
$R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.$
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
$\mbox{Area} = \frac{1}{2}a b\sin C.$
All of the trigonometric functions of an angle ? can be constructed geometrically in terms of a unit circle centered at O.
### Law of cosines
The law of cosines (known as the cosine formula, or the “cos rule”) is an extension of the Pythagorean theorem to arbitrary triangles:
$c^2=a^2+b^2-2ab\cos C ,\,$
or equivalently:
$\cos C=\frac{a^2+b^2-c^2}{2ab}.\,$
### Law of tangents
The law of tangents:
$\frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{2}(A+B)\right]}$
### Euler’s formula
Euler’s formula, which states that eix = cosx + isinx, produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
$\sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8524771928787231, "perplexity": 518.2024916242897}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141747887.95/warc/CC-MAIN-20201205135106-20201205165106-00518.warc.gz"} |
https://math.meta.stackexchange.com/questions/1801/i-just-noticed-my-latex-error-in-a-comment-but-its-been-5-minutes-how-can-i-fi | # I just noticed my LaTeX error in a comment but it's been 5 minutes. How can I fix it?
Arghhh! I messed up some LaTeX in a comment, but didn't notice until after 5 minutes had passed and so I can't fix it. Is that it? Is it always going to look ugly and I can't do anything about it? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8746159672737122, "perplexity": 207.2667281796725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655888561.21/warc/CC-MAIN-20200705184325-20200705214325-00281.warc.gz"} |
http://www.edurite.com/kbase/different-parts-of-cartesian-plane | • Class 11 Physics Demo
Explore Related Concepts
different parts of cartesian plane
From Wikipedia
Cartesian product
In mathematics, a Cartesian product (or product set) is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept.
Specifically, the Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X× Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y (e.g., the whole of the x–y plane):
X\times Y = \{(x,y) | x\in X \ \text{and} \ y\in Y\}.
For example, the Cartesian product of the 13-element set of standard playing card ranks {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2} and the four-element set of card suits {♠, ♥, ♦, ♣} is the 52-element set of all possible playing cards: ranks× suits = {(Ace, ♠), (King, ♠), ..., (2, ♠), (Ace, ♥), ..., (3, ♣), (2, ♣)}. The corresponding Cartesian product has 52 = 13 × 4 elements. The Cartesian product of the suits× ranks would still be the 52 pairings, but in the opposite order {(♠, Ace), (♠, King), ...}. Ordered pairs (a kind of tuple) have order, but sets are unordered. The order in which the elements of a set are listed is irrelevant; you can shuffle the deck and it's still the same set of cards.
A Cartesian product of two finite sets can be represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.
Basic properties
Let A, B, C, and D be sets.
In cases where the two input sets are not the same, the Cartesian product is not commutative because the ordered pairs are reversed.
Although the elements of each of the ordered pairs in the sets will be the same, the pairing will differ.
A \times B \neq B \times A
For example:
{1,2} x {3,4} = {(1,3), (1,4), (2,3), (2,4)}
{3,4} x {1,2} = {(3,1), (3,2), (4,1), (4,2)}
One exception is with the empty set, which acts as a "zero", and for equal sets.
A \times \emptyset = \emptyset \times A = \emptyset
and, supposing G,T are sets and G=T:
(G) \times (T) = (T) \times (G).
Strictly speaking, the Cartesian Product is not associative.
(A\times B)\times C \neq A \times (B \times C)
The Cartesian Product acts nicely with respect to intersections.
(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)
Notice that in most cases the above statement is not true if we replace intersection with union.
(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)
However, for intersection and union it holds for:
(A) \times (B \cap C) = (A \times B) \cap (A \times C)
and,
(A) \times (B \cup C) = (A \times B) \cup (A \times C).
n-ary product
The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn:
X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) : x_i \in X_i \}.
It is a set of n-tuples. If tuples are defined as nested ordered pairs, it can be identified to (X1× ... × Xn-1) × Xn.
Cartesian square and Cartesian power
The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X2 = X× X. An example is the 2-dimensional planeR2= R × R where R is the set ofreal numbers - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The cartesian power of a setX can be defined as:
X^n = \underbrace{ X \times X \times \cdots \times X }_{n}= \{ (x_1,\ldots,x_n) \ | \ x_i \in X \ \text{for all} \ 1 \le i \le n \}.
An example of this is R3= R × R × R, with R again the set of real numbers, and more generally Rn.
The n-ary cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Infinite products
It is possible to define the Cartesian product of an arbitrary (possibly infinite) family of sets. If I is any index set, and {Xi | i∈ I} is a collection of sets indexed by I, then the Cartesian product of the sets in X is defined to be
\prod_{i \in I} X_i = \{ f : I \to \bigcup_{i \in I} X_i\ |\ (\forall i)(f(i) \in X_i)\},
that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of Xi .
For each j in I, the function
\pi_{j} : \prod_{i \in I} X_i \to X_{j},
defined by πj(f) = f(j) is called the j -th projection map.
An important case is when the index set is N the natural numbers: this Cartesian product is the set of all infinite sequences with the i -th term in its corresponding set Xi . For example, each element of
\prod_{n = 1}^\infty \mathbb R =\mathbb{R}^\omega= \mathbb R \times \mathbb R \times \cdots,
can be visualized a
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.
The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768-1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745-1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.
The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates– the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
Notational conventions
In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this:
z = x + iy\, for example: z = 4 + i5,
where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.
In the Cartesian plane the point (x, y) can also be represented in polar coordinates as
In the Cartesian plane it may be assumed that the arctangent takes values from −Ï€/2 to Ï€/2 (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x≤ 0. In the complex plane these polar coordinates take the form
z = x + iy = |z|\left(\cos\theta + i\sin\theta\right) = |z|e^{i\theta}\,
where
|z| = \sqrt{x^2+y^2}; \quad \theta = \arg(z) = -i\ln\frac{z}.\,
Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ< 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Notice that the argument of z is multi-valued, because the complex exponential function is periodic, with period 2πi. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠0.
The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.
Almost all of complex analysis is concerned with complex functions– that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In symbols we write
z = x + iy;\qquad f(z) = w = u + iv\,
and often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).
Stereographic projections
It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place it's center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.
We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped
Horizontal plane
In geometry, physics, astronomy, geography, and related sciences and contexts, a planeis said to be horizontal at a given point if it is locally perpendicular to thegradient of the gravityfield, i.e., with the direction of the gravitational force (per unit mass) at that point.
In radio science, horizontal plane is used to plot an antenna's relative field strength in relation to the ground (which directly affects a station's coverage area) on a polar graph. Normally the maximum of 1.000 or 0 dB is at the top, which is labeled 0o, running clockwise back around to the top at 360°. Other field strengths are expressed as a decimal less than 1.000, a percentage less than 100%, or decibels less than 0 dB. If the graph is of an actual or proposed installation, rotation is applied so that the top is 0otrue north. See also the perpendicular vertical plane.
In general, something that is horizontal can be drawn from left to right (or right to left), such as the x-axis in the Cartesian coordinate system.
Discussion
Although the word horizontal is common in daily life and language (see below), it is subject to many misconceptions. The precise definition above and the following discussion points will hopefully clarify these issues.
• The concept of horizontality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being horizontal loses its meaning.
• In the presence of a simple, time-invariant, rotationally symmetric gravity field, a plane is horizontal only at the reference point. The horizontal planes with respect to two separate points are not parallel, they intersect.
• In general, a horizontal plane will only be perpendicular to a vertical direction if both are specifically defined with respect to the same point: a direction is only vertical at the point of reference. Thus both horizontality and verticality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to. Note that (1) the same restriction applies to the straight lines contained within the plane: they are horizontal only at the point of reference, and (2) those straight lines contained in the plane but not passing by the reference point are not horizontal anywhere.
• In reality, the gravity field of a heterogeneous planet such as Earth is deformed due to the inhomogeneous spatial distribution of materials with different densities. Actual horizontal planes are thus not even parallel even if their reference points are along the same vertical direction.
• At any given location, the total gravitational force is a function of time, because the objects that generate the reference gravity field move relative to each other. For instance, on Earth, the local horizontal plane at a given point (as materialized by a pair of spirit levels) changes with the relative position of the Moon (air, sea and land tides).
• Furthermore, on a rotating planet such as Earth, there is a difference between the strictly gravitational pull of the planet (and possibly other celestial objects such as the Moon, the Sun, etc.), and the apparent net force applied (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is due to the centrifugal force associated with the planet's rotation. This is a fictitious force: it only arises when calculations or experiments are conducted in non-inertial frames of reference.
Practical use in daily life
The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life.
This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy.
In graphical contexts, such as drawing and drafting on rectangular paper, it is very common to associate one of the dimensions of the paper with a horizontal, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the horizontal direction is typically from the left side of the paper to the right side. This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.
Projective plane
In mathematics, the projective plane is a geometric construction that extends the concept of a plane. In the ordinary plane, two lines typically intersect in a single point, but there are some pairs of lines — namely, parallel lines — that do not intersect. The projective plane is, in one view, the ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in the projective plane intersect in one and only one point.
The projective plane has two common definitions. The first comes from linear algebra; it produces planes that are homogeneous spaces for some of the classical groups. Important examples include the real projective plane \mathbb{RP}^2 and the complex projective plane \mathbb{CP}^2. The second, more general definition comes from axiomatic geometry and finite geometry; it is suitable for study of the incidence properties of plane geometry.
The projective plane generalizes to higher-dimensional projective spaces; that is, a projective plane is a 2-dimensional projective space.
Linear-algebraic definition
In one view, the projective plane is the set of lines through the origin in 3-dimensional space, and a line in the projective plane arises from a plane through the origin in 3-dimensional space. This idea can be made precise as follows.
Let K be any division ring. Let K3 denote the set of all triples x = (x0, x1, x2) of elements of K (a Cartesian product). For any nonzero x in K3, the line in K3 through the origin and x is the subset
\{k x : k \in K\}
of K3. Similarly, let x and y be linearly independent elements of K3, meaning that if k x + l y = 0 then k = l = 0. The plane through the origin, x, and y in K3 is the subset
\{k x + l y : k, l \in K\}
of K3. The plane contains various lines.
The projective plane over K, denoted K\mathbb{P}^2, is the set of all lines in K3. A subset L of K\mathbb{P}^2 is a line in K\mathbb{P}^2 if there exists a plane in K3 whose set of lines is exactly L.
A slightly different definition is as follows. The projective plane is the set K3 - {(0, 0, 0)} modulo the equivalence relation
x \sim k x, k \in K.
Lines in the projective plane are defined exactly as above. If K is a topological space, then K\mathbb{P}^2 inherits a topology via the product, subspace, and quotient topologies.
The coordinates (x0, x1, x2) on K\mathbb{P}^2 are called homogeneous coordinates. Each triple (x0, x1, x2) represents a well-defined point in K\mathbb{P}^2, except for the triple (0, 0, 0), which represents no point. Each point in K\mathbb{P}^2 is potentially represented by many triples.
Examples
The real projective plane \mathbb{RP}^2 arises when K is taken to be the real numbers. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.
The complex projective plane \mathbb{CP}^2 arises when K is taken to be the complex numbers. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields serve as fundamental examples in algebraic geometry.
The quaternionic projective plane is also of independent interest. The Cayley plane is considered to be a projective plane over the octonions, but the preceding construction does not suffice to describe it, because the octonions do not form a division ring.
Taking K to be the finite field of pn elements produces a projective plane of p2 n + pn + 1 points. The Fano plane, discussed below, is the example with pn = 2.
Relationship to the ordinary plane
The ordinary plane K2 over K embeds into K\mathbb{P}^2 via the map
(x_1, x_2) \mapsto (1, x_1, x_2).
The complement of the image is the set of points of the form (0, x1, x2). From the point of view of the embedding just given, these points are points at infinity. They constitute a line in K\mathbb{P}^2 — namely, the line arising from the plane
\{k (0, 0, 1) + l (0, 1, 0) : k, l \in K\}
in K3. Intuitively, the points at infinity are the "extra" points where parallel lines intersect; the point (0, x1, x2) is where all lines of slope x2 / x1 intersect. Consider for example the two lines
a = \{(x_1, 0) : x_1 \in K\},
b = \{(x_1, 1) : x_1 \in K\}
in the ordinary plane K2. These lines have slope 0 and do not intersect. They can be regarded as subsets of K\mathbb{P}^2 via the embedding above, but these subsets are not lines in K\mathbb{P}^2. Add the point (0, 1, 0) to each subset; that is, let
\bar a = \{(1, x_1, 0) : x_1 \in K\} \cup \{(0, 1, 0)\},
\bar b = \{
Question:To me questions like these force me to think outside the box. Our history and our universe is on one timeline. We wake up, live our day, then go to sleep. This is what we know and we discover all of what we know in this timeline about the world surrounding us and inside of us. I consider this timeline to be viewed in one dimension. Has anyone considered the possibility of more than one timeline on seperate axis' existing all around our timeline? i.e. Parallel dimensions. What is the best theories on parallel dimensions? Are gaps between timelines considered? Inter-timeline travel? -Properties concerning each timeline - A timeline (us) One time interval, Days are 24-hours - B timeline, objects could dissapear and reappear in one time interval, while living organisms exist in another time interval. -C timeline The universe is growing and collapsing at the same time.
Answers:Time is one dimension, one of the famous four: height, width, depth, time, using commonplace terms. I could also write (i, j, k, t); where i, j, and k are unit vectors (e.g., i dot i = 1) designating the three spatial dimensions. And t would be the fourth dimension. Time is real...it's not just the passage of events, like rising, showering, breakfasting, etc. Time passes even if no events take place. Time can be stretched out so that, for example, it would take 2 Earth seconds for 1 second to tick off on a very fast spaceship. Such a stretch is called dilation and this phenomenon demonstrates that time can be manipulated. That is a prime clue that time is real. If it were not real, we wouldn't be able to dilate it. This dilation can be used to travel into the future. For instance, if a star trekker in the above example traveled one year according to his clock on the spaceship and then returned to Earth, he would find Earth time had advanced two years. In other words, the spaceman would find himself one year into his future when he stepped out of the ship. As to "parallel dimensions" you are mixing concepts. In fact it's higher dimensions and parallel universes. [See source.] String theory posits up to 11 dimensions instead of the conventional four we know and love. One aspect of the theory suggests the other seven (all spatial) are simply curled up so tiny (1 Planck length = 10^-33 cm) that we can't see them. But strings, because they are also tiny, can see the extra dimensions and are constrained by them just as we are constrained by the four dimensions of our universe. One WAG of string theory is the parallel universe. Each universe is like a slice of bread in a mega universe loaf. Each slice is separated by 1 Planck length and 1 Planck time (which is also very tiny but I've forgotten the number). One SWAG resulting from the WAG is that two or more of the parallel universes collided and rebound. That change in momentum over time gave rise to the tremendous energy we call the Big Bang. Thus, there would be a BB in our universe as well as another BB in the universe that collided with us. And so, if we count the BB as t = 0, the timeline of the two BBs starts at the same time the two parallel universes. That is not to say the chains of events are identical...it's unlikely they are. But time, a real dimension, will be identical. There would be a gap between the timelines of the two parallel universes. The time length of that discontinuity would be 1 Planck time. And there would be a spatial gap of 1 Planck length between the two after they rebound from the collision. Both these gaps result because, theorectically, the 1 Planck time and the 1 Planck length are the smallest possible intervals in time and space. Unless the makeup of the two colliding universes was significantly different, the makeup of the two BBs ought to be about the same. So the uniform initial energies of both would go through the same evolutions and end up with the same kind of galaxies, planets, and energies. This suggests there might be living, intelligent beings living out their lives in the parallel universe...wondering if there is life out there.
Question:the cartesian plane, specifies that both the x-coordinate and the y-coordinate in quadrant one, are always positive. Explain and prove why the points in the other thee quadrants have a different outcome?
Answers:The points in the other three quadrants have a different outcome, because in all of these, at least one co-ordinate is negative. In quadrant II, x is negative, y is positive. In quadrant iii, both co-ordinates are negative. In the fourth quadrant, x is positive, y is negative. NB: For a better answer, ask in the math section.
Question:Can someone show me how I would do this? It asks me to calculate the distance between the points F = (2,2) and M = (5,-4) in the cartesian plane. And to write the answer in radical form. I'm just not sure how I would go about doing this. Could someone please help me out? 10 points will be awarded to the best answer!
Answers:it's a lot like computing slope in that you find the differences between the x's and between the y's, but then you square them, add them, and square root the result: d = [ (x1 - x2) + (y1 - y2) ] d = [ (5-2) + (-4-2) ] d = ( 9 + 36) d = 45 = 3 5
Question:read title i need it today
Answers:Hilarious... I think. I know you needed this yesterday, or the day before, but here's an idea or two. A parody of the X-Men - the "X" is really a slanted Cartesian plane anyways, isn't it? Passion of the Cartesian Plane - Jesus being crucified on the C-plane is already causing controversy, push the envelope a bit farther. C-Plane Soaps - create torrid love triangles (Oblique, Equilateral, Isosceles, and Scalene) where drama erupts between different points. That's all I got for now - hope I helped a bit! :) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9507756233215332, "perplexity": 481.6459735509411}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098990.43/warc/CC-MAIN-20150627031818-00211-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.thejournal.club/c/paper/344/ | #### Solving Constraint Satisfaction Problems through Belief Propagation-guided decimation
##### Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian
Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after each step. Its outcome provides an heuristic to make choices at next step. This approach has been referred to as `decimation,' with reference to analogous procedures in statistical physics. The behavior of decimation procedures is poorly understood. Here we consider a simple randomized decimation algorithm based on belief propagation (BP), and analyze its behavior on random k-satisfiability formulae. In particular, we propose a tree model for its analysis and we conjecture that it provides asymptotically exact predictions in the limit of large instances. This conjecture is confirmed by numerical simulations.
arrow_drop_up | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8822713494300842, "perplexity": 925.6630744453371}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303779.65/warc/CC-MAIN-20220122073422-20220122103422-00115.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=91953 | ## Water, oxygen, etc.
Miranda 1J
Posts: 51
Joined: Fri Sep 29, 2017 7:06 am
### Water, oxygen, etc.
When we have a balanced equation and we have to split it into its reduction and its oxidation do we always put the oxygens, waters, etc. in the reduction equation and why?
Justin Lau 1D
Posts: 51
Joined: Sat Jul 22, 2017 3:00 am
### Re: Water, oxygen, etc.
Water, H+ and OH- are only included in the half reactions when the half reactions occur in an acidic or basic solution, or if the reaction itself such as the electrolysis of water involves these factors. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.849902868270874, "perplexity": 3262.8604740575197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669454.33/warc/CC-MAIN-20191118053441-20191118081441-00233.warc.gz"} |
https://www.clutchprep.com/chemistry/practice-problems/82705/how-many-moles-and-how-many-ions-of-each-type-are-present-in-each-of-the-followi-5 | # Problem: How many moles and how many ions of each type are present in each of the following?1.65 L of a solution containing 8.83×1021 formula units of cesium nitrate per liter
⚠️Our tutors found the solution shown to be helpful for the problem you're searching for. We don't have the exact solution yet.
###### Problem Details
How many moles and how many ions of each type are present in each of the following?
1.65 L of a solution containing 8.83×1021 formula units of cesium nitrate per liter | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9126970767974854, "perplexity": 1362.728137485117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657172545.84/warc/CC-MAIN-20200716153247-20200716183247-00561.warc.gz"} |
http://teaching-direction.blogspot.com/2011/04/what-is-paragraph-chapter-1.html | ## Saturday, April 30, 2011
### 0What is Paragraph...? (Chapter 1)
Paragraph Development,
What is a paragraph?
Paragraphs are the building blocks of papers. Many students define paragraphs in terms of length: a paragraph is a group of at least five sentences, a paragraph is half a page long, etc. In reality, though, the unity and coherence of ideas among sentences is what constitutes a paragraph. A paragraph is defined as "a group of sentences or a single sentence that forms a unit" (Lunsford and Connors 116). Length and appearance do not determine whether a section in a paper is a paragraph. For instance, in some styles of writing, particularly journalistic styles, a paragraph can be just one sentence long. Ultimately, a paragraph is a sentence or group of sentences that support one main idea. In this handout, we will refer to this as the "controlling idea," because it controls what happens in the rest of the paragraph.
How do I decide what to put in a paragraph?
Before you can begin to determine what the composition of a particular paragraph will be, you must first decide on a working thesis for your paper. What is the most important idea that you are trying to convey to your reader? The information in each paragraph must be related to that idea. In other words, your paragraphs should remind your reader that there is a recurrent relationship between your thesis and the information in each paragraph. A working thesis functions like a seed from which your paper, and your ideas, will grow. The whole process is an organic one—a natural progression from a seed to a full-blown paper where there are direct, familial relationships between all of the ideas in the paper.
The decision about what to put into your paragraphs begins with the germination of a seed of ideas; this "germination process" is better known as brainstorming. There are many techniques for brainstorming; whichever one you choose, this stage of paragraph development cannot be skipped. Building paragraphs can be like building a skyscraper: there must be a well-planned foundation that supports what you are building. Any cracks, inconsistencies, or other corruptions of the foundation can cause your whole paper to crumble.
So, let's suppose that you have done some brainstorming to develop your thesis. What else should you keep in mind as you begin to create paragraphs? Every paragraph in a paper should be
* Unified—All of the sentences in a single paragraph should be related to a single controlling idea (often expressed in the topic sentence of the paragraph).
* Clearly related to the thesis—The sentences should all refer to the central idea, or thesis, of the paper (Rosen and Behrens 119).
* Coherent—The sentences should be arranged in a logical manner and should follow a definite plan for development (Rosen and Behrens 119).
* Well-developed—Every idea discussed in the paragraph should be adequately explained and supported through evidence and details that work together to explain the paragraph's controlling idea (Rosen and Behrens 119).
How do I organize a paragraph?
There are many different ways to organize a paragraph. The organization you choose will depend on the controlling idea of the paragraph. Below are a few possibilities for organization, with brief examples.
* Narration: Tell a story. Go chronologically, from start to finish. (See an example.)
* Description: Provide specific details about what something looks, smells, tastes, sounds, or feels like. Organize spatially, in order of appearance, or by topic. (See an example.)
* Process: Explain how something works, step by step. Perhaps follow a sequence—first, second, third. (See an example.)
* Classification: Separate into groups or explain the various parts of a topic. (See an example.)
* Illustration: Give examples and explain how those examples prove your point. (See the detailed example in the next section of this handout.) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8196592330932617, "perplexity": 1317.530906419579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119655159.46/warc/CC-MAIN-20141024030055-00197-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://theoval.cmp.uea.ac.uk/~nlct/latex/packages/ueaexam/ueaexam-manual.html | ueaexam v4.14: LaTeX 2"Class for Writing UEA Exam Sheets
ueaexam v4.14: LATEX2" Class for Writing UEA Exam Sheets
2012-01-22
1 Introduction
This is the documentation for ueaexam, a class le for typesetting exam sheets. It is loosely based on the sys-exam package and on the standard article class le1.
Top
2 Required Packages
The following packages are automatically loaded:
• amsmath
• etoolbox
• probsoln allows questions and answers to be stored in and loaded from databases. Must be at least version 3.02. Earlier versions will cause an error.
• drawwatermark provides facility to put DRAFT or SOLUTIONS across the background of each page.
• geometry sets up page layout.
• ifthen provides high-level conditional commands.
• calc used for counter calculations.
• fp used for adding up the marks for each question.
• graphicx images can be included using \includegraphics[hkey=value optionsi]{hlenamei}.
• xcolor text colour may be changed using \textcolor{hcolouri}{htexti}.
• enumerate provides optional argument to enumerate environment to change the item labels.
• paralist provides inparaenum enviroment for numbered lists within a paragraph. Note that probsoln also provides textenum for in-line enumerations.
• fmtcount provides command to convert numerics into words.
If the times class option is specied, the following packages are also loaded:
• mathptmx
• helvet
• courier
• lmodern
The exsizes bundle is required if any of the class options 14pt, 17pt or 20pt are used.
Top
3 Class Options
times
Use Times/Helvetica/Courier fonts instead of Computer Modern. (Default.)
notimes
Use Computer Modern fonts.
solutions
Display solutions given by
\Solution \Solution{htexti}
and put SOLUTIONS banner across every page.
nosolutions
Hide solutions given by \Solution.
watermarkfirst
Only put watermark on rst page instead of on all pages. Has no eect if neither solutions nor draft used.
scorecheck
Check that the scores for each question add up to the correct total given by \questiontotal. (Default.)
notesallowed
Species that notes are allowed in the exam.
nonotesallowed
Species that notes are not allowed in the exam. (Default.)
noscorecheck
Don't check the scores.
10pt
Set the normal font size to 10pt.
11pt
Set the normal font size to 11pt.
12pt
Set the normal font size to 12pt. (Default.)
14pt
Set the normal font size to 14pt.
17pt
Set the normal font size to 17pt.
20pt
Set the normal font size to 20pt.
a4paper
Set the paper size to A4. (Default.)
a3paper
Set the paper size to A3.
oneside
Set the page formatting for one-sided printing.
twoside
Set the page formatting for double-sided printing. (Default.)
draft
Draft mode on and puts DRAFT banner across every page. This also sets draft mode for all loaded packages, included graphicx, so you won't see included images. You can override this with
\setkeys{Gin}{draft=false}
final
Draft mode o. (Default.)
leqno
Left equation numbers.
fleqn
Flush left equations.
\ueaexamhook \ueaexamhook
If this command has been dened before ueaexam is loaded, it will be executed during the option processing stage. This can be used to set up a script to generate exam sheet and solution sheet without having to modify the document. For example, suppose the document is called exam2011.tex and loads ueaexam without any class options:
\documentclass{ueaexam}
then the exam sheet (exam2011.pdf) can be create using:
pdflatex exam2011
and the solution sheet (exam2011-solutions.pdf) can be created using (no line breaks):
pdflatex -jobname exam2011-solutions
Top
4 Large Font Sizes for Partially Sighted Students
As from version 2.01, there are extra class options 14pt, 17pt and 20pt in order to produce larger font sizes. Note that to use these options you must have the extsizes bundle installed2.
Returning to the example in the previous section where the document is called exam2011.tex, the large-font exam sheet (exam2011-large.pdf) can be created using (no line breaks):
pdflatex -jobname exam2011-large
"\\def\\ueaexamhook{\setptsize{20}\\setexampapersize{a3}}"
\\input{exam2011}
Alternatively, you can use an external application to resize the A4 document. For example (no line breaks):
gs -q -dNOPAUSE -dBATCH -sDEVICE=pdfwrite -sPAPERSIZE=a3
-dFIXEDMEDIA -dPDFFitPage -sOutputFile=exam2011-large.pdf
exam2011.pdf
Top
5 Available Commands
Top
5.1 Preamble Commands
The following commands may be used in the preamble:
\questiontotal \questiontotal{hnumberi}
Used to specify the total number of marks for each question. (Default: 15). If the marks for a question don't add up to this value, a warning is issued in draft mode and an error is issued in nal mode. You can suppress the score check using the noscorecheck class option.
\gquestiontotal \gquestiontotal{hnumberi}
As \questiontotal but has a global eect.
\university \university{huniversity namei}
Used to specify the name of the university. Defaults to University of East Anglia.
\school \school{hschool namei}
Used to specify the name of the school. Defaults to School of Computing Sciences.
\sitting \sitting{htexti}
Used to specify the sitting of the exam. For example:
\sitting{May/June UG}
or
\sitting{UG Reassessment/Delayed First Sit}
\examyear \examyear{hyeari}
Used to specify the academic year (e.g. 2010/11). Defaults to the current academic year.
\course \course{hcodei}{hnamei}
Used to specify the course code and name. For example:
\course{CMP-1A4Y}{Programming --- Languages and Software Construction}
\timeallowed \timeallowed{htime limiti}
Used to specify the time allowed. For example:
\timeallowed{2 hours}
\version \version{hnumberi}
Used to specify the exam version number, as required by the new regulations. For example:
\version{1}
\contact \contact{hnamei}
Used to specify the module contact name, as required by the new regulations. For example:
\contact{Dr A.N. Other, CMP}
\rubric \rubric{htexti}
The rubric on the front page regarding the number of questions to be done in each section can be automatically generated using the \section command (see below), however additional information can be added using the command \rubric. For example:
\rubric{Use separate answer books for each section.}
\turnovertrue \turnovertrue
This will put the words TURN OVER at the bottom of each page3 except for the last page. (This is the default.)
\turnoverfalse \turnoverfalse
Prevents the words TURN OVER appearing at the bottom of each page.
Top
5.2 Document Commands
The following commands may be used within the document:
\maketitle \maketitle
This makes the title page. It should be the rst command in the document environment.
\score \score{hnumberi}
This is used to indicate the maximum number of marks for a question or part of question. For example:
Find the derivative of $f(x)=x^2 - x + 1$.\score{5}
This adds hnumberi to the running total for the current question and displays the mark using
\scoreformat \scoreformat{hnumberi}
which has the default denition: \marginpar{[\marklabel{hmarki}]} where
\marklabel \marklabel{hmarki}
prints hmarki followed by either mark (where hmarki is 1) or marks.
The \score command must be used within an enumerate environment.
Warning: don't attempt to use or redene \mark or \marks to specify the mark. These commands are TEX primitives and should not be meddled with.
\section \section{hNumber of Questions to Be Answeredi}
This is the only sectioning command dened within this class le. This command issues a \clearpage, prints SECTION followed by the section letter (e.g. A), and on the following line it prints the argument. Both lines are centrally aligned. As usual, the section can be referenced using \label and \ref. For example, the input:
\section{2}
would produce the output:
SECTION A
Answer TWO questions from this Section.
The \section command may occur within an enumerate or any of the other list-making environments, provided at least one \item precedes it. This means that all the exam questions can be placed within a single enumerate environment, ensuring consistent numbering throughout the document. Note that there is no starred version of this command.
If the argument hNumber of Questions to Be Answeredi is a number or the strings all or the, the full section title and rubric information will be generated automatically. For example:
\section{all}
will produce:
SECTION A
Answer ALL questions from this Section.
and it will add ALL questions from Section A to the rubric information on answering questions.
Alternatively, the section title and the rubric information can be explicitly entered using:
\section[hrubric infoi]{htitle texti}
For example:
Note: It is best not to have any commands within the optional argument of \section, unless they expand to a simple text string4. At best, this will cause LATEX to keep complaining that the title page is not up to date, at worse it will cause a TEX capacity exceeded error.
This may be used to insert any additional text to the rubric. The above note also applies to this command.
\Solution \Solution{htexti}
(Note the initial capital letter.) This may be used to specify the solution to the problem. The solution is only displayed if the solutions class option is used. Figures and tables within the argument of \Solution will have dierent numbering to those outside of \Solution. This ensures that the gures and tables that form part of the questions retain the same numbering in the solution sheet. The solution text is formatted according to
\solutionfont \solutionfont
This defaults to slanted sans-serif.
As from v4.08, the argument of \Solution may contain verbatim text.
Top
6 Dening Problems and their Solutions in External Files
It's possible to dene the exam questions in a dierent le. This makes it easier to change the ordering of the questions within the exam. Questions can be dened using:
defproblem \begin{defproblem}{hlabeli}
htexti
\end{defproblem}
For example:
\begin{defproblem}{compute-factorial}
Compute $4!$, $5!$ and $6!$.\score{3}
\Solution
{%
\begin{align*}
4! &= 4 \times 3 \times 2 \times 1 = 24\\
5! &= 5 \times 4! = 5 \times 24 = 120\\
6! &= 6 \times 5! = 6 \times 120 = 720
\end{align*}
}
\end{defproblem}
There is a short-cut command
For example, the above can also be written as:
\newproblem{compute-factorial}
{%
Compute $4!$, $5!$ and $6!$.\score{3}
}
{%
\begin{align*}
4! &= 4 \times 3 \times 2 \times 1 = 24\\
5! &= 5 \times 4! = 5 \times 24 = 120\\
6! &= 6 \times 5! = 6 \times 120 = 720
\end{align*}
}
textenum \begin{textenum}
You can use the textenum environment for in-line numbered lists. This environment uses the same counter as the corresponding enumerate level, so answers that require longer passages than the question can use the same numbering system. For example:
\newproblem{compute-factorial}
{%
Compute
\begin{textenum}
\item $4!$, \item $5!$ and \item $6!$.
\end{textenum}\score{3}
}
{%
\begin{enumerate}
\item $4! = 4 \times 3 \times 2 \times 1 = 24$
\item $5! = 5 \times 4! = 5 \times 24 = 120$
\item $6! = 6 \times 5! = 6 \times 120 = 720$
\end{enumerate}
}
In your document, you can load the questions using one of the following:
Loads all problems dened in hlei into a database named hdb namei.
Loads the problems dened in hlei whose labels are given in hlabel listi into a database named hdb namei.
Loads the problems dened in hlei whose labels are not given in hlabel listi into a database named hdb namei.
Loads hni randomly selected problems dened in hlei into a database named hdb namei.
Once you have loaded the required problems, you can either explicitly select problems using:
\useproblem \useproblem[hdb namei]{hlabeli}
or you can iterate over all problems using:
\foreachproblem \foreachproblem[hdb namei]{hbodyi}
Within hbodyi, you can use
\thisproblem \thisproblem
to use the current problem and
\thisproblemlabel \thisproblemlabel
to access the current label.
Top
7 Example Documents
The following is a short sample document illustrating the use of this class le:
\documentclass{ueaexam}
\course{ABC-1XY}{SAMPLE COURSE}
\timeallowed{2 hours}
\version{1}
\contact{Dr A.N. Other}
\begin{document}
\maketitle
\section{the}
\begin{enumerate}
\item This is the first question, it has two parts.
\begin{enumerate}
\item The first part \score{20}
\Solution{This is the solution to the first part.}
\item The second part \score{20}
\Solution{This is the solution to the second part.}
\end{enumerate}
\section{2}
\item This is the first question of the second part,
but because we are still in the same enumerate environment,
this question is question number 2. If the last line of this
paragraph is long, it will run into the marks so in this
case, we can put the marks on the following line to make it neater.
\par\mbox{}\score{30}
\Solution{This is the solution.}
\item This is question number 3.\score{30}
\Solution{This is the solution to question number 3.}
\item This is the last question.\score{30}
\Solution{This is the solution to the last question.}
\end{enumerate}
\end{document}
In the following example, the exam is made up of three sections, where each section is written by a dierent lecturer (call them Dr A, Dr B and Dr C.) Rather than the lecturers trying to determine who has the most up-to-date version of the le, the questions for each section are dened in three separate les, say A.tex, B.tex and C.tex. In this way, each lecturer can independently edit their own questions.
Suppose Dr A is providing questions on counting, then A.tex might look like:
\newproblem{compute-factorial}
{%
Compute
\begin{textenum}
\item $4!$, \item $5!$ and \item $6!$.
\end{textenum}\score{15}
}
{%
\begin{enumerate}
\item $4! = 4 \times 3 \times 2 \times 1 = 24$
\item $5! = 5 \times 4! = 5 \times 24 = 120$
\item 6! = 6 \times 5! = 6 \times 120 = 720$\end{enumerate} } \newproblem{factorial-terms}% {% Write in terms of factorials: \begin{textenum} \item$21\times20$, \item$\frac{1}{9\times8}$, \item$42$. \end{textenum}\score{15} }% {% \begin{enumerate} \item$21\times20 = \frac{21!}{19!}$\item$\frac{1}{9\times8} = \frac{7!}{9!}$\item$42 = \frac{42!}{41!}$\end{enumerate} } Suppose Dr B is providing questions on dierentiation, then B.tex might look like: \begin{defproblem}{diff-f} Differentiate each of the following functions with respect to$x$: \begin{enumerate} \item$f(x) = x^2$.\score{5} \Solution{$f’(x) = 2x$.} \item$f(x) = 3x^3$.\score{5} \Solution{$f’(x) = 9x^2$.} \item$f(x) = 2x^2 + x$.\score{5} \Solution{$f’(x) = 4x + 1$.} \end{enumerate} \end{defproblem} Suppose Dr C is providing questions on set theory, then C.tex might look like: \newproblem{sets-showequal}% {% Which of these sets are equal:$\{a, b, c\}$,$\{c, b, a\}$,$\{c, b, b, a\}$,$\{a, c, b, c\}\$?\score{15}
}%
{%
They are all equal. Order and repetition do not change a set.
}
The main le might then look as follows:
\documentclass{ueaexam}
\course{ABC-2XY}{SAMPLE COURSE II}
\timeallowed{3 hours}
\rubric{Use a separate answer book for each section.}
\version{1}
\contact{Dr A. Other, CMP}
\begin{document}
\maketitle
\section{2}
\begin{enumerate}
\foreachproblem[counting]{\item\thisproblem}
\section{1}
\foreachproblem[differentiation]{\item\thisproblem}
\section{all}
\foreachproblem[sets]{\item\thisproblem}
\end{enumerate}
\end{document}
Top
Acknowledgements
Some of the code was amended by G. Janacek.
Index
A
amsmath package 2
article class 3
article.cls class 4
C calc package 5
class options:
10pt 6
11pt 7
12pt 8
14pt 9, 10, 11
17pt 12, 13, 14
20pt 15, 16, 17
a3paper 18
a4paper 19
draft 20, 21
final 22
fleqn 23
leqno 24
nonotesallowed 25
noscorecheck 26
nosolutions 27
notesallowed 28
notimes 29
oneside 30
scorecheck 31
solutions 32, 33, 34
times 35, 36
twoside 37
watermarkfirst 38
\contact 39
courier package 40
\course 41
D defproblem (environment) 42
drawwatermark package 43
E enumerate (environment) 44, 45, 46, 47, 48
enumerate package 49
environments:
defproblem 50
enumerate 51, 52, 53, 54, 55
inparaenum 56
textenum 57, 58, 59
etoolbox package 60
\examyear 61
exsizes package 62
extsizes package 63
F fmtcount package 64
\foreachproblem 65
fp package 66
G geometry package 67
\gquestiontotal 68
graphicx package 69, 70
H helvet package 71
I ifthen package 72
inparaenum (environment) 73
\item 74
L lmodern package 75
M \maketitle 80
\marklabel 81
mathptmx package 82
N \newproblem 83
P package options:
noscorecheck 84
paralist package 85
probsoln package 86, 87
Q \questiontotal 88, 89
R \rubric 90
S \school 91
\score 92
\scoreformat 93
\section 94, 95
\sitting 96
\Solution 97, 98, 99
\solutionfont 100
sys-exam package 101
T textenum (environment) 102, 103, 104
\thisproblem 105
\thisproblemlabel 106
\timeallowed 107
\turnoverfalse 108
\turnovertrue 109
U ueaexam class 110, 111, 112
\ueaexamhook 113
\university 114
\useproblem 115
V \version 116
X xcolor package 117
1although not all the commands dened in the article.cls article are available in this class le
2If you don't have this package it can be download from http://www.ctan.org/pkg/extsizes
3odd pages only, if two sided printing specied
4This shouldn't be much of a problem as it's unlikely that there will be any, although it does mean that you can't use any spacing macros (such as ˜) either. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8806079626083374, "perplexity": 1206.6446456857295}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687333.74/warc/CC-MAIN-20170920161029-20170920181029-00194.warc.gz"} |
http://www.scholarpedia.org/article/Belykh_map | # Belykh map
Post-publication activity
The Belykh map is a piecewise linear map possessing a chaotic attractor, called the Belykh attractor. The Belykh attractor contains only hyperbolic (saddle) orbits and belongs to the class of quasi-hyperbolic chaotic attractors.
## The Map
Figure 1: First image of the unit square $$S=S_1 \cup S_2$$ under the map $$f\ ,$$ where $$S_1$$ and $$S_2$$ are separated by the discontinuity line $$L(x,y)=0.$$ One iterate of $$f$$ transforms trapezoids $$S_1$$ (light brown) and $$S_2$$ (light blue) into $$fS_1$$ (dark brown) and $$fS_2$$ (dark blue), respectively.
The two-dimensional Belykh map $$f$$ is defined by $$\tag{1} \begin{cases} \bar{x}= \lambda x\\ \bar{y}= \gamma y \end{cases} \mbox{if} \;\;L(x,y)\le 0$$
and $$\tag{2} \begin{cases} \bar{x}= \lambda (x-1)+1\\ \bar{y}= \gamma (y-1)+1 \end{cases} \mbox{if} \;\;L(x,y)>0,$$
where $$L(x,y)=k(2x-1)+2y-1 \ .$$ The map $$f$$ is defined on a square $$S=\{(x,y): 0\le x\le 1,\;\; 0\le y\le 1\}\ .$$ The square $$S$$ is cut into two parts $$S_1$$ and $$S_2$$ by a line $$L(x,y)=k(2x-1)+2y-1=0$$ (see Figure 1). The dynamics of the map $$f$$ is governed by (1) on the lower part $$S_1 \ ,$$ where $$L(x,y) \le 0 \ ,$$ and by (2) on the upper part $$S_2 \ .$$
The set $$A=\bigcap \limits_{n=1}^{\infty} f^n(S)$$ is called the attractor of $$f\ ,$$ or the Belykh attractor.
The map was first introduced and studied in (Belykh, 1976; Belykh, 1980; Belykh, 1982) as a simple model of digital phase locked loop (PLL). The Belykh map is a special case of a hyperbolic map with singularities. Another examples of such a map are the Lozi map (Lozi, 1978) and a dispersing dynamical billiard (Sinai, 1970; Bunimovich, 1974).
The Belykh map is invertible if $$0<\lambda<1/2\ .$$ The ergodic properties of the Belykh map in the parameter range of invertibility were studied in (Pesin, 1992; Afraimovich et al, 1995; Sataev, 1992; Belykh 1995; Sataev, 1999). It has been proved that under the conditions $$\tag{3} 0<\lambda<1/2, \;\;1<\gamma \le \frac{2}{1+|k|}, \;\;\mbox{and}\;\; |k|<1$$
the map $$f$$ has a hyperbolic attractor in the sense that it contains only hyperbolic (saddle) orbits. However, the attractor might be structurally unstable as its saddle periodic orbits bifurcate or disappear when at least one point on the periodic orbit hits the discontinuity line $$L(x,y)=0.$$ Such structurally unstable attractors containing only saddle orbits are called quasi-hyperbolic chaotic attractors (see, for example, (Afraimovich and Hsu, 2003)). The famous Lorenz attractor (Lorenz, 1963) also belongs to this class.
Figure 2: Chaotic Belykh attractor generated by the map $$f$$ with parameters $$\lambda=0.48,$$ $$\gamma=1.3,$$ and $$k=0.5$$ for which the map is invertible.
Figure 3: Fat Belykh attractor generated by the non-invertible map $$f\ .$$ Parameters are $$\lambda=0.8,$$ $$\gamma=1.3,$$ and $$k=0.5\ .$$
Pesin (1992) and Sataev (1992) proved the existence of the Sinai-Ruelle-Bowen measure for the chaotic map $$f\ .$$
Schmeling and Troubetzkoy (1998) considered the Belykh map in a wider range of parameters ($$1/2<\lambda <1$$) where the map is non-invertible. In analogy with the fat baker's transformation (Alexander and Yorke, 1984), they called the map $$f$$ within the wider range of parameters the fat Belykh map and proved that the fat map is chaotic and has a continuous invariant measure (Schmeling, 1998; Schmeling and Troubetzkoy, 1998; Persson, 2008).
Figure 2 demonstrates the chaotic structure of the Belykh attractor in the parameter range of invertibility (3). The fat Belykh attractor in the parameter range of non-invertibility is depicted in Figure 3.
## Embedding of the Belykh attractor into 3-D phase space
Figure 4: Embedding of the Belykh attractor into the 3-D phase space of the hybrid ODE system (9). The parameters $$\delta=-{\rm ln}\, 0.48=0.73,$$ $$\sigma={\rm ln}\, 1.3=0.262,$$ and $$k=0.5$$ correspond to those of Figure 2.
The Belykh map and its planar attractor can be embedded into the phase space of a system of ordinary differential equations (ODEs). The motivation for finding possible flow embeddings of planar discrete-time chaotic attractors is two-fold. First, it gives specific examples of ODE systems, possessing strange attractors with rigorously proven chaotic properties. While planar attractors of discrete-time maps such as, for example, the baker's map, the Lozi map, and the Belykh map have been analytically shown to exhibit quasi-hyperbolic strange attractors, rigorous proofs of chaoticity in ODE systems are rare and often require computer assistance. In light of this, finding a systematic way of planar attractors embedding has its own value for the theory of dynamical systems. Second, these embeddings are an excellent way of visualizing discrete-time attractors and creating graphically appealing images.
The construction that generates the Belykh attractor is formed by two saddle-focus equilibria of two 3-D linear systems. The details of the construction are given below. The same technique can be used for embedding various piecewise-linear chaotic maps into the 3-D phase space by means of hybrid ODE systems.
Consider the following linear system of three ODEs that has a saddle-focus equilibrium at the origin$\tag{4} \left \{\begin{array}{l} \dot{x}=-\delta x\\ \dot{y}=\sigma y+ 2 \pi z,\\ \dot{z}=-2\pi y+ \sigma z \end{array} \right.$
where $$\delta$$ and $$\sigma$$ are positive parameters to be defined.
System (4) has a solution $$\tag{5} x=x_0 e^{-\delta t},\;\;y=y_0 e^{\sigma t}\cos 2\pi t,\;\;z=-y_0 e^{\sigma t}\sin 2\pi t$$
for the initial conditions $$x(0)=x_0 \ge 0,$$ $$y(0)=y_0 \ge 0,$$ and $$z(0)=0.$$ Therefore, the system (4) generates a Poincaré map of the quarter plane $$P^{+}=\{(x,y): x\ge 0,\;\; y\ge 0,\;\;z=0\}$$ into itself as the solution (5) returns to $$P^{+}$$ at every instant $$t=i\in \mathbb{Z}.$$ The Poincaré map on the cross-section $$P^{+}$$ reads $$\tag{6} \bar{x}=e^{-\delta}x,\;\;\bar{y}=e^{\sigma}y.$$
Setting $$e^{-\delta}=\lambda$$ and $$e^{\sigma}=\gamma,$$ where $$\lambda$$ and $$\gamma$$ are the original parameters of the Belykh map (1)-(2), one transforms the Poincare map (6) into the first part of the Belykh map (1) defined for the region $$S_1$$ where $$L(x,y)\le 0.$$
Changing the variables $$(x,y)\rightarrow (1-x,1-y)$$ in the system (4) yields another linear system with the saddle-focus equilibrium shifted to $$(1,1,0)\ .$$ The new system reads $$\tag{7} \left \{\begin{array}{l} \dot{x}=-\delta (x-1)\\ \dot{y}=\sigma (y-1)- 2 \pi z\\ \dot{z}=2\pi (y-1)+ \sigma z. \end{array} \right.$$
The corresponding Poincaré map on the cross-section $$P^{-}=\{(x,y): x\le 1,\;\; y\le 1,\;\;z=0\}$$ takes the form $$\tag{8} \bar{x}=e^{-\delta}(x-1)+1,\;\;\bar{y}=e^{\sigma}(y-1)+1$$
which coincides with the second part of the Belykh map (2) for $$L(x,y)>0.$$
Therefore, two ODE systems (4) and (7) act on a common cross-section $$P=P^{+} \cap P^{-}=\{(x,y):0 \le x\le 1,\;\; 0 \le y\le 1,\;\;z=0\}$$ according to the following rule $$\tag{9} \left \{\begin{array}{l} \dot{x}=-\delta [x-H(L(x_i,y_i))]\\ \dot{y}=\sigma [y-H(L(x_i,y_i))]+2\pi(-1)^{H(L(x_i,y_i))}z\\ \dot{z}=-2\pi (-1)^{H(L(x_i,y_i))}[y-H(L(x_i,y_i))]+ \sigma z, \end{array} \right.$$
where $$H \left (L(x_i,y_i) \right )$$ is the Heaviside step function with $$H(0)=0$$ and $$L(x_i,y_i)=k(2x_i-1)+2y_i-1.$$ The system (9) must be integrated over the time intervals $$t\in [i,i+1],$$ $$i=0,1,2,...$$ The Heaviside step function $$H(L(x_i,y_i))$$ can only switch its value at time $$t=i$$ when the trajectory leaves the cross-section $$P$$ to come back at time $$t=i+1\ .$$ Note that the system (9) becomes (4) when $$H(L(x_i,y_i))=0\ .$$ This happens when the trajectory hits the cross-section $$P$$ at a point for which $$L(x_i,y_i) \le 0.$$ At the same time, $$H(L(x_i,y_i))=1$$ yields the system (7).
Figure 5: Strange attractor of the ODE system (9) generating the Belykh map on the Poincaré cross-section $$P=\{(x,y):0 \le x\le 1,\;\; 0 \le y\le 1,\;\;z=0\}$$ (bright yellow). The parameters are the same as in Figure 4. Figure 6: Corresponding $$xy$$-projection of the ODE attractor. Points of intersections with the Poincaré cross-section $$P$$ yield the Belykh attractor (cf. Figure 7). Figure 7: Belykh attractor on the Poincaré cross-section $$P$$ is identical to the one of Figure 2. The parameters $$\delta=-{\rm ln}\, 0.48=0.73,$$ $$\sigma={\rm ln}\, 1.3=0.262,$$ and $$k=0.5$$ correspond to those of Figure 2.
In short, the hybrid ODE system (9) is a way of gluing trajectories of two linear systems (4) and (7) periodically in time such that the trajectory of one system is continued by the other system if the trajectory returns to the cross-section $$P$$ on the other side of the line $$L(x_i,y_i).$$
To ensure the one-to-one correspondence between the original Belykh map (1)-(2) and the Poincaré map (6)-(8) on the cross-section $$P\ ,$$ the parameters of the system (9) must be recalculated via $$\lambda$$ and $$\gamma$$ as follows$\delta=-{\rm ln} \lambda$ and $$\sigma={\rm ln} \gamma\ .$$ The parameter $$k,$$ present in $$L(x_i,y_i),$$ remains the same.
Under these conditions, the hybrid ODE system (9) with initial conditions $$\{0 \le x(0) \le 1,\;\; 0 \le y(0) \le 1,\;\;z(0)=0\}$$ acts as the Belykh map on the cross-section $$P$$ ( Figure 5-Figure 6) and generates a quasi-hyperbolic Belykh attractor ( Figure 7).
## References
• Afraimovich, V.S., Chernov, N.I., and Sataev, E.A. (1995). Statistical properties of 2-D generalized hyperbolic attractors. Chaos 5: 238-252.
• Afraimovich, V.S., Hsu, S.-B. (2003). Lectures on chaotic dynamical systems. American Mathematical Society and International Press. Studies in Advanced Mathematics, 28.
• Alexander, J.C. and Yorke, J.A. (1984). Fat baker's transformations. Ergodic Theory and Dynamical Systems 4: 1-23.
• Belykh, V.N. (1976). On models of phase synchronization systems and their study. Dinamika System, Gorky State University Press 11: 23-32 (in Russian).
• Belykh, V.N. (1980). Qualitative methods of nonlinear oscillation theory in point systems. Gorky State University Press, Gorky (in Russian).
• Belykh, V.N. (1982). Models of discrete systems of phase synchronization. In Systems of Phase Synchronization, V.V. Shakhildyan and L.N. Belyustina, eds., Radio i Svyaz, Moscow, 161-216.
• Belykh, V.N. (1995). Chaotic and strange attractors of a two-dimensional map. Math. USSR Sbornik 186: 311-326.
• Bunimovich, L.A. (1974). On billiards close to dispersing. Math. USSR Sbornik 95: 49-73.
• Lorenz, E.N. (1963). Deterministic nonperiodic flow. Journal of Atmospheric Science 20: 130–141.
• Lozi, R. (1978). Un attracteur étrange du type attracteur de Hénon. J. Phys., Paris 39: 9-10.
• Pesin, Ya.B. (1992). Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergodic Theory and Dynamical Systems 12: 123-151.
• Sataev, E.A. (1992). Invariant measures for hyperbolic maps with singularities. Russian Math. Surveys 47: 191-251.
• Sataev, E.A. (1999). Ergodic properties of the Belykh map. Journal of Mathematical Sciences 95: 2564-2575.
• Schmeling, J. (1998). A dimension formula for endomorphisms — the Belykh family. Ergodic Theory and Dynamical Systems 18: 1283-1309.
• Schmeling, J. and Troubetzkoy, S. (1998). Dimension and invertibility of hyperbolic endomorphisms with singularities. Ergodic Theory and Dynamical Systems 18: 1257-1282.
• Schmeling, J. (2001). Dimension theory of smooth dynamical systems. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler, ed., Springer, 108-129.
• Sinai, Ya.G. (1970). Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Mathematical Surveys 25: 137-189.
• Persson, T. (2008). Absolutely continuous invariant measures for some piecewise hyperbolic affine maps. Ergodic Theory and Dynamical Systems 28: 211-228. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9107604622840881, "perplexity": 1139.484385861798}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542323.80/warc/CC-MAIN-20161202170902-00211-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/34264/why-is-a-finite-cw-complex-compact | Why is a finite CW complex compact?
Hatcher explains on page 5 how a CW complex can be constructed inductively by attaching $n$-cells i.e. open $n$-dimensional disks.
On page 520 in the appendix he writes "A finite CW complex, ... , is compact since attaching a single cell preserves compactness."
Now my question: why is this obvious? An open disk is not compact, so how can I see that sticking two together is?
-
You are attaching closed discs in a CW complex (In the notation of hatcher $D^n$ is the closed $n$-disc cf. page XII). Each closed disc is compact.
But on page 5 he writes "open $n$-disk" where he explains how to construct a CW complex in step (2). He is attaching open disks! – Rudy the Reindeer Apr 21 '11 at 10:25
@Matt- So usually the open disk' will correspond to what's called a cell,' but if you look carefully the way you actually construct the CW-complex is to glue the cell in along its boundary... i.e. look at a map $f: S^{n-1} \rightarrow X^{n-1}$ and use that to add an $n$-cell. – Dylan Wilson Apr 21 '11 at 10:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857032299041748, "perplexity": 595.0034125168601}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783402479.21/warc/CC-MAIN-20160624155002-00033-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_206_Precalculus/6%3A_Periodic_Functions/6.2%3A_Graphs_of_the_Other_Trigonometric_Functions | $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
# 6.2: Graphs of the Other Trigonometric Functions
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
Skills to Develop
• Analyze the graph of $$y=\tan x$$.
• Graph variations of $$y=\tan x$$.
• Analyze the graphs of $$y=\sec x$$ and $$y=\csc x$$.
• Graph variations of $$y=\sec x$$ and $$y=\csc x$$.
• Analyze the graph of $$y=\cot x$$.
• Graph variations of $$y=\cot x$$.
We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions.
## Analyzing the Graph of $$y =\tan x$$
We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that
$\tan \, x=\dfrac{\sin \, x}{\cos \, x}$
The period of the tangent function is $$\pi$$ because the graph repeats itself on intervals of $$k\pi$$ where $$k$$ is a constant. If we graph the tangent function on $$−\frac{\pi}{2}$$ to $$\frac{\pi}{2}$$, we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.
We can determine whether tangent is an odd or even function by using the definition of tangent.
\begin{align*} \tan(-x)&= \dfrac{\sin(-x)}{\cos(-x)} \qquad \text{Definition of tangent}\\ &= \dfrac{-\sin \, x}{\cos \, x} \qquad \text{Sine is an odd function, cosine is even}\\ &= -\dfrac{\sin \, x}{\cos \, x} \qquad \text{The quotient of an odd and an even function is odd}\\ &= -\tan \, x \qquad \text{Definition of tangent} \end{align*}
Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in Table $$\PageIndex{1}$$.
$$x$$ $$\tan x$$ $$−\dfrac{\pi}{2}$$ $$−\dfrac{\pi}{3}$$ $$−\dfrac{\pi}{4}$$ $$−\dfrac{\pi}{6}$$ 0 $$\dfrac{\pi}{6}$$ $$\dfrac{\pi}{4}$$ $$\dfrac{\pi}{3}$$ $$\dfrac{\pi}{2}$$ undefined $$-\sqrt{3}$$ $$–1$$ $$-\dfrac{\sqrt{3}}{3}$$ 0 $$\dfrac{\sqrt{3}}{3}$$ 1 $$\sqrt{3}$$ undefined
These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when $$\frac{\pi}{3}<x<\frac{\pi}{2}$$, we can use a table to look for a trend. Because $$\frac{\pi}{3}≈1.05$$ and $$\frac{\pi}{2}≈1.57$$, we will evaluate $$x$$ at radian measures $$1.05<x<1.57$$ as shown in Table $$\PageIndex{2}$$.
$$x$$ $$\tan x$$ 1.3 1.5 1.55 1.56 3.6 14.1 48.1 92.6
As $$x$$ approaches $$\dfrac{\pi}{2}$$, the outputs of the function get larger and larger. Because $$y=\tan \, x$$ is an odd function, we see the corresponding table of negative values in Table $$\PageIndex{3}$$.
$$x$$ $$\tan x$$ −1.3 −1.5 −1.55 −1.56 −3.6 −14.1 −48.1 −92.6
We can see that, as $$x$$ approaches $$−\frac{\pi}{2}$$, the outputs get smaller and smaller. Remember that there are some values of $$x$$ for which $$\cos \, x=0$$. For example, $$\cos \left (\frac{\pi}{2} \right)=0$$ and $$\cos \left (\frac{3\pi}{2} \right )=0$$. At these values, the tangent function is undefined, so the graph of $$y=\tan \, x$$ has discontinuities at $$x=\frac{\pi}{2}$$ and $$\frac{3\pi}{2}$$. At these values, the graph of the tangent has vertical asymptotes. Figure $$\PageIndex{1}$$ represents the graph of $$y=\tan \, x$$. The tangent is positive from $$0$$ to $$\frac{\pi}{2}$$ and from $$\pi$$ to $$\frac{3\pi}{2}$$, corresponding to quadrants I and III of the unit circle.
Figure $$\PageIndex{1}$$: Graph of the tangent function
## Graphing Variations of $$y = \tan \, x$$
As with the sine and cosine functions, the tangent function can be described by a general equation.
$y=A\tan(Bx) \nonumber$
We can identify horizontal and vertical stretches and compressions using values of $$A$$ and $$B$$. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.
Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant $$A$$.
FEATURES OF THE GRAPH OF $$y = A \tan(Bx)$$
• The stretching factor is $$|A|$$.
• The period is $$P=\dfrac{\pi}{|B|}$$.
• The domain is all real numbers $$x$$,where $$x≠\dfrac{\pi}{2| B |}+\dfrac{π}{| B |}k$$ such that $$k$$ is an integer.
• The range is $$(−\infty,\infty)$$.
• The asymptotes occur at $$x=\dfrac{\pi}{2| B |}+\dfrac{π}{| B |}k$$ where $$k$$ is an integer.
• $$y=A\tan(Bx)$$ is an odd function.
### Graphing One Period of a Stretched or Compressed Tangent Function
We can use what we know about the properties of the tangent function to quickly sketch a graph of any stretched and/or compressed tangent function of the form $$f(x)=A\tan(Bx)$$. We focus on a single period of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function’s domain if we wish. Our limited domain is then the interval $$\left (−\dfrac{P}{2},\dfrac{P}{2} \right )$$ and the graph has vertical asymptotes at $$\pm \dfrac{P}{2}$$ where $$P=\dfrac{\pi}{B}$$. On $$\left (−\dfrac{\pi}{2},\dfrac{\pi}{2} \right )$$, the graph will come up from the left asymptote at $$x=−\dfrac{\pi}{2}$$, cross through the origin, and continue to increase as it approaches the right asymptote at $$x=\dfrac{\pi}{2}$$. To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use
$f \left (\dfrac{P}{4} \right )=A\tan \left (B\dfrac{P}{4} \right )=A\tan \left (B\dfrac{\pi}{4B} \right )=A \nonumber$
because $$\tan \left (\dfrac{\pi}{4} \right )=1$$.
Howto: Given the function $$f(x)=A \tan(Bx)$$, graph one period.
1. Identify the stretching factor, $$| A |$$.
2. Identify B and determine the period, $$P=\dfrac{\pi}{| B |}$$.
3. Draw vertical asymptotes at $$x=−\dfrac{P}{2}$$ and $$x=\dfrac{P}{2}$$.
4. For $$A>0$$, the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for $$A<0$$).
5. Plot reference points at $$\left (\dfrac{P}{4},A \right )$$, $$(0,0)$$, and $$\left (−\dfrac{P}{4},−A \right )$$, and draw the graph through these points.
Example $$\PageIndex{1}$$: Sketching a Compressed Tangent
Sketch a graph of one period of the function $$y=0.5\tan \left (\dfrac{\pi}{2}x \right )$$.
Solution
First, we identify $$A$$ and $$B$$.
Figure $$\PageIndex{2}$$
Because $$A=0.5$$ and $$B=\dfrac{\pi}{2}$$, we can find the stretching/compressing factor and period. The period is $$\dfrac{\pi}{\dfrac{\pi}{2}}=2$$, so the asymptotes are at $$x=±1$$. At a quarter period from the origin, we have
\begin{align*} f(0.5)&= 0.5\tan \left (\dfrac{0.5\pi}{2} \right )\\ &= 0.5\tan \left (\dfrac{\pi}{4} \right )\\ &= 0.5 \end{align*}
This means the curve must pass through the points $$(0.5,0.5)$$, $$(0,0)$$,and $$(−0.5,−0.5)$$. The only inflection point is at the origin. Figure $$\PageIndex{3}$$ shows the graph of one period of the function.
Figure $$\PageIndex{3}$$
Exercise $$\PageIndex{1}$$
Sketch a graph of $$f(x)=3\tan \left (\dfrac{\pi}{6}x \right )$$.
Figure $$\PageIndex{4}$$
### Graphing One Period of a Shifted Tangent Function
Now that we can graph a tangent function that is stretched or compressed, we will add a vertical and/or horizontal (or phase) shift. In this case, we add $$C$$ and $$D$$ to the general form of the tangent function.
$f(x)=A\tan(Bx−C)+D \nonumber$
The graph of a transformed tangent function is different from the basic tangent function $$\tan x$$ in several ways:
FEATURES OF THE GRAPH OF $$y = A\tan(Bx−C)+D$$
• The stretching factor is $$| A |$$.
• The period is $$\dfrac{\pi}{| B |}$$.
• The domain is $$x≠\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an integer.
• The range is $$(−∞,−| A |]∪[| A |,∞)$$.
• The vertical asymptotes occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an odd integer.
• There is no amplitude.
• $$y=A \tan(Bx)$$ is and odd function because it is the qoutient of odd and even functions(sin and cosine perspectively).
Howto: Given the function $$y=A\tan(Bx−C)+D$$, sketch the graph of one period.
1. Express the function given in the form $$y=A\tan(Bx−C)+D$$.
2. Identify the stretching/compressing factor, $$| A |$$.
3. Identify $$B$$ and determine the period, $$P=\dfrac{\pi}{|B|}$$.
4. Identify $$C$$ and determine the phase shift, $$\dfrac{C}{B}$$.
5. Draw the graph of $$y=A\tan(Bx)$$ shifted to the right by $$\dfrac{C}{B}$$ and up by $$D$$.
6. Sketch the vertical asymptotes, which occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{2| B |}k$$,where $$k$$ is an odd integer.
7. Plot any three reference points and draw the graph through these points.
Example $$\PageIndex{2}$$: Graphing One Period of a Shifted Tangent Function
Graph one period of the function $$y=−2\tan(\pi x+\pi)−1$$.
Solution
• Step 1. The function is already written in the form $$y=A\tan(Bx−C)+D$$.
• Step 2.$$A=−2$$, so the stretching factor is $$|A|=2$$.
• Step 3. $$B=\pi$$, so the period is $$P=\dfrac{\pi}{| B |}=\dfrac{\pi}{pi}=1$$.
• Step 4. $$C=−\pi$$, so the phase shift is $$CB=\dfrac{−\pi}{\pi}=−1$$.
• Step 5-7. The asymptotes are at $$x=−\dfrac{3}{2}$$ and $$x=−\dfrac{1}{2}$$ and the three recommended reference points are $$(−1.25,1)$$, $$(−1,−1)$$, and $$(−0.75,−3)$$. The graph is shown in Figure $$\PageIndex{5}$$.
Figure $$\PageIndex{5}$$
Analysis
Note that this is a decreasing function because $$A<0$$.
Exercise $$\PageIndex{2}$$
How would the graph in Example $$\PageIndex{2}$$ look different if we made $$A=2$$ instead of $$−2$$?
It would be reflected across the line $$y=−1$$, becoming an increasing function.
Howto: Given the graph of a tangent function, identify horizontal and vertical stretches.
1. Find the period $$P$$ from the spacing between successive vertical asymptotes or x-intercepts.
2. Write $$f(x)=A\tan \left (\dfrac{\pi}{P}x \right )$$.
3. Determine a convenient point $$(x,f(x))$$ on the given graph and use it to determine $$A$$.
Example $$\PageIndex{3}$$: Identifying the Graph of a Stretched Tangent
Find a formula for the function graphed in Figure $$\PageIndex{6}$$.
Figure $$\PageIndex{6}$$: A stretched tangent function
Solution
The graph has the shape of a tangent function.
• Step 1. One cycle extends from $$–4$$ to $$4$$, so the period is $$P=8$$. Since $$P=\dfrac{\pi}{| B |}$$, we have $$B=\dfrac{π}{P}=\dfrac{\pi}{8}$$.
• Step 2. The equation must have the form $$f(x)=A\tan \left (\dfrac{\pi}{8}x \right )$$.
• Step 3. To find the vertical stretch $$A$$,we can use the point $$(2,2)$$. \begin{align*} 2&=A\tan \left (\dfrac{\pi}{8}\cdot 2 \right )\\ &=A\tan \left (\dfrac{\pi}{4} \right ) \end{align*}
Because $$\tan \left (\dfrac{\pi}{4} \right )=1$$, $$A=2$$.
This function would have a formula $$f(x)=2\tan \left (\dfrac{\pi}{8}x \right )$$.
Exercise $$\PageIndex{3}$$
Find a formula for the function in Figure $$\PageIndex{7}$$.
Figure $$\PageIndex{7}$$
$$g(x)=4\tan(2x)$$
## Analyzing the Graphs of $$y = \sec x$$ and $$y = \csc x$$
The secant was defined by the reciprocal identity $$sec \, x=\dfrac{1}{\cos x}$$. Notice that the function is undefined when the cosine is $$0$$, leading to vertical asymptotes at $$\dfrac{\pi}{2}$$, $$\dfrac{3\pi}{2}$$ etc. Because the cosine is never more than $$1$$ in absolute value, the secant, being the reciprocal, will never be less than $$1$$ in absolute value.
We can graph $$y=\sec x$$ by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure $$\PageIndex{8}$$. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
The secant graph has vertical asymptotes at each value of $$x$$ where the cosine graph crosses the $$x$$-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.
Note that, because cosine is an even function, secant is also an even function. That is, $$\sec(−x)=\sec x$$.
Figure $$\PageIndex{8}$$: Graph of the secant function, $$f(x)=\sec x=\dfrac{1}{\cos x}$$
As we did for the tangent function, we will again refer to the constant $$| A |$$ as the stretching factor, not the amplitude.
FEATURES OF THE GRAPH OF $$y = A \sec(Bx)$$
• The stretching factor is $$| A |$$.
• The period is $$\dfrac{2\pi}{| B |}$$.
• The domain is $$x≠\dfrac{\pi}{2| B |}k$$, where $$k$$ is an odd integer.
• The range is $$(−∞,−|A|]∪[|A|,∞)$$.
• The vertical asymptotes occur at $$x=\dfrac{\pi}{2| B |}k$$, where $$k$$ is an odd integer.
• There is no amplitude.
• $$y=A\sec(Bx)$$ is an even function because cosine is an even function.
Similar to the secant, the cosecant is defined by the reciprocal identity $$\csc x=\dfrac{1}{\sin x}$$. Notice that the function is undefined when the sine is $$0$$, leading to a vertical asymptote in the graph at $$0$$, $$\pi$$, etc. Since the sine is never more than $$1$$ in absolute value, the cosecant, being the reciprocal, will never be less than $$1$$ in absolute value.
We can graph $$y=\csc x$$ by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure $$\PageIndex{7}$$. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.
The cosecant graph has vertical asymptotes at each value of $$x$$ where the sine graph crosses the $$x$$-axis; we show these in the graph below with dashed vertical lines.
Note that, since sine is an odd function, the cosecant function is also an odd function. That is, $$\csc(−x)=−\csc x$$.
The graph of cosecant, which is shown in Figure $$\PageIndex{9}$$, is similar to the graph of secant.
Figure $$\PageIndex{9}$$: The graph of the cosecant function, $$f(x)=\csc x=\frac{1}{\sin x}$$
FEATURES OF THE GRAPH OF $$y = A \csc(Bx)$$
• The stretching factor is $$| A |$$.
• The period is $$\dfrac{2\pi}{|B|}$$.
• The domain is $$x≠\dfrac{\pi}{|B|}k$$, where $$k$$ is an integer.
• The range is $$(−∞,−|A|]∪[|A|,∞)$$.
• The asymptotes occur at $$x=\dfrac{\pi}{| B |}k$$, where $$k$$ is an integer.
• $$y=A\csc(Bx)$$ is an odd function because sine is an odd function.
## Graphing Variations of $$y = \sec x$$ and $$y= \csc x$$
For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.
$y=A\sec(Bx−C)+D$
$y=A\csc(Bx−C)+D$
FEATURES OF THE GRAPH OF $$y = A\sec(Bx−C)+D$$
• The stretching factor is $$|A|$$.
• The period is $$\dfrac{2\pi}{|B|}$$.
• The domain is $$x≠\dfrac{C}{B}+\dfrac{\pi}{2| B |}k$$,where $$k$$ is an odd integer.
• The range is $$(−∞,−|A|]∪[|A|,∞)$$.
• The vertical asymptotes occur at $$x=\dfrac{C}{B}+\dfrac{π}{2| B |}k$$,where $$k$$ is an odd integer.
• There is no amplitude.
• $$y=A\sec(Bx)$$ is an even function because cosine is an even function.
FEATURES OF THE GRAPH OF $$y = A\csc(Bx−C)+D$$
1. The stretching factor is $$|A|$$.
2. The period is $$\dfrac{2\pi}{|B|}$$.
3. The domain is $$x≠\dfrac{C}{B}+\dfrac{\pi}{2| B |}k$$,where $$k$$ is an integer.
4. The range is $$(−∞,−|A|]∪[|A|,∞)$$.
5. The vertical asymptotes occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{|B|}k$$,where $$k$$ is an integer.
6. There is no amplitude.
7. $$y=A\csc(Bx)$$ is an odd function because sine is an odd function.
HOWTO: Given a function of the form $$y=A\sec(Bx)$$, graph one period
1. Express the function given in the form $$y=A\sec(Bx)$$.
2. Identify the stretching/compressing factor, $$|A|$$.
3. Identify $$B$$ and determine the period, $$P=\dfrac{2\pi}{| B |}$$.
4. Sketch the graph of $$y=A\cos(Bx)$$.
5. Use the reciprocal relationship between $$y=\cos \, x$$ and $$y=\sec \, x$$ to draw the graph of $$y=A\sec(Bx)$$.
6. Sketch the asymptotes.
7. Plot any two reference points and draw the graph through these points.
Example $$\PageIndex{4}$$: Graphing a Variation of the Secant Function
Graph one period of $$f(x)=2.5\sec(0.4x)$$.
Solution
• Step 1. The given function is already written in the general form, $$y=A\sec(Bx)$$.
• Step 2. $$A=2.5$$ so the stretching factor is $$2.5$$.
• Step 3. $$B=0.4$$ so $$P=\dfrac{2\pi}{0.4}=5\pi$$. The period is $$5\pi$$ units.
• Step 4. Sketch the graph of the function $$g(x)=2.5\cos(0.4x)$$.
• Step 5. Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.
• Steps 6–7. Sketch two asymptotes at $$x=1.25\pi$$ and $$x=3.75\pi$$. We can use two reference points, the local minimum at $$(0,2.5)$$ and the local maximum at $$(2.5\pi,−2.5)$$. Figure $$\PageIndex{10}$$ shows the graph.
Figure $$\PageIndex{10}$$
Exercise $$\PageIndex{4}$$
Graph one period of $$f(x)=−2.5\sec(0.4x)$$.
This is a vertical reflection of the preceding graph because $$A$$ is negative.
Figure $$\PageIndex{11}$$
Q&A: Do the vertical shift and stretch/compression affect the secant’s range?
Yes. The range of $$f(x)=A\sec(Bx−C)+D$$ is $$(−∞,−|A|+D]∪[|A|+D,∞)$$.
Howto: Given a function of the form $$f(x)=A\sec(Bx−C)+D$$, graph one period.
1. Express the function given in the form $$y=A \sec(Bx−C)+D$$.
2. Identify the stretching/compressing factor, $$| A |$$.
3. Identify $$B$$ and determine the period, $$\dfrac{2\pi}{|B|}$$.
4. Identify $$C$$ and determine the phase shift, $$\dfrac{C}{B}$$.
5. Draw the graph of $$y=A \sec(Bx)$$. but shift it to the right by $$\dfrac{C}{B}$$ and up by $$D$$.
6. Sketch the vertical asymptotes, which occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{2| B |}k$$, where $$k$$ is an odd integer.
Example $$\PageIndex{5}$$: Graphing a Variation of the Secant Function
Graph one period of $$y=4\sec \left (\dfrac{\pi}{3x}−\dfrac{\pi}{2} \right )+1$$.
Solution
• Step 1. Express the function given in the form $$y=4\sec \left (\dfrac{\pi}{3x}−\dfrac{\pi}{2} \right )+1$$.
• Step 2. The stretching/compressing factor is $$| A |=4$$.
• Step 3. The period is
\begin{align*} \dfrac{2\pi}{|B|}&= \dfrac{2\pi}{\dfrac{\pi}{3}}\\ &= 2\pi \cdot \dfrac{3}{\pi}\\ &= 6 \end{align*}
• Step 4. The phase shift is
\begin{align*} \dfrac{C}{B}&= \dfrac{\dfrac{\pi}{2}}{\dfrac{\pi}{3}}\\ &= \dfrac{\pi}{2}\cdot \dfrac{3}{\pi}\\ &= 1.5 \end{align*}
• Step 5. Draw the graph of $$y=A\sec(Bx)$$, but shift it to the right by $$\dfrac{C}{B}=1.5$$ and up by $$D=6$$.
• Step 6. Sketch the vertical asymptotes, which occur at $$x=0$$, $$x=3$$, and $$x=6$$. There is a local minimum at $$(1.5,5)$$ and a local maximum at $$(4.5,−3)$$. Figure $$\PageIndex{12}$$ shows the graph.
Figure $$\PageIndex{12}$$
Exercise $$\PageIndex{5}$$
Graph one period of $$f(x)=−6\sec(4x+2)−8$$.
Figure $$\PageIndex{13}$$
Q&A: The domain of $$\csc \, x$$ was given to be all $$x$$ such that $$x≠k\pi$$ for any integer $$k$$. Would the domain of $$y=A\csc(Bx−C)+D$$ be $$x≠\dfrac{C+k\pi}{B}$$?
Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.
Howto: Given a function of the form $$y=A\csc(Bx)$$, graph one period.
1. Express the function given in the form $$y=A\csc(Bx)$$.
2. $$|A|$$.
3. Identify $$B$$ and determine the period, $$P=\dfrac{2\pi}{| B |}$$.
4. Draw the graph of $$y=A\sin(Bx)$$.
5. Use the reciprocal relationship between $$y=sin \, x$$ and $$y=\csc \, x$$ to draw the graph of $$y=A\csc(Bx)$$.
6. Sketch the asymptotes.
7. Plot any two reference points and draw the graph through these points.
Example $$\PageIndex{6}$$: Graphing a Variation of the Cosecant Function
Graph one period of $$f(x)=−3\csc(4x)$$.
Solution
• Step 1. The given function is already written in the general form, $$y=A\csc(Bx)$$.
• Step 2. $$| A |=| −3 |=3$$,so the stretching factor is $$3$$.
• Step 3. $$B=4$$,so $$P=\dfrac{2\pi}{4}=\dfrac{\pi}{2}$$. The period is $$\dfrac{\pi}{2}$$ units.
• Step 4. Sketch the graph of the function $$g(x)=−3\sin(4x)$$.
• Step 5. Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function.
• Steps 6–7. Sketch three asymptotes at $$x=0$$, $$x=\dfrac{\pi}{4}$$, and $$x=\dfrac{\pi}{2}$$. We can use two reference points, the local maximum at $$\left (\dfrac{\pi}{8},−3 \right )$$ and the local minimum at $$\left (\dfrac{3\pi}{8},3 \right )$$. Figure $$\PageIndex{14}$$ shows the graph.
Figure $$\PageIndex{14}$$
Exercise $$\PageIndex{6}$$
Graph one period of $$f(x)=0.5\csc(2x)$$.
Figure $$\PageIndex{15}$$
Howto: Given a function of the form $$f(x)=A \csc(Bx−C)+D$$, graph one period
1. Express the function given in the form $$y=A\csc(Bx−C)+D$$.
2. Identify the stretching/compressing factor, $$|A|$$.
3. Identify $$B$$ and determine the period, $$\dfrac{2\pi}{| B |}$$.
4. Identify $$C$$ and determine the phase shift, $$\dfrac{C}{B}$$.
5. Draw the graph of $$y=A\csc(Bx)$$ but shift it to the right by and up by $$D$$.
6. Sketch the vertical asymptotes, which occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an integer.
Example $$\PageIndex{7}$$: Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant
Sketch a graph of $$y=2\csc \left (\dfrac{\pi}{2}x \right )+1$$. What are the domain and range of this function?
Solution
• Step 1. Express the function given in the form $$y=2\csc \left (\dfrac{\pi}{2}x \right )+1$$.
• Step 2. Identify the stretching/compressing factor, $$| A |=2$$.
• Step 3. The period is $$\dfrac{2\pi}{| B |}=\dfrac{2\pi}{\dfrac{\pi}{2}}=2\pi⋅\dfrac{2}{\pi}=4$$.
• Step 4. The phase shift is $$\dfrac{0}{\dfrac{\pi}{2}}=0$$.
• Step 5. Draw the graph of $$y=A\csc(Bx)$$ but shift it up $$D=1$$.
• Step 6. Sketch the vertical asymptotes, which occur at $$x=0$$, $$x=2$$, $$x=4$$.
The graph for this function is shown in Figure $$\PageIndex{16}$$.
Figure $$\PageIndex{16}$$: A transformed cosecant function
Analysis
The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of $$f(x)=2\sin \left (\frac{\pi}{2}x \right )+1$$,shown as the orange dashed wave.
Exercise $$\PageIndex{7}$$
Given the graph of $$f(x)=2\cos \left (\frac{\pi}{2}x \right )+1$$ shown in Figure $$\PageIndex{17}$$, sketch the graph of $$g(x)=2\sec \left (\dfrac{\pi}{2}x \right )+1$$ on the same axes.
Figure $$\PageIndex{17}$$
Figure $$\PageIndex{18}$$
## Analyzing the Graph of $$y = \cot x$$
The last trigonometric function we need to explore is cotangent. The cotangent is defined by the reciprocal identity $$cot \, x=\dfrac{1}{\tan x}$$. Notice that the function is undefined when the tangent function is $$0$$, leading to a vertical asymptote in the graph at $$0$$, $$\pi$$, etc. Since the output of the tangent function is all real numbers, the output of the cotangent function is also all real numbers.
We can graph $$y=\cot x$$ by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure $$\PageIndex{19}$$. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.
The cotangent graph has vertical asymptotes at each value of $$x$$ where $$\tan x=0$$; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, $$\cot x$$ has vertical asymptotes at all values of $$x$$ where $$\tan x=0$$, and $$\cot x=0$$ at all values of $$x$$ where $$\tan x$$ has its vertical asymptotes.
Figure $$\PageIndex{19}$$: The cotangent function
FEATURES OF THE GRAPH OF $$y = A \cot(BX)$$
• The stretching factor is $$|A|$$.
• The period is $$P=\dfrac{\pi}{|B|}$$.
• The domain is $$x≠\dfrac{\pi}{|B|}k$$, where $$k$$ is an integer.
• The range is $$(−∞,∞)$$.
• The asymptotes occur at $$x=\dfrac{\pi}{| B |}k$$, where $$k$$ is an integer.
• $$y=A\cot(Bx)$$ is an odd function.
## Graphing Variations of $$y =\cot x$$
We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.
$y=A\cot(Bx−C)+D$
PROPERTIES OF THE GRAPH OF $$y = A \cot(Bx-c)+D$$
• The stretching factor is $$| A |$$.
• The period is $$\dfrac{\pi}{|B|}$$
• The domain is $$x≠\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an integer.
• The range is $$(−∞,−|A|]∪[|A|,∞)$$.
• The vertical asymptotes occur at $$x=\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an integer.
• There is no amplitude.
• $$y=A\cot(Bx)$$ is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)
Howto: Given a modified cotangent function of the form $$f(x)=A\cot(Bx)$$,graph one period.
1. Express the function in the form $$f(x)=A\cot(Bx)$$.
2. Identify the stretching factor, $$|A|$$.
3. Identify the period, $$P=\dfrac{\pi}{|B|}$$.
4. Draw the graph of $$y=A\tan(Bx)$$.
5. Plot any two reference points.
6. Use the reciprocal relationship between tangent and cotangent to draw the graph of $$y=Acot(Bx)$$.
7. Sketch the asymptotes.
Example $$\PageIndex{8}$$: Graphing Variations of the Cotangent Function
Determine the stretching factor, period, and phase shift of $$y=3\cot(4x)$$, and then sketch a graph.
Solution
• Step 1. Expressing the function in the form $$f(x)=A\cot(Bx)$$ gives $$f(x)=3\cot(4x)$$.
• Step 2. The stretching factor is $$|A|=3$$.
• Step 3. The period is $$P=\dfrac{\pi}{4}$$.
• Step 4. Sketch the graph of $$y=3\tan(4x)$$.
• Step 5. Plot two reference points. Two such points are $$\left (\dfrac{\pi}{16},3 \right )$$ and $$\left (\dfrac{3\pi}{16},−3 \right )$$.
• Step 6. Use the reciprocal relationship to draw $$y=3\cot(4x)$$.
• Step 7. Sketch the asymptotes, $$x=0$$, $$x=\dfrac{\pi}{4}$$.
The orange graph in Figure $$\PageIndex{20}$$ shows $$y=3\tan(4x)$$ and the blue graph shows $$y=3\cot(4x)$$.
Figure $$\PageIndex{20}$$
Howto: Given a modified cotangent function of the form $$f(x)=A\cot(Bx−C)+D$$, graph one period.
1. Express the function in the form $$f(x)=A\cot(Bx−C)+D$$.
2. Identify the stretching factor, $$| A |$$.
3. Identify the period, $$P=\dfrac{\pi}{|B|}$$.
4. Identify the phase shift, $$\dfrac{C}{B}$$.
5. Draw the graph of $$y=A\tan(Bx)$$ shifted to the right by $$\dfrac{C}{B}$$ and up by $$D$$.
6. Sketch the asymptotes $$x=\dfrac{C}{B}+\dfrac{\pi}{| B |}k$$,where $$k$$ is an integer.
7. Plot any three reference points and draw the graph through these points.
Example $$\PageIndex{9}$$: Graphing a Modified Cotangent
Sketch a graph of one period of the function $$f(x)=4\cot \left (\dfrac{\pi}{8}x−\dfrac{\pi}{2} \right )−2$$.
Solution
• Step 1. The function is already written in the general form $$f(x)=A\cot(Bx−C)+D$$.
• Step 2. $$A=4$$,so the stretching factor is $$4$$.
• Step 3. $$B=\dfrac{\pi}{8}$$, so the period is $$P=\dfrac{\pi}{| B |}=\dfrac{\pi}{\dfrac{\pi}{8}}=8$$.
• Step 4. $$C=\dfrac{\pi}{2}$$,so the phase shift is $$CB=\dfrac{\dfrac{\pi}{2}}{\dfrac{\pi}{8}}=4$$.
• Step 5. We draw $$f(x)=4\tan \left (\dfrac{\pi}{8}x−\dfrac{\pi}{2} \right )−2$$.
• Step 6-7. Three points we can use to guide the graph are $$(6,2)$$, $$(8,−2)$$, and $$(10,−6)$$. We use the reciprocal relationship of tangent and cotangent to draw $$f(x)=4\cot \left (\dfrac{\pi}{8}x−\dfrac{\pi}{2} \right )−2$$.
• Step 8. The vertical asymptotes are $$x=4$$ and $$x=12$$.
The graph is shown in Figure $$\PageIndex{21}$$.
Figure $$\PageIndex{21}$$: One period of a modified cotangent function
## Using the Graphs of Trigonometric Functions to Solve Real-World Problems
Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let’s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function.
Example $$\PageIndex{10}$$: Using Trigonometric Functions to Solve Real-World Scenarios
Suppose the function $$y=5\tan(\dfrac{\pi}{4}t)$$ marks the distance in the movement of a light beam from the top of a police car across a wall where $$t$$ is the time in seconds and $$y$$ is the distance in feet from a point on the wall directly across from the police car.
1. Find and interpret the stretching factor and period.
2. Graph on the interval $$[0,5]$$.
3. Evaluate $$f(1)$$ and discuss the function’s value at that input.
Solution
1. We know from the general form of $$y=A\tan(Bt)$$ that $$| A |$$ is the stretching factor and $$\dfrac{\pi}{B}$$ is the period.
Figure $$\PageIndex{22}$$
We see that the stretching factor is $$5$$. This means that the beam of light will have moved $$5$$ ft after half the period.
The period is $$\dfrac{\pi}{\tfrac{\pi}{4}}=\dfrac{\pi}{1}⋅\dfrac{4}{\pi}=4$$. This means that every $$4$$ seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.
1. To graph the function, we draw an asymptote at $$t=2$$ and use the stretching factor and period. See Figure $$\PageIndex{23}$$
Figure $$\PageIndex{23}$$
1. period: $$f(1)=5\tan(\frac{\pi}{4}(1))=5(1)=5$$; after $$1$$ second, the beam of has moved $$5$$ ft from the spot across from the police car.
Media
Access these online resources for additional instruction and practice with graphs of other trigonometric functions.
## Key Equations
Shifted, compressed, and/or stretched tangent function $$y=A \tan(Bx−C)+D$$ Shifted, compressed, and/or stretched secant function $$y=A \sec(Bx−C)+D$$ Shifted, compressed, and/or stretched cosecant function $$y=A \csc(Bx−C)+D$$ Shifted, compressed, and/or stretched cotangent function $$y=A \cot(Bx−C)+D$$
## Key Concepts
• The tangent function has period $$π$$.
• $$f( x )=A\tan( Bx−C )+D$$ is a tangent with vertical and/or horizontal stretch/compression and shift. See Example $$\PageIndex{1}$$, Example $$\PageIndex{2}$$, and Example $$\PageIndex{3}$$.
• The secant and cosecant are both periodic functions with a period of $$2\pi$$. $$f( x )=A\sec( Bx−C )+D$$ gives a shifted, compressed, and/or stretched secant function graph. See Example $$\PageIndex{4}$$ and Example $$\PageIndex{5}$$.
• $$f( x )=A\csc( Bx−C )+D$$ gives a shifted, compressed, and/or stretched cosecant function graph. See Example $$\PageIndex{6}$$ and Example $$\PageIndex{7}$$.
• The cotangent function has period $$\pi$$ and vertical asymptotes at $$0,±\pi,±2\pi$$,....
• The range of cotangent is $$( −∞,∞ )$$, and the function is decreasing at each point in its range.
• The cotangent is zero at $$±\dfrac{\pi}{2},±\dfrac{3\pi}{2}$$,....
• $$f(x)=A\cot(Bx−C)+D$$ is a cotangent with vertical and/or horizontal stretch/compression and shift. See Example $$\PageIndex{8}$$ and Example $$\PageIndex{9}$$.
• Real-world scenarios can be solved using graphs of trigonometric functions. See Example $$\PageIndex{10}$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9864782094955444, "perplexity": 409.8227704508521}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998986.11/warc/CC-MAIN-20190619123854-20190619145854-00275.warc.gz"} |
http://mathoverflow.net/questions/84224/multiple-transversal-pullback | # Multiple Transversal Pullback
Suppose we have three smooth manifolds $M_1$, $M_2$ and $N$ and two smooth maps $f_1:M_1 \rightarrow N$ and $f_2:M_2 \rightarrow N$. Than an important and central construction in differential topology is the $transversal$ $pullback$
$$M_1 \times_{f_1Nf_2} M_2 = \\lbrace\left(x_1,x_2 \right) \in M_1 \times M_2 |f_1(x_1)=f_2(x_2) \rbrace$$
A proof that it is a manifold goes like:
$M_1 \times_{f_1Nf_2} M_2 = (f_1 \times f_2 )^{−1}(\Delta)$, where $f_1 \times f_2 : M_1 \times M_2 \rightarrow N \times N$ and where $\Delta$ is the diagonal of $N \times N$ , and $f_1 \times f_2$ is transversal to $\Delta$ if and only if $f_1$ and $f_2$ are transversal.
================================================================================
Now the question is, can we extend this to multiple transversal pullbacks? For example a "three times pullback":
$M_1 \times_{f_1Nf_2} M_2 \times_{f_2Nf_3} M_3 = \lbrace \left(x_1,x_2,x_3 \right) \in M_1 \times M_2 \times M_3 |f_1(x_1)=f_2(x_2); f_2(x_2)=f_3(x_3) \rbrace$
is this well defined as a smooth manifold and if yes how is it proofed?
And is there another generalization to the $n$-times transversal pullback?
-
Nothing at all ? – Mirco Dec 26 '11 at 14:01
You could define $f_1: M_1 \to Z, f_2: M_2 \to Z, f_3: M_3 \to Z$ to be transversal if $T_zZ \cong \Im(df_1) + \Im(df_2) + \Im(df_3)$, for all $x \in M_1, y \in M_2, w \in M_3$ such that $f_1(x)=f_2(y)=f_3(w)=z$. The obvious definition of transversal pullback would be as the limit of the diagram with legs $f_i$, that is, a manifold $M$ with smooth $\pi_i: M \to M_i$ such that $f_1\pi_1=f_2\pi_2=f_3\pi_3$, and universal with this property, i.e if $(N,\phi_1, \phi_2, \phi_3)$ satisfy similar identities, then there is a unique smooth map $\psi: N \to M$ such that $\pi_i\psi=\phi_i$, ($i=1,2,3$). – José Siqueira Mar 5 at 10:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9940063953399658, "perplexity": 213.4505422997579}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121776.48/warc/CC-MAIN-20160428161521-00059-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/does-2-spring-30-degrees-1-90.541753/ | # Does 2 spring @ 30 degrees = 1 @ 90?
1. Oct 18, 2011
### alaix
Let's say you have 2 identical springs
Let's say lenght = 1m
Spring constant = 0.5
You attach each spring to the ceiling and a mass. The springs make a 30 degree angle with the ceiling. Basically the system forms a triangle...
Is it OK to say that the effective spring constant of the system is equal to the spring constant of one spring, since 2*sin(30) = 1?
2. Oct 18, 2011
### gsal
effective constant,,,for how long? I mean, as soon as you move a bit, that 30 degree angle is not going to be there...
...the truth is that I have not look into it, just yet, I figure I shoot you back a quick reply to make you think about your own statement, for now.
3. Oct 18, 2011
Yes, I think this is correct... for the vertical spring constant, in a small neighbourhood around the equilibrium.
Horizontally it is different, and it will change as the end moves, as gsal mentioned.
4. Oct 20, 2011
### anigeo
for two springs connected in a series, the effective spring constant K is given by-
1/K=1/k1+1/k2.
in parallel -
K=k1+k2.
this is a question of parallel connection so accordingly use the right expression.
(note this is righty the opposite to series and parallel connections of resistors as in electricity) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8717473149299622, "perplexity": 1137.655276425942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209165.16/warc/CC-MAIN-20180814150733-20180814170733-00086.warc.gz"} |
https://chem.libretexts.org/Courses/Arkansas_Northeastern_College/CH14133%3A_Chemistry_for_General_Education/09%3A_Electrons_in_Atoms_and_the_Periodic_Table/9.02%3A_Light_is_Visible_Electromagnetic_Radiation | # 9.2: Light is Visible Electromagnetic Radiation
Learning Objectives
• Define the terms wavelength and frequency with respect to wave-form energy.
• State the relationship between wavelength and frequency with respect to electromagnetic radiation.
During the summer, almost everyone enjoys going to the beach. Beach-goers can swim, have picnics, and work on their tans. But if a person gets too much sun, they can burn. A particular set of solar wavelengths are especially harmful to the skin. This portion of the solar spectrum is known as UV B, with wavelengths of $$280$$-$$320 \: \text{nm}$$. Sunscreens are effective in protecting skin against both the immediate skin damage and the long-term possibility of skin cancer.
## Waves
Waves are characterized by their repetitive motion. Imagine a toy boat riding the waves in a wave pool. As the water wave passes under the boat, it moves up and down in a regular and repeated fashion. While the wave travels horizontally, the boat only travels vertically up and down. The figure below shows two examples of waves.
A wave cycle consists of one complete wave—starting at the zero point, going up to a wave crest, going back down to a wave trough, and back to the zero point again. The wavelength of a wave is the distance between any two corresponding points on adjacent waves. It is easiest to visualize the wavelength of a wave as the distance from one wave crest to the next. In an equation, wavelength is represented by the Greek letter lambda $$\left( \lambda \right)$$. Depending on the type of wave, wavelength can be measured in meters, centimeters, or nanometers $$\left( 1 \: \text{m} = 10^9 \: \text{nm} \right)$$. The frequency, represented by the Greek letter nu $$\left( \nu \right)$$, is the number of waves that pass a certain point in a specified amount of time. Typically, frequency is measured in units of cycles per second or waves per second. One wave per second is also called a Hertz $$\left( \text{Hz} \right)$$ and in SI units is a reciprocal second $$\left( \text{s}^{-1} \right)$$.
Figure B above shows an important relationship between the wavelength and frequency of a wave. The top wave clearly has a shorter wavelength than the second wave. However, if you picture yourself at a stationary point watching these waves pass by, more waves of the first kind would pass by in a given amount of time. Thus the frequency of the first wave is greater than that of the second wave. Wavelength and frequency are therefore inversely related. As the wavelength of a wave increases, its frequency decreases. The equation that relates the two is:
$c = \lambda \nu$
The variable $$c$$ is the speed of light. For the relationship to hold mathematically, if the speed of light is used in $$\text{m/s}$$, the wavelength must be in meters and the frequency in Hertz.
Example $$\PageIndex{1}$$: Orange Light
The color orange within the visible light spectrum has a wavelength of about $$620 \: \text{nm}$$. What is the frequency of orange light?
Solution
Steps for Problem Solving Example $$\PageIndex{1}$$
Identify the "given" information and what the problem is asking you to "find."
Given :
• Wavelength $$\left( \lambda \right) = 620 \: \text{nm}$$
• Speed of light $$\left( c \right) = 3.00 \times 10^8 \: \text{m/s}$$
Find: Frequency (Hz)
List other known quantities. $$1 \: \text{m} = 10^9 \: \text{nm}$$
Identify steps to get the final answer.
1.Convert the wavelength to $$\text{m}$$.
2. Apply the equation $$c = \lambda \nu$$ and solve for frequency. Dividing both sides of the equation by $$\lambda$$ yields:
$$\nu = \frac{c}{\lambda}$$
Cancel units and calculate.
$$620 \: \text{nm} \times \left( \frac{1 \: \text{m}}{10^9 \: \text{nm}} \right) = 6.20 \times 10^{-7} \: \text{m}$$
$$\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \: \text{m/s}}{6.20 \times 10^{-7}} = 4.8 \times 10^{14} \: \text{Hz}$$
Think about your result. The value for the frequency falls within the range for visible light.
Exercise $$\PageIndex{1}$$
What is the wavelength of light if its frequency is 1.55 × 1010 s−1?
0.0194 m, or 19.4 mm
## Summary
All waves can be defined in terms of their frequency and intensity. $$c = \lambda \nu$$ expresses the relationship between wavelength and frequency.
## Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
• CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.
• Henry Agnew (UC Davis) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9297037720680237, "perplexity": 715.1319796193134}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487643703.56/warc/CC-MAIN-20210619051239-20210619081239-00467.warc.gz"} |
https://ftp.aimsciences.org/journal/1937-1632/2022/15/7 | # American Institute of Mathematical Sciences
ISSN:
1937-1632
eISSN:
1937-1179
All Issues
## Discrete and Continuous Dynamical Systems - S
July 2022 , Volume 15 , Issue 7
Issue on advances in discontinuous control systems and applications. Part I
Select all articles
Export/Reference:
2022, 15(7): i-ii doi: 10.3934/dcdss.2022119 +[Abstract](106) +[HTML](51) +[PDF](83.64KB)
Abstract:
2022, 15(7): 1615-1631 doi: 10.3934/dcdss.2021144 +[Abstract](364) +[HTML](319) +[PDF](1288.0KB)
Abstract:
In this article, an adaptive asymptotic tracking control scheme is proposed for fractional order nonlinear systems (FONSs) with time-varying disturbance. By introducing some well defined smooth functions and the bounded estimation approach, the effects caused by the unknown virtual control coefficients (UVCC) and unknown nonlinear functions are counteracted. For the UVCC, we only need to assume that their lower bounds are positive constants. Fuzzy logic systems (FLSs) are applied to approximate unknown nonlinear functions. Moreover, the fractional directed Lyapunov method is used to prove that the tracking error asymptotically converges to zero. Finally, an illustrative simulation example is applied to verify the superior performance of the presented control algorithms.
2022, 15(7): 1633-1650 doi: 10.3934/dcdss.2021145 +[Abstract](528) +[HTML](500) +[PDF](667.74KB)
Abstract:
The hydraulic servo actuators (HSA) are often used in the industry in tasks that request great powers, high accuracy and dynamic motion. It is well known that HSA is a highly complex nonlinear system, and that the system parameters cannot be accurately determined due to various uncertainties, inability to measure some parameters, and disturbances. This paper considers control problem of the HSA with unknown dynamics, based on adaptive dynamic programming via output feedback. Due to increasing practical application of the control algorithm, a linear discrete model of HSA is considered and an online learning data-driven controller is used, which is based on measured input and output data instead of unmeasurable states and unknown system parameters. Hence, the ADP based data-driven controller in this paper requires neither the knowledge of the HSA dynamics nor exosystem dynamics. The convergence of the ADP based control algorithm is also theoretically shown. Simulation results verify the feasibility and effectiveness of the proposed approach in solving the optimal control problem of HSA.
2022, 15(7): 1651-1667 doi: 10.3934/dcdss.2021160 +[Abstract](387) +[HTML](181) +[PDF](393.51KB)
Abstract:
In this study, the stable dynamics of a kind of high-order cellular neural networks accompanying \begin{document}$D$\end{document} operators and mixed delays are analyzed. The global existence of bounded positive solutions is substantiated by applying some novel differential inequality analyses. Meanwhile, by exploiting Lyapunov function method, some sufficient criteria are gained to validate the positiveness and globally exponential stability of pseudo almost periodic solutions on the addressed networks. In addition, computer simulations are produced to test the derived analytical findings.
2022, 15(7): 1669-1683 doi: 10.3934/dcdss.2021161 +[Abstract](591) +[HTML](268) +[PDF](940.49KB)
Abstract:
This paper considers the 3D printing process as a discontinuous control system and gives a simple and coherent bond stress-slip model for a new and intelligent building 3-D printed concrete. The previous models focused on either the maximal stress or the maximal slip, however, the novel model uses an energy approach by the dimension analysis, so that the main factors affecting the bond stress-slip relationship can be clearly revealed, mainly including the concrete's properties (its porous structure and its strength), the steel bar's properties (its printing direction, its strength, its surface roughness and its geometrical property) and the printing process. It is confirmed that the proposed model, similar to the constitutive relationship in elasticity, plays a key role in concrete mechanics, and it can conveniently explain the observed phenomena from the experiment.
2022, 15(7): 1685-1697 doi: 10.3934/dcdss.2021162 +[Abstract](451) +[HTML](217) +[PDF](307.46KB)
Abstract:
Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.
2022, 15(7): 1699-1712 doi: 10.3934/dcdss.2021167 +[Abstract](455) +[HTML](215) +[PDF](441.6KB)
Abstract:
In order to solve the control problem of Underwater Vehicle with Manipulator System (UVMS), this paper proposes a finite-time sliding mode control strategy via T-S fuzzy approach. From the general dynamic model of UVMS and considering the influence between the manipulator and the underwater vehicle, hydrodynamic damping, buoyancy and gravity as the fuzzy items, we establish global fuzzy dynamic model and design a closed-loop fuzzy sliding mode controller. We prove the model in theory from two aspects: the reachability of sliding domain and the finite-time boundedness. We also give the solution of the controller gain. A simulation on the actual four joint dynamic model of UVMS with two fuzzy subsystems is carried out to verify the effectiveness of this method.
2022, 15(7): 1713-1731 doi: 10.3934/dcdss.2021168 +[Abstract](435) +[HTML](177) +[PDF](2153.75KB)
Abstract:
This paper investigates the switching mechanism-based event-trig-gered fuzzy adaptive control issue of multi-input and multi-output (MIMO) nonlinear systems with prescribed performance (PP). Utilizing fuzzy logic systems (FLSs) to approximate unknown nonlinear functions. By using the switching threshold strategy, the system has more flexibility in strategy selection. The proposed control scheme can better solve the communication resource limitation. On account of the Lyapunov stability theory, the stability of the controlled system is proved. And all signals of the controlled system are bounded. Moreover, the tracking errors are controlled in a diminutive realm of the origin within the PP bounded. Simultaneously, the Zeno behavior is avoided. Finally, illustrate the effectiveness of the control scheme that has been proposed by demonstrating some simulation consequences.
2022, 15(7): 1733-1748 doi: 10.3934/dcdss.2021169 +[Abstract](348) +[HTML](153) +[PDF](1348.49KB)
Abstract:
The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function \begin{document}$f(r) = (r-d_0)^{-\theta }$\end{document}. For \begin{document}$\theta \ge 1$\end{document}, collision will not appear in the system if agents' initial positions are different. For the case \begin{document}$\theta \in [0,1)$\end{document} that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of \begin{document}$\psi (r) = (1+r^2)^{-\beta }$\end{document} and \begin{document}$f(r)$\end{document} on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.
2022, 15(7): 1749-1765 doi: 10.3934/dcdss.2022004 +[Abstract](409) +[HTML](176) +[PDF](1912.94KB)
Abstract:
This paper focuses on the state bounding problem for the time-delay impulsive and switching genetic regulatory networks (ISGRNs) with exogenous disturbances. Firstly, a sufficient criterion for the state bounding is obtained such that all the trajectories of ISGRNs under consideration converge exponentially into a sphere on the basis of an average dwell time (ADT) switching. Besides, globally exponential stability conditions for the considered system are further stated when the exogenous disturbance vanishes. As a special case, the equivalent state bounding criteria are established by using the properties of some special matrices when there exist no impulses at the switching instants in ISGRNs. Finally, an illustrating example is given to demonstrate the derived results. Compared with the existing literatures, the considered genetic regulatory networks (GRNs) have more general structure and the approach adopted in the present paper is more simple than Lyapunov-Krasovskii functional (LKF) approach.
2022, 15(7): 1767-1776 doi: 10.3934/dcdss.2022005 +[Abstract](395) +[HTML](119) +[PDF](291.16KB)
Abstract:
A class of fractional instantaneous and non-instantaneous impulsive differential equations under Dirichlet boundary value conditions with perturbation is considered here. The existence of classical solutions is presented by using the Weierstrass theorem. An example is given to verify the validity of the obtained results.
2022, 15(7): 1777-1795 doi: 10.3934/dcdss.2022006 +[Abstract](550) +[HTML](172) +[PDF](499.15KB)
Abstract:
As a new retail model, the front distribution center (FDC) has been recognized as an effective instrument for timely order delivery. However, the high customer demand uncertainty, multi-item order pattern, and limited inventory capacity pose a challenging task for FDC managers to determine the optimal inventory level. To this end, this paper proposes a two-stage distributionally robust (DR) FDC inventory model and an efficient row-and-column generation (RCG) algorithm. The proposed DR model uses a Wasserstein distance-based distributional set to describe the uncertain demand and utilizes a robust conditional value at risk decision criterion to mitigate the risk of distribution ambiguity. The proposed RCG is able to solve the complex max-min-max DR model exactly by repeatedly solving relaxed master problems and feasibility subproblems. We show that the optimal solution of the non-convex feasibility subproblem can be obtained by solving two linear programming problems. Numerical experiments based on real-world data highlight the superior out-of-sample performance of the proposed DR model in comparison with an existing benchmark approach and validate the computational efficiency of the proposed algorithm.
2022, 15(7): 1797-1821 doi: 10.3934/dcdss.2022080 +[Abstract](327) +[HTML](69) +[PDF](794.41KB)
Abstract:
This survey addresses stability analysis for impulsive systems with delayed impulses, which constitute an important generalization of delayed impulsive systems. Fundamental issues such as the concept of a solution to an impulsive system with delayed impulses and methods to determine impulse instants are revisited and discussed. In view of the types of delays in impulses, impulsive systems with delayed impulses are classified into two categories including systems with time-dependent delayed impulses and systems with state-dependent delayed impulses. Then more efforts are devoted to the stability analysis of these two classes of impulsive systems, where corresponding Lyapunov-function-based sufficient conditions for Lyapunov stability, asymptotic stability, exponential stability, input-to-state stability and finite-time stability are presented, respectively. Moreover, the double effects of time-dependent delayed impulses on system performance are reemphasized, and recent applications of delayed impulses in synchronization control are discussed in detail. Several challenges are suggested for future works.
2022, 15(7): 1823-1837 doi: 10.3934/dcdss.2022010 +[Abstract](400) +[HTML](162) +[PDF](1078.66KB)
Abstract:
This paper considers the attitude tracking control problem for a rigid body. In order to avoid the complexity and ambiguity associated with other attitude representations (such as Euler angles or quaternions), the attitude dynamics and the proposed control system are represented globally on special orthogonal groups. An adaptive controller based on a Lie subgroup of SO(3) is developed such that the rigid body can track any given attitude command asymptotically without requiring the exact knowledge of the inertia moment. In the presence of external disturbances, the adaptive controller is enhanced with an additional robust sliding mode term by following the same idea within the framework of SO(3). Finally, simulation results are presented to demonstrate efficiency of the proposed controllers.
2022, 15(7): 1839-1858 doi: 10.3934/dcdss.2022014 +[Abstract](484) +[HTML](152) +[PDF](762.1KB)
Abstract:
This paper discusses the problem of stabilization of interval type-2 fuzzy systems with uncertainties, time delay and external disturbance using a dynamic sliding mode controller. The sliding surface function, which is based on both the system's state and control input vectors, is used during the control design process. The sliding mode dynamics are presented by defining a new vector that augments the system state and control vectors. First, the reachability of the addressed sliding mode surface is demonstrated. Second, the required sufficient conditions for the system's stability and the proposed control design are derived by using extended dissipative theory and an asymmetric Lyapunov-Krasovskii functional approach. Unlike some existing sliding mode control designs, the one proposed in this paper does not require the control coefficient matrices of all linear subsystems to be the same, reducing the method's conservatism. Finally, numerical examples are provided to demonstrate the viability and superiority of the proposed design method.
2022, 15(7): 1859-1870 doi: 10.3934/dcdss.2022019 +[Abstract](233) +[HTML](115) +[PDF](455.07KB)
Abstract:
In this paper, the composite anti-disturbances control problem is considered for a class of stochastic systems with multiple disturbances. The states of the system are assumed to be unavailable. A state observer and a disturbance observer are constructed to estimate the states and the matched disturbance respectively. Based on the estimated values of state observer and disturbance observer, a non-fragile composite controller is designed to achieve disturbance attenuation and rejection. By means of the technique of the disturbance compensation control and stochastic control theory, some sufficient conditions are obtained to guarantee that the closed-loop system is asymptotically bounded in mean square or asymptotically stable in probability. Finally, a numerical example is given to verify the validity of the obtained results.
2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8105915188789368, "perplexity": 847.3014863878793}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104375714.75/warc/CC-MAIN-20220704111005-20220704141005-00186.warc.gz"} |
https://web2.0calc.com/questions/consider-the-line-with-equation-2-i-z-2-i-overline | +0
# Consider the line with equation $(2-i)z + (2+i)\overline{z} = 20.$For each complex number in the following list, \[2, 4+2i, 3i, 1-4i, 5, 3
0
149
1
Consider the line with equation
$$(2-i)z + (2+i)\overline{z} = 20.$$
For each complex number in the following list,
$$2, 4+2i, 3i, 1-4i, 5, 3, 10i,$$
figure out whether each one is on the line, then enter "yes" or "no" in the blank corresponding to each option.
Feb 7, 2019
#1
+22188
+8
Consider the line with equation
$$\large (2-i)z + (2+i)\overline{z} = 20.$$
For each complex number in the following list,
$$\large 2,\ 4+2i,\ 3i,\ 1-4i,\ 5,\ 3,\ 10i,$$
figure out whether each one is on the line.
$$\text{Let z = a+bi} \\ \text{Let \overline{z} = a-bi}$$
$$\begin{array}{|rcll|} \hline (2-i)z + (2+i)\overline{z} &=& 20 \quad & | \quad z = a+bi, \ \overline{z} = a-bi \\ (2-i)(a+bi) + (2+i)(a-bi) &=& 20 \\ 2a+2bi-ia-bi^2 +2a -2bi +ia -bi^2 &=& 20 \\ 2a-bi^2 +2a -bi^2 &=& 20 \\ 4a-2bi^2 &=& 20 \quad & | \quad i^2 = -1 \\ 4a+2b &=& 20 \quad & | \quad : 2 \\ \mathbf{2a+b} &\mathbf{=}& \mathbf{10} \\ \hline \end{array}$$
$$\begin{array}{|l|l|c|l|c|} \hline &\text{list } z = a+bi:&& \mathbf{2a+b = 10} & \text{on the line} \\ \hline 1) & 2: & a= 2 & 2\cdot 2 + 0 \ne 10 \\ & & b= 0 \\\\ \hline 2) & 4+2i: & a= 4 & 2\cdot 4 + 2 \mathbf{= 10} & \checkmark \\ & & b= 2 \\\\ \hline 3) & 3i: & a= 0 & 2\cdot 0 + 3 \ne 10 \\ & & b= 3 \\\\ \hline 4) & 1-4i: & a= 1 & 2\cdot 1 -4 \ne 10 \\ & & b= -4 \\\\ \hline 5) & 5: & a= 5 & 2\cdot 5 + 0 \mathbf{= 10} & \checkmark \\ & & b= 0 \\\\ \hline 6) & 3: & a= 3 & 2\cdot 3 + 0 \ne 10 \\ & & b= 0 \\\\ \hline 7) & 10i: & a= 0 & 2\cdot 0 + 10 \mathbf{= 10} & \checkmark \\ & & b= 10 \\\\ \hline \end{array}$$
Feb 7, 2019
edited by heureka Feb 7, 2019
edited by heureka Feb 7, 2019 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9620824456214905, "perplexity": 2942.453939350928}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232259327.59/warc/CC-MAIN-20190526165427-20190526191427-00325.warc.gz"} |
http://math.stackexchange.com/questions/276340/image-of-a-normal-space-under-a-closed-and-continuous-map-is-normal | # Image of a normal space under a closed and continuous map is normal
$p : X \to Y$ is continuous closed and surjective, and $X$ is a normal space. Show $Y$ is normal.
There is a hint, which I'm trying to prove: show that if $U$ is open in $X$ and $p^{-1}(\{y\}) \subset U$, $y \in Y$, then there is a neighbourhood W of y s.t. $p^{-1}(W) \subset U$.
I have a candidate for W namely $W=Y\setminus p(X \setminus U)$. I did prove that this W is open, and that $p^{-1}(W) \subset U$, but I don't see how $y \in W$...I think this would require injectivity of p...
I have also shown that $y \in p(U)$ and that $W \subset p(U)$, so if also $W \supset p(U)$, then $y \in W$.
Can anyone help me? Thank you in advance.
-
I'm actually not so sure how helpful the hint you've been given is. I think the following might get you to the proof quite quickly.
Hint: Note that essentially by de Morgan's Laws, normality of a topological space $X$ is equivalent to the following: given open $U , V \subseteq X$ such that $U \cup V = X$ there are closed $E \subseteq U$ and $F \subseteq V$ such that $E \cup F = X$.
As for the hint you have been given, you have given the correct set $W$. Note that as $p^{-1} \{ y \} \subseteq U$, then $p(x) \neq y$ for all $x \in X \setminus U$, which implies that $y \notin p [ X \setminus U ]$, or, equivalently, $y \in Y \setminus p [ X \setminus U ] = W$.
-
Thanks. I failed to use that $p^{-1}(\{y\}) \subset U$. – fuente Jan 12 '13 at 15:16
Deleted. Misprint. – fuente Jan 12 '13 at 15:18
@fuente: It was no problem to flesh out the last remaining detail for the hint. – Arthur Fischer Jan 12 '13 at 15:21
No, apparently not :) – fuente Jan 12 '13 at 15:27
Why go through all that hassle, when you can do as follows:
Let $p: X \rightarrow Y$ be a closed, continuous surjection. Now let $A,B$ be two disjoint closed subsets of $Y$. Because $X$ is normal, we can separate the closed disjoint sets $p^{-1}(A), p^{-1}(B)$ in $X$ by respective neighborhoods $U_1, U_2$. Now choose neighborhoods $V_1$ of $A$, and $V_2$ of $B$ s.t. $p^{-1}(V_1) \subset U_1$, and $p^{-1}(V_2) \subset U_2$. Then it follows that $V_1, V_2$ are disjoint. Hence, $Y$ is normal.
Note that in general, a continuous image of a normal space is not necessarily normal.
-
This in fact means that the image of a Hausdorff space under a closed, continuous surjection is Hausdorff. This can be proved easily: just replace your closed subsets with points instead. – Libertron Jan 12 '13 at 19:34
does it work same for images of T4 space under closed continuous map is T4 – math Feb 21 '13 at 17:01
Yes, I believe it does. This sounds like a good exercise. Haha, actually a $T_4$-space is normal and Hausdorff, so of course it should work! – Libertron Apr 11 '13 at 21:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9811374545097351, "perplexity": 239.13090314878966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273663.2/warc/CC-MAIN-20140728011753-00070-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://encyclopediaofmath.org/wiki/Lebesgue-Stieltjes_integral | # Lebesgue-Stieltjes integral
A generalization of the Lebesgue integral. For a non-negative measure $\mu$ the name "Lebesgue–Stieltjes integral" is used in the case when $X=\mathbf R^n$ and $\mu$ is not the Lebesgue measure; then the integral $\int_Xfd\mu$ is defined in the same way as the Lebesgue integral in the general case. If $\mu$ is of variable sign, then $\mu=\mu_1-\mu_2$, where $\mu_1$ and $\mu_2$ are non-negative measures, and the Lebesgue–Stieltjes integral
$$\int\limits_Xfd\mu=\int\limits_Xfd\mu_1-\int\limits_Xfd\mu_2,$$
under the condition that both integrals on the right-hand side exist. For $X=\mathbf R^1$ the fact that $\mu$ is countably additive and bounded is equivalent to the fact that the measure is generated by some function $\Phi$ of bounded variation. In this case the Lebesgue–Stieltjes integral is written in the form
$$\int\limits_a^bfd\Phi.$$
For a discrete measure the Lebesgue–Stieltjes integral is a series of numbers.
#### References
[1] E. Kamke, "Das Lebesgue–Stieltjes-Integral" , Teubner (1960) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9992225170135498, "perplexity": 155.63723276910028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655890181.37/warc/CC-MAIN-20200706191400-20200706221400-00192.warc.gz"} |
https://mathoverflow.net/questions/167407/monge-amp%C3%A8re-type-equation | # Monge–Ampère type equation
Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function). I am looking for the solutions among of such functions of the following differential equation: $$2B''_{xx}B''_{yy}-(B''_{xy})^{2}=0$$
Besides of the trivial solutions, I know that these type of functions $x^{\alpha}y^{\beta}$ solve the problem (for appropriate choice of $\alpha, \beta$). I am wondering if there are other solutions and if somebody can describe all possible solutions.
I know how to solve homogeneous Monge–Ampère equation, and unfortunately any technique used in that case does not apply here (as far as I understand).
Thanks.
Update 1, Q2:
Describe all nonnegative functions $B(x_{1},x_{2},x_{3})$, defined for $x_{1},x_{2},x_{3} \geq 0$, such that $B$ is concave function with respect to each variable i.e. $B_{x_{j}x_{j}}\leq 0$ and $B$ satisfies the following PDE:
\begin{align*} \det \left( {\begin{array}{ccc} 2B_{x_{1}x_{1}} & B_{x_{1}x_{2}} &B_{x_{1}x_{3}} \\ B_{x_{2}x_{1}} & 2B_{x_{2}x_{2}} &-B_{x_{2}x_{3}}\\ B_{x_{3}x_{1}} & -B_{x_{3}x_{2}} &2B_{x_{3}x_{3}}\\ \end{array} } \right)=0 \end{align*}
Note that besides the trivial solutions, the function $B(x_{1},x_{2},x_{3})=Cx_{1}^{2/3}x_{2}^{2/3}x_{3}^{2/3}$ satisfies the above conditions.
• What does the notation $B''_{xx}$ mean? Is it just $B_{xx}$? – Robert Bryant May 17 '14 at 9:30
• yes, it is just second derivative $B''_{xx}=\frac{\partial^{2} B}{\partial x^{2}}$ – Paata Ivanishvili May 17 '14 at 9:40
I had a little time on a flight today to think about your problem, and so I applied the standard integration method to see whether or not your equation could be explicitly integrated (in the sense that the Monge-Ampère equation $u_{xx}u_{yy}-{u_{xy}}^2=1$ can be integrated by transforming it to Laplace's equation). The answer is that it cannot be integrated that way. However, one can make the problem equivalent to a linear one, at least locally, that has an explicit solution in series, and this may or may not help you. I'll record the results here, just in case you find it useful.
Suppose that we have a solution $B(x,y)$ on some simply connected domain $D$ in the $xy$-plane and suppose that it satisfies $B_{xx} = -2p^2 < 0$, $B_{yy} = -q^2 < 0$, and $B_{xy} = 2pq$ for some positive functions $p$ and $q$. (One can also treat the case when one of $p$ or $q$ is negative, but that's a minor variation that I'll leave to you.) Then, we have $$\mathrm{d} B_x = -2p^2\,\mathrm{d} x + 2pq\,\mathrm{d} y \quad\text{and}\quad \mathrm{d} B_y = 2pq\,\mathrm{d} x - q^2\,\mathrm{d} y$$ Thus, the functions $p$ and $q$ must satisfy $$\mathrm{d}\bigl(-2p^2\,\mathrm{d} x + 2pq\,\mathrm{d} y\bigr) =\mathrm{d}\bigl(2pq\,\mathrm{d} x - q^2\,\mathrm{d} y\bigr) = 0,$$ and, conversely, if $p$ and $q$ satisfy these two conditions (which are first order PDE), then the above equations determine $B_x$ and $B_y$ (up to an additive constant) and then $\mathrm{d}B = B_x\,\mathrm{d}x + B_x\,\mathrm{d}y$ determines $B$ up to an additive constant, so the two PDE on $p$ and $q$ are essentially equivalent to the original equations.
Next, make a change of variables: Set $s = px$ and $t = qy - px$ and then $p= e^u$ and $q = e^v$. Then set $w = s + it$ and $z = -\tfrac12(v-iu)$. Then the above two PDE relating $(x,y,p,q)$ simply become the real and imaginary parts of the linear elliptic complex equation $$\frac{\partial w}{\partial \bar z} = \bar w$$ Now, this equation is known not to be integrable by the method of Darboux, so, in particular, you cannot write down its general solution in terms of a single holomorphic function of $z$. However, all the $C^1$ solutions are real analytic (because the equation is elliptic), and there is an explicit representation of the analytic solution in terms of power series: $$w(z)=\sum_{k=0}^\infty c_k\ f^{(k)}(z\bar z) z^k+\overline{c_k}\ f^{(k+1)}(z\bar z) \bar z^{k+1}$$ where $f^{(k)}$ is the $k$-th derivative of the modified Bessel $I$-function whose series representation is $$f(r) = 1 + \sum_{j=1}^\infty \frac{r^j}{(j!)^2}$$ Using this representation, you can trace back through and integrate by parts to get series representations of $B(x,y)$, $x$, and $y$ in terms of $u$ and $v$, which gives you a graphical representation of the local solutions in this case.
• Here is the equation: \begin{align*} \det \left( {\begin{array}{ccc} 2B_{x_{1}x_{1}} & B_{x_{1}x_{2}} &B_{x_{1}x_{3}} \\ B_{x_{1}x_{2}} & 2B_{x_{2}x_{2}} &-B_{x_{2}x_{3}}\\ B_{x_{3}x_{1}} & -B_{x_{3}x_{2}} &2B_{x_{3}x_{3}}\\ \end{array} } \right) =0 \end{align*} where $B(x_{1},x_{2},x_{3})$ is concave function with respect to each variable and it is given for $x_{1},x_{2},x_{3} \geq 0$. Assume $B_{x_{i}x_{j}}\geq 0$ for $i\neq j$ and $B=0$ if one of the coordinate is zero. One can see that the function $B(x_{1},x_{2},x_{3})=Cx_{1}^{2/3}x_{2}^{2/3}x_{3}^{2/3}$ works. – Paata Ivanishvili May 27 '14 at 11:49
• This is about Brascamp-Lieb inequality which is better than Holder's inequality. Right now we are trying to prove that in the case $n=3$ there is only one (or maybe not, any answer is interesting) Brascamp-Lieb inequality which corresponds to the function $B(x,y,z)=x^{1/p_{1}}y^{1/p_{2}}z^{1/p_{3}}$ where $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}} = 2$ and $1 \leq p_{j} \leq \infty$. I will add my second question to my first question. We also have homogeneity for the function $B$, it is homogeneous of degree 2. – Paata Ivanishvili May 27 '14 at 17:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9873269200325012, "perplexity": 146.07290047122777}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251796127.92/warc/CC-MAIN-20200129102701-20200129132701-00223.warc.gz"} |
http://tex.stackexchange.com/questions/8092/placing-equation-number-on-left-with-reqno-option-in-amsart-document-class | # Placing equation number on left with reqno option in amsart document class
Is there a way to force individual equations to place their equation numbers (or more specifically the result of a \tag*{blah}) on a specific side of the page regardless of the options to amsart.
I sometimes wish to use the equation enviornment to define 'requirements' in my papers which should look like:
R_e: equation stuff
Where R_e is the equation tag. R_e should be on the left even when reqno is passed to the documentclass. How can I accomplish this?
-
If you just want to do this for the equation environment, define a command
\newcommand{\LeftEqNo}{\let\veqno\@@leqno}
and use it as
$$\LeftEqNo x^2 + y^2 = 1.$$
The align environment is implemented quite differently, and you will need the environ package used like
\usepackage{environ}
\NewEnviron{Lalign}{\tagsleft@true\begin{align}\BODY\end{align}}
Then use \begin{Lalign} ... \end{Lalign} for left-numbered align environments.
I'm afraid that various hacks of the same kind will be necessary for the other ams environments, but I can't list everything without knowing which ones you need :).
-
equation or possibly align is fine. I plan to wrap this in a special environment for this construct and it will be one of those two. – Peter Gerdes Jan 2 '11 at 0:18
When I try this (using pdflatex) I get an error: ! Undefined control sequence. \env@Lalign@process ->\tagsleft – Sam Nead Sep 9 '11 at 14:59
@Sam Nead: I assume you might have a different version of the environ package. Can you post somewhere the result of \listfiles? – Bruno Le Floch Sep 11 '11 at 17:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8203745484352112, "perplexity": 2749.724527585801}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345769121/warc/CC-MAIN-20131218054929-00001-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://non.copyriot.com/general-formula-for-the-digital-and-the-analog/ | ## General Formula for the Digital and the Analog
By
21 Feb , 2019
Euclid is remembered as a geometer, when he is remembered at all. But Euclid’s Elements was an omnibus compendium of all mathematical knowledge known to him at the time, beginning with the first mathematics, geometry, then addressing ratio and proportion–that is, logos and analogos–and ultimately arithmetic, irrationality, and other topics. “There is hardly anything in mathematics more beautiful than [Euclid’s] wondrous fifth book,” wrote British mathematician Arthur Cayley. Indeed the definitions that begin book five of the treatise furnish a series of important concepts, first the mathematical ratio, then proportion, understood as an equality of ratios.
Definition 3: “A ratio is a sort of relation in respect of size between two magnitudes of the same kind” [“Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα ποιὰ σχέσις“].
Definition 6: “Let magnitudes which have the same ratio be called proportional” [“Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον καλείσθω“].
Digital and analog appear here on the same page, perhaps for the first time, at least so under the guise of logos and analogos. Of immediate interest is the expression “two magnitudes of the same kind” (“δύο μεγεθῶν ὁμογενῶν“), or, to mimic Euclid’s terminology even more closely, two homogenous magnitudes. What does it take for two magnitudes to be homogeneous, to be “of the same genus”? They must contain a “part” or submultiple [μέρος] out of which each are measured without remainder. Hence 4 and 3 may form the ratio 4:3 because each is measurable by a shared, discrete submultiple, the simple arithmetical unit more commonly known as 1. But, apples and oranges are not comparable, as the old saying goes, and may form no discrete ratio, because they share no submultiple as a common basis for measurement. (This is one indication for why aesthetics and digitality belong to fundamentally different paradigms; perception easily accommodates qualitative difference while digitality constitutionally prohibits it.) The logos ratio is thus a strange beast, both multiple and homogenous. The digital begins with a differential cut, the cut of distinction. But beyond the initial cut all future differentiation is based on the same genus (the homogenous). Later in the treatise, Euclid expands this basic insight by stipulating that logos ratios are symmetric [σύμμετρα], literally “with measure” or commensurable through a shared, common part.
Definition 6 (above) shifts the discussion slightly. While the previous definition concerned a single ratio, itself defined as a relation of two discrete numbers, this definition duplicates the ratio, bringing two ratios into a relation of equality. When two ratios are the same they are analogos, or proportional.
The general formula for logos is thus a/b, or the ratio between two homogenous elements. Whereas the general formula for analogos is a/b = c/d, or the equation of two existing ratios.
General formula of the digital: a/b
General formula of the analog: a/b = c/d
These two expressions are revealing. At the outset, they confirm that analogos is not the negation or inversion of logos–and thus, by extrapolation, the analog is not the opposite of the digital–but rather, in some fundamental sense, its twin or echo. Yet even as the former is shown to be a reduplication of the latter, the two terms diverge dramatically in their connotations and effects. The two expressions may look similar, and they may be composed the one out of the other, but they ultimately produce two very different technologies.
First, the digital or logos relies on a homogeneous substrate of elements that are differentiated quantitatively. Those famous “zeros and ones” get the most attention, but the rest of the integers are just as digital, as are the natural numbers overall and the rational number line as a whole. And the discussion need not be limited to number, since the alphabet is an advanced digital technology too, as influential as the integers if not more so. (Indeed in languages like Hebrew or Greek, letters of the alphabet are deployed as counting numbers.) Any other system of mediation constructed from quantitative difference will likewise earn the monicker digital.
In this way, the digital follows what might be called the rule of two in that it entails an ever present discretization into two or more parts–the two, the three, the multiple. These parts are brought into relation and assembled into a combinatory whole. Examples of this combinatory mechanism include rational numbers like 3/4 or 5/8, or the composition of words and phrases from simple linguistic elements. Yet even as combinatory wholes, such ratios never elide the two elements that form them. Two voices may sing the harmony of the fifth interval, yet they will forever remain two voices. There is no fifth outside of the two, just as there is no more reduced form of 3:2 than the two arithmetical atoms that compose it.
Finally–and this will be the hardest to demonstrate–the digital generates a transcendental essence within a symbolic order, that is, something significant that supersedes the merely homogeneous substrate of elements. The simple terms of the digital ratio are more or less useless. Alone the number “5” or the letter “g” carries little meaning. Nevertheless, as combinatory wholes, logos ratios contain symbolic value. Such is the magic of language. The letters of the alphabet are not inherently meaningful, and indeed many simple words are not particularly meaningful either, nevertheless constructions may be made of them that bear deep signification. The digital is the site, in other words, of what in poststructuralism was called the symbolic order, where a system of regularly interoperable terms (letter/number, signifier/signified, ego/superego, self/other, etc.), themselves “empty” or generic structures, recombine in complex ways to produce rich compositions, from novels and poems all the way up to human beings and entire societies. In sum, the digital is differential and homogenous, but also transcendental. Digital atoms may be standardized, but that does not preclude them from transcending their own empty consistency. Indeed it mandates that they do.
taken from here | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.833672285079956, "perplexity": 2504.1449770409617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232258453.85/warc/CC-MAIN-20190525224929-20190526010929-00306.warc.gz"} |
https://www.physicsforums.com/threads/integration-by-residual-theorem.715171/ | # Integration by residual theorem
1. Oct 7, 2013
### AlonsoMcLaren
1. The problem statement, all variables and given/known data
Integrate $$\int_{-\infty}^{\infty} dx \ x^{-3}(sin x -x )$$
2. Relevant equations
Residual theorem, Jordan's lemma
3. The attempt at a solution
$$\int_{-\infty}^{\infty} dx \ x^{-3}(sin x -x ) = Im \ (PV\int_{-\infty}^{\infty} dx \ x^{-3}(e^{ix} -ix ))$$
In the contour, if we let R goes to infinity and ρ goes to zero, then we have
$$\oint_{Total}^{ } dz \ z^{-3}(e^{iz} -iz) = \int_{C_{1}}^{ } dz \ z^{-3}(e^{iz} -iz) + \int_{C_{2}}^{ } dz \ z^{-3}(e^{iz} -iz) + PV \int_{-\infty}^{\infty} dx \ x^{-3}(e^{ix} -ix)$$
Because the only singularity is 0, not inside the contour. By residual theorem (or Cauchy's theorem),
$$\oint_{Total}^{ } dz \ z^{-3}(e^{iz} -iz) = 0$$
By Jordan's lemma, as R goes to infinity,
$$\int_{C_{2}}^{ } dz \ z^{-3}(e^{iz} -iz) =0$$
For the C1 contour
$$\int_{C_{1}}^{ } dz \ z^{-3}(e^{iz} -iz)= \int_{\pi}^{0 } d(\rho e^{i\theta }) \frac{e^{i\rho e^{i\theta }} -i\rho e^{i\theta }}{(\rho e^{i\theta })^{3}}= i \int_{\pi}^{0 }\frac{e^{i\rho e^{i\theta }}-i\rho e^{i\theta}}{\rho ^{2}e^{2i\theta }} d \theta$$
But as ρ goes to 0, $$\frac{e^{i\rho e^{i\theta }}-i\rho e^{i\theta}}{\rho ^{2}e^{2i\theta }}$$
goes to infinity. So I am stuck.
The final answer, according to WolframAlpha, is -π/2
File size:
5.4 KB
Views:
72
2. Oct 8, 2013
### jackmell
No.
Your mission, should you choose to accept it, is to show:
$$\lim_{\rho\to 0} \int_{\pi}^{0}\frac{e^{i\rho e^{it}}-i\rho e^{it}}{\rho^2 e^{2 i t}} dt =\pi/2$$
and you can't just say it is cus' I told you so because I could be wrong. I've not been able to do so myself yet. Ain't that the fun part of math?
Edit: Ok, I think I have it. Kinda' rickety and not sure about the legitimacy of applying L'Hopital's rule to integral functions but it's something. You try to get it and let me know what you get.
Last edited: Oct 8, 2013
3. Oct 9, 2013
### jackmell
A PF member was kind enough to help me with a part of this analysis in the Calculus sub-forum and according to his analysis, I am justified in applying L'Hopital's rule under the integral sign in this particular case:
Fun problem. Thanks voko!
4. Oct 9, 2013
### brmath
It is clear that the original integral converges since $x^{-3}$ will govern. And I suppose you are taking a course in complex variables and are supposed to use contour integrals.
But in a contrarian spirit, allow me to suggest that you can find an antiderivative for this integrand. Integrate by parts several times.
And in an inquiring spirit, could you enlighten me as to what PV means. I know the Cauchy Integral formula but have not seen the notation PV.
Re applying L'Hospital's theorem, that theorem is just expressing some facts about the Taylor's polynomial. No reason why you can't expand your function under an integral sign (we do it all the time).
5. Oct 9, 2013
### Office_Shredder
Staff Emeritus
How would you do this? I assume you intend to integrate the x-3 but when you run into a log you are going to be in trouble.
It means principal value, and means instead of taking the true double sided limit you take
$$\lim_{R\to \infty} \int_{-R}^{R}$$
which is what you actually get when you do contour integration (meaning you can do contour integration to calculate the principal value of something where the true integral doesn't exist). If the integral converges then it converges to the same value as the principal value of course.
6. Oct 9, 2013
### AlonsoMcLaren
But $$\lim_{\rho\to 0} \int_{\pi}^{0}\frac{e^{i\rho e^{it}}-i\rho e^{it}}{\rho^2 e^{2 i t}} dt =\pi/2$$ is indeed infinity
If you do taylor expansion:
$$e^{i\rho e^{it}} = 1+ i\rho e^{it}-\frac{1}{2}\rho^{2} e^{2it}+....$$
$$e^{i\rho e^{it}}-i\rho e^{it} = 1-\frac{1}{2}\rho^{2} e^{2it}+....$$
$$\frac{e^{i\rho e^{it}}-i\rho e^{it}}{\rho^2 e^{2 i t}} = \rho^{-2}e^{-2it}-1/2+..$$
It blows up as ρ goes to zero!
7. Oct 9, 2013
### brmath
Sorry about my contrarian spirit, which both saw that the $x^{-3}$ would govern, and viewed it as $x^3$ for integration purposes. This is too contrarian even for me.
I've used lim$\int_{-R}^R$ in contour integration, but didn't know it was called the Principal Value.
8. Oct 10, 2013
### jackmell
Outstanding Alonso. I see you've completed your mission and taught me a very simple way to show the limit:
\begin{align*} \lim_{\rho\to 0}\int_{\pi}^0 \frac{e^{i\rho e^{it}}- i\rho e^{it}}{\rho^2 e^{it}}dt&=\lim_{\rho\to 0}\int_{\pi}^0 \left(-\frac{1}{2}+\frac{e^{-2 i t}}{\rho ^2}-\frac{1}{6} i e^{i t} \rho +\frac{1}{24} e^{2 i t} \rho ^2+\frac{1}{120} i e^{3 i t} \rho ^3+\cdots\right)dt \\ &=\pi/2+\lim_{\rho\to 0} \frac{1}{\rho^2}\int_{\pi}^0 e^{-2 i t}dt+0 \\ &=\pi/2+\lim_{\rho\to 0} \frac{1}{\rho^2}(0)+0 \\ &=\pi/2 \end{align*}
Last edited: Oct 10, 2013
Draft saved Draft deleted
Similar Discussions: Integration by residual theorem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9952171444892883, "perplexity": 1605.9362654325348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105334.20/warc/CC-MAIN-20170819085604-20170819105604-00098.warc.gz"} |
https://www.physicsforums.com/threads/show-that-a-definite-integral-is-positive.177857/ | # Show that a definite integral is positive
1. Jul 22, 2007
### danago
Show that $$\int_0^1 {x^n \sin \left( {\frac{{\pi x}}{2}} \right)} {\kern 1pt} dx > 0$$ for all $${\rm{n}} \ge {\rm{0}}$$
Basically, what i want to show is that over the interval (0,1), the integrand is above the x-axis, or positive. I will show that over the interval, both functions, $$x^n$$ and $$sin(\frac{\pi x}{2})$$ are both positive.
For the trig function, this is how i proceeded:
$$\begin{array}{l} 0 < x < 1 \\ 0 < \frac{{\pi x}}{2} < \frac{\pi }{2} \\ \sin 0 < \sin \left( {\frac{{\pi x}}{2}} \right) < \sin \left( {\frac{\pi }{2}} \right) \\ 0 < \sin \left( {\frac{{\pi x}}{2}} \right) < 1 \\ \end{array}$$
Is that a valid way of doing it?
For the xn function, i took the same approach:
$$\begin{array}{l} 0 < x < 1 \\ 0^n < x^n < 1^n \\ 0 < x^n < 1 \\ \end{array}$$
Since, over the interval of (0,1) of which the function is being integrated, the integrand is positive, the definite integral will be positive for all values of n.
My main question though; Is my approach a good way of doing such a problem? It seemed to work fine for this question, but are there cases where i cant use such a method?
My book has this for an answer:
Let $$g(x)=x^n$$ and $$h(x)=sin(\frac{\pi x}{2})$$.
On the interval 0 < x < 1 , g(x)>0. On the same interval, h(x)>0, therefore, the definite integral is greater than zero.
That is basically the concept i used, but i really dont like that method, since it really doesnt show much, and in exams, depending on the marker, i may lose marks for such a bland proof, which is why im wanting to use a more solid proof.
Dan.
EDIT: Just realised that this method using the inequalities does not work for all cases. But it should work if the function is strictly increasing, or strictly decreasing over the interval, right?
Last edited: Jul 22, 2007
2. Jul 22, 2007
### nrqed
No. Consider using your logic to get the following obviosuly wrong result:
$$\begin{array}{l} 0 < x < 1 \\ 0 < \pi x < \pi \\ \sin 0 < \sin \left( \pi x\right) < \sin \left( \pi \right) \\ 0 < \sin \left( \pi x \right) < 0 \\ \end{array}$$
Don't try to do a proof with usual manipulation of inequalities when you have trig functions like this because trig functions are not monotonous, they increase and then decrease and so on.
You should simply state it as an obvious fact that for an argument between 0 and pi/2, the sin function is always positive.
Similar Discussions: Show that a definite integral is positive | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9710361361503601, "perplexity": 387.75504989030327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00270.warc.gz"} |
https://lms.gckottayam.ac.in/course/index.php?categoryid=95 | ### PROBABILITY DISTRIBUTIONS
This course introduce the basics of elementary proabaility distributions, law of large numbers and important sampling distributions. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9896414875984192, "perplexity": 3734.7230078784987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500158.5/warc/CC-MAIN-20230205000727-20230205030727-00245.warc.gz"} |
http://acm.sdibt.edu.cn/JudgeOnline/problem.php?id=1052 | ## Welcome To SDIBT ACM-ICPC Online Judge
VIRTUAL JUDGE Recent Contest F.A.Qs Discuss Home ProblemSet Status Ranklist Contest LoginRegister Exam
Problem 1052. -- The Stern-Brocot Number System
## The Stern-Brocot Number System
Time Limit: 1 Sec Memory Limit: 64 MB
Submit: 21 Solved: 13
[Submit][Status][Discuss]
## Description
The Stern-Brocot tree is a beautiful way for constructing the set of all non-negative fractions where m and n are relatively prime. The idea is to start with two fractions , and then repeat the following operation as many times as desired:
Insert between two adjacent fractions and .
For example, the first step gives us one new entry between and ,
,,
and the next gives two more:
,,,,
The next gives four more:
,,,,,,,,
The entire array can be regarded as an infinite binary tree structure whose top levels look like this-
This construction preserves order, and thus we cannot possibly get the same fraction in two different places.
We can, in fact, regard the Stern-Brocot tree as a number system for representing rational numbers, because each positive, reduced fraction occurs exactly once. Let us use the letters L'' and R'' to stand for going down the left or right branch as we proceed from the root of the tree to a particular fraction; then a string of L's and R's uniquely identifies a place in the tree. For example, LRRL means that we go left from down to , then right to , then right to , then left to . We can consider LRRL to be a representation of . Every positive fraction gets represented in this way as a unique string of L's and R's.
Well, almost every fraction. The fraction corresponds to the empty string. We will denote it by I, since that looks something like 1 and stands for identity."
In this problem, given a positive rational fraction, represent it in the Stern-Brocot number system.
## Input
The input file contains multiple test cases. Each test case consists of a line containing two positive integers m and n, where m and n are relatively prime. The input terminates with a test case containing two 1's for m and n, and this case must not be processed.
## Output
For each test case in the input file, output a line containing the representation of the given fraction in the Stern-Brocot number system.
## Sample Input
5 7
878 323
1 1
## Sample Output
LRRL
RRLRRLRLLLLRLRRR
## Source
[Submit][Status][Discuss]
HOME Back
한국어 中文 English
All Copyright Reserved 2008-2010 SDIBT TEAM
GPL2.0 2003-2010 HUSTOJ Project TEAM | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8675686120986938, "perplexity": 1087.4057535005184}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945584.75/warc/CC-MAIN-20180422100104-20180422120104-00143.warc.gz"} |
https://testbook.com/question-answer/the-process-in-which-liquid-changes-to-gaseous-sta--5f451c579fd0e22a5a5b4f86 | # The process in which liquid changes to gaseous state at the surface, below the boiling point due to the transfer of heat is known as ________
This question was previously asked in
KPSC JE 2016: Specific Paper
View all KPSC JE Papers >
1. Boiling
2. Evaporation
3. Run off
4. Drainage
Option 2 : Evaporation
Free
ST 22: Geotechnical Engineering
2176
20 Questions 20 Marks 15 Mins
## Detailed Solution
Evaporation:
Evaporation is a process in which a liquid changes to a gaseous state at the free surface of the liquid.
Evaporation is a cooling process in which latent heat of vaporization must be provided by water bodies.
When a solute is dissolved in water, there is a reduction in the rate of evaporation.
The rate of evaporation is an inversely proportional to atmospheric pressure. Keeping all the other remaining factors the same, an increase in pressure will decrease the evaporation. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8301095366477966, "perplexity": 1634.1889637139443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304959.80/warc/CC-MAIN-20220126162115-20220126192115-00334.warc.gz"} |
https://www.arxiv-vanity.com/papers/1707.05284/ | arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.
Date: January 5, 2020
Proposal for JLab PAC45
PR12–17–001
Strange Hadron Spectroscopy with a Secondary Beam at GlueX
S. Adhikari, H. Al Ghoul, A. Ali, M. J. Amaryan, E. G. Anassontzis, A. V. Anisovich, A. Austregesilo, M. Baalouch, F. Barbosa, A. Barnes, M. Bashkanov, T. D. Beattie, R. Bellwied, V. V. Berdnikov, T. Black, W. Boeglin, W. J. Briscoe, T. Britton, W. K. Brooks, B. E. Cannon, E. Chudakov, P. L. Cole, V. Crede, M. M. Dalton, A. Deur, P. Degtyarenko, S. Dobbs, G. Dodge, A. G. Dolgolenko, M. Döring, M. Dugger, R. Dzhygadlo, R. Edwards, H. Egiyan, S. Eidelman, A. Ernst, A. Eskandarian, P. Euginio, C. Fanelli, S. Fegan, A. M. Foda, J. Frye, S. Furletov, L. Gan, A. Gasparian, G. Gavalian, V. Gauzshtein, N. Gevorgyan, D. I. Glazier, K. Goetzen, J. Goity, V. S. Goryachev, L. Guo, H. Haberzettl, M. Hadžimehmedović, H. Hakobyan, A. Hamdi, S. Han, J. Hardin, A. Hayrapetyan, T. Horn, G. M. Huber, C. E. Hyde, D. G. Ireland, M. M. Ito, B. C. Jackson, N. S. Jarvis, R. T. Jones, V. Kakoyan, G. Kalicy, M. Kamel, C. D. Keith, C. W. Kim, F. J. Klein, C. Kourkoumeli, S. Kuleshov, I. Kuznetsov, A. B. Laptev, I. Larin, D. Lawrence, M. Levillain, W. I. Levine, K. Livingston, G. J. Lolos, V. E. Lyubovitskij, D. Mack, M. Mai, D. M. Manley, U.-G. Meißner, H. Marukyan, V. Mathieu, P. T. Mattione, M. Matveev, V. Matveev, M. McCaughan, M. McCracken, W. McGinley, J. McIntyre, C. A. Meyer, R. Miskimen, R. E. Mitchell, F. Mokaya, V. Mokeev, K. Nakayama, F. Nerling, Y. Oh, H. Osmanović, A. I. Ostrovidov, R. Omerović, Z. Papandreou, K. Park, E. Pasyuk, M. Patsyuk, P. Pauli, R. Pedroni, M. R. Pennington, L. Pentchev, K. J. Peters, W. Phelps, E. Pooser, B. Pratt, J. W. Price, N. Qin, J. Reinhold, D. Richards, D.-O. Riska, B. G. Ritchie, J. Ritman, L. Robison, D. Romanov, H-Y. Ryu, C. Salgado, E. Santopinto, A. V. Sarantsev, R. A. Schumacher, C. Schwarz, J. Schwiening, A. Semenov, I. Semenov, K. K. Seth, M. R. Shepherd, E. S. Smith, D. I. Sober, D. Sokhan, A. Somov, S. Somov, O. Soto, N. Sparks, J. Stahov, M. J. Staib, J. R. Stevens, I. I. Strakovsky, A. Subedi, A. Švarc, A. Szczepaniak, V. Tarasov, S. Taylor, A. Teymurazyan, A. Tomaradze, A. Tsaris, G. Vasileiadis, D. Watts, D. Werthmüller, N. Wickramaarachchi, T. Whitlatch, M. Williams, B. Wojtsekhowski, R. L. Workman, T. Xiao, Y. Yang, N. Zachariou, J. Zarling, Z. Zhang, B. Zou, J. Zhang, X. Zhou, B. Zihlmann
Arizona State University, Tempe, AZ 85287, USA
National and Kapodistrian University of Athens, Athens 15771, Greece
Institut für Experimentalphysik I - Ruhr-Universität, Bochum 44780, Germany
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
California State University, Dominguez Hills, Carson, CA 90747, USA
Carnegie Mellon University, Pittsburgh, PA 15213, USA
The Catholic University of America, Washington, DC 20064, USA
Institute of Theoretical Physics CAS, Beijing 100190, People’s Republic of China
University of Connecticut, Storrs, CO 06269, USA
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
Finnish Society of Science and Letters, Helsinki 00130, Finland
Florida International University, Miami, FL 33199, USA
Florida State University, Tallahassee, FL 32306, USA
National Research Centre "Kurchatov Institute", Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
The George Washington University, Washington, DC 20052, USA
University of Glasgow, Glasgow G12 8QQ, United Kingdom
GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt 64291, Germany
University of Georgia, Athens, GA 30602, USA
II. Physikalisches Institut, Justus Liebig-University of Giessen, Giessen 35392, Germany
Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, Bonn 53115, Germany
Hampton University, Hampton, VA 23668, USA
University of Houston, Houston, TX 77204, USA
Idaho State University, Pocatello, ID 83209, USA
Indiana University, Bloomington, IN 47403, USA
I.N.F.N. Sezione di Genova, Genova 16146, Italy
Institute für Kernphysik & Jülich Center für Hadron Physics, Jülich 52425, Germany
Kent State University, Kent, OH 44242, USA
Kyungpook National University, Daegu 702-701, Republic of Korea
National Research Centre "Kurchatov Institute", Institute for Theoretical and Experimental Physics, Moscow 117218, Russia
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Novosibirsk State University, Novosibirsk 630090, Russia
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Institute of Theoretical Physics, University of Tübingen, Tübingen 72076, Germany
University of Massachusetts, Amherst, MA 01003, USA
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Institut für Kernphysik, University of Mainz, Mainz 55099, Germany
National Research Nuclear University Moscow Engineering Physics Institute, Moscow 115409, Russia
Norfolk State University, Norfolk, VA 23504, USA
North Carolina A&T State University, Greensboro, NC 27411, USA
Old Dominion University, Norfolk, VA 23529, USA
University of North Carolina at Wilmington, Wilmington, NC 28403, USA
Northwestern University, Evanston, IL 60208, USA
Pusan National University, Busan 46241, Republic of Korea
University of Regina, Regina, SA S4S 0A2, Canada
Rudjer Bošković Institute, Zagreb 10002, Croatia
Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile
Tomsk State University, Tomsk 634050, Russia
Tomsk Polytechnic University, Tomsk 634050, Russia
University of Tuzla, Tuzla 75000, Bosnia and Herzegovina
Yerevan Physics Institute, Yerevan 0036, Armenia
College of William and Mary, Williamsburg, VA 23185, USA
University of Virginia, Charlottesville, VA 22904, USA
Wuhan University, Wuhan, Hubei 430072, People’s Republic of China
Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Jülich 52425, Germany
Contact Person:
Spokesperson
(The GlueX Collaboration)
Abstract
We propose to create a secondary beam of neutral kaons in Hall D at Jefferson Lab to be used with the GlueX experimental setup for strange hadron spectroscopy. A flux on the order of /s will allow a broad range of measurements to be made by improving the statistics of previous data obtained on hydrogen targets by three orders of magnitude. Use of a deuteron target will provide first measurements on the neutron which is terra incognita.
The experiment will measure both differential cross sections and self-analyzed polarizations of the produced , , , and hyperons using the GlueX detector at the Jefferson Lab Hall D. The measurements will span c.m. from to 0.95 in the c.m. range above MeV and up to 3500 MeV. These new GlueX data will greatly constrain partial-wave analyses and reduce model-dependent uncertainties in the extraction of strange resonance properties (including pole positions), and provide a new benchmark for comparisons with QCD-inspired models and lattice QCD calculations.
The proposed facility will also have an impact in the strange meson sector by providing measurements of the final-state system from threshold up to 2 GeV invariant mass to establish and improve on the pole positions and widths of all P-wave states as well as for the S-wave scalar meson .
## 1 Executive Summary
We propose to establish a secondary beam line at JLab Hall D for scattering experiments on both proton and neutron (for the first time) targets in order to determine the differential cross sections and the self-polarization of strange hyperons with the GlueX detector to enable precise partial wave analysis in order to determine all the resonances up to 2400 MeV in the spectra of the , and hyperons.
In addition, we intend to do strange meson spectroscopy by studies of the interaction to locate the pole positions in the and channels.
The beam will be generated by directing a high energy, high intensity photon beam onto a Be target in front of the GlueX detector. The flux of the beam will be of the order /s on a liquid hydrogen/deuterium cryotarget within the GlueX detector, which has a large acceptance with coverage of both charged and neutral particles. This flux will allow statistics in the case of hydrogen targets to exceed that of earlier experiments by almost three orders of magnitude. The main components of the experimental setup are the compact photon source, the Be target with a beam plug, sweeping magnet and a pair spectrometer.
The physics case for the experiments is aligned with the 2015 Long Range Plan for Nuclear Science [1]: “…a better understanding of the role of strange quarks became an important priority". Knowledge of the hyperon spectra is an important component for this. The empirical knowledge of the low lying spectra of the and hyperons remains very poor in comparison with that of the nucleon, and in the case of the hyperons extremely poor. The structure of these hyperon resonances cannot be understood without empirical determination of their pole positions and decays, which is the goal of the proposed experiments. The determination of the strange hyperon spectra in combination with the current measurements of the spectra of the charm and beauty hyperons at the LHCb experiment at CERN should allow a clear understanding of soft QCD matter and the approach to heavy quark symmetry.
As the first stage of the GlueX program the focus will be on two-body and quasi-two-body: elastic and charge-exchange reactions, then on two-body reactions producing hyperons as , , and , as well as three body .
For analyzing the data a coupled channel partial wave analysis will be done of the GlueX data in parallel with an analysis of the data from the J-PARC measurements, when available. The best fit will determine the partial wave amplitudes and the resonance pole positions, residues and Breit-Wigner parameters. These will provide a benchmark for results of forthcoming QCD lattice calculations and lead to the desired understanding of the structure of the strange hyperons.
Our timeline is to begin beam experiments at the completion of the current GlueX physics program.
## 2 Scope of the Proposal
The nature of QCD confinement continues to provide a challenge to our understanding of soft QCD. Studies of the baryon spectrum provide one obvious avenue to understand this region since the location and properties of excited states reflect the dynamics and relevant degrees-of-freedom of hadrons.
Through analyses of decades worth of data, from both hadronic and electromagnetic (EM) scattering experiments, numerous baryon resonances have been observed, with their masses, widths, and quantum numbers fully determined. There are 109 baryons in the PDG2016 listings but only 58 of them are or [2]. Many more states are predicted by quark models (QMs). For example, in the case of , 434 resonances would be required, if all partly revealed multiplets were completed (three 70 and four 56).
Three light quarks can be arranged in six baryonic families, , , , , , and . The possible number of members in a family is not arbitrary [3]. If the symmetry of QCD is controlling, then for the octet: , , and , and for the decuplet: , , , and . The number of experimentally identified resonances in each baryon family in PDG2016 summary tables is 17 , 24 , 14 , 12 , 7 , and 2 . Constituent QMs, for instance, predict the existence of no fewer than 64 and 22 states with mass less than 3 GeV. The seriousness of the “missing-states" problem [4] is obvious from these numbers. To complete multiplets, one needs no fewer than 17 s, 41 s, 41 s, and 24 s.
If these “missing resonances" exist, they have either eluded detection or have produced only weak signals in the existing data sets. The search for such resonances provides a natural motivation for future measurements at Jefferson Lab. As stated in the 2015 Long Range Plan for Nuclear Science [1]: For many years, there were both theoretical and experimental reasons to believe that the strange sea-quarks might play a significant role in the nucleon’s structure; a better understanding of the role of strange quarks became an important priority.
Low-lying baryon resonances, both hyperons and non-strange states, are usually considered to be three-quark systems; however, those quarks are constituent, not current ones. This prevents their description by the well-understood perturbative QCD. It seems, however, that some qualitative consequences of perturbative QCD still apply even for the non-perturbative constituent quarks. One of them is the suppression of the effective strong interaction for the heavier strange quark in comparison with the lighter up and down flavored quarks (due to the asymptotic freedom). This is revealed, e.g., in smaller widths of hyperon resonances as compared with similar non-strange baryon resonances. The same phenomenon is seen also for meson resonances (compare the widths of and meson resonances). Further investigation of this and other similar properties may help to improve our understanding of the nature of the constituent quarks and other non-perturbative effects.
The JLab 12 GeV energy upgrade, with the new Hall D, is an ideal tool for extensive studies of non-strange and, specifically, strange baryon resonances [5, 6]. Our plan is to take advantage of the existing high-quality photon beam line and experimental area in the Hall D complex at Jefferson Lab to deliver a beam of particles onto a liquid hydrogen/deuterium cryotarget (LH/LD) within the GlueX detector.
The recently constructed GlueX detector in Hall D is a large acceptance spectrometer with good coverage for both charged and neutral particles that can be adapted to this purpose. Obviously, a beam facility with good momentum resolution is crucial to provide the data needed to identify and characterize the properties of hyperon resonances. The masses and widths of the lowest and baryons were determined mainly with kaon beam experiments in the 1970s [2]. First determinations of the pole position in the complex-energy plane for a hyperon, for instance for the , has been made only recently [7]. An intense beam would open a new window of opportunity, not only to locate “missing resonances", but also to establish their properties by studying different decay channels systematically.
The recent white paper, dedicated to the physics with meson beams and endorsed by a broad physics community, summarized unresolved issues in hadron physics, and outlined the vast opportunities and advances that only become possible with a “secondary beam facility" [8]. The Hall D GlueX K-long Facility (KLF) measurements will allow studies of very poorly known multiplets of , , , and even hyperons with unprecedented statistical precision. These measurements also have the potential to observe dozens of predicted (but heretofore unobserved) states and to establish the quantum numbers of already observed hyperons listed in PDG2016 [2]. Interesting puzzles exist for PDG-listed excited hyperons that do not fit into any of the low-lying excited multiplets, and these need to be further revisited and investigated. Excited s, for instance, are very poorly known. Establishing and discovering new states is important, in particular, for determination of the multiplet structure of excited baryons.
We have organized three Workshops: Physics with Neutral Kaon Beam at JLab (KL2016) (February 2016) [9], Excited Hyperons in QCD Thermodynamics at Freeze-Out (YSTAR2016) (November 2016) [10], and New Opportunities with High-Intensity Photon Sources (HIPS2017) (February 2017) [11]. They were dedicated to the physics of hyperons produced by the neutral kaon beam. The KL2016 Workshop [12] followed our LoI–12–15–001 [13] to help address the comments made by PAC43 and to prepare the full proposal for PAC45. The proposed GlueX KLF program is complementary, for instance, to the CLAS12 baryon spectroscopy experiments [14, 15] and would operate in Hall D for several years. The YSTAR2016 Workshop [16] was a successor to the recent KL2016 Workshop and considered the influence of possible “missing" hyperon resonances on QCD thermodynamics, on freeze-out in heavy ion collisions and in the early universe, and in spectroscopy. Finally, the HIPS2017 Workshop [17] aimed at producing an optimized photon source concept with potential increase of scientific output at Jefferson Lab, and at refining the science for hadron physics experiments benefitting from such a high-intensity photon source.
Additionally, the proposed facility will also have a great impact in the strange meson sector by measurements of the final-state system from threshold up to 2 GeV in invariant mass to establish and improve on pole positions and widths of all -wave states and the -wave scalar meson . In particular, the meson has been under discussion for decades and still remains to be unequivocally confirmed with corresponding quantum numbers by doing detailed phase-shift analysis with high statistics data [18]. A detailed study of the system is very important to extract the so-called vector and scalar form factors to be compared with decay and can be used to constrain the Cabibbo-Kobayashi-Maskawa (CKM) matrix element as well as to be used in testing CP violation in decays of heavy and mesons into final states.
The proposal is organized in the following manner. We give an Executive Summary in Sec. 1 and the Scope of the proposal in Sec 2. Then the Brief Case of Hyperon Spectroscopy is given in Sec. 3 while Hyperons in Lattice studies are presented in Sec. 4. An overview of the Interest of the RHICH/LHC community in Hyperon measurements is summarized in Sec. 5. The short overview of previous bubble chamber measurements is given in Sec. 6. Partial-wave phenomenology is considered in Sec. 7 and Theory for the “Neutron" Target in Sec. 8. A short overview for Strange Meson Spectroscopy is given in Sec. 9. Our Proposed measurements are reported in Sec. 10. It describes a Compact Photon Source, production and beam properties, Start Counter Resolution, measurements of flux, and cryotarget description. Running conditions are described in Sec. 11. Sec. 12 contains a Cover Letter for the KLF proposal submission. The Appendixes contain many technical details for our proposal: Analysis of Three-Body Final States in Appendix A1 13, Determination of Pole Positions in Appendix A2 14, Statistics Tools for Spectroscopy of Strange Resonances in Appendix A3 15, Neutron Background in Appendix A4 16, Details of Monte Carlo Study in Appendix A5 17, Current Hadronic Projects in Appendix A6 18, Additional Physics Potential with a Beam Appendix in A7 19, and List of New Equipment and of Changes in Existing Setup Required in Appendix A8 20.
## 3 The Brief Case for Hyperon Spectroscopy
Our present experimental knowledge of the strange hyperon spectrum is deplorably incomplete, despite the fact that the ground states of the strange hyperons have been known since the 1960s. In the case of the hyperon resonance spectrum, only the lowest negative-parity doublet is well established even though the structure of these resonances remains under discussion. In the case of the and hyperons, only the lowest decuplet resonance states and are well understood.
The masses of the lowest positive-parity resonances in the spectrum of the and hyperons, the and are experimentally known, but their structure is not. In the case of the hyperon, the lowest positive-parity resonance remains unobserved.
To settle the nature of the hyperon resonances, their main decay modes have to be determined by experiment. A clear example of how the decay modes can settle the structure of the resonances is provided by the -decay widths of the decuplets , , and . The ratio of these decay widths is 13:4:1, whereas if they were simple three-quark states, with 3, 2, and 1 light quarks each, the ratio should be 9:4:1. A comparison of these ratios indicates that the and appear to be three-quark states, while the is more complex and formed by a three-quark core with a surrounding meson (or multiquark) cloud. This conclusion is well supported by extensive theoretical calculations [19, 20].
### 3.1 The Λ(1405)−Λ(1520)1/2−−3/2− Doublet
In the simplest constituent quark model, the most natural and the oldest interpretation, is that the doublet is a low-lying flavor singlet multiplet of three quarks (uds). Dynamical versions of this model, with two-body interactions between the quarks can describe the low mean energy of this multiplet, but not the 115 MeV splitting between them. This has led to suggestions that there may even be two different 1/2 states one dynamical low resonance at 1405 MeV, and an unresolved higher state close to 1520 MeV [21]. If so, it is high time that the “missing" 1/2 higher-energy state be empirically identified. This problem indicates that the has a more complex multiquark structure. This structure is tested in modern theoretical approached, including contraints from unitarity and chiral symmerty. Confirmed by multiple calculations later on, a two pole structure of was found in Ref. [22]. The narrow pole lies slightly below threshold fixed by the scattering data rather well, see Ref. [23] for the comparison of different modern coupled-channel approaches. However, the position of the second pole is determined less precisely, lying much further below threshold and deeper in the complex plane. Recent photoproduction data on by CLAS [24] can be used to reduce the theoretical ambiguity on this (second) pole of as demonstrated in Ref. [25]. Modern lattice QCD (LQCD) calculations also support the view that its structure is a state [26]. In Skyrme’s topological soliton model for the baryons, the low-lying state also appears naturally as a mainly 5-quark state [27, 28]. That model is consistent with QCD in the large color number () limit.
There are similar low-lying flavor-singlet parity doublets in both the charm and bottom hyperon spectra: and doublets [2]. The ratio between the splittings in these three doublets are 8.2:2.1:1, which is not far from the corresponding inverse ratios of the , , and meson masses: 10.7:2.8:1. The latter is what one should expect from the gradual approach to heavy-quark symmetry with increasing meson (or constituent quark) mass if the quark structure of these three multiplets is similar. This pattern is also consistent with the large N limit of QCD.
### 3.2 The Low-Lying Positive-Parity Resonances
In the spectra of the nucleon and the and hyperons, the lowest positive-parity resonances all lie below the lowest negative-parity multiplets except for the flavor singlet doublet . This reversal of normal ordering cannot be achieved in the constituent quark model with purely color-spin-dependent quark interactions. These low-lying positive-parity resonances are the , , and the 1/2 states. Their low masses do however appear naturally, if the interactions between the quarks are flavor dependent [29].
Present day LQCD calculations have not yet converged on whether these low-lying states can be described as having a mainly three-quark structure [30]. This may reflect that there is a collective nature in the quark content of all these resonances, which have a low soft vibrational mode Skyrme’s topological soliton model for the baryons, which represents one version of the large limit of QCD, describes these low-lying states as such vibrational states.
In the spectrum of the , the may be such a 1/2 state as well, although the quantum numbers of that state are yet to be determined.
In the corresponding decuplet spectra, a similar low-lying positive-parity state has so far only been definitely identified in the spectrum: namely, the . The resonance very likely represents the corresponding positive-parity state. It should be important to identify the corresponding state in the spectrum of the .
It is of course very probable that corresponding low-lying positive-parity states will be found in the spectra of the and hyperons, given the fact that they have low-lying negative-parity states akin to those of the hyperon as described above. The experimental identification of those is an important task. Even if the still tentative resonance turns out to be a 1/2 state, its energy appears to be too high for being the equivalent of the in the charm hyperon spectrum.
In the spectrum of the , the decuplet state is well established. The tentative resonance may, should it turn out to be a 1/2 state, correspond to the in the strange hyperon spectrum.
### 3.3 The Negative-Parity Hyperon Resonances
In the spectrum of the nucleon, two well-separated groups of negative-parity resonances appear above the 1/2 state . In the three-quark model, the symmetry of the lowest energy group is [21][21][21]; i.e., it has mixed flavor (F) and spin (S) symmetry as well as mixed flavor-spin (FS) symmetry [29, 31]. This group consists of the and the resonances. There is a direct correspondence in the and the resonances. There is also a repeat of this group in the spectrum of the hyperon in the two resonances (tentative) and .
These spin and states in the spectum of the nucleon have intriguing decay patterns. The resonance has a large (32-52%) decay branch to , even though its energy lies very close to the threshold. This pattern repeats in the case of the the , which also has a substantial (10-25%) decay branch to the corresponding state, even though it lies even closer to the threshold for that decay. As the still uncertain resonance is located almost exactly at the threshold for , there is naturally no signal for an decay from it. The ratio of the decay widths of the and the is about 6:1, which suggests that the decay might involve a pair of quarks rather than a single constituent quark as in the decay of the decuplet resonances.
In the spectrum of the hyperon, none of the negative-parity multiplets is complete. The state may be the analog in the spectrum of the states , , and . It should be important to identify the lowest resonance in the spectrum. If that resonance lacks an decay branch, it would demonstrate that the decay of the resonances in the spectra of the nucleon, and involves two quarks.
It should also be important to determine whether the uncertain “bumps" referred to in the Particle Data Tables labelled , , and represent true resonances.
About 120 MeV above the pair of nucleon resonances and , the nucleon spectrum has three negative-parity resonances close in energy to one another. This multiplet is formed of the , , and resonances. In the three-quark model the symmetry configuration of these states are [21][21][21]; i.e., their spin configuration is completely symmetric.
The analogs in the spectrum of the of the first and last of these nucleon resonances are the and the resonances. This correspondence remains uncertain, however, because the missing 3/2 state in this resonance multiplet has not yet been identified.
A common feature of all the 1/2 resonances in these multiplets is their substantial decay branches.
Our present knowledge of the spectrum of the hyperons remains too incomplete to identify any member of the negative-parity multiplet with the symmetry structure [21][21][21].
### 3.4 Summary for the Brief Case
This overview shows that the present empirical knowledge of the spectrum of the strange hyperons remains remarkably incomplete. As a consequence, the quark structure of even the lowest-energy resonances remains uncertain. Only an experimental determination of the lowest-energy positive- and negative-parity hyperon resonances and their decay branches would settle the main open issues.
In the spectrum of the hyperon, there remains a question of the existence of a 1/2 partner to the resonance. In addition, it should be important to search for the missing 3/2 resonance near 1700 MeV. Equally important would be the search for the apparently “missing" 3/2 state near 1750 MeV in the spectrum of the hyperon.
Our present knowledge of the spectrum of the hyperons remains too incomplete to identify any member of the corresponding negative-parity multiplet formed of 1/2, 3/2, and 5/2 resonances.
It should also be important to determine, whether the uncertain “bumps" referred to in the Particle Data Tables labelled , , and represent true resonances [2].
## 4 Strange Hadrons from the Lattice
Our knowledge of the excited-state spectrum of QCD through the solution of the theory on a Euclidean space-time lattice has undergone tremendous advances over the past several years. What we characterize as excited states are resonances that are unstable under the strong interaction, and their properties are encapsulated in momentum-dependent scattering amplitudes. The means of calculating such momentum-dependent phase shifts for elastic scattering on a Euclidean lattice at finite volume was provided many years ago [32] and extended to systems in motion [33], but its implementation for QCD remained computationally elusive until recently. A combination of theoretical, algorithmic, and computational advances has changed this situation dramatically, notably in the case of mesons. There have been several lattice calculations of the momentum-dependent phase shift of the mesons [34, 35, 36, 37, 38, 39, 40]. More recently, the formulation to extract amplitude information has been extended to the coupled-channel case[41, 42, 43, 44, 45], and applied to the case of the coupled [46] system describing the resonance, and to the [47, 48].
The application to baryons is far more limited but, nonetheless, important insights have been gained. In an approach in which the excited-state hadrons are treated as stable particles, a spectrum of baryons at least as rich as that of the quark model is revealed [49, 50], and evidence has been presented for “hybrid" baryon states, beyond those of the quark model, in which gluon degrees of freedom are essential [51]. Notably, this picture extends to the spectrum of , and states where the counting of states relects symmetry, and the presence of hybrids is common across the spectrum. As indicated above, these calculations are incomplete in that the momentum-dependent scattering amplitudes remain to be extracted. In Fig. 1, baryon spectra from [52] are presented in units of mass from LQCD calculations with ensemble MeV (not yet at physical ). However, in comparison with the case of mesons cited above, the challenges are more computational than theoretical or conceptual, and the progress made in the meson sector will be reflected for the case of baryons in the coming years.
## 5 The Interest of the RHIC/LHC Community in Excited Hyperon Measurements
The relativistic heavy-ion community at RHIC and the LHC has recently embarked on specific analyses to address the issue of strangeness hadronization. LQCD calculations in the QCD crossover transition region between a deconfined phase of quark and gluons and a hadronic resonance gas have revealed a potentially interesting sub-structure related to the hadronization process. Studies of flavor-dependent susceptibilities, which can be equated to experimental measurements of conserved quantum-number fluctuations, seem to indicate a slight flavor hierarchy in the three-quark sector (u,d,s) in thermalized systems. Specifically, the ratios of higher-order susceptibilities in the strange sector show a higher transition temperature than in the light sector [53]. Both pseudo-critical temperatures are still within the error bars of the quoted transition temperature based on all LQCD order parameters [54, 55], which is 1549 MeV, but the difference of the specific susceptibilities is around 18 MeV and well outside their individual uncertainties.
This difference seems to be confirmed by statistical thermal-model calculations that try to describe the yields of emitted hadrons from a QGP based on a common chemical freeze-out temperature. Although the yields measured by ALICE at the LHC in 2.76 TeV PbPb collisions can be described by a common temperature of 1562 MeV, with a reasonable , the fit improves markedly if one allows the light quark baryons to have a lower temperature than the strange quark baryons [56]. A similar result has been found when the thermal fluctuations of particle yields as measured by STAR Collaboration [57, 58], which can be related to the light quark dominated susceptibilities of the electric charge and the baryon number on the lattice, have been compared to statistical model calculations [59].
If one assumes that strange and light quarks indeed prefer different freeze-out temperatures, then the question arises how this could impact the hadronization mechanism and abundance of specific hadronic species. In other words, is the production of strange particles, in particular excited resonant states, enhanced in a particular temperature range in the crossover region? Strange ground-state particle production shows evidence of enhancement, but the most likely scenario is that the increased strange quark abundance will populate excited states; therefore, the emphasis of any future experimental program trying to understand hadron production is shifting towards strange baryonic resonance production. Furthermore, recent LHC measurements in small systems, down to elementary proton-proton collisions, have revealed that even in these small systems there is evidence for deconfinement, if the achieved energy density, documented by the measured charged particle multiplicity is large enough [60]. Therefore, future measurements of elementary collisions in the K-Long Facility experiment at JLab might well provide the necessary link to future analysis of strange resonance enhancements in heavy-ion collisions at RHIC and the LHC and a deeper understanding of the hadronization process.
An interesting conclusion that arises from these studies is that the improvement in the listing of strange resonances between PDG-2008 [72] and PDG-2016 definitely brought the HRG calculations closer to the LQCD data. By looking at details in the remaining discrepancy, which is in part remedied by including one-star rated resonances in PDG-2016, it seems that the effect is more carried by singly strange resonances rather than multi-strange resonances, also in light of comparisons to quark models that include di-quark structures [73] or enhanced quark interactions in the baryon (hypercentral models [71]). This is good news for the experiments since the and resonances below 2 GeV/ are well within reach of the KLF experiment and, to a lesser significance, the RHIC/LHC experiments. In this context it is also important to point out that the use of both hydrogen and deuterium targets in KLF is crucial since it will enable the measurement of charged and neutral hyperons. A complete spectrum of singly strange hyperon states is necessary to make a solid comparison to first-principle calculations.
In summary: Any comparisons between experimentally verified strange quark-model states from YSTAR and LQCD will shed light on a multitude of interesting questions relating to hadronization in the non-perturbative regime, exotic particle production, the interaction between quarks in baryons and a possible flavor hierarchy in the creation of confined matter.
## 6 Previous Measurements
While a formally complete experiment requires the measurement, at each energy, , and angle, , of at least three independent observables, the current database for and is populated mainly by unpolarized cross sections. Figure 3 illustrates this quite clearly.
The initial studies of the KLF program at GlueX will likely focus on two-body and quasi-two-body processes: elastic and charge-exchange reactions, then two-body reactions producing () hyperons as , , and (). Most of the previous measurements induced by a beam, were collected for MeV and with some data up to MeV. Experiments were performed between 1961 and 1982 with mostly hydrogen bubble chambers at ANL, BNL, CERN, DESY, KEK, LRL, NIMROD, NINA, PPA, and SLAC. Note that some of data were taken at EM facilities at NINA [75] (a short overview about NINA experiments is given by Albrow recently [76]) and SLAC [77]. The goal of the Manchester University group that worked at the Daresbury 5-GeV electron synchrotron NINA was CP-violation, which was a hot topic back to the mid 1960s. The main physics topics that the SLAC group addressed were studies of the systematics for particle/anti-particle processes through the intrinsic properties of the K-longs.
The first paper that discussed the possibility of creating a practical neutral kaon beam at an electron synchrotron through photoproduction was an optimistic prediction for SLAC by Drell and Jacob in 1965 [78]. With significant developments in technology, high-quality EM facilities, such as JLab [13], are now able to realize a complete hyperon spectroscopy program.
The overall systematics of previous p experiments varies between 15% and 35%, and the energy binning is much broader than hyperon widths. The previous number of -induced measurements (2426 , 348 , and 115 observables) [74] was very limited. Additionally, we are not aware of any measurements on a “neutron" target.
Our knowledge about the non-strange sector is more advanced vs. the strange one [2]. For the non-strange case, for instance, phenomenology has access to 51k data of and 39k data of below GeV [79].
## 7 Phenomenology / Partial-Wave Analysis
Here, we summarize some of the physics issues involved with such processes. Following Ref. [80], the differential cross section and polarization for scattering are given by
dσdΩ=λ-2(|f|2+|g|2), (1)
PdσdΩ=2λ-2Im(fg∗), (2)
where , with the magnitude of c.m. momentum for the incoming meson. Here and are the usual spin-nonflip and spin-flip amplitudes at c.m. energy and meson c.m. scattering angle . In terms of partial waves, and can be expanded as
f(W,θ)=∞∑l=0[(l+1)Tl++lTl−]Pl(cosθ), (3)
g(W,θ)=∞∑l=1[Tl+−Tl−]P1l(cosθ), (4)
where is the initial orbital angular momentum, is a Legendre polynomial, and is an associated Legendre function. The total angular momentum for the amplitude is , while that for the amplitude is . For hadronic scattering reactions, we may ignore small CP-violating terms and write
KL=1√2(K0−¯¯¯¯¯¯¯K0), (5)
KS=1√2(K0+¯¯¯¯¯¯¯K0). (6)
We may generally have both and amplitudes for and scattering, so that the amplitudes can be expanded in terms of isospin amplitudes as
Tl±=C0T0l±+C1T1l±, (7)
where are partial-wave amplitudes with isospin and total angular momentum , with the appropriate isospin Clebsch-Gordon coefficients.
We plan to do a coupled-channel partial-wave analysis (PWA) with new GlueX data in combination with available new J-PARC measurements when they will come. Then the best fit will allow determine model-independent (data-driven) partial-wave amplitudes and associated resonance parameters (pole positions, residues, Breit-Wigner (BW) parameters, etc.) as the SAID group does, for instance, for the analysis of -elastic, charge-exchange, and data [81].
### 7.1 Kn and ¯¯¯¯¯KN Final States
The amplitudes for reactions leading to and final states are
T(K−p→K−p) = 12T1(¯¯¯¯¯KN→¯¯¯¯¯KN)+12T0(¯¯¯¯¯KN→¯¯¯¯¯KN), (8) T(K−p→¯¯¯¯¯¯¯K0n) = 12T1(¯¯¯¯¯KN→¯¯¯¯¯KN)−12T0(¯¯¯¯¯KN→¯¯¯¯¯KN), (9) T(K+p→K+p) = T1(KN→KN), (10) T(K+n→K+n) = 12T1(KN→KN)+12T0(KN→KN), (11)
T(KLp→KSp)=12(12T1(KN→KN)+12T0(KN→KN))−12T1(¯¯¯¯¯KN→¯¯¯¯¯KN), (12)
T(KLp→KLp)=12(12T1(KN→KN)+12T0(KN→KN))+12T1(¯¯¯¯¯KN→¯¯¯¯¯KN), (13)
T(KLp→K+n)=1√2(12T1(KN→KN)−12T0(KN→KN)). (14)
No differential cross-section data are available for below MeV. A fair amount of data are available for the reaction, , measured on a deuterium target. Figure 4 shows a sample of available differential cross sections for compared with predictions determined from a recent PWA of data [83, 84], combined with amplitudes from the SAID database [79]. The predictions at lower and higher energies tend to agree less well with the data.
### 7.2 πΛ Final States
The amplitudes for reactions leading to final states are
T(K−p→π0Λ) = 1√2T1(¯¯¯¯¯KN→πΛ), (15) T(KLp→π+Λ) = −1√2T1(¯¯¯¯¯KN→πΛ). (16)
The and amplitudes imply that observables for these reactions measured at the same energy should be the same except for small differences due to the isospin-violating mass differences in the hadrons. No differential cross-section data for are available at c.m. energies MeV, although data for are available at such energies. At 1540 MeV and higher energies, differential cross-section and polarization data for the two reactions are in fair agreement, as shown in Figs. 5 and 6.
### 7.3 πΣ Final States
flavor symmetry allows as many baryon resonances as there are and resonances combined (); however, until now only three states, , , and , have their quantum numbers assigned and only a few more states have been observed [2].
The amplitudes for reactions leading to final states are
T(K−p→π−Σ+) = −12T1(¯¯¯¯¯KN→πΣ)−1√6T0(¯¯¯¯¯KN→πΣ), (17) T(K−p→π+Σ−) = 12T1(¯¯¯¯¯KN→πΣ)−1√6T0(¯¯¯¯¯KN→πΣ), (18) T(K−p→π0Σ0) = 1√6T0(¯¯¯¯¯KN→πΣ), (19) T(K0Lp→π+Σ0) = −12T1(¯¯¯¯¯KN→πΣ), (20) T(K0Lp→π0Σ+) = 12T1(¯¯¯¯¯KN→πΣ). (21)
Figure 7 shows a comparison of differential cross-section data for and reactions leading to final states at MeV (or MeV/). The curves are based on energy-dependent isospin amplitudes from a recent PWA [83, 84]. No differential cross-section data are available for . As this example shows, the quality of the data is comparable to that for the data. It would, therefore, be advantageous to combine the data in a new coupled-channel PWA with available data. Note that the reactions and are isospin selective (only amplitudes are involved) whereas the reactions and are not. New measurements with a beam would lead to a better understanding of states and would help constrain the amplitudes for scattering to final states
### 7.4 KΞ Final States
The amplitudes for reactions leading to final states are
T(K−p→K0Ξ0) = 12T1(¯¯¯¯¯KN→KΞ)+12T0(¯¯¯¯¯KN→KΞ), (22) T(K−p→K+Ξ−) = 12T1(¯¯¯¯¯KN→KΞ)−12T0(¯¯¯¯¯KN→KΞ), (23) T(KLp→K+Ξ0) = −1√2T1(¯¯¯¯¯KN→KΞ). (24)
The threshold for and reactions leading to final states is fairly high ( MeV). In Fig. 8(left), we present the cross section for production using a -beam [85]. There are no differential cross-section data available for and very few (none recent) for or . Measurements for these reactions would be very helpful, especially for comparing with predictions from dynamical coupled-channel (DCC) models [86, 87] and other effective Lagrangian approaches [88]. The Review of Particle Physics [2] lists only two states with branching fractions (BF) to , namely, (BF 3%) and (BF 2%)
### 7.5 KKΩ Final States
The experimental situation with s is even worse than for the case – there are very few data for excited states. The main reason for such a scarce dataset is the very low cross section for their indirect production with pion or photon beams. In Fig. 8(right), we present the cross section for production using a beam [85].
A major effort in LQCD calculations involves the determination of inelastic and multi-hadron scattering amplitudes, and the first calculation to study an inelastic channel was recently performed [89, 90]. For lattice calculations involving baryons that contain one or more strange quarks an advantage is that the number of open decay channels is generally smaller than for baryons comprised only of the light and quarks.
### 7.6 Summary for PWA
Pole positions have been determined (no uncertainties) for several s and s but information about pole positions has not been determined for or hyperons [2]. Our plan is to do a coupled-channel PWA with new GlueX KLF data in combination with available and new J-PARC measurements when they will be available. Then the best fit will allow the determination of data-driven (model independent) partial-wave amplitudes and associated resonance parameters (pole positions, residues, BW parameters, and so on. Additionally, PWAs with new GlueX data will allow a search for “missing" hyperons via looking for new poles in complex plane positions. It will provide a new benchmark for comparisons with QCD-inspired models and LQCD calculations.
## 8 Theory for “Neutron" Target Measurements
So-called coupled-channel chiral unitary approaches (UChPT) successfully describe the properties of the sub-threshold resonance . Furthermore, such models lead to the prediction that the scattering amplitude has two poles in the complex-energy plane for the quantum numbers of this resonance (). This coins the so-called the two-pole structure of the ; see the current Review of Particle Physics [2] for more details.
In the most recent formulation, the aforementioned UChPT approaches rely on a chiral amplitude for the meson-baryon scattering up to next-to-leading chiral order. Whereas the unitarity constraint is usually imposed via the Bethe-Salpeter equation either in the full off-shell formulation [94, 95] or in the so-called on-shell approximation, e.g, [25, 91, 92]. For the analysis of data the former is quite intricate, while as it was shown in Ref. [94] the off-shell effects are rather small. Therefore, it is meaningful to use the latter formulation. Recently, a direct quantitative comparison of the on-shell models [25, 91, 92, 93] was performed in Ref. [23]. It was found there that various models, which typically have many free parameters, adjusted to the same experimental data, predict very different behavior of the scattering amplitude on and off the real-energy axis. This systematic uncertainty becomes evident, when comparing the pole positions of the in these models (see Fig. 9). The position of the narrow (first) pole seems to be constrained at least in the real part rather well, while the predictions for the position of the broad (second) pole cover a very wide region of the complex-energy plane. This uncertainty is present even within models of the same type. This ambiguity can be traced back to the fact that the experimental data used to fix the parameters of the models are rather old and imprecise. It is expected that the proposed KLF experiment will lead to an improvement of this situation, as described below.
The beam can be scattered on a “neutron" target, while measuring the strangeness final meson-baryon states (see, e.g., Sec. 7). In such a setup, the proposed experiment can become a new and very strongly desired new source of experimental data to pinpoint the properties of the scattering amplitude. To make this statement more quantitative, we compare predictions of both solutions of the model111The choice of this model for the present analysis is justified by the fact that it includes the -wave interaction in the kernel of the Bethe-Salpeter equation explicitly. from Ref. [25]. These solutions agree with all presently available scattering, threshold as well as the photoproduction data for the line shapes by the CLAS Collaboration [24]. The predicted differential cross sections () as well as polarized ones () for the scattering with the final states , , , and are presented in Figs. 10 and 11, respectively. There is very little agreement on the prediction of these observables in the energy range aimed to study in the proposed experiment. The latter is very encouraging, meaning that the actual data can sort out one (or maybe both) solutions as unphysical, which was not possible based on present experimental data.
In summary: The proposed KLF experiment will lead to new constraints on models; thus, these data will sharpen our understanding of the long-debated nature of strangeness resonances.
## 9 Strange Meson Spectroscopy: Kπ Interaction
Below we present current status of K-pi scattering summarized in Ref. [47]: “The bulk of our knowledge of kaon scattering amplitudes comes from kaon beam experiments at Stanford Linear Accelerator Center (SLAC) in the 1970s and 1980s. scattering amplitudes were extracted from reactions using a proton target by extrapolating to small momentum transfer, , where nearly on-shell pion exchange dominates. Phaseshift analysis of the flavor-exotic isospin-3/2 amplitudes as extracted from and by Estabrooks et al. [96] indicates a weak repulsive interaction in the S-wave and very weak interactions in the P-wave and higher.
In isospin 1/2, as well as the phase-shift analysis of Estabrooks et al., there is a considerable set of scattering results provided by the LASS experiment-of particular relevance here are the final states [97], [98], and [99]. In the partial-wave analysis of , a peaking amplitude in the S-wave is interpreted as a broad resonance which appears to saturate unitarity. The narrow elastic vector resonance, , presents itself as a rapid rise in the P-wave phase shift. The D-wave amplitude has a peak, well below the unitarity limit, that can be interpreted as an inelastic resonance. Further resonances in the "natural-parity" series () are observed at higher energies.
is the first inelastic channel to open, but LASS reports no significant amplitude into for GeV in S-, P-, and D-waves. Indeed the inelasticity in P- and D-waves and higher appears to come first from the final state, where a significant amplitude is seen in above 1.3 GeV and a peak in at the , also couples to the "unnatural-parity" series, notably to , where peaking behavior is observed that is commonly described in terms of two axial resonances, , ."
Recently LQCD studies with MeV were performed to search for resonances in coupled and scattering [89]. Scalar and form factors have been calculated within a variety of approaches using (unitarized) chiral perturbation theory [100, 101, 102, 103, 104, 105, 106, 107] and dispersion relations [106, 108, 109], in many cases using the former to constrain polynomial ambiguities of the latter.
Measuring scattering provides a possibility for studying scalar and vector states, including the S-wave state (see [110, 111]), which is not yet well established. Such studies are also necessary to get precise vector and scalar form factors as an input for the extraction of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element from decay. scattering amplitudes with high precision are needed to study CP violation from Dalitz plot analyses of both open charm -mesons [112] and the charmless decay of -mesons [113] into final state.
In Fig. 12, we present the phase of the form factor with experimental results of LASS Estabrooks [96, 97] together with the fit of Boito et al. to decay data [114].
As one can see, all experimental data obtained at SLAC have very poor statistics above 1.2 GeV; furthermore, the data do not extend to higher energies, which are even more important for -meson decays. Moreover, direct comparison of charged with assumes isospin invariance as in the decay one has final state depending on the sign of lepton.
Similarly, as one can see from Fig. 13, the and S-wave and P-wave phase shifts are very poorly measured and need more experimental data.
The intensive beam flux of the proposed beam will provide high statistics data on both charged as well as with final-state neutral kaon in the reactions:
• (simultaneousely measurable with beam).
• on a proton target (for the first time).
• on a deuteron target (for the first time).
In summary: Experimental data obtained in the proposed KLF experiment at JLab will provide valuable data to search for yet not well understood and possibly incomplete scalar, vector, and tensor resonances in the strange sector through a phase-shift analysis of and scattering amplitudes.
## 10 Proposed Measurements
We propose to use the KL Facility with the GlueX spectrometer, in JLab Hall D, to perform precision measurements of from liquid hydrogen and deuterium cryotarget (LH/LD) in the resonance region, – 3500 MeV and c.m. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8992666006088257, "perplexity": 1743.0114106816638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897707.23/warc/CC-MAIN-20200708211828-20200709001828-00272.warc.gz"} |
http://export.arxiv.org/abs/2109.04475 | ### Current browse context:
cond-mat.stat-mech
(what is this?)
# Title: Many-Body Quantum Chaos and Space-time Translational Invariance
Abstract: We study the consequences of having translational invariance in space and in time in many-body quantum chaotic systems. We consider an ensemble of random quantum circuits, composed of single-site random unitaries and nearest neighbour couplings, as a minimal model of translational invariant many-body quantum chaotic systems. We evaluate the spectral form factor (SFF) as a sum over many-body Feynman diagrams, which simplifies in the limit of large local Hilbert space dimension $q$. At sufficiently large $t$, diagrams corresponding to rigid translations dominate, reproducing the chaotic behavior of random matrix theory (RMT). At finite $t$, we show that translational invariance introduces additional mechanisms via two novel Feynman diagrams, known as the crossed and deranged diagrams, which delay the emergence of RMT. Our analytics suggests the existence of exact scaling forms which describe the approach to RMT behavior in the scaling limit where both $t$ and $L$ are large while the ratio between $L$ and $L_\mathrm{Th}(t)$, the many-body Thouless length, is fixed. We numerically demonstrate, with simulations of two distinct circuit models, that in such a scaling limit, most microscopic details become unimportant, and the resulting scaling functions are largely universal, remarkably being only dependent on a few global properties of the system like the spatial dimensionality, and the space-time symmetries.
Comments: 8+13 pages, 5+24 figures Subjects: Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph) Cite as: arXiv:2109.04475 [cond-mat.stat-mech] (or arXiv:2109.04475v2 [cond-mat.stat-mech] for this version)
## Submission history
From: Amos Chan [view email]
[v1] Thu, 9 Sep 2021 18:00:00 GMT (571kb,D)
[v2] Mon, 11 Apr 2022 06:08:57 GMT (6376kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8382644057273865, "perplexity": 2105.760633559651}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510117.12/warc/CC-MAIN-20220516104933-20220516134933-00147.warc.gz"} |
http://etna.mcs.kent.edu/volumes/2011-2020/vol40/abstract.php?vol=40&pages=489-506 | ## Vector extrapolation applied to algebraic Riccati equations arising in transport theory
### Abstract
We apply the reduced rank extrapolation method (RRE) to an iterative method for computing the minimal positive solution of a nonsymmetric algebraic Riccati equation that arises in transport theory. The computations yield the minimal positive solution of a vector equation, which is derived from the special form of solutions of the Riccati equation and by exploiting the special structure of the coefficient matrices of the Riccati equation. Numerical experiments and comparisons illustrate the effectiveness of the new approach.
Full Text (PDF) [235 KB]
### Key words
nonsymmetric algebraic Riccati equation, transport theory, minimal positive solution, iterative methods, vector sequences, polynomial vector extrapolation methods, convergence acceleration, reduced rank extrapolation.
### AMS subject classifications
15A24, 65F10, 65B05
< Back | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9594138860702515, "perplexity": 904.0658824487509}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583722261.60/warc/CC-MAIN-20190120143527-20190120165527-00520.warc.gz"} |
http://mathhelpforum.com/calculus/111740-another-comparison-thereom-problem.html | # Math Help - another comparison thereom problem
1. ## another comparison thereom problem
Hi
I am suppose to use the comparison thereom to determine if the following integral is convergent or divergent
integral sign (S) a= e and b= infinity (2010 + cos 2011 x)/ 2009x dx
these numbers are huge and I don't know where to start - especially with a cos in the middle of it.
Help!
Calculus beginner
2. $I=\int_0^\infty \frac{2010+\cos(2011x)}{2009x}dx$
The range for cosine is $[-1,1]$, so the numerator $N\in[2009,2011]$. The entire fraction $f(x)\in[\frac{1.001}x,\frac1x]$.
So, since $f(x)>\frac{1.001}x, I>\int_0^\infty\frac{1.001}{x}dx$. Can you get it from here? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.983018696308136, "perplexity": 1383.8233646454444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398449160.83/warc/CC-MAIN-20151124205409-00139-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/unit-n-box-notation.256837/ | # Unit n - box notation
1. Sep 16, 2008
### javi438
1. The problem statement, all variables and given/known data
can someone please explan what the line segment [-1,0]x{0} means? i don't understand the notation. if someone could explain it to me it would help me a lot!!
thanks!
2. Relevant equations
3. The attempt at a solution
2. Sep 16, 2008
### Dick
It means the set of all ordered pairs (x,0) where -1<=x<=0.
3. Sep 16, 2008
### javi438
so what would [1,0]x[1,0] mean?
the set of (x,y) such that 1<=x<=0 and 1<=y<=0?
does the x in between the two [1,0]'s mean anything?
4. Sep 16, 2008
### Dick
Sure. If you interpret (x,y) as a point in the plane that's a square. The 'x' is called the cartesian product. AxB is the set of all (x,y) such that x is in A and y is in B. (BTW you usually want to write the unit interval as [0,1], not [1,0]).
5. Sep 16, 2008
### javi438
S = {(x,y) : x and y are rational numbers in [0,1]}
the closure and boundary of S = [0,1]x[0,1]
what would it mean in this case, word for word?
6. Sep 16, 2008
### Dick
You know what the sets mean, right? It says that the closure and boundary of the set of all rational points in the filled unit square are equal to ALL points in the filled unit square. (By filled I mean it includes the interior of the square). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8562512397766113, "perplexity": 1019.7086169352774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721555.36/warc/CC-MAIN-20161020183841-00359-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://ayandas.me/blog-tut/2020/01/01/variational-autoencoder.html | Foundation of Variational Autoencoder (VAE)
In the previous article, I started with Directed Probabilitic Graphical Models (PGMs) and a family of algorithms to do efficient approximate inference on them. Inference problems in Directed PGMs with continuous latent variables are intractable in general and require special attention. The family of algorithms, namely Variation Inference (VI), introduced in the last article is a general formulation of approximating the intractable posterior in such models. Variational Autoencoder or famously known as VAE is an algorithm based on the principles on VI and have gained a lots of attention in past few years for being extremely efficient. With few more approximations/assumptions, VAE eshtablished a clean mathematical formulation which have later been extended by researchers and used in numerous applications. In this article, I will explain the intuition as well as mathematical formulation of Variational Autoencoders.
## Variational Inference: A recap
A quick recap would make going forward easier.
Given a Directed PGM with countinuos latent variable $$Z$$ and observed variable $$X$$, the inference problem for $$Z$$ turned out to be intractable because of the form of its posterior
$\mathbb{P}(Z|X) = \frac{\mathbb{P}(X,Z)}{\mathbb{P}(X)} = \frac{\mathbb{P}(X,Z)}{\sum_Z \mathbb{P}(X,Z)}$
To solve this problem, VI defines a parameterized approximation of $$\mathbb{P}(Z\vert X)$$, i.e., $$\mathbb{Q}(Z;\phi)$$ and formulates it as an optimization problem
$\mathbb{Q}^*(Z) = arg\min_{\phi}\ \mathbb{K}\mathbb{L}[\mathbb{Q}(Z;\phi)\ ||\ \mathbb{P}(Z|X)]$
The objective can further be simplified as
$\mathbb{K}\mathbb{L}[\mathbb{Q}(Z;\phi)\ \vert\vert \ \mathbb{P}(Z\vert X)]$ $\let\sb_ = \mathbb{E}_{\mathbb{Q}} [\log \mathbb{Q}(Z;\phi)] - \mathbb{E}\sb{\mathbb{Q}} [\log \mathbb{P}(X, Z)] \triangleq - ELBO(\mathbb{Q})$
$$ELBO(\mathbb{Q})$$ is precisely the objective we maximize. The $$ELBO(\cdot)$$ can best be explained by decomposing it into two factors. One of them takes care of maximizing the expected conditional log-likelihood (of the data given latent) and the other arranges the latent space in a way that it matches a predifined distribution.
$ELBO(\mathbb{Q}) = \mathbb{E}\sb{\mathbb{Q}} [\log \mathbb{P}(X\vert Z)] - \mathbb{K}\mathbb{L}[\mathbb{Q}(Z;\phi)\ ||\ \mathbb{P}(Z)]$
For a detailed explanation, go through the previous article.
## Variational Autoencoder
Variational Autoencoder (VAE) is first proposed in the paper titled “Auto-Encoding Variational Bayes” by D.P.Kingma & Max Welling. The paper proposes two things:
1. A parameterized inference model instead of just $$\mathbb{Q}(Z;\phi)$$
2. The reparameterization trick to achieve efficient training
As we go along, I will try to convey the fact that these are essentially developments on top of the general VI framework we learnt earlier. I will focus on how each of them is related to VI in the following (sub)sections.
### The “Inference Model”
The idea is to replace the generically parameterized $$\mathbb{Q}(Z;\phi)$$ in the VI framework by a data-driven model $$\mathbb{Q}(Z\vert X; \phi)$$, named as Inference model. What does it mean ? It basically means, we are no longer interested in the unconditional distribution on $$Z$$ but instead we want to have a conditional distribution on $$Z$$ given observed data. Please recall our “generative view” of the model
$z^{(i)} \sim \mathbb{P}(Z)$ $x^{(i)} \sim \mathbb{P}(X|Z=z^{(i)})$
With the inference model in hand, we now have an “inference view” as follows
$z^{(i)} \sim \mathbb{P}(Z\vert X=x^{(i)})$
It means, we can do inference just by ancestral sampling after our model is trained. Of course, we don’t know the real $$\mathbb{P}(Z\vert X)$$, so we consider a parameterized approximation $$\mathbb{Q}(Z\vert X; \phi)$$ as I already mentioned.
These two “views”, when combined, forms the basis of Variational Autoencoder (See Fig.1: Subfig.1).
$z^{(i)} \sim \mathbb{P}(Z\vert X=x^{(i)})$ $x^{(i)} \sim \mathbb{P}(X\vert Z=z^{(i)})$
The “combined model” shown above gives us insight about the training process. Please note that the model starts from $$x^{(i)}$$ (a data sample from our dataset) - generates $$z^{(i)}$$ via the Inference model - and then maps it back to $$x^{(i)}$$ again using the Generative model (See Fig.1: Subfig.2). I hope the reader can now guess why its called an Autoencoder ! So, we clearly have a computational advantage here: we can perform training on per-sample basis; just like Inference. This is not true for many of the approximate inference algorithms of pre-VAE era.
So, succinctly, all we have to do is a “forward pass” through the model (yes, the two sampling equations above) and maximize $$\log \mathbb{P}(X=x^{(i)}\vert Z=z^{(i)}; \theta)$$ where $$z^{(i)}$$ is a sample we got from the Inference model. Note that we need to parameterize the generative model as well (with $$\theta$$). In general, we almost always choose $$\mathbb{Q}(\cdot;\phi)$$ and $$\mathbb{P}(\cdot;\theta)$$ as a fully-differentiable functions like Neural Network (See Fig.1: Subfig.3 for a cleaner diagram). Now we go back to our objective function from VI framework. To formalize the training objective for VAE, we just need to replace $$\mathbb{Q}(Z; \phi)$$ by $$\mathbb{Q}(Z\vert X; \phi)$$ in the VI framework (please compare the equations with the recap section)
$\mathbb{Q}^*(Z\color{red}{\vert X}) = arg\min_{\phi}\ \mathbb{K}\mathbb{L}[\mathbb{Q}(Z\color{red}{\vert X};\phi)\ ||\ \mathbb{P}(Z|X)]$
And the objective
$\mathbb{K}\mathbb{L}[\mathbb{Q}(Z\color{red}{\vert X};\phi)\ \vert\vert \ \mathbb{P}(Z\vert X)]$ $\let\sb_ = \mathbb{E}_{\mathbb{Q}} [\log \mathbb{Q}(Z\color{red}{\vert X};\phi)] - \mathbb{E}\sb{\mathbb{Q}} [\log \mathbb{P}(X, Z)] \triangleq - ELBO(\mathbb{Q})$
Then,
$ELBO(\mathbb{Q}) = \mathbb{E}\sb{\mathbb{Q}} [\log \mathbb{P}(X\vert Z; \theta)] - \mathbb{K}\mathbb{L}[\mathbb{Q}(Z\color{red}{\vert X};\phi)\ ||\ \mathbb{P}(Z)]$
As usual, $$\mathbb{P}(Z)$$ is a chosen distribution which we want the structure of $$\mathbb{Q}(Z\vert X; \phi)$$ to be; which is often Standard Gaussian/Normal (i.e., $$\mathbb{P}(Z) = \mathcal{N}(0, I)$$)
$\mathbb{Q}(Z\vert X; [ \phi_1, \phi_2 ]) = \mathcal{N}(Z; \mu (X; \phi_1), \sigma (X; \phi_2))$
The specific parameterization of $$\mathbb{Q}(Z\vert X; \left[ \phi_1, \phi_2 \right])$$ reveals that we predict a distribution in the forward pass just by predicting its parameters.
The first term of $$ELBO(\cdot)$$ is relatively easy, its a loss function that we have used a lot in machine learning - the log-likelihood. Very often it is just the MSE loss between the predicted $$\hat{X}$$ and original data $$X$$. What about the second term ? It turns out that we can have closed-form solution for that. Because I don’t want unnecessary maths to clutter this post, I am just putting the formula for the readers to look at. But, I would highly recommend looking at the proof in Appendix B of the original VAE paper. Its not hard, believe me. So, putting the proper values of $$\mathbb{Q}(Z\vert \cdot)$$ and $$\mathbb{P}(Z)$$ into the KL term, we get
$\mathbb{K}\mathbb{L}\bigl[\mathcal{N}(\mu (X; \phi_1), \sigma (X; \phi_2))\ ||\ \mathcal{N}(0, I)\bigr]$
$= \frac{1}{2} \sum_j \bigl( 1 + \log \sigma_j^2 - \mu_j^2 - \sigma_j^2 \bigr)$
Please note that $$\mu_j, \sigma_j$$ are the individual dimensions of the predicted mean and std vector. We can easily compute this in forward pass and add it to the log-likelihood (first term) to get the full (ELBO) loss.
Okay. Let’s talk about the forward pass in a bit more detail. Believe me, its not as easy as it looks. You may have noticed (Fig.1: Subfig.3) that the forward pass contains a sampling operation (sampling $$z^{(i)}$$ from $$\mathbb{P}(Z\vert X=x^{(i)})$$) which is NOT differentiable. What do we do now ?
### The reparameterization trick
I showed before that in forward pass, we get the $$z^{(i)}$$ by sampling from our parameterized inference model. Now that we know the exact form of the inference model, the sampling will look something like this
$z^{(i)} \sim \mathcal{N}(Z\vert \mu (X; \phi_1), \sigma (X; \phi_2))$
The idea is basically to make this sampling operation differentiable w.r.t $$\mu$$ and $$\sigma$$. In order to do this, we pull a trick like this
$z^{(i)} = \mu^{(i)} + \epsilon^{(i)} * \sigma^{(i)}\text{ , where } \epsilon^{(i)} \sim \mathcal{N}(0, I)$
This is known as the “reparameterization”. We basically rewrite the sampling operation in a way that separates the source of randomness (i.e., $$\epsilon^{(i)}$$) from the deterministic quantities (i.e., $$\mu$$ and $$\sigma$$). This allows the backpropagation algorithm to flow derivatives into $$\mu$$ and $$\sigma$$. However, please note that it is still not differentiable w.r.t $$\epsilon$$ but .. guess what .. we don’t need it ! Just having derivatives w.r.t $$\mu$$ and $$\sigma$$ is enough to flow it backwards and pass it to the parameters of inference model (i.e., $$\phi$$). Fig.2 should make everything clear if not already.
### Wrap up
That’s pretty much it. To wrap up, here is the full forward-backward algorithm for training VAE:
1. Given $$x^{(i)}$$ from the dataset, compute $$\mu(x^{(i)}, \phi_1), \sigma(x^{(i)}, \phi_1)$$.
2. Compute a latent sample as $$z^{(i)} = \mu^{(i)} + \epsilon^{(i)} * \sigma^{(i)}\text{ , where } \epsilon^{(i)} \sim \mathcal{N}(0, I)$$
3. Compute the full loss as $$L = \log \mathbb{P}(x^{(i)}\vert Z = z^{(i)}) + \frac{1}{2} \sum_j \bigl( 1 + \log \sigma_j^2 - \mu_j^2 - \sigma_j^2 \bigr)$$.
4. Update parameters as $$\left\{ \phi, \theta \right\} := \left\{ \phi, \theta \right\} - \eta \frac{\delta L}{\delta \left\{ \phi, \theta \right\}}$$
5. Repeat.
That’s all for this article. Wait for more probabilistic models .. umm, maybe the next one is Normalizing Flow. See you. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9206326007843018, "perplexity": 571.6499930817414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103516990.28/warc/CC-MAIN-20220628111602-20220628141602-00721.warc.gz"} |
http://openstudy.com/updates/4f9c8f65e4b000ae9ed18a66 | Here's the question you clicked on:
55 members online
• 0 viewing
## mmbuckaroos 3 years ago Use the rational roots theorem to form a list of possible rational roots of 3x^7+5x^3-4=0 I have started this problem but am stuck... Delete Cancel Submit
• This Question is Closed
1. satellite73
• 3 years ago
Best Response
You've already chosen the best response.
1
make all possible fractions where the numerator divides 4 and the denominator divides 3
2. satellite73
• 3 years ago
Best Response
You've already chosen the best response.
1
they look like this $\pm1,\pm2,\pm4,\pm\frac{1}{3},\pm\frac{2}{3},\pm\frac{4}{3}$
3. mmbuckaroos
• 3 years ago
Best Response
You've already chosen the best response.
0
hmm ok I was close I had the right idea. thanks how did you decide to use those numbers?
4. satellite73
• 3 years ago
Best Response
You've already chosen the best response.
1
grind it til you find it start with the easy ones like 1 and -1 btw no one says the actual roots have to be any of these just the POSSIBLE RATIONAL ROOTS
5. mmbuckaroos
• 3 years ago
Best Response
You've already chosen the best response.
0
oh ok I see what you did, makes much more sense then my book! Thank you!
6. Not the answer you are looking for?
Search for more explanations.
• Attachments:
Find more explanations on OpenStudy
##### spraguer (Moderator) 5→ View Detailed Profile
23
• Teamwork 19 Teammate
• Problem Solving 19 Hero
• You have blocked this person.
• ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9984010457992554, "perplexity": 4442.863009541104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701158609.98/warc/CC-MAIN-20160205193918-00232-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://science.sciencemag.org/content/170/3961/977 | Reports
Alkalinity and Formation of Zeolites in Saline Alkaline Lakes
See allHide authors and affiliations
Science 27 Nov 1970:
Vol. 170, Issue 3961, pp. 977-980
DOI: 10.1126/science.170.3961.977
Abstract
The solubility of rhyolitic glass increases with increasing alkalinity, whereas the ratio of silicon to aluminum decreases with increasing alkalinity. The strong correlation observed between alkalinity and zeolite mineralogy in saline, alkaline lakes is thought to be a function of this relationship between pH and the Si/Al ratio. It is suggested that this function is a result of the reaction between silicic glass and alkaline solution whereby (i) a gel forms, whose Si/Al ratio is controlled by the Si/Al ratio of the solution, and (ii) a zeolite forms from the gel, whose Si/Al ratio is, in turn, controlled by the composition of the gel. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9772307872772217, "perplexity": 4689.989549536381}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202698.22/warc/CC-MAIN-20190322220357-20190323002357-00453.warc.gz"} |
https://physics.stackexchange.com/questions/185789/reflectivity-of-metals-real-and-imaginary-part-of-k-vector-and-complex-dielect | # Reflectivity of Metals - real and imaginary part of k-vector and complex dielectric function
De electric field component of an electromagnetic wave that is traveling in the $x$-direction is given as $$E(x,t) = E_0 e^{i(kx - \omega t)}$$ with $E_0$ the amplitude and $k$ the wave vector.
From what I understand, this wave vector, since it is not the same as the wave number, has both a real and an imaginary part.
The imaginary part is responsible for the damping of the wave and dissipation of energy? And what does the real part mean physically? That, if it is large enough, the wave gets entirely transmitted by the medium, and nothing is reflected?
Also, I was wondering what's exactly the physical meaning of the complex dielectric function for metals. For metals it is given as : $$\epsilon_r (\omega) = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}$$ with $\omega_p$ the plasmafrequency and $\gamma$ a damping factor.
I know a dielectric constant describes the behaviour of some material in the presence of an electric field, but why do we need a function here?
• The dielectric "constant" is always a function. It depends at least on material, temperature, frequency and pressure. – Sebastian Riese May 24 '15 at 22:28
• I've made an addition to my post, don't know if it helps you. – Constantine Black May 25 '15 at 16:08
First, note that the way you have written the electrical wave-field isn't anything more than exactly a way to write a wave function in general. This is because the term $e^{i(kx-ωt)}$ can be written as: $$e^{i(kx-ωt)} = cos(kx-ωt) +i sin(kx-ωt)$$, and from here you can keep in general the real or the imaginary part as you wish.
As for $κ$, the imaginary part is a term that expresses absorption(the amplitude decreases as we get inside the material).That is: $$E(x,t)=E_0e^{-k_{im} z}e^{i(kx-ωt)}$$ , with $$k_{im} =ω \sqrt{ {εμ \over 2}}[\sqrt{1+ {σ \over εω}^2}-1]^{1/2}$$. The distance at which the amplitude decreases by a factor of $1/e$ is defined as $$d={1 \over k_{im}}$$ Note that $σ$ is the special electrical conductivity.
The electric susceptibility $χ_e$ of a dielectric material is a measure of how easily it polarizes in response to an electric field. This, in turn, determines the electric permittivity of the material and thus influences many other phenomena in that medium, from the capacitance of capacitors to the speed of light.
It is defined as the constant of proportionality (which may be a tensor) relating an electric field E to the induced dielectric polarization density P such that: $$P=ε_0 χ_ε Ε$$
where $ε_0$ is the electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity $ε_0$ by $$χ_ε = ε_r -1$$.
Note, as commented at your question by Sebastian Riese, that in general $ε_r$ is a function of the properties of the material.
Hope this helps with your question.
EDIT
As for reflectivity, it depends on the angle of incident on the surfaco of the material and properties as $ε and μ$. You can prove, as the most simple example that that the Reflection and Transmission factors are: $$R={n_1 -n_2 \over n_1 +n_2}^2$$ $$T={4n_1 n_2 \over (n_1 +n_2)^2}$$ where n is the index of refraction defined as: $$n=\sqrt{εμ \over ε_0 μ_0}$$ and it is also: $$k=k_{re} +i k_{im}$$ with: $$k^2 =μεω^2 +ι μσω$$, so there is a connection between $k$ and the reflection factor.
That's all. Hope I helped. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9391697645187378, "perplexity": 215.12870988327998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986670928.29/warc/CC-MAIN-20191016213112-20191017000612-00107.warc.gz"} |
https://www.physicsforums.com/threads/a-quesiton-on-derivatives.543428/ | A Quesiton on Derivatives
1. Oct 23, 2011
stonecoldgen
1. The problem statement, all variables and given/known data
Find a point P on y=x3 such that the tangent at P intersects the original curve again at point Q so that the slope of the tangent at Q is 4 times the slope of the tangent at P.
2. Relevant equations
y'=3x2
and the slope of the what I think it should be a secant line=(y2-y1)/(x2-x1)
3. The attempt at a solution
I figured that 2P=Q because 3P2=(3/4)Q2 and then by using algebra I expressed Q in terms of P.
I know this leads to somewhere, but im not sure what should I do next.
2. Oct 23, 2011
LCKurtz
But P and Q are points. So P2 and Q2 don't make any sense. Try writing P = (a,a3) and Q= (c,c3) so you have two unknowns a and c. Write the equation of the tangent line at P and use the two facts that it must intersect Q and the slope condition you are given.
Similar Discussions: A Quesiton on Derivatives | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.896787703037262, "perplexity": 426.996636277959}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814101.27/warc/CC-MAIN-20180222101209-20180222121209-00462.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=3866036 | Physical space properties questions
by neginf
Tags: physical, properties, space
P: 56 Is it known if space: 1. is "grainy" or smooth ?, 2. has singular points ?, 3. is like R^3 ?
P: 3 the physical space i think is amazing, you know its exist but sometimes it is so abstract!
P: 1,261
Remember that space is an element/subset of space-time.
Quote by neginf 1. is "grainy" or smooth ?
For all intents and purposes, space-time is very smooth. On the smallest scales, we don't know; its possible that space-time is riddled with vacuum fluctuations just like quantum fields. We might need a quantum theory of gravity to find out.
Quote by neginf 2. has singular points ?
General relativity (GR) says that space-time does have singular points, but most people believe thats just a sign of GR's incompleteness---and once we have a good quantum theory of gravity, those singularities will be smoothed out.
Quote by neginf 3. is like R^3 ?
I'm not sure what you mean here. If you mean, is space euclidean---then the answer is 'asymptotically yes', but locally no---thats why we need general relativity.
If you're asking if it has three dimensions, then---at least macroscopically---yes. But there may be 'hidden' microscopic dimensions.
P: 56 Physical space properties questions Thank you both for the replies. Sorry question 3 wasn't specific enough. Could physical space have a different topology than the usual R^3 ?
P: 2,193
Quote by neginf Could physical space have a different topology than the usual R^3 ?
Yes, the simplest of them being a 3-torus.
P: 555
Quote by aimilvping the physical space i think is amazing, you know its exist but sometimes it is so abstract!
Yeah; well, apparently its invisible, transparent to light, and even weightless ! And even though it is very hard to grasp, I have managed to get some excellent samples of deep space for further experimentation.
I found space to be very stiff. Nevertheless I was able to make massive objects pass right through it unhindered ! and was even able to make it appear to bend in a graviational field ! ;))
So you can have hours of fun and experimentation....we are offering these excellent untouched samples of space for the unheard of low low price of \$12.95 per cubic centimeter...(plus s & h).
Call 1-800-vacuum; hurry before the supply runs out. :)
Creator :))
P: 1,261
Quote by Nabeshin Yes, the simplest of them being a 3-torus.
How does this interplay with measurements of flatness? The universe is flat to some high percentage, so does that place limits on the curvature of such a torus (i.e. analogous to the toroidal radius of a 2-torus)? Then, if such limits were placed, would that provide limits on the size of the universe? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8409996032714844, "perplexity": 1821.0719032161455}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921318.10/warc/CC-MAIN-20140909044836-00190-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://convexoptimization.com/wikimization/index.php?title=Moreau%27s_decomposition_theorem&oldid=1277 | Moreau's decomposition theorem
Characterization of the projection
Let $LaTeX: (\mathcal H,\langle\cdot,\cdot\rangle)$ be a Hilbert space, $LaTeX: \mathcal C$ a closed convex set in $LaTeX: \mathcal H,\,u\in\mathcal H$ and $LaTeX: v\in\mathcal C$. Then, $LaTeX: v=P_{\mathcal C}u$ if and only if $LaTeX: \langle u-v,w-v\rangle\leq0$ for all $LaTeX: w\in\mathcal C$.
Proof
Suppose that $LaTeX: v=P_{\mathcal C}u$ and let $LaTeX: w\in\mathcal C$ be arbitrary. By using the convexity of $LaTeX: \mathcal C$, it follows that $LaTeX: (1-t)v+tw\in\mathcal C$, for all $LaTeX: t\in (0,1)$. Then, by using the definition of the projection, we have
$LaTeX: \|u-v\|^2\leq\|u-[(1-t)v+tw]\|^2=\|u-v-t(w-v)\|^2=\|u-v\|^2-2t\langle u-v,w-v\rangle+t^2\|w-v\|^2$.
Hence,
$LaTeX: \langle u-v,w-v\rangle\leq\frac t2\|w-v\|^2.$
By tending with $LaTeX: t$ to $LaTeX: 0$, we get $LaTeX: \langle u-v,w-v\rangle\leq0$.
Conversely, suppose that $LaTeX: \langle u-v,w-v\rangle\leq0,$ for all $LaTeX: w\in\mathcal C$. Then,
$LaTeX: \|u-w\|^2=\|u-v-(w-v)\|^2=\|u-v\|^2-2\langle u-v,w-v\rangle+\|w-v\|^2\geq \|u-v\|^2,$
for all $LaTeX: w\in\mathcal C$. Hence, by using the definition of the projection, we get $LaTeX: v=P_{\mathcal C}u$.
Moreau's theorem
Moreau's theorem is a fundamental result characterizing projections onto closed convex cones in Hilbert spaces.
Let $LaTeX: \mathcal K$ be a closed convex cone in the Hilbert space $LaTeX: (\mathcal H,\langle\cdot,\cdot\rangle)$ and $LaTeX: \mathcal K^\circ$ its polar. For $LaTeX: x,y,z\in\mathcal H$ the following statements are equivalent:
1. $LaTeX: z=x+y,\,x\in\mathcal K,\,y\in\mathcal K^\circ$ and $LaTeX: \langle x,y\rangle=0$
2. $LaTeX: x=P_{\mathcal K}z$ and $LaTeX: y=P_{\mathcal K^\circ}z$
Proof of Moreau's theorem
• 1$LaTeX: \Rightarrow$2: For all $LaTeX: p\in K$ we have
$LaTeX: \langle z-x,p-x\rangle=\langle y,p-x\rangle=\langle y,p\rangle\leq0$.
Then, by the characterization of the projection, it follows that $LaTeX: x=P_{\mathcal K}z$. Similarly, for all $LaTeX: q\in K^\circ$ we have
$LaTeX: \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0$
and thus $LaTeX: y=P_{\mathcal K^\circ}z$.
• 2$LaTeX: \Rightarrow$1: Let $LaTeX: x=P_{\mathcal K}z$. By the characterization of the projection we have $LaTeX: \langle z-x,p-x\rangle\leq0,$ for all $LaTeX: p\in\mathcal K$. In particular, if $LaTeX: p=0,$ then $LaTeX: \langle z-x,x\rangle\geq0$ and if $LaTeX: p=2x,$ then $LaTeX: \langle z-x,x\rangle\leq0$. Thus, $LaTeX: \langle z-x,x\rangle=0$. Denote $LaTeX: y=z-x$. Then, $LaTeX: \langle x,y\rangle=0$. It remained to show that $LaTeX: y=P_{\mathcal K^\circ}z$. First, we prove that $LaTeX: y\in\mathcal K^\circ$. For this we have to show that $LaTeX: \langle y,p\rangle\leq0$, for all $LaTeX: p\in\mathcal K$. By using the characterization of the projection, we have
$LaTeX: \langle y,p\rangle=\langle y,p-x\rangle=\langle z-x,p-x\rangle\leq0,$
for all $LaTeX: p\in\mathcal K$. Thus, $LaTeX: y\in\mathcal K^\circ$. We also have
$LaTeX: \langle z-y,q-y\rangle=\langle x,q-y\rangle=\langle x,q\rangle\leq0,$
for all $LaTeX: q\in K^\circ$, because $LaTeX: x\in K$. By using again the characterization of the projection, it follows that $LaTeX: y=P_{\mathcal K^\circ}z$.
References
• J. J. Moreau, Décomposition orthogonale d'un espace hilbertien selon deux cones mutuellement polaires, C. R. Acad. Sci., volume 255, pages 238–240, 1962. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 59, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9942528605461121, "perplexity": 103.80513337004948}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711126.30/warc/CC-MAIN-20221207021130-20221207051130-00801.warc.gz"} |
http://rfcwalters.blogspot.com/2014/05/the-algebra-of-processes-viii.html | Thursday, May 15, 2014
The algebra of processes VIII - the distributive laws
I want to first say something about the abstract setting. In the arXiv paper (arXiv:0909.4136 ) we considered a more complicated notion of process (mentioned in the second post of this series) with nine graphs, and we described many operations. We now believe that this definition was too general, and instead a process should consist of five graphs A,B,X,Y, G and four morphisms δ0: G → A, δ1: G → B, γ0: X → G, γ1: Y → G as we have been discussing in these posts.
What I will describe is essentially what we have written in the paper A Formalization of the IWIM Model, Coordination 2000, SLN in CS. That paper was however written for a computer science conference and was not sufficiently abstract, and was without data types.
I have said that such a thing is an arrow in Cospan(Graph/AxB) and the operations there are composition (denoted G;H)and sum (denoted G+H).
But such a thing also belongs to another category. Let T be the three object category which is a model of a cospan (that is, has a central object and two arrows one from each of the other objects into the central object). The a process is also and arrow in Span(Graph T) in the following way:
The central arrows are the deltas and gammas; the lefthand side and the right hand side consist of identities. The whole diagram is a span of cospans, that is, an arrow in Span(Graph T).
The composition in this category will be denoted G•H and the tensor G⊗H.
Now the parallel operations distribute over the sequential ones in a particular way. I describe one:
Consider the processes G with left parallel interface A, right parallel interface B, and processes H,K with left parallel interface B, right parallel interface C. Let G=G0;G1 be the tabulation of G considered as a cospan (of spans) (that is, G0 is the half of G pointing backward; G1 is the part pointing forward).
Then G•(H;K)=(G0 •H);(G1•K).
I will try to say something of the meaning of this law in the next post.
Suffice it to say here that this law (and the other similar ones) allow the flattening of a hierarchical system.
See the Coordination paper above for an example.
Another paper with the span-cospan algebra in a probabilistic context is:
Luisa de Francesco Albasini, Nicoletta Sabadini, Robert F. C. Walters: The compositional construction of Markov processes II. RAIRO - Theor. Inf. and Applic. 45(1): 117-142 (2011)
An early version available at arXiv:1005.0949v1.
The operations corresponding to the ones we have discussed are called parallel, parallel-with-communication, local sum, and local sequential.
Labels: , | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9022655487060547, "perplexity": 1906.2845786468808}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247484772.43/warc/CC-MAIN-20190218074121-20190218100121-00221.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=sign&L=F | sign - Maple Programming Help
sign
sign of a number or a polynomial
Calling Sequence sign(expr, [x1, x2, ...], 'y')
Parameters
expr - multivariate polynomial [x1, ...] - (optional) list of indeterminates y - (optional) unevaluated name
Description
• The sign function computes the sign of the leading coefficient of expr. The sign function works for polynomials with coefficients of type numeric. It does not take assumptions into account.
• The leading coefficient of expr is determined with respect to the indeterminates given. If none are given, the leading coefficient is taken with respect to all its indeterminates. Note therefore that the leading coefficient is dependent on the order of the indeterminates which may vary from one Maple session to another, but not within a session.
• The unevaluated name specified as the optional third argument is assigned the leading term.
• The sign command is thread-safe as of Maple 15 .
Examples
> $\mathrm{sign}\left(0\right)$
${1}$ (1)
> $\mathrm{sign}\left(-\frac{2}{3}\right)$
${-1}$ (2)
> $\mathrm{expr}≔3{x}^{2}{y}^{4}-2x{y}^{5}+x$
${\mathrm{expr}}{≔}{3}{}{{x}}^{{2}}{}{{y}}^{{4}}{-}{2}{}{x}{}{{y}}^{{5}}{+}{x}$ (3)
> $\mathrm{indets}\left(\mathrm{expr}\right)$
$\left\{{x}{,}{y}\right\}$ (4)
> $\mathrm{sign}\left(\mathrm{expr}\right)$
${1}$ (5)
> $\mathrm{sign}\left(\mathrm{expr},\left[x,y\right]\right)$
${1}$ (6)
> $\mathrm{sign}\left(\mathrm{expr},\left[y,x\right]\right)$
${-1}$ (7)
> $\mathrm{sign}\left(\mathrm{expr},\left[y,x\right],'a'\right)$
${-1}$ (8)
> $a$
${x}{}{{y}}^{{5}}$ (9)
You can also plot the sign function. A first attempt:
> $\mathrm{plot}\left(\mathrm{sign}\left(x\right),x=-1..1\right)$
Notice that the plot results in the line $y=1$. This occurs because it computes the constant sign(x)=1 and plots that. To get the expected plot, enclose sign(x) in right-single quotes. Since the function has a discontinuity, include the option discont to get a better plot.
> $\mathrm{plot}\left('\mathrm{sign}\left(x\right)',x=-1..1,\mathrm{discont}=\mathrm{true}\right)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 21, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833399653434753, "perplexity": 1599.358448771854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400221980.49/warc/CC-MAIN-20200925021647-20200925051647-00015.warc.gz"} |
https://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/ | Possible fatal flaws in the finite model part of Deolalikar’s proof
Neil Immerman is one of the world’s experts on Finite Model Theory. He used insights from this area to co-discover the great result that ${\mathsf{NLOG}}$ is closed under complement.
Today I had planned not to discuss the proof, but I just received a note from Neil on Vinay Deolalikar “proof” that P${\neq}$NP. Neil points out two flaws in the finite model part that sound extremely damaging to me. He has already shared them with Vinay, and suggested that I highlight them here. The comments from Neil are in the next section—I have only edited it slightly to make it “compile.”
Two Flaws?
Dear Vinay Deolalikar,
Thank you very much for sharing your paper with me. I find your approach and your ideas fascinating, but I am afraid that there is currently a serious hole in your paper as I now describe. For page numbers, I refer to the 102 page, 12 pt. version of your paper.
Your main idea for the lower bound is to show that FO(LFP) is not powerful enough to express SAT, by using Hanf-Gaifman locality to limit the connectivity of the graphs you consider at successive stages of the fixed point computation. As you point out, if a total ordering is given as part of the input structure, then the Gaifman graph has diameter one, so locality is meaningless. Thus you restrict to a successor relation and as you point out, it is still true that FO(LFP) is equal to P in the presence of a successor relation. However, you make two assertions that are not true.
You use the relation ${R_E}$ as your successor relation. On page 67 you write, “The reason for the relation ${R_E}$ that creates the chain is that on such structures, polynomial time queries are captured by FO(LFP) [EF06, S11.2].” This is a technicality. Recall that an order on the structure enables the LFP computation (or the Turing machine the runs this computation) to represent tuples in a lexicographical ordering. In our problem ${k}$-SAT, it plays no further role. Specifically, the assignments to the variables that are computed by the LFP have nothing to do with their order. They depend only on the relation ${R_C}$ which encodes the clauses and the relation ${R_P}$ that holds the initial partial assignment that we are going to ask the LFP to extend. In other words, each stage of the LFP is order invariant. It is known that the class of order invariant queries is also Gaifman local [GS00].
Unfortunately, it is not true that each stage of the fixed point must be order invariant. In particular, consider the definition of ordering from successor, easily defined by a least fixed point and thus the reason that successor suffices to capture P. The ordering is defined by taking the transitive closure of the successor relation. At each stage, the successor distance is doubled, so in ${\log n}$ stages we have the whole ordering. Note that all these stages contain the order dependent information that is part of the original successor relation. It is true that the final aim of your computation is the SAT property which is order independent. But that definitely does not imply that each stage of the induction is order independent.
The other problem is that you restrict your attention to monadic fixed points. You write,
“Remark 7.4. The relation being constructed is monadic, and so it does not introduce new edges into the Gaifman graph at each stage of the LFP computation. When we compute a ${k}$-ary LFP, we can encode it into a monadic LFP over a polynomially (${n^k}$) larger product space, as is done in the canonical structure, for instance, but with the linear order replaced by a weaker successor type relation. Therefore, we can always chose to deal with monadic LFP. This is really a restatement of the transitivity principle for inductive definitions that says that if one can write an inductive definition in terms of other inductively defined relations over a structure, then one can write it directly in terms of the original relations that existed in the structure [Mos74, p. 16].”
It is not the case that you can freely assume that your fixed points are monadic. If you actually work on the canonical structure, then you require the multiple arity relations that can take us from a tuple to its individual elements and back again. These would again make the diameter of the whole graph bounded. However, in your proof you do not include these relations. Thus, your restriction to only have successor and to restrict to monadic fixed points is fundamental. In this domain—only monadic fixed points and successor—FO(LFP) does not express all of P!
Currently, as I see it, the strongest tool we have in descriptive complexity — rather than the locality theorems — is the Håstad Switching Lemma. Paul Beame and Johan Håstad used this to shown that Mike Sipser’s hierarchy theorem extends all the way to FO[${\log n/\log \log n}$]. As you know, FO(LFP) is the same thing as FO[${n^{O(1)}}$]—properties expressible by the polynomial iteration of a fixed block of restricted quantifiers. We know that ${\mathsf{NC^1}}$ is contained in FO[${\log n/\log \log n}$] and this is tight. Furthermore, L and NL are contained in ${AC^1 = FO[\log n]}$, and it remains open whether ${NC^1}$ is equal to NP. A state of the art paper that I very much recommend is Ben Rossman’s result that expressing the existence of a ${k}$ clique requires ${k/4}$ variables in FO, and even in FO[${c\log n/\log \log n}$] for appropriate ${c}$. (In Rossman’s result, as in all results that use the switching lemma, the lower bound is true in the presence not just of order, but of any numeric relations including addition and multiplication.)
—Neil
Open Problems
Is Neil right? It seems that the most damaging statement of Neil is:
Thus, your restriction to only have successor and to restrict to monadic fixed points is fundamental. In this domain—only monadic fixed points and successor—FO(LFP) does not express all of P!
Most of us have focused on the issues of 2-SAT and XOR-SAT compared to ${k}$-SAT, but Neil’s comment points out serious flaws in the finite model part. It was the connection of this work with the random structure of ${k}$-SAT that excited many of us in the first place. Unless Neil is wrong, it seems that his points are very serious.
328 Comments leave one →
1. Milind Bandekar "Milz" permalink
August 12, 2010 11:24 pm
Here’s my 2 cents (again, first post, sorry!):
I’ve previously worked for HP Bangalore as a software engineer. HP kind of had 3 divisions in Bangalore: Labs, Software and Consulting/ Outsourcing.
The division I worked for, Software had quite an emphasis on innovation. We had the option of research disclosures, or patents. Some of the walls in our building were covered with framed certificates of innovations made by local employees. I found the environment to be tremendously empowering for innovation, especially my boss was really encouraging!
To extend this, I can only imagine what the environment would be like at HP LABS in the US, where Vinay hails from! So it would be for HP Labs Bangalore!
I have graduated with an M.A. in CS at Wayne State before.
But I’ve never felt as empowered to publish anything, even though I think learnt an irreplaceable lot there!
So in fact, I published my first paper, a research disclosure while at HP. And, I was brimming with so much confidence that I published a book AFTER I left HP.
http://amzn.com/1452828687
Even though Vinay’s (and possibly mine) attempts may turn out to be duds, I would encourage you to keep an open mind. Don’t hang us for trying!
The state of affairs is reflected in the fact that, even though there are a ton of brilliant people in US schools, somehow they do not seem to have the confidence or initiative to do great stuff. Maybe its the government funded central research that’s killing innovation. Everything is mired in “String theory” to please the NSF and NIH stuffed suits.
While I was studying in the US, I just thought of my school’s low ranking and that I could possibly never do anything considering my low perch! But I now realize that even people sitting high up in US universities are too strait-jacketed to do anything. No wonder, an outsider, Grigoriy Perelman could develop on Richard Hamilton’s Ricci flow to confidently give his solution. On arXiv.
In the end, I’m a strong believer in the future of corporate innovation.
Go HP!
Milz
• August 13, 2010 5:07 am
I’m not quite sure where you get these strange ideas about research and “initiative”. As far as I can tell, most of the tools and techniques that Deolalikar used in his attempt came from the “strait-jacketed” people in universities. And where in the world did you get the idea that the NIH funds anything with “String theory”? Or that the majority of NSF grants involve “String theory”? Or that even the majority of physics grants involve it?
And if you think that Perelman came from “outside”, you’ve been reading the tabloids too much. You should take the time to actually learn who Perelman is and where he came from.
• August 13, 2010 5:49 pm
He’s right. If you need education on the prevalence of string theorists as a political (rather than academic) bloc in universities, ISBN-13: 9780618551057. There’s a narrow line between visionaries and devotees.
• August 18, 2010 6:44 am
Dear SN,
Thanks for your support!
Cheers – Milz
• Phil Miller permalink
August 13, 2010 8:49 am
Perhaps you’ve misunderstood the character of a response like this. A leading researcher taking the time to read a paper and identify the weaknesses in it is a mark of great esteem, and is generally very encouraging. Most papers that are this bold are simply ignored as “less than serious attempts” or even “cranks”.
• August 14, 2010 4:41 am
Milind,
The above review is not about/against HP research, Corporate Researchers, or Indians. It is about scientists working together to make sure that Mathematics and Science are heading in the correct direction on a solid foundation.
>>”Even though Vinay’s (and possibly mine) attempts may turn out to be duds, I would encourage you to keep an open mind. Don’t hang us for trying!”
Please do not put yourself in the same boat as Vinay – this discussion is not about rest of mathematical community vs HP researchers.
Also, your comments seem to be out of context, so it makes me wonder if you are just trying to market your book.
Lastly, the complaint about brilliant minds in the US not having confidence / initiative is plain wrong (and so does the funding part). Either you had a unfortunately bad sub sample of peers during your university days, or seem to be generally discontent with the whole US higher education system. Either way, the view is skewed and is not supported by facts.
I would advice you to look at the scientific community as a valuable ally in refining your research work and not as an obstacle to overcome. For an excellent example, follow the history of the “Primes in P” by AKS.
August 14, 2010 8:06 am
Dear Milz,
I don’t know where you get this idea that all the elite academics are focused on string theory, or that they are too straight jacketed to do great things. Gregory Perlman too was a postdoctoral researcher at top American Universities- NYU, Berkeley and Stony Brook and had developed a very good reputation as a serious scholar. Had he not developed such a reputation, no one would have taken his papers seriously. So please remember, that half knowledge is dangerous. In fact it seems like this whole incident may highlight this fact more than anything else.
August 16, 2010 4:20 am
Milz,
My understanding, perceptions, and experiences do not agree with your comments even in the slightest manner. Intellectual debate is an essential part of advancing human knowledge in any culture and is not limited to the scientific culture. Although, many-a-times, some people might have had historical distastes with intellectual debate, we need to realize that it can only benefit humanity in the long run especially when intelligence is modulated with the human spirit for collaboration and for understanding people and things. In fact, the lack of it may be contributing to the current ills of modern society.
So many people from academia and industry are devoting their precious time and energy to his paper. Some of them have been supported by bodies such as NSF, which you mention. Effort is what is important and Vinay’s efforts have been rewarded already. I hope that everybody’s effort in this forum and outside are also rewarded by this paper. This is synergy, is it not?
PS: I am with HPL and I feel that this is the spirit that is the foundation of HP. http://www.hpalumni.org/hp_way.htm
Best regards,
Kapali
• August 18, 2010 7:06 am
Dear Kapali,
Thanks for reminding me about the HP alumni network! I forgot about it, but not about HP – its’ employees keep making the news for funny reasons – Carly Fiorina (Sarah Palin), Mark Hurd (Bill Clinton) and now Vinay Deolalikar (as?!) … :D
Maybe that’s the real HP Way it shouldn’t stop ;) The chaos was great fun!
Cheers – Milz
August 12, 2010 11:45 pm
Hi Dick
The wiki contains two very similar points raised by Steven Lindell and Albert Atserias (Atserias recently expanded on his earlier comments, and made essentially the same comments that Neil makes in the part you refer to as the ‘fatal flaw’.
It seems therefore that independently, three FMT researchers have isolated the same problem in the LFP portion of the argument, and strengthens the weight of this objection.
• August 13, 2010 12:43 am
Neil’s point about order being needed at intermediate stages of an induction seems to me to be new. I read the comments by Atserias and scrolled thru every use of “order” on the wiki and Lindell’s Critique. The latter touches on this at the end, but perhaps in a different vein—I didn’t get Neil’s point from it.
Is it known exactly what “FO(LFP) in the domain of only monadic fixed points and successor” captures? Anything more than AC^0 or some sub-log(n)/loglog(n) extension of it? The closest paper I’ve found in a brief look is this by Nicole Schweikardt in 2005, but the systems there seem to be richer.
• August 13, 2010 1:48 am
Following up my query, maybe good to place on the wiki an entry listing classes C that are already known to be properly contained in NP?
• August 13, 2010 3:12 am
That Deolalikar assumes that each stage of the induction is order-invariant is a point that I raised on this blog (see my posting of 11 August, 12.07pm).
And while it is clear, as “FO(LFP) in the domain of only monadic fixed points and successor” is to weak to capture P, when you also add the restriction that every stage of the induction is order-invariant, then it is weaker still. It can still express 2-SAT, but not XOR-SAT, as I’ve pointed out, and not even parity.
• August 13, 2010 10:21 am
Anuj, yes indeed, thanks! Here’s the link to what you wrote. I didn’t do a similar search-pass on the comemnts here, just the wiki—which this may help document as it’s (pre-)confirmation of Neil’s point.
3. August 13, 2010 12:39 am
期待 Vinay Deolalikar的回复 新的博弈开始
August 13, 2010 1:06 am
Every time I read about a potential flaw in the paper, it reminds me of Russel Impagliazzo’s comments on the lack of a tradional lemma->lemma-> theorem style structure in the paper. Perhaps that’d unravel some of the issues to the author himself in the first place.
5. Robert Solovay permalink
August 13, 2010 1:07 am
I posted the text of Professor Immerman’s letter to the Wiki. (I also included a pointer on the main page). The URL for this Wiki page is:
http://michaelnielsen.org/polymath1/index.php?title=Immerman%27s_letter
OR
http://bit.ly/aIcWBe
August 13, 2010 1:34 am
My understanding of finite model theory is minimal, but the following highlighted comment of Neil raised a question (probably silly) in my mind–
“Thus, your restriction to only have successor and to restrict to monadic fixed points is fundamental. In this domain—only monadic fixed points and successor—FO(LFP) does not express all of P! ”
The question is what exactly does this restriction capture? Any ideas.
7. August 13, 2010 2:04 am
So if Neil is right (I do not know enough to judge that) then Deolalikar does not cover ALL P algorithms. Is it possible that simply Gaussian elimination does not belong to the class he covers? That would resolve the XOR-SAT issue.
So maybe Deolalikar’s work is not a proof for P!=NP, but maybe it is a proof for something like: No polynomial algorithms of type X will solve random k-SAT if the space of solutions had property A. If X covers more than say resolution type of algorithms then such result would still be really interesting!
• August 13, 2010 6:38 am
Precisely matches my first comment on first thread. After all, asking how it proves that answers to SAT are not encoded starting from $10^{10^{10}}$ digit of $\pi$ was a good sanity check idea.
8. August 13, 2010 2:13 am
Neil’s critique does seem to align quite consistently with the consensus that we had just been reaching today, that the problems with the argument are originating from the finite model theory component of the argument and then being indirectly detected also at the random SAT and complexity theory side of things. With reference to the three levels of questions from the previous thread, it seems we can now answer both question #1 and question #2 as a definite “No.”.
The status of question #3 still seems to me, though, to be a more equivocal “Probably not”. Ryan’s examples demonstrating how solution space structure seems to be almost decoupled from worst case complexity is admittedly rather compelling, but there could still _conceivably_ be an elusive “Property A” that is obeyed by solution spaces to easy problems (possibly after a reduction), but is not obeyed by a standard problem suspected to be difficult, such as k-SAT, thus splitting a complexity separation task into the Claim 1 and Claim 2 components discussed previously (provided that reduction issues are dealt with properly). While it is now clear that the type of finite model theory tools being deployed in the Deolalikar paper are not powerful enough by themselves to handle Claim 1, it could still be that there exists a reasonable candidate for Property A along the lines of the “polylog-parameterisable” intuition of Deolalikar for which Claim 1 and Claim 2 could still be conjecturally true for some complexity separation problem (perhaps one weaker than P != NP, but still nontrivial). While Ryan’s examples (and also, to some extent, the natural proofs barrier) are certainly significant obstructions to this occuring (and the whole SAT/XORSAT business would also significantly obstruct any attempt to prove the Claim 2 side of things), perhaps there is still some loophole that may offer a way to salvage the strategy at least of Deolalikar’s argument, if not an actual result?
• August 13, 2010 3:01 am
I think Ryan’s 2-SAT example does not have hard structure of solutions in the sense k-SAT or XOR-SAT do. In the 2-SAT example there is only finite number (as opposed to exponentially large in n) of clusters. And more importantly if I condition on the value of one variable, one whole loop of variables is implied by the 2-SAT formula. One just needs to do this n/k times. On the other hand both k-SAT and XOR-SAT are exponentially hard for such kind of “resolution” algorithms. This may be what is called “conditional independence” in the paper.
My current take is that if the answer on question #3 is positive, then the separation must be X!=NP, where X does not even include Gaussian elimination (this may be not very interesting). But said another way: Class X of algorithms cannot solve random K-SAT, XOR-SAT etc. this would be interesting if X is at least a bit non-trivial.
August 13, 2010 3:08 am
TC^0?
• Jan Krajicek permalink
August 13, 2010 4:31 am
There are already results of this form in terms of
‘proof complexity’. A SAT algorithm can be thought of
as a proof system for propositional tautologies
(a proof of a tautology is the run of the algorithm
failing to find a falsifying assignment). There
are known exponential size lower bounds for proof systems
strictly stronger than resolution, e.g. for systems
operating with AC^0 circuits, with integer linear
inequalities, with polynomials over finite fields,
with OBDDs, and more. Wiki page for ‘proof complexity’
links to some surveys.
Jan
August 13, 2010 5:38 am
Since your research is in the statistical physics related part, could you perhaps give an explanation about the number of parameters issue?
Do you know how is his number of parameters defined? Is there some way to define this so that it corresponds to number of clusters (or something in the field). Does this part make sense to you?
I think this would be very important to know. Never in the paper he described precisely what his number of parameters is. Does he refer to something which is known, or did he left the point undefined as I suspect he has?
Please answer this, if you do not have idea what he is talking about than this is significant, if you do than please let us know, as this would clarify a lot of things.
• August 13, 2010 9:49 am
Hello Jan,
You are of course correct that all these proof systems have exponential lower bounds; but as far as I know, not for random k-CNF’s. For random k-CNF’s only resolution (and Res(k) and polynomial calculus, together with some of their weak extensions) have known super-polynomial lower bounds.
So perhaps, as Lenka Zdeborova has noted above, the proof strategy might be somewhat helpful for establishing some random k-CNF’s lower bounds in proof complexity (in proof complexity it is fairly interesting to prove such lower bounds also on proof systems that apparently cannot efficiently simulate Gaussian Elimination; e.g., bounded depth Frege).
-I
• August 13, 2010 12:25 pm
To vloodin: I have no idea what he means exactly by the “number of parameters”. Trivially all the solutions can be described by the k-SAT formula itself, that is linear number of parameters in n. But he must mean something else. I do not know what.
The solid non-trivial results about the space of solutions are of the kind cited as theorem 5.1 [ART06] in Deolalikar’s paper. But the same theorem (with different constants) is known for k-XOR-SAT. In statistical physics the non-rigorous analytic predictions go beyond that theorem. But we never speak about the “number of parameters needed to describe the distribution”. It is more all about characterizing some statistical properties of the distribution – distribution of distances, correlations between variables, marginals over clusters, number of clusters, etc. And then we have a number empirical results about algorithms, for instance: “Known polynomial algorithms empirically never find frozen solutions in random k-SAT” (where frozen is defined as having a non-trivial core in the sense defined in Deolalikar’s paper just before he states theorem 5.1 [ART06]). I believe that frozen variables in k-SAT pose some kind of non-trivial algorithmic barrier, and if this “barrier” could be formalized that would be great.
August 13, 2010 1:12 pm
Thanks for the reply. In his new survey of the proof, he promises a proof that exponential number of parameters is needed in the hard phase in his new draft.
Apparently, number of parameters has to do with Gibbs potential representation and Hammersen Clifford theorem
“(Hammersley-Clifford). X is Markov random field with respect to a
neighborhood system NG on the graph G if and only if it is a Gibbs random field with respect to the same neighborhood system. ”
It is good to know that it is something he defines, rather than a standard thing in the field.
According to him, his definition of a number of parameters would separate k-XORSAT from k-SAT. But he neither gave definition, nor explained the link to the clustering or other phenomena. He might do it in his new draft.
Another hint about what is meant by number of parameters is his comment that if particles are interacting with at most m-ary potentials, then number of parameters for n-particle distribution is O(n^m). So, it is something like number of potentials in the expression for thermodynamic distribution.
So, I can try to make sense of this in the following way: suppose we have an n-(spin) particle system, with interaction potential U. Then partition function Z=\sum {x is in n-cube} e^{-U(x)/T} can be computed efficiently if number of “parameters” is low.
Number of parameters of thermodynamic distribution would correspond to number minimal number of computations needed so that partition function (it is a function of T) can be computed using product and sum, starting from terms of the form e^{-V(x)/T}.
Now does this make k-XORSAT a 2^poly-log parametrizable, while k-SAT needs exponential number of such parameters?
The energy of the system is given by
U(x_1,… x_n)= \sum delta (\sum (C_{li}S_i),-k)
where first sum is taken over all clauses and second for each clause measures
weather it is satisfied (cf. 5.1 of old or 6.1 of third draft). Hence, energy of the system is equal to the number of unsatisfied clauses. If total number of clauses is m, then energy is between 0 and m.
In k-XORSAT Kronecker delta can be replaced by XOR sum of corresponding elements. To get the partition function, we need to consider cases when energy is equal to 0,1, 2… m. We are interested in number of satisfied equations. Supposedly, simple (linear) structure of solution space would allow us to compute this number as a linear sum of a few parameters, and this would than give simple expression for partition function.
• August 13, 2010 1:39 pm
Computing e^{-U(x)/T} takes polynomial number of parameters, computing \sum {x is in n-cube} e^{-U(x)/T} in k-SAT is #P-complete, and hence we all believe (but do not know how to prove) that exponential number of steps is needed, but counting is #P-complete even in polynomial problems like matching. So I am still confused about the definition of “parameters”.
August 13, 2010 2:29 pm
parameters are not x_1… x_n, so in computing sum e^{-U(x)/T} he more likely counts number of terms – basic potentials;
Complexity of the problem Z(T) is not so relevant – it might be #P complete in general, but he uses it to determine (if I am right) the number of parameters/terms, does not need to solve it. It is the other way around – when he CAN compute Z(T), then number of “parameters” is small.
9. Micki St James permalink
August 13, 2010 2:16 am
I don’t understand Immerman’s first objection. He points out that in an LFP computation of ordering from successor, each stage of LFP computation is far from order invariant. However, Deolalikar is not doing that computation but rather is computing an extension to a partial assignment of variables to try to satisfy an instance of 3-SAT, right? Does Deolalikar need to make a claim about the former computation to get what he needs for the latter computation? Can some stages of some LFP computations be order invariant even if others aren’t?
• August 13, 2010 8:46 am
As I understand it, Neil is just pointing out that the stages of an LFP induction need not be order-invariant even if the result of the induction is.
The relevance of this observation is that Deolalikar assumes that the stages *are* order-invariant. So, he only proves (if even that) that LFP inductions of this type are unable to express k-SAT. This does not cover all polynomial-time computations.
August 13, 2010 3:09 am
TC^0?
August 13, 2010 3:16 am
Does not Neil’s comment explain why 2-SAT and XORSAT might have ‘complicated’ solution space?
12. Concerned Citizen permalink
August 13, 2010 3:17 am
I would like to reiterate what I said a few days ago:
“It is quite clear that there is something very wrong in the model theory part of the paper, which is where all the interesting action has to be happening. After having a very weak (and incorrect) characterization of P, the proof could end in a variety of ways. The author has chosen to reach a contradiction using random k-SAT. And this, apparently, is leaving some people (who are unfamiliar with the work on random CSPs) hailing it as a “new approach.”
Yes, it is great that people have isolated some incorrect claims in the model theory sections and thus have “found the bug.” But the point is this: It was obvious from the beginning that this was the problem. All the “power” is coming from the incredibly weak “characterization” of poly-time algorithms, and random k-SAT is just a guise (that got people excited because… they were bored during summer vacation?)
How could I (and, I’m pretty sure tens of other people) know this without pinpointing a specific bug? Because there is no sensical exportable statement from that section. There is no formally defined property “Q” such that all assignments generated by a P-time procedure satisfy “Q.” If the author had even once tried to write down this property formally, the flaw would have become immediately clear. So I will say again: people are being too kind; the author is either crazy or disrespectful.
That’s not a mean or nasty thing to say. It’s just the opposite. It’s protecting the community and its integrity by requiring some basic rules and courtesies in mathematical discourse.
• John Meercat permalink
August 13, 2010 4:32 am
Give him a break. This is just unnecessary.
August 13, 2010 5:07 am
His lack of proper definitions is another red flag. He hand waves with “parametrizations”, central to his approach, but the way he counts parameters is never defined, and not a single person among many experts here does not have ANY idea what he exactly means.
So, a perhaps more objective assessment of this piece of text (over 100 pages in 12pt, yet more like 70 pages in 10pt) might be:
(1) Author never defines his key notion, number of parameters needed for a distribution description, that supposedly separates P from NP in his strategy.
(2) He than claims that this undefined number of parameters is exponential in a certain phase of k-SAT, as described by work of various statistical physicists. However, he confuses the phases, which is not surprising, since he nowhere explains how is his fictional undefined number of parameters is linked to the statistical physics conclusions. Namely, he points out that in certain phases there are exponential number of clusters. But never does he explain how that means that number of his parameters is exponential. On a rough reading one is perhaps deceived into thinking that this is OK, but it is not – not the least because he never defined his number of parameters. His confusion of phases is another indicator in the lack of proper rigor (or rather meaning) in his vague claim, which seems plausible only if we don’t go much beneath the superficial surface.
(3) Then he proceeds to prove that this fictional undefined number of parameters is O(2^poly(log(n))) for polynomial algorithms. Since no one understands what he means, people here were at least able to analyze this part of the proof, where he constructs certain graph of limited vertex index. He claims that this means the number of parameters is bounded, but since we do not know how he counts these, one can only take this claim for granted, and analyze the reduction to the graph, because at last there we know what he is talking about (graph with bounded index of vertices).
(4) It turns out that even this reduction to a graph is deeply flawed. It does not capture P. He uses monadic (unary) relations, hand waves about reduction to this case, which turns out not to be possible.
He then proceeds to conclusion. Yet as we see, every step is dubious. It is only the point (4) that had claims that were clear enough to be analyzed, so for this reason, clear flaws have been found there (and it is not a single flaw, but several problems).
On another level, other restricted CSP (as I have pointed out early on), including here many times mentioned k-XORSAT etc. have same phases, but a P algorithm. So, even if he was able to provide some reasonable definition of parameter number (that would be the real contribution from his attempt), either this number cannot be concluded to be exponential from phase properties, or this number cannot be concluded to be c^polylog bounded for all P algorithms. But as it seems now, he did not really prove any of these.
He did not even define his central property- number of parameters.
So, the question is not if this paper has any chance of being fixed. The question is not weather this approach is promising. The question is not weather this paper will be published in a peer review journal. These are all answered by a most emphatic NO, no way, not in a million years.
The real question is why would someone who clearly has enough background to understand relevant material (and hence not a complete quack) do something like this, and how can he save face given this embarrassment. Another question is was he aware that his proof is wrong or incomplete when he e-mailed it out.
My guess is that he did not think a proof strategy is so wrong, but probably was aware that proof was incomplete. Perhaps he thought that details can be fixed, and his vague ideas (here we have more a sketch or a strategy) can be made precise. Perelman also published a draft, though his draft was small (30+ pages) packed with the essential points in it, standard (precise) mathematical formulations, and there were many of the breakthrough ideas there. People analyzed and supplied the details. Perhaps he was hoping something of the kind. But Perelman did work out his details (when people asked him during his US tour, he was able to answer all queries), and here we have serious reasons to think that many details are far from being worked out, and hence we have a lot of troubles due to hand waving as opposed to troubles in understanding due to skipped proofs. Here, precise statements are missing. So a manuscript was sent out to colleagues early. I wonder if the reason/timing has anything to do with ICM 2010 held in his native India this month.
Also, the response of the community was perhaps more than such papers usually get. It is rare that these huge claims are made by mathematicians/cs people with any degree of real competence, so this is perhaps the reason he was treated differently (his background is better than that of most of people who make such claims). But this only underlines one other case, where response of the community was far less kind, though merits were clearly much greater than that of this middle age middle range HP researcher with a couple of average publications.
It is the case of Louis de Branges. This guy was attempting to prove Riemann conjecture his whole life. In the process he developed a whole theory, deep on its own right. He was able to prove the famous Bieberbach conjecture using it, in the 80s. But he had to go to Leningrad to get his proof checked. In the meantime, his life turned to hell as math community didn’t want to take him seriously, his wife left him as a failure, and all sorts of bad things befell on this noble soul. His proof turned out to be correct, and this was reluctantly recognized (Bieberbach conjecture was THE major problem for a big class of complex analysts), but professional jealousy further alienated him from the math community (it didn’t help that he was a bit difficult/uncompromising/unwilling to trade/uncorrupted in dealing with them and not part of the social crowd in the field). So this black sheep of complex analysis continued to work on his own, outcasted by the wonderful mathematics community of the time. This person was a top mathematician, expert of his field, professor at Pen-State (I think), who proved some major theorems and solved major conjectures. Yet when he published his proof of the Riemann conjecture on the web a few years ago, he was completely ignored. There was no Soviet Union at this time, no alternative community to talk to. I am sure that his theory and proof attempt have a lot more merit than this one we are all devoting our energies to, and the problem is no less famous, and the guy has a whole insightful theory constructed for that purpose over a lifetime, yet he was deliberately not helped. Too many hurt vanities stood in the way. I can only say, compared to this, his treatment was far from fair. On the other side, and in comparison, the attention this guy is getting is far from what it is worth. Way off.
• Quantum Tennis Referee permalink
August 13, 2010 6:06 am
Vloodin: I think you are reading your preconceptions into the person’s motives. Be careful! This blog is not about judging personal motives, but only for verifying the manuscript.
• August 13, 2010 6:51 am
Louis de Branges is a distinguished professor of mathematics at Purdue (and not
Penn State as claimed).
• August 13, 2010 7:49 am
Every mathematical proof can be broken down into smaller pieces. Eventually these pieces become small enough to be sent to friends, or if you don’t have friends in that area, conferences and journals. Then they are verified and accepted. Then you have your proof verified completely. Win, success!
• steve uurtamo permalink
August 13, 2010 8:04 am
just some online advice:
nice people generally use the following “emergency” rule of thumb: the more hostile an otherwise nice person’s comment, the more attention it should be given (because it’s such unusual behavior).
however, because it’s online in a weblog, without your full name, your comment is hostile without a known lifetime of nice behavior to back it up. so it has the unfortunate characteristic of being difficult to distinguish from that of “just another internet jerk”.
why not think of a way to phrase your concerns that doesn’t make you sound like you have a mouth full of sour grapes and are trying to denigrate one person’s reputation in order to prop up the reputation of someone else?
it’s really bizarre and antisocial behavior.
if it concerns you that this proof idea was given more time and energy and friendly lookings-over than that of some other proof idea by some other person for some totally different problem, then maybe you could quietly and privately reflect on that and try to figure out why. perhaps you’ll never know, but it’s possible that one doesn’t have a whole lot to do with the other.
August 13, 2010 8:13 am
Oh, come on, stop coming down so hard on the poor guy! He did NOT submit the paper to a journal, he just emailed it to a few people, so what’s your problem? Stop your personal insults.
Having said that, working alone is a bit foolish, no matter whether you are a genius. It helps to have a research soul-mate to discuss and iron out things, to battle out your arguments.
And I did not find any acknowledgment in the paper (except for the dedication page at the front). Perhaps this explains the end result. He could have just driven down the road to Stanford or Berkeley and had it checked it by someone there.
August 13, 2010 8:54 am
I don’t know what is wrong with this paper getting the attention it did. Are you implying that is not worthy of expert’s time to discuss a serious attempt at solving a very important problem? Do you think the mathematical community won’t gain anything from this discussion even if the proof is wrong? Surely many worthy papers haven’t received this same attention, but is it really a bad thing that this paper did?
August 13, 2010 11:59 am
Well you are of course entitled to your opinions, but fail to see why you want to denigrate Vinay’s nationality and age in your criticism. Wrong proofs have happened to people of all nationalities and ages- history of mathematics is full of these examples.
• Jeffrey Stopple permalink
August 13, 2010 12:20 pm
In regards to DeBranges’ attempt to prove the Riemann Hypothesis, it’s worth noting that what he claimed is actually stronger than RH, and that Conrey and Li proved and published a result that contradicts DeBranges, see MR1792282 (2001h:11114)
August 16, 2010 11:37 pm
You make your view very clear by a lot of arguments. But think about the extremely small possibility of some math-amateur giving the correct proof for some really big problem…Riemann, Goldbach, Collatz, whatever…how easy would it be, to see if such a proof is pure nonsense (probably minutes to hours) but how tremendous would be its impact, if it would be correct, and how equally tremendous would be the loss, if this yet unknown guy does not dare to publish anything, because professional mathematicians tend to be somewhat uncomfortable and unpleasant when dealing with some semi-professionals work? Surely it is the rigour, that makes a proof mathematical, but sometimes ideas are more important, even the ideas someone may takes from a wrong proof to devise a better one on this basis.
August 21, 2010 2:46 pm
“Yet when he published his proof of the Riemann conjecture on the web a few years ago, he was completely ignored.”
This is not correct. He was taken seriously, but did not – in contrast to Perelman – give explanations of specific obscure points in his paper when they were requested. His general approach to technical inquisition was “Go figure”.
Mathematicians do not like this.
• Quantum Tennis Referee permalink
August 13, 2010 6:05 am
As if the community needs protecting! ha! What a wild notion!
August 13, 2010 2:12 pm
It’s comments like these that show that public and anonymous mathematical collaboration cannot work.
13. August 13, 2010 3:51 am
The end-quote (”) in the third paragraph of Immerman’s letter is misplaced: it should be at the end of that paragraph, not at the end of the first sentence of the quote. So the sentences starting with “This is a technicality.” are also from Deolalikar’s proof attempt; they are quoted by Immerman, not said by Immerman.
14. August 13, 2010 4:05 am
Since the process evolving on this blog may be interesting as a social phenomenon(independently of the mathematical contents), it seems important to track the publicity following from it, as is well done on in the Media section of the wiki. I should therefore like to point you to
http://www.zeit.de/wissen/2010-08/milleniumsproblem-beweis-mathematik
where the process was noted in (the online version of) one of the most respected newspapers of Germany, “Die Zeit”. I propose including the link in the Media section of the wiki.
15. Citizen II permalink
August 13, 2010 4:25 am
You say
> Yes, it is great that people have isolated some incorrect claims in the model theory sections and thus have “found the bug.” But the point is this: It was obvious from the beginning that this was the problem.
How is that a point? Of what? It is not hard to check that both the lack of rigorous definitions and the specific model theory flaws were pointed out very early on. It just took long until they got attention or were explained in a way that were understood by everyone.
August 13, 2010 4:34 am
The fish in my aquarium cannot tell by visual inspection the difference between food and their own excrements, so they put the thing in their mouth and after a fraction of a second they spit it if it is not food. I particularly like to watch them when I referee papers.
This said, I do not agree with the previous post. It seems, as others noted elsewhere, that in addition to the social/web/1M$context, what attracted some really great minds to study the paper was the putting together of different threads from different domains. Certainly, the style of the manuscript cannot be recommended for a standard publishable math paper, but mathematics is *not* only about theorems and proofs, especially in the creative phase: “All beginnings are obscure. Inasmuch as the mathematician operates with his conceptions along strict and formal lines, he, above all, must be reminded from time to time that the origin of things lie in greater depths than those to which his methods enable him to descend. Beyond the knowledge gained from the individual sciences, there remains the task of *comprehending*. In spite of the fact that the views of philosophy sway from one system to another, we cannot dispense with it unless we are to convert knowledge into a meaningless chaos.” (Hermann Weyl). • om57 permalink August 13, 2010 9:01 am “previous post” in my post referred to the concerned citizen. 17. August 13, 2010 5:37 am Concerned Citizen: I well know Vinay to be far from crazy and disrespectful. He asked a few for comments and uploaded the manuscript on the net. Unfortunately, too many people got into it too soon. Although active in theoretical computer science, he is not a mathematician by training but is a PhD in Electrical Engg. So, the manuscript might lack the pure math rigor. If his results and/or arguments are wrong, the constructive criticism given by Terrence and others has a hope of salvaging a useful result. Some others may prefer the safe “see no evil” approach though. 18. Janos Simon permalink August 13, 2010 8:04 am It seems that what remains is Terry Tao’s 3. — can these ideas / strategy be used to advance the field? More precisely, can we get a nice characterization, (using finite model theory?) that must be true of all languages in P, but false for some in NP? The kind of property suggested in the paper as a candidate would be related to the statistical physics landscape of the solution space, which is intriguing to most complexity theorists: it is a subarea where we do not see (yet?) difficult barriers to further progress. An unreasonable (but perhaps not totally crazy) goal in such a strategy would be showing separation from NL (or, from L), instead of P. Such a result would be also tremendously impressive. Is there a sensible definition of “number of parameters” or of some property of the solution space where a characterization theorem for what algorithms in L can produce could be proven? Further, if we are to study the solution space of NP predicates, why not go to PP (unbounded probabilistic polynomial time)? The intuitive “distance” from L to #P is bigger than from P to NP, so “it should be easier”. For example, a logspace Turing machine cannot even write down on its workspace the number of satisfying solutions to 3-SAT. Also, the class PP is directly related to the geometry of the solution space: loosely speaking, #P counts the number of absolute minima. Dealing with PP has an “aesthetic” advantage: it is well known (since the 70s) that the difficulty of counting solutions does not relate nicely to the difficulty of finding solutions (the number of satisfying assignments to 2-SAT is #P-complete, as is the number of satisfying assignments to 3-SAT. Counting the number of spanning trees of a graph is in P, counting the number of perfect matchings is #P-complete). Admittedly, the number of solutions is the simplest statistics of the solution space. It may well be that, as Russel pointed out, the big problem is to find a way to relate the “shape” of the solution space to algorithmic difficulty. • Istvan permalink August 13, 2010 8:13 am Honestly, I doubt that even something like that will come out from this. There are #P-complete problems for which polynomial delay enumeration algorithms exist. Existence of PDE algorithm indicates that the state space is not as hard as one might think… • Cristopher Moore permalink August 13, 2010 10:26 am But the XORSAT problem suggests that we have to be very low in the hierarchy, somewhere where we can’t tell what the rank of a matrix is or invert it. This puts us below DET, right? 19. Milos Hasan permalink August 13, 2010 9:14 am I have a naive question, please forgive my simple non-expert view of things: So maybe there is a mistake in Deolalikar’s characterization of P – but there are certainly other correct characterizations of P. There are P-complete problems, e.g. Horn-SAT. So the question (actually questions): 1. If the “solution spaces” (however defined) of Horn-SAT and SAT are different, is it enough to separate P from NP? (My guess is no – unless the translation from any P algorithm to a Horn-SAT instance preserves the “solution space”, which might be impossible.) 2. Is Deolalikar’s approach similar to the above naive idea (just much more technical) or fundamentally different? 20. Mauricio Karchmer permalink August 13, 2010 9:20 am I am convinced that there is a proof here that P != NP. Indeed, browsing at the paper, and reading the various comments published here and elsewhere, is proof enough that verifying the correctness of the proof is exponentially hard. Therefore, NP = EXPTIME and, as a result, NP >> P. Seriously, the paper is in fact ‘poetic’. But, can someone tell me, where is the meat? P = FO[LFP] is a beautiful theorem (thanks Neil, et. al.) but if using LFP, you might as well stay with the RAM model. Locality of computation? Reminds me of myself in my first year as a graduate student. And this ‘statistical mechanics’ business? It is also beautiful, and even more poetic than the ‘proof’ itself. Frozen variables, yummy, especially in summer. It used to be called random graph theory in my days. Or am I missing something? Russell, you are the greatest. Which reminds me, I have a great ‘get rich quick scheme’. Goes like this. Write some ‘poetic’ proof that P != NP. No less than 100 pages, please. Ask Steve to write that the proof is ‘relatively serious’. Have the community at large work hard filling the details. Collect the Clay award. Anyway, it was great being part of the community again, after 15 years. I better get back to work now, or I might get fired.. Cheers, Mauricio Karchmer • V Vinay permalink August 13, 2010 10:48 am Did not the several rounds of monotone conversation convince you of the depth of the paper!? Alas :-) Cannot agree with you more on the LFP front though. Yes, it has all turned out to be very poetic and Zen; peel and peel and then nothing! Tao elsewhere has checked all his three questions to a “no.” But then who knows; there might still be a twist. D has promised a new version over the weekend … • Someone permalink August 13, 2010 10:59 am > And this ‘statistical mechanics’ business? It is also beautiful, > and even more poetic than the ‘proof’ itself. Frozen variables, > yummy, especially in summer. It used to be called random graph > theory in my days. Or am I missing something? Not directly related to the “proof”, but if you want a challenge try to beat the survey propagation algorithm they invented. The “fusion method” sounds poetic to me. 21. August 13, 2010 9:25 am Cynical people might think that the attention given to this proof was not so much because the proof looked promising but rather because it was an opportunity to promote the field, attract graduate students, and increase theory funding. • anon permalink August 13, 2010 9:54 am You’re right, that does sound pretty damn cynical. I think several of us only read the paper because our colleagues in other areas were asking “but you’re the complexity guy/gal, how can you not have read it?” Some put$200k on it being wrong and went back to vacation. Others tried to understand things, airing their concerns in a public forum. But I think most of us ignored it entirely after reading the first few comments on this blog and the wiki.
Somehow this particular paper spread very rapidly. I wouldn’t “blame” complexity theory for that. If anything, blame Clay Math for putting $1M bounty on the problem. • August 13, 2010 11:04 am Do you think the publicity from this will have an impact on complexity theory? If so, what sort of impact? I find it hard to believe that those who commented on the proof gave no thought whatsoever to the fact that the world is watching. This has become a spectator sport. • Random permalink August 13, 2010 11:24 am Paradoxically, I think a lot of people who are still paying attention are non-specialists (like me). For people who know enough about barriers, it must have been easy to take a cursory glance, convince oneself that the strategy was flawed, and go back to one’s own research. • Sasha Razborov permalink August 13, 2010 1:03 pm Yes (to Random). • Sasha Razborov permalink August 13, 2010 1:10 pm P.S. But I have to admit I am a little bit harassed by the Russian media these days (I am sure many colleagues have similar problems), and it is *extremely* handy to have this blog around as a pointer. Dick, the Group et. all — thank you *very* much for investing effort into this. • Anonymous permalink August 16, 2010 7:40 am Amir: People are commenting here *precisely* because they know the world is watching them :-) • Anonymous permalink August 13, 2010 10:39 am Amir: LOL. • Fnord permalink August 13, 2010 10:40 am The reason the proof got so much attention was because Stephen Cook was quoted characterizing it as a “serious attempt.” Otherwise it would have just been another entry appended to the P vs NP page. Clearly, the proof is now dead, and we may all move on. • Random permalink August 13, 2010 11:25 am a “relatively serious attempt”. The adverb changes everything. • SubtleLisp permalink August 14, 2010 2:32 am Although I see this as a bit cynical, there is a certain truth aspect. In the Wolfram Prize discussion, for instance, Vaughan Pratt intertwined some commentary about funding, and it seemed almost non sequitur to me. Not to flog this too much, but for context, see http://cs.nyu.edu/pipermail/fom/2007-November/012253.html > This entire episode seems to me a massive perversion of the scientific process. Pratt: In the grand scheme of things I find “perversion of the scientific process” far less destructive than the wholesale starvation of computer science funding that the US government has been indulging itself in during the past decade or so. So you can see where some priorities are. • August 14, 2010 7:08 am Not intending to hijack the thread, but funding need to be balanced between basic and applied research. Applied research should be funded by corporations, and incorporate the basic results from government funded research. Some CS research is basic, some is applied, but there needs to be more flexibility so that scientists can move freely between domains, carrying their passions and ideas wherever their thoughts lead them. 22. August 13, 2010 9:36 am Tim Gowers has just made a post at http://gowers.wordpress.com/2010/08/13/could-anything-like-deolalikars-strategy-work/ which I think makes a very strong case that there is a natural proofs-like barrier that prevents the answer to question #3 from being “Yes”. Basically, he considers a _random_ (or more precisely, _pseudorandom_) P problem, basically by getting a monkey to design randomly (or pseudorandomly) a polynomial length logical circuit, and looking at the solution space to that problem. Intuitively, this solution space should look like a random subset of {0,1}^n: a pseudorandom circuit should give a pseudorandom function (after selecting parameters suitably to ensure enough “mixing’). In such a setting, any reasonable “property A” should not be able to distinguish that function from a random function. But if “property A” has any strength at all, it should be constraining the structure of the solution space to the extent that a random solution space should not obey property A. (Certainly this is heavily suggested by terminology such as “polylog parameterisable” by basic counting arguments.) As such, the Claim 1 + Claim 2 strategy cannot work. To summarise, I think we now have at least four significant obstructions to the Claim 1 + Claim 2 strategy working: 1. Ryan’s examples showing that complexity and solution space structure are decoupled. 2. The point made by Russell (and others) that the Claim 1 + Claim 2 strategy ought to also separate non-uniform complexity classes as well, thus possibly triggering the natural proofs barrier or something similar. 3. The difficulty in getting a argument to distinguish “difficult” solution spaces such as SAT from “easy” solution spaces such as XORSAT; 4. The difficulty in getting an argument to distinguish the solution space of a random polynomial length circuit from the solution space of a random function. Based on this, I think I’m ready to change my answer on Q3 to “No”. • August 13, 2010 10:15 am I wonder if it might be possible to obtain something constructive by going the other way. For example, suppose the conjecture is true that the hard instances of k-SAT are exactly when the clusters are frozen. Then maybe “all” we need to do in order to produce a pseudorandom generator is to build a function whose clusters “appear” frozen. (It’s not clear to me whether k-XORSAT freezes in a similar way to k-SAT.) Or, perhaps, there are limits on how frozen something can appear, so there are limits on what kind of PRNGs can be built. • August 13, 2010 12:08 pm XORSAT freezes much in the way K-SAT does… No differences! • August 13, 2010 10:19 am One minor clarification: Gowers’ pseudorandom circuit is made out of _reversible_ logic gates, and so one cannot distinguish this circuit from random functions simply by looking at the influence of individual bits, etc. • Istvan permalink August 13, 2010 1:25 pm I cannot see the barrier. As far as I can see, Gower’s pseudorandom circuit makes a pseudorandom function from problem instances to {0,1} (or {0,1}^n), while Deolalikar is talking about the distribution of the solution space of one problem instance. If Gower’s pseudorandom circuit is implemented as a NDTM, why the distribution of acceptance paths should look pseudorandom? Or I just miss the point, sorry if saying something stupid… • August 13, 2010 1:37 pm Yes, we got a bit confused on this point. But Tim’s argument can be modified to deal with solution spaces of individual instances also; see my second response to Jun Tarui below. 23. August 13, 2010 9:40 am In the midst of such intense academic discussion — here’s a note on the lighter side :). Maybe we should use the football(/soccer) world cup Oracle: Paul the octopus to answer if this proof is ok or not [or any other related binary predicates] :). (in case you havent heard about Paul the octopus, here’s the link http://en.wikipedia.org/wiki/Paul_the_octopus) • Random permalink August 13, 2010 11:28 am Why not use it on P=NP and P!=NP while we’re at it? It wouldn’t give us a proof, but it could at last tell us if the majority is right. 24. August 13, 2010 9:48 am Now that a consensus seems to have formed that this P !NP paper is deeply flawed, and skepticism is even beginning to form as to whether it contains any new credible idea (though opinion might shift around on this), I thought I would put forward my impressions on some general issues suggested by this entire episode. Imagine two quite different actions. 1. Someone with quite reasonable credentials circulates a well written (not in the technical sense) manuscript suggesting what they regard as “promising new approaches to P !NP”, with many details included. 2. Someone with quite reasonable credentials circulates a well written (not in the technical sense) manuscript claiming that they have proved P !NP, with many details included. The effect of these two actions are radically different. In the case of 2, a large number of very strong theoretical computer scientists and interested mathematicians are rightly compelled to pretty much drop whatever they are doing and look at this seriously NOW – or, in some cases, be compelled to make public statements as to why they are not dropping whatever they are doing to look at this seriously. So 2 amounts to forcing one’s ideas to the top of the queue for a very large number of very powerful people who are not going to be able to continue doing the great things they normally do until at least a reasonable consensus forms. Under 1, eventually some very powerful people will take a look, but when it is convenient for them. The cost of this is, on balance, considerable enough to be of some concern. I’m not thinking of retribution, or anything negative like that. But considerable enough so that the question of how this came about, and how to prevent this kind of thing from happening in the future, becomes pretty interesting – both from the theoretical and practical points of view. From the theoretical point of view, broadly speaking, we have a very satisfactory understanding of what an ultimately rigorous mathematical proof is. Furthermore, ultimately rigorous mathematical proofs have actually been constructed – with the help of computers – for a substantial number of theorems, including rather deep ones. E.g., see http://www.ams.org/notices/200811/tx081101408p.pdf This also has a careful discussion at the end of what major advances are needed in order for the construction of ultimately rigorous mathematical proofs to become attractive for more than a few specialists. On the practical side, I think that there is a reasonably clear notion of “completely detailed proof”, written in ordinary friendly mathematical language, which definitely falls well short of “ultimately rigorous proof”. I have, from time to time, felt compelled to construct “completely detailed proofs” – and it is rather exhausting. I don’t do this as often as I should – perhaps because I don’t prove something important enough as often as I should or would like. But there is some satisfaction that comes from writing up a completely detailed proof. Bringing this to the matter at hand, I am under the impression that many people do not have a clear idea of what a completely detailed proof is, and rely on some sort of general intuition – which is required to come up with just about any serious proof in any form – even for proofs for publication. Since the refereeing process generally consists mainly of whether the referee feels that they can “see” a correct proof based on what is written, authors are normally never compelled to get their hands dirty with anything remotely approaching a completely detailed proof. This works well as long as they are in an arena where their intuitions are really good enough. Of course, (most) really strong researchers know just how dangerous it is to operate this way, particularly with notorious problems. But obviously reasonably strong people may not realize this. In fact, I have seen this kind of thing many times before in and around logic – and there is often a defiance stage. So it might be valuable to a. Flesh out both theoretically and practically what a “completely detailed proof” is – still well short of “ultimately rigorous proof” which we know a lot about already. b. Give examples of “completely detailed proofs”. c. Instill this in the educational process. E.g., all Ph.D.s in theoretical mathematical areas must have written a substantial “completely detailed proof”. I would be delighted to hear what people think of these comments. • rjlipton permalink* August 13, 2010 9:56 am Harvey, Thanks for these comments. • August 13, 2010 12:22 pm Professor Lipton, When the dust finally settles you should write a CACM article on the whole episode :-). • darshan permalink August 13, 2010 11:08 am Would somebody tell what approaches have been used by people who proved significant results in the past: example cases 1. Andrew Wiles (First Announcement) 2. Andrew Wiles (Correct Proof) 3. Grisha Perelman • darshan permalink August 13, 2010 11:14 am I meant approaches to make announcements of significant proof • vloodin permalink August 13, 2010 11:59 am I don’t know about Wiles, other than he was very careful and organized a graduate course where students were checking parts of his proof (without being aware of that). He made first announcement in a talk, directly to a high level public, being cryptic about it. Yet a flaw was nevertheless found, but he was able to fix it. Grisha Perelman just posted his results on archive. They were written in normal math (read – he had precise statements) and then people started to check it. There was for instance a seminar at MIT devoted to figuring out the details. He was then invited to a tour to USA, to Stony Brook, MIT, California. He communicated directly to the experts, who were already prepared with detailed questions. Grisha was a top researcher and did never hand wave. Unfortunately, after returning to St. Petesburg, he had a row with local math community and quit mathematics for mushroom collecting, a much more profound activity on many levels, including not having to deal with colleagues. For some reason, he started to despise the system, and as one of the reasons he refused Fields medal he said he did not want to validate the system, which is inherently unfair (interestingly, another math genius, Alex Grothendieck, abandoned mathematics for healthy living in Andorra for the very similar reasons). • SubtleLisp permalink August 14, 2010 2:53 am The opposite technique, aiming for media hype, has become common. “PRIMES IN P” was like that, though supposedly it was Pomerance (I think?) who “notified the press” after he read the version of the manuscript for which the proof finally worked. At the end of the conference on “PRIMES IN P” at AIM, Goldston gave a talk about his new work on prime gaps (this was the earlier wrong version, before Soundararajan found the flaw), again in part to generate press, which is an AIM speciality. Perelman’s work was odd in that there was a HEAVY demand for COMPLETE details, with some experts saying that if this wasn’t the Poincare conjecture, it would have been accepted much more easily. I don’t think Wiles was affected by the Miyaoka FLT/1988 incident, as he was already working in his attic. But crying wolves do have some effect on communities. As for something (fixable) caught in a close review (but not by the referee!), I heard that the famed Friedlander-Iwaniec paper on primes of the form x^2+y^4 was given by Granville to some student to look at, and, in a flurry of estimates in one of the middle technical chapters involving bounds with Mellin transforms(?), there was something incorrect. I think the fault was that they claimed it worked like the other estimates in a previous section, which was not true. The method was robust enough that they could easily rewrite it, moving some of the workload to a different portion, but they did have to make a nontrivial change. Friedlander and Iwaniec are also technical geniuses, and all the ideas were already clearly there, so the final capitulation was easily foreseen. • August 13, 2010 11:14 am I fully agree with the opinion expressed. Mathematical theorems require proofs which meets the current standards in mathematics — for this TCS needs to adopt standard mathematical publishing practices, journal publications, careful refereeing etc . The current publication practices in TCS in my opinion falls short in this regard, and there seems to be no consensus in the TCS community how to remedy the situation (despite periodic attempts by various people to raise this issue — for example, Neal Koblitz in the AMS notices a couple of years ago). • sand permalink August 13, 2010 12:17 pm Yes, I think the real problem is w/ current publishing practices in TCS. Most conference submissions require only proof sketches and additional details may be included in appendices (and viewed at the discretion of the reviewers). This is a serious problem for two reasons, one mathematical and the other political. The mathematical as mentioned above is that authors do not necessarily go through all the details in writing the paper. (I have read papers that have had errors later found in them for this reason.) Second, without everyone being required to support and present their work carefully and rigorously, there is a bias in the review process whereby some researchers (i.e. better reputation, school, insert your favorite bias here etc) are given more serious consideration and more generous slack that their work is correct. It is seriously harmful to the TCS community in many ways. • SubtleLisp permalink August 14, 2010 2:57 am There is also a strong impact from “cryptography”, perhaps just as a buzzword for funding purposes. Springer Lecture Notes in Computer Science aren’t even counted as publications by some departments any more because of the decline in level. This said, there are still a number of quality researchers in these fields (and many of them, at least privately, are dismayed at the lack of quality content that has emerged in some places). • August 13, 2010 12:08 pm Harvey, My feelings on this are that complete detail at the “high-level” scale (i.e. a clear synopsis of the argument, a sense that the problem is being factored into strictly simpler problems, high-level compatibility with other results and insights, generation of new insight and understanding for this problem and for related problems, etc.) is more important for establishing the convincing and robust nature of a proof than complete detail at the “low-level” scale (at the level of individual steps in the proof), though of course the latter is also valuable. See for instance this earlier comment of Gowers, and this earlier comment of Impagliazzo. It’s also worth mentioning an older observation of Thurston that mathematicians ultimately are not after proofs; they are after understanding. • August 13, 2010 3:34 pm Terry, As you well know, depending on the mathematical context, the high level and the low level are more critical or less critical. The high level is always of first importance, because it is next to impossible to construct a fully detailed proof of something interesting without a good understanding at the high level. But I am addressing the issue of how to prevent the present situation from arising in the first place (assuming it is as it now appears). I.e., a flat out apparently bogus claim of P != NP – sufficient to attract the press, and a whole host of stars and superstars from their normal brilliant activity. We all know the essential importance of “high level” criteria. Yet this obviously did not prevent a claim being made from a credible looking scholar, with a large number of yet more important scholars dropping what they are doing in order to spend serious time looking at this! And involving the press!! When the dust settles, the best guess will be that the whole episode was a waste of a lot of people’s time – relative to what else they would be doing. For those of us who will have learned something from this, we probably will have learned a lot more from continuing the wonderful things we were doing before this occurred. But adherence to my “low level” criteria would have almost certainly prevented the claim from being made. Instead of claiming P != NP, it would have read something like “an approach to P != NP” and then interested parties could take a look according to their inclinations and schedules. There would have been no press coverage. So I repeat my suggestion, with c) below as an *addition* to the usual process for getting a Ph.D. in pure mathematics, theoretical computer science, and heavily mathematical areas: a. [Those of us interested in logical issues] Flesh out both theoretically and practically what a “completely detailed proof” is – still well short of “ultimately rigorous proof” which we know a lot about already [but which is generally too cumbersome to create under current technology by nonprofessionals]. b. Give [a wide palette of] examples of “completely detailed proofs”. c. Instill this in the educational process. E.g., all Ph.D.s in heavily mathematical areas must have written a modest sized “completely detailed proof” in normal mathematical language. Exceptions always should be made for “geniuses”. This is my suggestion for prevention. • Conan B. permalink August 13, 2010 4:12 pm For those of us who will have learned something from this, we probably will have learned a lot more from continuing the wonderful things we were doing before this occurred. I think you exaggerate a bit. Even the greatest scientists, most of the time, don’t do “wonderful things”. They just live. Especially that now it’s summer. So I don’t think there is any harm if they invest several days in this forum (even if the paper is completely bogus); and I believe that most of the non-experts readers here certainly learned a lot just by witnessing the discussion. • August 13, 2010 4:24 pm While these suggestions may be laudable in their own right, I doubt that they will do much to prevent the problem of a proof that is unlikely to work being nevertheless being taken seriously by a large number of people. For instance, a typical flawed proof written by a serious mathematician may well resemble a completely detailed proof in 98% of the manuscript, and it is only in a crucial 2% of the ms in which there is enough hand-waving that a serious error can slip through that ends up dooming the entire argument. Such a paper may well resemble a “completely detailed proof” to a superficial or even a moderately careful inspection, and in particular in the mind of the author, while being nothing of the sort. (Now this can be remedied in principle by writing the proof in a PCP format, but clearly this is not going to be feasible in practice.) In contrast, a proof with a complete high-level description can be gauged for plausibility much more effectively. One of the criticisms of the current paper was that a precise top-level description of the proof (in particular, splitting the main result into key sub-propositions) was lacking, and it is only in the last day or so that we are able to reverse engineer such a description (and not coincidentally, we are now also beginning to find significant obstructions to the method). If such a high-level description was in place in the initial ms then indeed I would imagine that a lot less time would have been spent to arrive at the current verdict. To put it another way: a paper with a careful high-level structure can often survive a lot of low-level sloppiness, whereas the converse is often not true once the complexity of the argument is sufficiently large. So if I had to drill my students on the importance of something, I would rather it be high-level structure than on getting every last detail right, though of course the latter should not be neglected either. But in either case, there is a limit as to how much good expository guidelines can help in preventing these sorts of events from occurring. There is an understandable (if unfortunate) tendency, in the excitement of thinking that one has solved a great problem, to rush out the paper and make the exposition mediocre rather than perfect, no matter how much they are drilled on the importance of good exposition in their education. And it is not as if one could enforce some sort of rule that badly written solutions to famous problems will not be read; if the attempt looks serious enough, people may well have to grit their teeth and plow through the bad exposition anyway, to settle the matter. • Random permalink August 13, 2010 10:44 pm In dealing with this high-level vs low-level problem, I have heard from a senior colleague who complains that he is expected to referee papers which are high-level only, the low level being riddled with holes (we’re talking here about CS papers in a very mathematized area, so you need both practical algorithms and theorems/proofs). He refuses to do that, and blames other referees who let people get away with that, as well as authors trying to expedite publication by letting the referee do the most unpleasant part of the job. I cannot comment on how widespread the problem is since I’m not in a similar position. But this suggests that sloppiness may arise more systematically than just from the excitement about having proved the next Big Theorem. • August 14, 2010 4:04 am re: Harvey Friedman’s call for heightened standards (or articulation of standards) so as to prevent “irregular” proofs from capturing researcher-time on a large scale —- is this really a repeatable episode, especially after happening once? In 15+ years of the World Wide Web this is the only instance where it occurred. The media also will be more skeptical for some time to come, and they were not credulous in this instance. • August 14, 2010 10:10 am Response to T. I like your point that I may be calling for , as you say, “heightened standards or articulation of standards so as to prevent irregular proofs from capturing researcher-time on a large scale”, where, as you suggest, this may happen extremely rarely. But perhaps there are enough famous open problems left in mathematics and mathematical computer science, so that this kind of reform would have greater applicability? Also, and most importantly, the kind of educational reform I proposed could really help in those much more common situations of people submitting papers for publication – or even internet publication – with erroneous proofs. I’m thinking that it would not only ease the burden on Journals, but also help unfortunate authors. What remains to be seen is whether this reform can really made sufficiently user friendly. I think it can, with the right kind of “logic engineering”. Perhaps we (or I) need to know more about what causes intelligent people to cling to erroneous mathematical arguments? I do think that unfamiliarity with what it would even look like to get to the “absolute bottom” of something, plays a big role. As a logician, I am more familiar with “absolute bottoms” than most mathematical people. 25. August 13, 2010 11:09 am A 3-page “SYNOPSIS OF PROOF” has been posted by Vinay Deolalikar on his HP site: http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_synopsis.pdf • Nimmy permalink August 13, 2010 11:48 am Based on the synopsis (see in particular the foonote) it seems the finite model theory part will be completely rewritten for the new version. Wait and see.. • vloodin permalink August 13, 2010 12:11 pm The synopsis seems promising enough. This guy is going to pull some noses a bit longer. Fixing his “proof” on the fly, he now claims that locality approach (that was so efficiently trashed from many angles on this very forum, and clearly in his private correspondence in 10pt and 12pt) was not really what he meant, but some other property of his obscure model of P computation. I guess he made a mistake of not obscuring some parts of his reasoning enough, but now that can be fixed. There are hints though of what he meant by number of parameters. These really have to do with Gibbs potential representation of a distribution. He even promises a proof that there are exponentially many of those (the part that is entirely missing in drafts so far). If he delivers that, than we will know what he had in mind. But he runs the danger of being precise, and hence allowing some proper scrutiny of his reasoning, and no doubt, new epic thrashing. • kew permalink August 13, 2010 2:31 pm vloodin: Hmm…judging by your many posts personally attacking Deolalikar, I must raise the question; do you have a personal grudge against him? This is no way to behave according to standards (which you apparently hold so dearly), and is frankly quite childish. • WTF permalink August 13, 2010 11:57 am “the final version should be up in 3-4 days here”. Can’t wait. • Conan B. permalink August 13, 2010 12:46 pm I think we are past that stage now. We now discuss the merits and limitations of the overall high-level strategy described by, e.g., Tao and Gowers, independently of the original paper. Maybe something can be done with this seemingly interesting strategy. (Maybe a new barrier?). 26. August 13, 2010 11:54 am It seems (to me) that Terry Tao is giving Vinay Deolalikar (and everyone else) a useful hint when he writes: “Errors are first originating in the finite model theory sections but are only really manifesting themselves when those sections are applied to random k-SAT and thence to the complexity conclusions.” Contemplating Tao’s insight, we see that one repair option for Deolalikar is to restrict P to a weaker model of computation, such that his proof (or a modified version of it) does go through. That suggests a pragmatical line of inquiry that is very natural for engineers: Can we weaken the mathematical definition of the complexity class P (we’ll call the weaker version P’), such that P’ still captures the practical engineering definition of PTIME computing, and yet Deolalikar’s technology—or indeed any proof technology—for P’/NP separation works? As a bonus, any reasonably natural definition of P’ (if such could be constructed) would help explain why P≠NP is so unexpectedly challenging, namely, it’s not quite the right question to ask. We engineers have embraced similar strategies quite often, and with considerable success, e.g., if quantum simulation is hard on Hilbert spaces, then move to a state-space whose geometry is more favorable. In disciplines as diverse as computational fluid dynamics, lattice gauge theory, and density functional theory, this strategy has led to practical and even mathematical successes … so why shouldn’t it work for P≠NP too? • August 13, 2010 12:00 pm That sounds like an interesting thing to try, but it also seems from the comments people have been making that it would be a big challenge to find a definition of P’ that allows you to do linear algebra. • August 13, 2010 12:32 pm Partly seriously and partly in jest, I will point that for us engineers there are two kinds of PTIME linear algebra codes: codes that are documented (or can be) versus codes that are more-or-less hopeless to document. This remark is partly serious in the sense that this distinction is very real to engineers … as the entries in the International Obfuscated C Code Contest (IOCCC) amply demonstrate … not to mention how difficult it proves to extinguish errors even in (to cite a painful example) LAPACK’s singular value routines. One shudders to contemplate what an adversarial version of LAPACK might look like. :) And the remark is partly in jest for simple reason that neither I nor any else (AFAIK) at present knows how to concretely formalize this none-the-less-real distinction. And finally, I should acknowledge that Tim Gowers’ weblog qualifies him among us engineers as yet another lamed-vavnik of computational complexity … meaning, that ofttimes we engineers imagine that we understand what he is writing about, and are grateful for it. • August 13, 2010 2:08 pm PS: Hmmmm … Tim, I will try to phrase my natural-to-engineers questions in the less-natural-to-engineers language of complexity theory … I ask your forgiveness in advance for any solecisms that arise. Q1: Are there algorithms in P that cannot be proven to be in P by any concrete computational resource (in effect, a fixed verification procedure) that is itself in P? Q2: Define P’ to be the exclusion of these (in effect, non-verifiable) algorithms from P; is P’ easier than P to separate from NP? Here the point is that for practical purposes of engineering, P’ is just as useful as P. • August 15, 2010 3:07 pm Like many folks, I’m winding up my own comments on this (wonderful) topic, until such time (in a few weeks?) when some kind of consensus emerges. Responding to (fun!) posts from Raoul Ohio and András Salamon on Scott Aaronson’s blog, I wrote-up a (clumsy) description of an Impagliazzo-style universe of ingeniarius worlds, that is, worlds in which all legal algorithms are provably in P. The bottom line was that Impagliazzo’s worlds—from algorithmica to cryptomania—are perhaps too smart for engineers; while in contrast, ingeniarius worlds are perhaps “dumb, dumber, uselessly dumb” … which describes our engineering conception of computational complexity all right! :) Especially, my sincere appreciation and thanks are extended t0 all who have helped to create, upon this forum and other forums, a community that has helped many folks (me especially) to a broader & better … and infinitely more enjoyable … appreciation of complexity theory. 27. August 13, 2010 12:16 pm 1) Why can’t a few competent theoretical computer scientists in the San Francisco bay area meet with Mr. D., and talk? 2) Raise the objections point blank 3) And report it to the larger community Seems like an effective strategy to figure out the truth, and the cost is minimal. The current strategy of raising objection without a careful reading of the paper seems to be counterproductive to me. The attempts by Terry Tao and others to see if the proof strategy will work at all are very honorable indeed. However, finding a very specific flaw is the most effective way to show that the proof is broken. The only people who have raised very carefully considered objections are Neil and Anuj on the finite model theory part, and Ryan Williams on how Mr. D tries to connect complexity and complicated structure of solution space. Vloodin has raised the objection that parameterization is not well-defined. However, nobody has reported on how Mr. D answered these objection, if at all. Let us ask him point blank in an email, and report his response on this blog. • rjlipton permalink* August 13, 2010 12:55 pm v, I offered early on to talk directly to VInay or to have him visit here (at our expense) so we could help. For a variety of reasons that did not happen. • Chris permalink August 13, 2010 9:52 pm Professor Lipton, That’s a great idea. Pick up the phone, talk to Vinay, then fly him down to Atlanta (and perhaps Terry Tao and a couple of others) — have a small workshop for 2-3 days to thrash out the issues. Great effort! Chris • Anonymous permalink August 13, 2010 10:49 pm I believe Vinay is giving a talk at HP Labs on Sep 1, 2010 on his paper. I am not sure if it is open to public. • sramanujan permalink August 14, 2010 4:11 pm Dr Lipton, Please allow me to implore you to lay down etiquette guidelines for posting opinions on your forum (analogous to those proposed for an unimpeachable mathematical proof by Dr Friedman above). Many self-appointed geniuses here (they know who they are) would have us believe that Vinay is either a Darth Vader or a Bernie Madoff or a parvenu. In UK, these folks could be sued for libel. Let us wait for Vinay to respond. If he has tried to con the math community, HP would fire him in a nanosecond or less – look what happened to their CEO for a far lesser offence. • Anonymous permalink August 13, 2010 1:18 pm Or even, Vinay Deolalikar can directly respond in this blog itself – at least in a hand-waving fashion (till he completes the update on his formal draft) – on why the objections could be overcome. To the majority, it doesn’t sound like they could be circumvented, but if the author himself thinks that way, everybody is happily willing to give him a ground (and he should also give others a chance as well) to bring his side of the arguments to the community’s disposition. Nothing personal going on here – after all, it is a search for one of the most elusive truths facing mankind as a whole for a few decades. • Chris permalink August 13, 2010 9:47 pm Yeah, sure! As if he will respond to emails! You must be crazy to think that he is going to read (and much less, answer) emails these days! Just pick up the phone and call him. 28. Jun Tarui permalink August 13, 2010 12:22 pm Some comments related to the comments by Terence Tao and Tim Gowers (and his blog): I am not sure if the paper’s claim is that for any poly-time-computable Boolean function f on the n-cube, the solution space of f, i.e., f^-1(1) is “poly-log parametrizable.” In fact, for the poly-time-computable predicate R(x, y) that is 1 iff y satisfies x, where x is a k-SAT instance and y is an n-bit vector, the paper seems to claim that the solution space of R(x, * ) is not “poly-log parametriable” for most x’s with respect to some distribution. The paper seems to be, or, may be claiming that (1) if the problem: “Given x, decide if y such that R(x, y) exists” is in P, then the solution space R(x, *) is poly-time samplable (by “extending a partial assignment”, LFP, etc), and (2) any poly-time samplable distribution is not “poly-log parametriable.” Possible claim (2) seems unplausible: Let f, g : {0,1}^n –> {0,1}^n respectively be a truly random function and the pseudorandom function obtained from a poly-time pseudorandom generator G: {0,1}^n –> {0,1}^2n in a standard way. Think of f and g as samplers defining distributions on n-cube: Think of the images of n-cube under f and g as the multisets of size 2^n; e.g, if an n-bit vector y appears 3 times, prob weight of y is 3/2^n. I don’t know what poly-log parametriability exactly means, but it seems reasonable to assume the following: Given a distribution D on n-cube, in terms of samples, deciding if D is (almost) poly-log parametrizable or not can be done in time 2^(n^epsilon). But then this amounts to distinguishing random f vs pseudorandom g, and can’t be done if G is a reasonably strong pseudorandom generator. • August 13, 2010 1:03 pm Ah, yes, my use of the term “solution space” was perhaps sloppy in that respect, it’s more like the fibers { y: R(x,y) = 1 } above the solution space { x: R(x,y)=1 for some y} than the solution space itself. Still, I think it is likely that Tim’s objection can be modified to concoct a polynomially solvable problem with pseudorandom fibres (I suppose this is similar to your point (2)). • Milos Hasan permalink August 13, 2010 1:11 pm I think you meant ‘(2) any poly-time samplable distribution *is* “poly-log parameterizable”’? • August 13, 2010 1:28 pm Actually, on thinking about it a bit I think that Tim’s objection can be trivially modified to deal with this more complicated notion of a solution space. Let f: {0,1}^{2n} -> {0,1} be a pseudorandom circuit. Let R(x,y) be the predicate “f(x,y) = 1″, where x, y are n-bit strings. Then one definitely expects by probabilistic heuristics that for all sufficiently large n that the satisfiability problem “Given x, does there exist y such that R(x,y)?” to have a positive answer, and so this problem is trivially in P (just return “Yes” all the time). But for typical x, the solution spaces {y: R(x,y)} of R(x,*) look pseudorandom, and so I think Tim’s objection still applies here (unless I am still misunderstanding something…) • August 13, 2010 1:48 pm P.S. one could argue in this case that the corresponding random satisfiability problem (in which f is replaced with a random function) is also trivially in P and there is thus no direct need to separate these two examples. But it is not hard to tinker with the problem a bit more to fix this also. For instance let R(x,y) be the predicate “f(x,y)=1 AND f'(x)=1″ where f': {0,1}^n -> {0,1} is another pseudorandom circuit. Then this problem is still in P and the fibers are still pseudorandom. But if we replace both f and f’ by random functions, then the problem is no longer in P, but the solution spaces look indistinguishable. • August 13, 2010 1:53 pm The high-level point here, I guess, is that once one possesses the primitive of a pseudorandom circuit whose output also behaves pseudorandomly, one can start constructing a very large number of examples of computationally feasible problems that are very difficult to distinguish from infeasible problems in their solution space geometry (regardless of how one chooses to define “solution space”) , and which look like an extremely robust barrier to any attempt to separate complexity through a structural analysis of solution spaces. • Istvan permalink August 13, 2010 1:54 pm isn’t (1-(1/2)^n)^(2^n) approximately 1/e? Well, still a fraction of the randomly generated pseudorandom circuits will be in P, as far as I can see, but what can we say about their pseudorandomness if they are selected for some feature (for each x have at least one y)??? • August 13, 2010 2:01 pm Well, OK, we can just give y 2n bits instead of n bits (so f is now a function on {0,1}^3n then), so that the probability of unsatisfiability goes exponentially to zero with n. As I said, there is a lot of flexibility here to design one’s problem to do more or less anything one wants in the solution space geometry… • August 13, 2010 2:04 pm But Deolalikar sais: “It is not only the geometry of the solution space, but the effect that it has on the number of independent parameters required to specify the distribution, that has to be taken into account.” This could in principle exclude the pseudo-random function example. But how? • Istvan permalink August 13, 2010 3:35 pm Thinking a bit more on it, it is still not clear. Actually, you need a pseudorandom generator that builds for each n a pseudorandom circuit, and then the R(x,y) predicate asks for a (whatever function in n) x if there is a (whatever function in n) long y for which the output of the circuit is 1. Is it in P? I doubt. I understand that there is a lot of flexibility here, on the other hand, our algorithm should work for any n, and I’m afraid the trivial solver that always returns with YES will not work. Then you have to build up a pseudorandom circuit, take a fixed x, and… How to continue? If we cannot be sure that the answer is yes, we are in trouble… • August 13, 2010 3:49 pm Actually, now that I look at it more carefully, one does not need to extend the length of y to 2n bits, because I don’t think the 1/e calculation was correct. If R behaves like a random function, then for any n-bit x and y, there is a 1/2 chance that R(x,y)=1, and so for any n-bit x, the solution space { y: R(x,y) = 1 } will be non-empty with probability 1 – 2^{-2^n}. Thus, for any fixed n, the probability that the solution space is non-empty for _all_ x is something like (1 – 2^{-2^n})^{2^n}, which is more than exponentially close to 1. From the Borel-Cantelli lemma we thus expect that the trivial YES solver will work for all sufficiently large n and all n-bit inputs x. (Note that we do not actually have to possess a proof that YES will work; we just need to believe the conjecture that it does in order to disbelieve that one can use the structure of solution spaces to separate complexity classes.) • Istvan permalink August 13, 2010 3:59 pm The point is that you cannot fix n You must have a solver that works for any n… • August 13, 2010 4:04 pm Yes, this is why I invoke the Borel-Cantelli lemma to ensure that (heuristically, at least), the YES solver is going to work for _all_ sufficiently large n. • Istvan permalink August 13, 2010 4:13 pm But you cannot apply the Borel-Cantelli lemma for pseudorandom events, I reckon… Sorry for my ignorance in analysis, and lots of apologizes if I am wrong, but I’m afraid, you even cannot set up a solver based on the Borel-Cantelli lemma, which first infers if n is sufficiently large, if not then do some brute-forse calculation and otherwise returns with YES. I’m afraid the problem is with the definition of sufficiently large. • August 13, 2010 4:37 pm Well, the Borel-Cantelli lemma is being applied at a heuristic level rather than a rigorous one. Consider for instance the standard heuristic argument that there should only be finitely many Fermat primes 2^2^n+1, because (using standard randomness heuristics about primes) each such candidate has a probability about 1/2^n of being prime (this is an oversimplification, but let us use it for sake of argument), and this is an absolutely summable sequence, so by Borel-Cantelli we heuristically conclude that there are only finitely many such primes. Indeed the same argument gives us quite good confidence that there are no such Fermat primes with n > 1000 (say). In a similar vein, the same heuristics give us quite a bit of confidence that the YES solver is going to work for all n > 100o and all inputs. Yes, this is only a heuristic conjecture rather than a rigorous argument, but so is the hypothesis that pseudorandom number generators exist in the first place. • Jun Tarui permalink August 13, 2010 2:44 pm 1. I meant to say: (2) any poly-time samplable distribution *is* “poly-log parameterizable, as pointed out/corrected by Milos Hasan (Thank you, Milos). 2. As to Terry’s comments, I still don’t understand: I think that the fact that pseudorandom fibers of poly-time random-like functions and random fibers of random functions looking very similar is consistent with the paper, as opposed to being a problem for it, because the paper claims that fibers of R(x, y) are *complex* (x: SAT instance, y: n-bit vector; R(x, y) = 1 iff y satisfies x). • August 13, 2010 2:52 pm Yes, but the point is that one can also make the fibers of R(x,y) complex if one replaces SAT by a P problem by using pseudorandom circuit constructions. So there does not appear to be a separation between SAT and P here. • Jun Tarui permalink August 13, 2010 4:37 pm I think the paper says “poly-log parametrizability” follows from the poly-time decidability of a question “Is y a solution?” *and* a question, for any partial assignment s fixing some bits to be 0/1, “Is there a solution extending s (ie, compatible with s)?” In other words, the claim for polylog-parametrizability is being made not for all solution spaces of P-problems, but only for their subclass where the existence of s-compatible solution is also poly-decidable for any partial assignment s. For random-like poly-time functions (and their fibers), compatibility existence questions do not seem be poly-time computable in general. I maybe missing something, and I’m a bit unsure if there is any point in analyzing the proof or the proof strategies further at this point… • August 13, 2010 5:12 pm Well, if one wanted to analyse the solvability of R(x,y)=1 for some pseudorandom circuit R, with x a fixed n-bit string and y already having a partial assignment s, then there are two cases. If there are less than 100 log n (say) bits remaining to assign to y, one can simply use brute force, which takes polynomial time. If there are more than 100 log n bits remaining, then a back-of-the-envelope heuristic computation suggests to me that the YES solver should always work once n is large enough (basically because there is a factor of 2^{-2^{100 log n}} that dominates all 2^{O(n)} entropy losses). I personally agree that there is not much doubt left that the proof strategies here are unlikely to lead to a viable approach to P != NP, but since we’ve come all this way already, we may as well finish the job and make a precise description of the obstruction… • Anonymous Coward permalink August 14, 2010 10:19 pm OK, along a similar line to various attacks on the paper, I will attack Tao’s argument: if a procedure like this worked, could it not be used on SAT itself, solving it in polynomial time? A solution to SAT can, after all, be trivially verified in P. I think the issue here is the density of the solution space–if this density gets exponentially lower in N, then an approach like this must become exponential. Tao’s argument relies on a solution (to the problem of extending a string) being verifiable in P. But being able to verify a solution implies that the problem is in NP, not in P. In other words, he has shown that the paper’s argument can’t substitute NP from NP… Or am I missing something here (I may very well be, since I don’t understand the paper at all). 29. vloodin permalink August 13, 2010 1:33 pm The question which I insisted on, how is number of parameters defined, seems to have been hinted on in his new survey. It is based on Gibbs potential. One idea of how this might work, based on the paper, is the following: Consider the formula f in k-SAT, with m clauses and n variables. Define the following function of T (a real parameter, temperature in stat. physics) Z(T)=\sum{2^n possible values of (x_1…x_n)} e^(-M(x)/T) where M(x) is number of unsatisfied clauses in f(x_1…x_n), 0<=M(x)<=m. We say that solutions are L-parametrizable if there are L terms of the form z_i(T)=e^(-c_i/T) so that we can express Z(T) from these using addition/multiplication, Z(T)=(z_1(T)+z_2(T)+…z_{l1-1}(T))*(z_l1(T)+…+z_{l2-1}(T))*…*(z_lj(T)+…+z_L(T)). Here c_i are some constants, depending on formula f. This defines L precisely, as a function of f, and is related to the solution space. Now apparently L is 2^polylog(n) for k-XORSAT but is exponential for k-SAT. • vloodin permalink August 13, 2010 2:03 pm I am not sure about the definition above, but I believe, at least conceptually, that the elusive “number of parameters” was encoded in the complexity of the partition function Z(T) (which depends on the formula). So, the following conceptual points can be made: (1) “Number of parameters” is something that depends on a k-SAT formula, not on an ensamble of formulas. Given a single instance of a k-SAT problem, we get a well defined function Z(T), and its complexity is measured by “number of parameters” (perhaps a bit more complicated than the one I suggested, but still, depending on function Z(T) of single variable) (2) Any polynomial algorithm gives rise to simple Z(T). Most generally, this means that we can compute Z(T) in polynomial time, but probably is much more than this. (3) General k-SAT formulas have Z(T) that are hard to compute. On a conceptual level, this is not necessarily a bad idea. We could define complexity of Z(T) in some way so that all polynomial algorithms, via sampling, would allow us to compute Z(T) easily. This should be manifested in a small number of “parameters” that measure complexity of Z(T). On the other side, complexity of Z(T) (“number of parameters”) should be defined in a way simple enough so that we can determine it for typical k-SAT formula in relevant phase to be exponential, i.e. greater than possible if P=NP. • August 13, 2010 2:11 pm As I commented above, I do not think one can define these “parameters” via “number of steps needed to compute Z(T)”. Z(T) is a counting problem, and many polynomial decision problems have #P-complete counting problems (e.g. matching). If he is using something like this, and if he is right, then the paper should be called P!=#P, isn’t it? • vloodin permalink August 13, 2010 2:23 pm He does say at one point in the paper that number of parameters in interactions with potentials which are k-ary, grows as c^k, rather than as c^n (n number of variables). So “parameters” seem to be more like these individual potentials, than variables x_1… x_n. Even if complexity of Z(T) were #P complete, that would not necessarily invalidate this approach. He wants to show that if we can solve formulas with P, THEN we can compute Z(T) efficiently (in essence, by computing contributions from each of these “parameters”). Than this would contradict the fact that in general Z(T) is hard to compute. But presumably, this is shown by mere counting of the “parameters” – there should be exponentially many of them. Yet in P algorithms, there are much fewer of these. So a question is: for k-XORSAT, can we compute Z(T) efficiently (is it in this case #P complete as well?). If we can, than there might be something in his approach. If we cannot, than one needs probably some other measure than complexity of Z(T). • August 13, 2010 3:15 pm Computing Z(T) for any value of T in XORSAT is indeed very difficult. In fact, when a XOR-SAT formula is UNSAT, finding the assignment with the lowest cost is NP-HARD… It is only the decision problem SAT/UNSAT that is easy for XORSAT formulas. • vloodin permalink August 13, 2010 4:24 pm This is why getting a precise definition is important. It would allow one to test the whole strategy in a straightforward way. It is possible that his definition of number of parameters is such that this number is exponentially large for k-XORSAT, despite of what he thinks. After all, this would only make sense since statistical behavior of k-SAT and k-XORSAT has same phases, and many similarities. That would mean that he has an error on the other side of the argument (where currently flaws are located), i.e. that having small number of parameters does not follow from P. He has some class much weaker than P that gives Z(T) simple structure. That is, if anything at all follows from his paper. We can wait and see what the new draft brings. Then it will be clear which of the speculations was right, but despite of what he says, it doesn’t seem likely that he has a way to capture all of P, and get small number of “parameters”, since this would probably give a simple Z(T). So, as we all suspected, there is little chance for this approach to be saved. • Istvan permalink August 14, 2010 12:57 pm Lenka, do we know that P != #P or would it be a new result? If it is new, and Deolalikar’s approach might save that it would be still nice! • Ørjan Johansen permalink August 15, 2010 6:11 pm Istvan: I believe that P != #P is stronger than P != PSPACE (you can count the solutions by iterating through all of them in polynomial space), which is not known. 30. August 13, 2010 1:59 pm From point 8. of Deolalikar’s synopsis: “We prove that each set of values the core takes in the exponentially many clusters must be specified as an independent parameter of the joint distribution”. If there is anybody out there having a clue of what is the definition of the “independent parameters”, then please save me the headache :)!!!!!! • vloodin permalink August 13, 2010 2:12 pm No one seems to know that. I have been asking this question for a while. Yet, it seems to me that this number of parameters measures complexity of the partition function, Z(T). It depends on a single formula. The distribution is just the Boltzmann distribution, but what matters here is complexity of Z(T). However, the details of this are not given. So, there is some measure of complexity of partition functions, the elusive “number of parameters”, call it P(Z). It is supposed to be 2^polylog on instances of k-XORSAT but exponential on typical hard instances of k-SAT. • August 13, 2010 11:36 pm After rereading section 2, independent parameters appear to refer to how many values are needed to describe the joint distribution. The standard analogue in directed graphical models would be the size of the tables that form the conditional probability distributions. For example, assuming three variables X,Y, and Z. If all are independent then P(x,y,z)=P(x)P(y)P(z) and would require three parameters because each is over one variable. This would be the case of a completely disconnected graph. But if this DAG was fully connected then, P(x,y,z)=P(x)P(y|x)P(z|xy), where P(z|xy) is over three variables and would require 4 parameters and the total distribution would take 7. So it takes 2^(c-1) parameters to specify each factor where c is the number of variables the factor is over. The analogue for an undirected graph would be, given size of the cliques in the graph, each clique of size c would require 2^c parameters. The analog for undirected models would b • vloodin permalink August 14, 2010 1:40 am Yes, but what probability distributions are parametrized/considered. I can understand in Markov network, you would have certain family of distributions satisfying Markov conditional independence property. But what about the solution space? What probability distribution family is parametrized there? He speaks of potentials, and Gibbs energy, so presumably you put Boltzmann distribution with parameter T. The parameters correspond to cliques or potentials that one can define arbitrarily. It is still unclear what he means, and given that this is the central issue in his “proof”, one has to see what he is going to say in his new draft. I suspect that his definition will imply that if there are few parameters for a given formula, then Z(T) can be computed easily. But then it would easily follow that he did not capture all of P, since k-XORSAT has NP-hard function Z(T). 31. Charanjit Jutla permalink August 13, 2010 2:05 pm Is it easier to prove Reimann hypothesis? I often tell people (not many, luckily) who want to prove P=NP or P/=NP that they are better off trying to prove the Reimann hypothesis. The problem with P/=NP is that it is not enough to prove great combinatorial results (which is most likely what is required to prove the Reimann hypothesis, whether it uses dazzling analytic tools or not), one has to first tackle the issue of what computation in P-time means. All known equivalent formulations of PTIME allow such generality, as well as the nature of computation to allow easy relativization, that some people think it is not even possible to separate P-time computation from others using proof techniques (and even very general fragments of logic) known to humans so far. As far as P=NP is concerned, suppose somone comes up with an algorithm for their favorite problem say SAT. Then, one realizes that they have a new algorithm for not just other NP-hard problems but also simpler problems like primality testing. Now, reducing Primality testing to NP requires just the minimal of axioms about numbers, for example associativity. So then, you have an algorithm for testing primality which uses only simple properties of numbers like associativity, and the new SAT algorithm. If the new SAT algorithm did not discover some profound properties of numbers but relied on logical formulations and combinatorics thereof, then, it means primality testing is at the same level…hence possibly you have a new insight for Reimann hypothesis. • August 14, 2010 1:10 am In order to argue that you are “better off trying to prove the Riemann hypothesis,” don’t you need to assume that P is equal to NP? If they’re not equal, how does that impact RH? • August 14, 2010 5:49 am In P/=NP case, I was just remarking that this problem has some results which lead us to believe it is independent of fragments of Number theory like bounded arithmetic (although, no definitive such theorem has yet be proven…but serious attempts and some progress has been made). This is not the case with RH. • August 14, 2010 6:04 am Paradoxically, I should add though that it is probably easier to prove that NP/=P is independent of certain fragments of bounded arithmetic than proving RH. 32. Ryan Williams permalink August 13, 2010 3:03 pm Hi all, I have posted what I think is a definitive objection to any proof separating P from NP that relies on the structure of the solution spaces of SAT, and tries to argue that no problem in P has this kind of solution space. The basic idea is that every satisfiable formula’s solution space can be mapped to the solution space of a formula satisfied by the all-zeroes assignment, which is always trivially solvable. This map exactly preserves the number of variables and all distances between satisfying assignments. It is of course non-uniform, but that doesn’t matter: for every infinite collection of satisfiable formulas there’s an infinite collection of formulas satisfied by all-zeroes which has exactly the same solution space structure. Details are here: http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper#Issues_with_random_k-SAT Thanks to Jeremiah Blocki for helping me sort this out. • Istvan permalink August 13, 2010 3:55 pm Isn’t it tautology? I mean if we know that there is a satisfying formula, then of course the answer is “yes, satisfiable”, without transforming the function… On the other hand, if you are given a formula that is not satisfiable, your predictor will “YES, satisfiable”, and gets wrong… The issue here is that we have a problem which is in P, meaning that for every problem instance there is an algorithm which solves the decision problem in polynom time. Deolalikar’s claim is that if the problem is in P, then every problem instance has a simple solution space. • Ryan Williams permalink August 13, 2010 4:06 pm A problem is in P when there is a polynomial time algorithm such that for every problem instance it is decided by the algorithm. Your quantifiers are backwards. I know what Deolalikar is claiming. I am proving that for any structure you have in the solution space of any SAT instance, you have identical structure in the solution space of some SAT0 instance. That is, there are problem instances with the SAME solution space structure. If you want more formality, the formal definition of “same structure” is that there is a distance-preserving isomorphism between solution spaces. So no proof of P vs NP can work by looking at the structure of solution spaces. • Istvan permalink August 13, 2010 4:27 pm Something is still fishy here. You force not to give hard problem instances in some sense. How could you achieve this? If you gave me a problem instance: !@£@£%$^!$^!^!£$QFS@£%^T@$I can say, NO, I am a linear equation solver, this is not a linear equation. How could your solver decide if it got a normal form that is not satisfiable or a normal form that you gave accidentally without transforming, otherwise satisfiable but the all-0 input is not accepted? • Ryan Williams permalink August 13, 2010 4:35 pm Istvan, there is nothing fishy going on. The map from SAT to SAT0 is definitely not polynomial-time computable, if that’s what you’re worried about! Before we even start talking about CNF satisfiability, we must agree on a formal encoding of CNF formulas. I assume that any prospective algorithm knows whatever encoding is agreed upon, and if it doesn’t see something in that encoding it just rejects. • Istvan permalink August 13, 2010 4:47 pm No, I am not worrying about the transformation. Rather, my concern is that what is SAT0 exactly? We agree in the encoding, fine. Then I gave you the problem instance: (x_1 OR NOT x_2 OR X_3 ) AND (X_123 OR … etc. And then your prospective algorithm will ask me: hey, if this formula is satisfiable, did not you forget to transform as we agree? And if I am rude, and will not answer, and might happen that I did not transform it, your prospective algorithm will be in trouble. I will be also in trouble if not going to bed, the kids get up early so do I have to. Let’s continue tomorrow, sorry. • Istvan permalink August 14, 2010 3:02 am OK, now I see why Ryan’s idea must work: Consider a NDTM solving SAT0 problems. It must accept all satisfiable SAT0 instances and reject all others. Then it first must infer if the all-0 assignment is a solution to the query, otherwise it could accept satisfiable normal forms coming from SAT\SAT0. At this point you can see that there might be several NDTM solving SAT0, and we are in trouble as we do not know whose acceptance path distribution we should consider. One might argue that there is an unlimited number of NDTMs solving SAT0, and one might come up with the idea to factorize the NDTMs based on they accept the same set of inputs. Then one might argue to look at the ‘simplest’ one. But amongst the equivalent NDTMs solving SAT0, the simplest one is the DTM solving SAT0. Hence one cannot prove that P!=NP by considering the solution space of the usual NDTM solving SATs, as one parallel must prove that there is no equivalent NDTM with a ‘simpler’ solution space. However, this later on its own would prove that SAT is not in P… So I think, the answer for Terry’s (3) point is a definite NO. • Albert Atserias permalink August 13, 2010 4:17 pm This is very very interesting. Thanks Ryan. • Conan B. permalink August 13, 2010 4:33 pm Yes, this is nice. This also seems to rule out even weaker separations by the same method: even uniform-AC^0 \neq NP seems to be ruled out! Am I correct? • Ryan Williams permalink August 13, 2010 4:36 pm Conan: Yep! See the wiki. • rjlipton permalink* August 13, 2010 4:42 pm Ryan, Thanks for this great comment. • Marko Amnell permalink August 13, 2010 5:17 pm Ryan Williams’s argument is very interesting, and Russell Impagliazzo made a related comment a couple of days ago that stuck in my mind: “Here’s why I think this paper is a dead-end, not an interesting attempt that will reveal new insights. As Terry says in his post, the general approach is to try to characterize hard instances of search problems by the structure of their solution spaces. This general issue has been given as an intuition many times, most notably and precisely by Aclioptas in a series of lectures I’ve heard over the past few years. (Probably co-authors of his should also be cited, but I’ve Aclioptas give the talks.) “The problem is that this intuition is far too ambitious. It is talking about what makes INSTANCES hard, not about what makes PROBLEMS hard. Since in say, non-uniform models, individual instances or small sets of instances are not hard, this seems to be a dead-end. There is a loophole in this paper, in that he’s talking about the problem of extending a given partial assignment. But still, you can construct artificial easy instances so that the solution space has any particular structure. That solutions fall in well-separated clusters cannot really imply that the search problem is hard. Take any instance with exponentially many solutions and perform a random linear transformation on the solution space, so that solution y is “coded” by Ay. Then the complexity of search hasn’t really changed, but the solution space is well-separated. So the characterization this paper is attempting does not seem to me to be about the right category of object.” http://tinyurl.com/349fbzu • August 13, 2010 5:19 pm Ryan, I think this shifting trick of yours and Jeremiah preserves the P-nature (or lack thereof) of the unrestricted SAT problem, but not for the restricted SAT problem in which some of the literals are already fixed. (In particular, the most extreme case in which _all_ the literals are fixed now morphs from a trivial problem to what looks to be an infeasible one, as it requires knowledge of the non-uniform shift (A_1,…,A_n).) As I just learned from Jun Tarui in his comment above, Deolalikar’s argument is only supposed to obtain a contradiction if both the unrestricted problem and its restrictions are both in P. But perhaps some variant of this trick can get around this issue… • rjlipton permalink* August 13, 2010 5:28 pm Ryan, Really like this simple but clever idea. • Albert Atserias permalink August 13, 2010 5:48 pm For the skeptical about Ryan’s idea, let me try to make my two cents: Ryan showed that there exists a mapping that takes an arbitrary satisfiable formula F with n (free) variables and maps it to a formula F’ also with n (free) variables with the following two properties: 1. the all-zero assignment satisfies F’, 2. there is a Hamming-distance-preserving permutation of {0,1}^n that maps the set of satisfying assignments of F to the set of satisfying assignments of F’. Let’s show that this rules out any proof-strategy of P != NP that depends only on distance-based landscapes of solutions. Let’s say a property of solution spaces A is landscape-invariant if the following holds: if S is the solution space of a SAT-instance with n variables that has property A and f is a Hamming-distance-preserving permutation of {0,1}^n, then f(S) also has property A. Now, suppose that we managed to show that P != NP by proving the following two claims: Claim 1: For every polynomial-time algorithm, every SAT-instance accepted by the algorithm has a solution space that has property A. Claim 2: There exists a satisfiable SAT-instance whose solution space does not have property A. In such a case we can argue that property A is not landscape-invariant. Here is how: Let F be the SAT-instance from Claim 2. Look at F’, the transformed F, and apply Claim 1 to the trivial algorithm that always says “yes” and F’, which is obviously accepted by the algorithm. Claim 1 tells us that the solution space of F’ has property A. Since there is a Hamming-distance-preserving permutation of {0,1}^n that maps the solution space of F to the solution space of F’, we must conclude that property A is not landscape-invariant. • Albert Atserias permalink August 13, 2010 6:22 pm I can see that there must be something ridiculously wrong in this thing I wrote, but I can’t find what. Sorry for that. I guess I am just too tired as it’s getting late here. I’ll stop here for today. • Russell Impagliazzo permalink August 13, 2010 10:12 pm I think you may be overlooking a subtlety. I think Vinod was claiming that the structure of a solution space implies that it is hard, given an arbitrary partial solution, to extend it to a complete solution. (Not that it is just hard to find a single solution.) This is more subtle to refute. If by partial solution, it means the values of consecutive bits from x_1,..x_i, then we can construct examples like you said. For example, let \Phi be from the distribution in question, and \Psi be any formula with a unique solution , say 0. Then the solution space for \Phi(x) \lor \Psi(y) is isomorphic to that for \Phi but given any consecutive partial solution, we can fill in the rest. (Simply assign unassigned bits 0. If we were not given all of x, this is a solution. If we are, it is a solution if and only if the given bits of y are all 0 if and only if a solution extending the partial solution exists.) But I haven’t worked out any example where, given any subset of bits, possibly non-consecutive, it is easy to extend it to a complete solution (if there is one). Russell • V Vinay permalink August 13, 2010 10:23 pm Ryan, a neat way to turn the tables, so to speak. Wonderful! • vloodin permalink August 14, 2010 2:08 am This trick is a tad too cheap. It doesn’t show anything. For in the paper, it was important to be able to extend partial assignments, and argument was from that. Yet SAT0 (which I guess is just a trivial algorithm checking weather assignment x_i=0 holds for the formula) has no such ability. In fact, there is no “solution space” for SAT0 (if SAT0 is anything other than checking assignment x_i=0, then I guess this whole comment is off the mark, but I assume that is what SAT0 is). What would a solution to a problem that checks a particular instance x_i=0 in a CNF be? It does not have any unassigned variables, it just checks one assignment and it is all that it does. We cannot determine all satisfying instances of a given CNF formula using SAT0. SAT0 is a closed formula, there is no existential quantifier there, and hence no solution space. This can be contrasted with k-SAT, that looks weather there is any satisfying assignment. There is existential quantifier and instances which give positive answer are the solutions. Hence, the analogy does not exist. For k-XORSAT we do have an analogy, since there is existential quantifier involved, and it gives rise to a solution space. But SAT0 is just a completely trivial algorithm that checks one particular assignment, and can not be used to replace k-SAT in the argument from the paper. It is of the wrong type and conceptually does not fit. • Istvan permalink August 14, 2010 3:21 am See my comment above. Yes, you are right, the problem is with the precise definition of solution space. For me, the easiest approach to consider solution spaces as the space of acceptance path for a NDTM. But it turns out that there are an unlimited number of equivalent NDTMs accepting the same inputs, and in each equivalent class, there are members with complicated solution space. To see this, consider the EDGE decision problem which asks if there is an edge in a graph. Surely, you can build up a NDTM that checks all pairs of vertices if they in the edge set. Fine, now extend this NDTM which does the following: if there is an edge connecting a pair of nodes, then do not stop, but split to two paths. The first is the safe ending, it ends with YES in the next step after the branching, the other continues as a NDTM that solves the maximum match problem. Then this NDTM has a complicated solution space, in fact it is in #P-complete. You might argue that it is sufficient to look at the solution space of the simplest NDTM, but in fact, if a problem is in P, then there is DTM which has only one acceptance path. So now, if you would like to separate P from NP based on the solution space of the ‘usual’ NDTM representation of the kSAT problem that has exactly one acceptance path for each satisfying assignment, then you also must prove that there is no equivalent NDTM with simpler solution space. However, this later on its own would prove that P!=NP. • Ryan Williams permalink August 14, 2010 11:52 am vloodin, you are missing the point. I am taking the solution space for SAT0 to be just the set of possible satisfying assignments to such a formula. That is the same solution space we have been talking about all along, for k-SAT, for XOR-SAT, for 2-SAT, etc. The goal has been to find polynomial time solvable problems with “hard” solution spaces. I am saying that for SAT0 that space can be just as “complex” as that of any usual SAT formula. I see no reason why one should “discriminate” against all the solution space structure arising in a SAT0 formula simply because it happens to be satisfied by all-zeroes. That said, the issue that Terry and Russell raised of extending partial assignments and keeping the problem inside P is tricky. I don’t know if this is necessary to show, though. There is some measure of how complicated the solution space of random k-SAT is, and all I am saying is that, under any reasonable measure, the “random SAT0″ solution space should require at least the same measure. Just because all-zeroes shows up among the satisfying assignments, I don’t see why that makes the entire distribution trivial. The fact that, if you extend a partial assignment to a SAT0 instance with some ones, then the problem becomes SAT again, seems to only help the case. You are right that much of the discussion has been hand-waving, but that is inevitable when you are trying to prove a “meta-theorem” showing that no approach to P vs NP can work in some generic manner. Think of this way: “you must hand-wave before you learn to fly”. If you would like a non-hand-waving comment, here goes. Strictly speaking, the “solution space” to an arbitrary problem inside of NP (and thus inside of P as well) is not well-defined, because it depends on which verifier you use to check witnesses. We are already giving significant liberties to the author in assuming that such a space does exist, independently of a verifier. This is related to what Istvan says above. • August 14, 2010 12:58 pm I tried to give some structure for “well-defining” things in my comment back here. If one steps back from a “property of solution spaces” to a hardness predicate, then one can quantify over verifier predicates R(x,y) defining the solution space. Likewise, quantifying over distributions D on the instance space may help pin down the “phases” the paper talks about. My effort doesn’t digest details like the “ENSP” model, but I think a “switch” from distributions on one space to induced distributions on the other is involved somewhere, maybe multiple times. One other idle thought, which perhaps Terry has picked up on: does working inside a proof where “P = NP” is taken as a for-contradiction assumption help with issues of “keeping the problem in P” as you say? Dick and I have been interested generally in whether/how-far this kind of device can matter. 33. Anonymous permalink August 13, 2010 4:22 pm The blame for wasting important people’s valuable time lies not only with the author of the bogus proof but also with the bloggers who prematurely made such a big deal of this story. Any reason for doing that? • Istvan permalink August 13, 2010 4:34 pm The only reason, IMO, is that nobody ever thought if the separation of P from NP can be based on looking at the solution space. We are not talking about Deolalikar’s work (at least, I got headache reading too much hand-wawing, so stopped considering reading it), but I guess the discussion here is about if any proof trying to separate P from NP could work by looking at the solution space. If so, it opens some avenue for brave guys trying to prove P?NP, if not, then we have another barrier here, which might be also interesting. • rjlipton permalink* August 13, 2010 4:43 pm I do not think that this is wasted time. I think we have learned some math, and also learned quite a bit about the social aspects of solving a big problem in the web world we now live in. • Conan B. permalink August 13, 2010 4:49 pm Yes, I agree. These are quite interesting and inspiring discussions. • Anonymous permalink August 13, 2010 4:59 pm I completely agree. This collaboration has been a great knowledge booster. It yet again showed what that P=NP beast. The wiki is a good collection of resource materials. Dick, you can’t be thanked enough for the polite initiative that you took. I am surprised to see no mention of you or your blog in any of VDs writeup/webpage. Not that you or me, for that matter, care much, but no mention of this big effort does not just sound right. • Go Vinay (and Dick)! permalink August 13, 2010 5:12 pm I also want to thank Dick Lipton for the monumental effort. I am very thankful not only for the things already said (things learned, putting N != NP in the spotlight, etc) but also for something else that many researchers here take for granted: this was a public showing of the rigorous peer-review process that many people, specially aspiring researchers, non graduate students and amateur scientists, are unfamiliar with. For somebody attending a good university in his/her country of origin but where top notch research is not conducted, this whole exercise provided a glimpse about what’s like doing cutting edge work. Hopefully that might be a motivation to pursue a career in math or science later on. Kudos Mr Lipton! • Robert permalink August 13, 2010 6:32 pm On the contrary, nobody’s time was wasted because THIS IS WHAT WE DO EVERYDAY. Exchanging ideas with colleagues in order to learn more about the nature of computation? CHECK Promoting our field to the public by showing how fascinating these questions can be? CHECK Collectively peer reviewing and correcting each other’s mistakes? CHECK This is probably the best promotion of theoretical computer science in a loong time, and unlike some dumb movie it wasn’t diluted or watered down. Thanks Vinay! • Random permalink August 13, 2010 10:58 pm Indeed, it must be an exciting time to be a student. (I find it fun even just being a bystander.) The internet has changed the way things are done a bit, but not so much after all, this is still refereeing and animated discussions etc. But it has opened up the process for outsiders to see, and I hope it will enlighten the next generations. (I was an undergrad during Wiles’s proof, and though we did hear about it some, I remained rather naive about research for a while after that.) 34. Istvan permalink August 13, 2010 4:38 pm Sorry guys, it is midnight here in Europe, so I have to go. Thanks for all participating in the discussion, it was really nice! 35. Paul Beame permalink August 13, 2010 5:46 pm One has to understand what Steve Cook meant by “relatively serious attempt” . The volume of obviously completely baseless “proofs” of P=NP or P!=NP we have all seen is enormous and you can imagine how much larger his inbox on the subject has been. Quick kills usually take a matter of seconds or minutes to find and there are some stock replies that can be sent out based on these. The last time I can recall that there was any argument, claiming either answer, that seemed worth exploring was one by Swart in 1984 which claimed that P=NP because he claimed that one could solve the TSP by creating an LP whose variables were indexed by 4-tuples (or later 6-tuples) of vertices, whose resulting extremal points were integral. It seemed highly unlikely but one could not dismiss it out of hand. There was considerable back and forth, with the number of indices growing. The final nail in the coffin was a lovely paper by Yannakakis, using communication complexity of all tools, to show that any such approach was bound to fail. This paper inspired people like Lovasz and Schrijver to consider the properties of of other methods for lifting LP-relaxations and SDP-relaxations of 01-IP problems. In the end, something useful did emerge from the community’s need to deal with the faulty argument. It remains to be seen if there is anything to be gained from the current paper, assuming a similar failure – which very strongly seems to be the case. (At least Swart’s paper had formal statements that could be checked, though it was fuzzy in some details.) I expect that it will focus some serious research, if only for the following reason: Given the current state of publicity, it is possible that it will generate an entirely new class of faulty arguments by a much broader group of people and we will need some like Yannakakis to provide a nail in the coffin in the approach in order to deal quickly with its inevitable imitators. • math grad student permalink August 13, 2010 6:40 pm Well, the problem is that, when one is making such (non-negative to say the least) comment, it is not terribly difficult to predict the reaction of all the journalists and pseudo academics like Greg Baker (whose research interest focuses on how to enforce his authority when dealing with “problem undergrads”). • Conan B. permalink August 13, 2010 7:32 pm Actually, the original statement was: “this appears to be a relatively serious claim to have solved…”. This puts two reservations before the word “serious”, i.e., “appears to be” and “relatively”. Let’s assume that each reservation decreases the statement coming after it by a factor of 1/3. So we end up with the statement saying that “the paper is 1/9-serious”. Which is somewhat accurate. • math grad student permalink August 13, 2010 7:51 pm It is indeed accurate — especially if his comment only stays within TCS. As anyone would not expect such proof to be confirmed within two days. But it became disastrous after it’s disclosed to the public (again thanks to Greg Baker…), you know what journalists are like, any slightest acknowledgement gets massively amplified… • Anonymous permalink August 14, 2010 12:03 am In case anybody is wondering, Greg Baker has: B.Sc., Queen’s University, 1998 M.Sc., Simon Fraser University, 2000 and is a lecturer at Simon Fraser. LOL. 36. Sanjay Chawla permalink August 13, 2010 6:15 pm Based on the synopsis it seems that the alternate perspective posted by Leonid Gurvits on the wiki captures Vinay’s ideas. Basically using FP(LFO) the proof generates a solution space of SAT whose distribution is described by poly(log n) parameters. Immerman has pointed out technical flaws in the use of FP(LFO) and it appears Vinay thinks he can address them. Coming from the NP side, the proof claim (roughly) that this violates the c^{n} lower bound for SAT for k >9. Ryan Williams has a “meta” argument against this. However, the claim made by Vinay is quite concrete: “In the hard phase of for random k-SAT, for k> 9, the number of independent parameters required to specify the distribution is c^n.” Is this true? • anon permalink August 13, 2010 6:34 pm What about the following question: Is there any new theorem proved correctly in Vinay’s paper? If yes, it could be useful to filter them out and put them on the wiki for future reference. • Ryan Williams permalink August 13, 2010 7:21 pm Suppose I assume “In the hard phase of for random k-SAT, for k> 9, the number of independent parameters required to specify the distribution is c^n.” What I don’t understand is: why doesn’t it also take c^n independent parameters to specify the distribution of a “random” SAT0 formula (a formula satisfied by the all-zeroes assignment), under the following distribution? First, I pick a random k-SAT formula. Whatever distribution you want to use, I will use it. Then I choose a satisfying assignment of that k-SAT formula uniformly at random, call it (A1…An). Then the transformation (seen in the wiki) to SAT0 is performed. We now have a random SAT0 instance, and its solution space is just as “complicated” as the original k-SAT formula… except for the little detail that the formula’s trivially satisfiable in polynomial time by trying the all-zero assignment. How can you “specify the distribution” of these SAT0 instances with less information? The fact that the all-zeroes assignment happens to be included in the distribution looks totally irrelevant to the complexity of specifying the distribution. • Sanjay Chawla permalink August 13, 2010 10:37 pm To finally dismiss the proof (from the NP side) it will have to be demonstrated that if F is the distribution of the solution space of random K-SAT and G is the distribution of solution space of SAT0, then F=G in distribution. Perhaps you can show that the Kullback-Liebler divergence which is the sum of F_iLog(F_i/G_i) indexed over the histogram bins is zero. At the moment your transformation is injective and not necessarily surjective so the distributions might not be the same (even though the solution space of K-SAT is contained in SATo) as the probabilities have to add up to 1). • vloodin permalink August 14, 2010 3:19 am This argument with SAT0 is really not showing anything. But it is perhaps a rebuttal that this paper deserves – sloppy paper with superficial arguments and lots of hand waving that are in fact wrong and off the mark should have at least some rebuttals of the similar quality. Cosmic justice, if you like. • vloodin permalink August 14, 2010 3:14 am First, we do not know what “number of parameters” means. Secondly, it seems that under any reasonable interpretation of “number of parameters”, not only k-SAT formulas will require exponential number of them, but k-XORSAT also. So, the part of the argument which fails is on the side that each algorithm in P gives rise to a small number of parameters. Then again, we do not know what is his number of parameters, and what are distributions that he parametrizes. That is one of the many (but perhaps the most important) places where the author waves his hands and confuses the naive audience. 37. Bill Kaminsky permalink August 13, 2010 6:18 pm Please pardon me if the following is an ignorant comment, but here goes: I’m a lowly PhD physics student at MIT. :) Specifically, I am a PhD student who is very concerned with how well classical and quantum annealing algorithms approximate the solutions to random constraint satisfaction problems. Now when one investigates random local search heuristics like classical simulated annealing or quantum annealing, one never expects a constrained-minimization problem to be easy just because it has a single global minimum. One needs to know how many local minima there are that could trap your search algorithm. As such, I agree with the many critiques of Deolalikar’s proof that say it’s a wrong-headed goal to separate P and NP by pointing to the “complex structure” in the “solution space” of random k-SAT for k \geq 9. While it’s still a little unclear to me what exactly Deolalikar means by “solution space”, it seems that he at least sometimes means “solution space” simply as the set of assignments that satisfy all the k-SAT clauses. If so, then as many others have mentioned, it’s quite obvious “complex structure” of “solution space” is neither necessary nor sufficient to prevent polynomial-time solution (e.g., one reason it’s unnecessary for computational intractability is Valiant and Vazirani’s randomized reduction of SAT to SAT-with-a-Guarantee-that-There’s-At-Most-1-Solution). But what if we generalize this notion of the complexity of parametrizing solution space? To be specific, what if we instead investigate the “complexity” of describing the distribution of the number of violated clauses over all possible assignments (thus caring not only about global minima but local minima too, and all those pesky maxima and saddles as well providing barriers between the local and global minima)? If we did this, how many of the random k-SAT critiques of Deolalikar’s proof could then be circumvented? (Granted, it might take major new ideas to figure out the “complexity” of parametrizing the distribution of violated clauses over all possible assignments). Thanks, Bill Kaminsky 38. August 13, 2010 7:41 pm With respect to the conversation that has unfolded on this blog, some people are still thinking in terms of what went wrong and how a repeat can be prevented. I can’t fathom this perspective. Although I am entirely unqualified to comment on Deolalikar’s paper, I have an overwhelmingly positive attitude toward the discussion that has transpired here in its wake. What I have seen on this blog is a process of immense power and beauty. Everyone should pay close attention to what Professor Lipton has facilitated, in hopes of making it happen repeatedly in the future. • asdf permalink August 14, 2010 2:43 pm Fritz, imagine a hiker is injured and stranded on a mountain top. A bunch of mountain climbers from the region drop everything they are doing and mount a brilliantly coordinated rescue effort which pulls off a number of amazing technical feats in a very short time in order to get the hiker off the mountain. Do we say it was a beautiful and exhilarating experience for the rescuers? OK, maybe we do. Would they say “let’s do it again” in the sense that people getting stranded (because they were improperly trained and didn’t bring the right equipment, say) is a good thing? Not the same question. • August 14, 2010 9:30 pm Nice analogy, except that mountain-climbing rescue efforts are a burden on society and benefit nobody other than the person being rescued. The present discussion, in contrast, benefits the participants and onlookers regardless of what it does for Deolalikar. At this point, with the proof seriously damaged if not completely busted, nobody is neglecting activities they would rather be doing in order to participate here out of a sense of social obligation. Clearly the participants are getting so much out of it that they volunteer their time gladly. It is misplaced for observers to fret about the “wasted” efforts of people are engaging in joyous exploration and producing concrete, valuable insights. • ravi permalink August 17, 2010 1:43 pm Sorry but this analogy is incorrect. Someone stranded atop a mountain is in mortal risk and it is ethically imperative for rescuers to make an effort to reach him. No such ethical compunction exists in the case of this proof. A better analogy would be to someone who, having reached the mountain top, sends photographs to other climbers suggesting that they come take a look as well. Should they choose to follow his call (which, once again, they are not compelled to do), and find later that the promised panorama is not in their opinion up to the billing, surely the blame does not lie with the author. Especially if, in the effort of scaling the mountain, much thought-provoking conversation (or mountaineering equivalent result) is obtained. In fact, the analogy is useful in a very different sense: The commentor (asdf) has correctly identified that what is being promised here is the mountaintop (from a mountaineer who is credentialed enough to have possibly gotten there). 39. TCS grad permalink August 14, 2010 12:39 am Assuming that D is able to answer all the questions raised here and else where, and that the proof does go through, what happens to complexity theory after that? I was just wondering… • Ryan Williams permalink August 14, 2010 3:14 pm Once P neq NP is proved, then true understanding in complexity theory will really begin, rather than end. It would make complexity theory an extremely hot area, because P neq NP is really among the “first” lower bound questions one can ask. Many people really think that SAT needs exponential time, and P neq NP is only the first step towards proving this. Right now one might say that complexity theory is in a kind of “prehistory”, where much of the work consists of proving many conditional statements of the form “If this lower bound holds up then this other lower bound holds”, “if this upper bound holds then this lower bound holds”, and so on. There are some unconditional lower bounds but they are quite rare. One idea behind building this catalog of relationships is that, with each new relationship found, the first gigantic breakthrough will have an even bigger “ripple effect” through complexity than before. Also, the complexity theory of polynomial-time solvable problems (which is covered by much of parameterized complexity theory) has hardly been touched. Fix k \geq 3. Does the k-clique problem really require n^{Omega(k)} time? It seems only harder to prove tight polynomial lower bounds on polynomial-time solvable problems. 40. August 14, 2010 1:33 am This will not be well received, and my late father would not approve (even if his old friend Steve Kleene would probably chuckle a bit), but I think it should just be put out there for the record. This “proof” is fundamentally non-constructive. As it is, it appears not to be correct anyway for reasons that Neil Immermann and various others have pointed out. But, even if their issues were somehow to be overcome, there would remain doubt from the point of view of a deeper critique. For the record, I have always found proofs by contradiction to be elegant and beautiful, even as I am unfortunately aware of their ultimate limitations. • August 14, 2010 5:37 am There are some powerful automatic conversion theorems in mathematical logic to the effect that if a sentence of a certain form is provable in the usual sense then it is provable constructively. Of course, these terms need to be defined for the purpose of this family of theorems. Generally speaking, this family of general results takes the following form, for a large variety of axiom systems T. Let T- be the constructive (intuitonistic) form of T, where we simply require that the proofs avoid the law of excluded middle (in the usual sense as formulated by Arend Heyting). It is known that under general conditions on T, any AE sentence provable in T is already provable in T-, and there is an explicit conversion method for this, known as the A-translation. Here AE means (for all integers n)(there exists integer m) such that something readily testable holds. The most commonly quoted case of this is where T = Peano Arithmetic = PA, and T- = Heyting Arithmetic = HA. HA is the constructive form of PA. But the result applies flexibly to many other systems, both much stronger and much weaker than Peano Arithmetic/Heyting Arithmetic. The statement P = NP takes the form EA, which means (numerical) existential universal, using SAT. So P !NP takes the form not(EA). This conversion theory tells us how to convert a proof of not(EA) in PA to a proof of not(EA) in HA, and more. Write P != NP in the form (En)(Am)(P(n,m)), where P is totally innocent logically. Here “E” is “exists”, and “A” is “for all”. If PA proves not(En)(Am)(P(n,m)), then HA proves not only not(En)(Am)(P(n,m)), but even (An)(Em)(not P(n,m)), and the conversion from PA proof to HA proof is rather explicit, using the A-translation. Returning to the specific case at hand, of P != NP, these considerations tell us, for example, that any (CORRECT!) proof in PA of P !=NP can be explicitly converted to a proof in HA of P !=NP. In fact, it can be converted to a proof in HA of the statement T(f) = “for all constants c, any algorithm of size at most n with built in running time <= (x+c)^c must give the wrong answer to SAT at some input of size x <= f(n,c)" where f is a so called <epsilon_0 recursive function. I am under the impression in this Deololikar P !=NP "proof", this statement above under quote signs is "proved" where f is an exponential(?). BTW, this raises some interesting questions that perhaps have been addressed(?) independently of any logical issues. What can we say about the statement T(f) when f is slow growing? I.e., what is the "strongest" statement that we can hope to prove, or at least can't refute, or believe, concerning the identification of, or size of, a mistake that any purported algorithm for SAT of a certain size running in a certain time must make? This question can obviously be raised in a very wide range of complexity contexts, and surely(?) has been addressed. • August 14, 2010 8:48 am For fast-growing f this was addressed by many people (including DeMillo-Lipton); then Deborah Joseph and Paul Young argued in the early 1980’s that f should be no worse than elementary. The latest effort in this line that I’ve tracked is by Shai Ben-David and Shai Halevi (ps.gz here). But all this stops short of a really concrete treatment of SAT, and maybe now that should be revisited. Scott Aaronson’s 2003 survey http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.6548 has more. Minor typo in paragraph beginning “The statement P=NP…”, should say later “Write P != NP in the form not(En)(Am)(P(n,m))” (missing “not”). • August 14, 2010 9:45 am I took a quick glance at the references you mention (Kenneth W. Regan), and these “arguments” seem to be exploiting the observation that current concrete independence results are connected with fast growing functions – e.g., no current concrete independence results for Pi-0-1 (purely universal sentences quantifying over integers). This is no longer the case. See the latest, http://www.cs.nyu.edu/pipermail/fom/2010-August/014972.html where natural (steadily increasingly natural) Pi-0-1 sentences are claimed to have been proved independent of ZFC (not just PA), and in fact stronger systems involving large cardinals. Here natural Pi-0-1 sentences (i.e., A sentences) relate to kernels in order invariant digraphs on the rationals. In any case, I didn’t spot an outright definite claim that, e.g., if P != NP is true in Pi-0-2 (i.e., AE) form, then it is true iterated exponentially. Did I misread these papers, and they do claim such a thing? (I’m better at thinking than reading – sometimes this is a handicap). • August 14, 2010 10:14 am Here is a TR version of the paper “Polynomial Time Computations in Models of ET” by Deborah Joseph—its essence is also mentioned in other papers by Joseph and (Paul) Young which can be found online. Maybe it speaks toward what your reference says about towers. (A slightly longer reply with 2 links got into the mod queue; I see that didn’t happen to my first reply because I muffed the Ben-David-Halevi link.) • August 14, 2010 12:00 pm Thank you Harvey and others. Very useful. • August 14, 2010 4:05 pm Just one other note—we mentioned right away last Sunday that the paper does not even state a concrete deterministic time lower bound for SAT. Among private friends I projected disappointment in the event that “P != NP” would be solved that way, though as Ryan Williams has just noted here, it would be a harbinger of more constructive complexity separations to come. 41. News Flash! permalink August 14, 2010 1:40 am http://www.topnews.in/p-vs-np-mathematical-problem-successfully-solved-indian-brain-2268953 P vs NP mathematical problem is successfully solved by an Indian brain Even though Vinay Deolalikar has solved this P vs NP mystery but many of the scientists have not accepted his way of solving the theory, they want him to solve it practically in the right way. Few of them also claim that if Vinay can prove its claim right then they promise to pay$ 2 million as a prize.
August 14, 2010 3:47 am
On the number of parameters issue. Take a look at this link:
http://www.nlpr.ia.ac.cn/users/szli/MRF_Book/Chapter_1/node12.html
Most interesting is the following:
For discrete labeling problems, a clique potential V_c(f) can be specified by a number of parameters. For example, letting f_c=(f_i1,f_i2,f_i3) be the local configuration on a triple-clique (i_1,i_2,i_3), f_c takes a finite number of states and therefore V_c(f) takes a finite number of values.
Thus, we see that to get this number of parameters, we need to express potential U as a sum of simpler potentials V_c(f), and then count parameters according to arity of these simpler parameters. However, if this were all there is, then the number of parameters would be proportional to number of clauses, so there is probably something else going on.
The goal is to compute Z(T). My guess is that number of parameters is somehow related to complexity of this function.
• August 14, 2010 10:23 am
Clique potentials V_c(f) are exponential in size of the number of configurations. For example if you have three nodes in the clique X,Y,Z which can take on binary values 0 and 1 then it would take 2^3 entries to fully specify V_c as something like so. The table of values could be something like
V_c(x=0,y=0,z=0) = .34 ( param 1)
V_c(x=0,y=0,z=1) = .34 ( param 2)
V_c(x=0,y=1,z=0) = .34 ( param 3)
V_c(x=0,y=1,z=1) = .34 ( param 4)
V_c(x=1,y=0,z=0) = .34 ( param 5)
V_c(x=1,y=0,z=1) = .34 ( param 6)
V_c(x=1,y=1,z=0) = .34 ( param 7)
V_c(x=1,y=1,z=1) = .34 ( param 8)
Therefore the number of parameters grows exponentially with the size of the clique. So in a fully connected graph with n nodes V_c would need 2^n parameters to specify. And a fully disconnected graph would only need 2 paramters for each node or 2*n values.
The key question is how does he go from FO(LFP) and k-sat to these graphical models and compare them in terms of size of cliques. Some of this is detailed in sections 4 and sections 7 but still fuzzy to me.
August 14, 2010 10:30 am
That is not the problem. The clique size is k, which is independent on n. In the expression for potential, we have m clauses, and if each is given by this number of parameters, number of parameters is not exponential. So he must mean something else.
I believe it is some number connected with complexity of Z(T), but it is never defined. Also it is not clear which set of distributions are parametrized/ allowed (Boltzmann under some variation of potential?)
• August 14, 2010 11:13 am
I don’t think Vinay is using a direct mapping of SAT into a graphical model with clauses represented by cliques of size k.
He talks about embedding into a larger graphical model and has a picture on p. 74 that seems to visualize some of the construction. Along with some details on p. 48 on how to go from LFP to a directed graphical model.
His new ideas, I believe are just these constructions that go from LFP described computations to graphical models. Hopefully, he clarifies these constructions in more detail.
43. August 14, 2010 7:58 am
“Thus, your restriction to only have successor and to restrict to monadic fixed points is fundamental. In this domain—only monadic fixed points and successor—FO(LFP) does not express all of P! ”
August 14, 2010 9:37 am
http://scottaaronson.com/blog/?p=458#comment-45448
Tim Gowers’ weblog has just posted an equally good one: “Deolalikar is making a move that I, for one, had not thought of, which is to assume that P=NP and derive a contradiction.”
Vinay said yesterday “I was able to fix the issues raised with the earlier versions – the final version should be up in 3-4 days here.”
3-4 days from yesterday is August 15, Independence Day in India. So is there a Phoenix rising from the ashes?
45. August 14, 2010 10:09 am
Here is a PDF of one of the Joseph-Young papers, and another TR version (http://ecommons.cornell.edu/bitstream/1813/6340/1/82-500.pdf) of the paper “Polynomial Time Computations in Models of ET” that goes the furthest to what you may be doing with “towers” in the item you gave (?—here I’m being better at posting than reading:-).
46. FIFA world cup permalink
August 14, 2010 10:18 am
Is this like the FIFA World Cup of the nerds.
47. August 14, 2010 12:48 pm
OK, after reading through the ms I think I have isolated the fundamental problem with Deolalikar’s argument, which was disguised to some extent by the language of FO(LFP). It is simply that he does not address the issue of how polylog parameterisability is affected by reductions (or more precisely, by “forgetting” a bunch of auxiliary variables). This appears to be an extremely major oversight that more or less means that the real difficulty in the P!=NP problem has not even been touched by the paper.
Let me explain in more detail. Consider a random k-SAT problem with a nonempty solution space S (in {0,1}^n), and let mu be the uniform distribution on S. As I now understand it, Deolalikar’s argument proceeds by asserting the following two claims:
1. If k-SAT was in P, then the measure mu would be “polylog parameterisable”.
2. For a certain phase of k-SAT, mu is not “polylog parameterisable”.
This is the contradiction that is supposed to give P != NP.
What does polylog parameterisable mean? Basically, it means that mu admits a Gibbs type factorisation into exp(polylog(n)) factors, each of which only involve polylog(n) of the variables, organised according to some directed acyclic graph. For instance, the uniform measure on {(0,..,0), (1,…,1)} can be factorised as
mu(x_1,…,x_n) = 1/2 prod_{i=2}^{n} 1_{x_i = x_{i-1}}
and thus qualifies as polylog parameterisable (due to the acyclic nature of the graph with edges from i-1 to i); more generally, the output of a straight line program (and apparently, certain types of monadic FLP programs) starting from uniformly distributed inputs would be polylog parameterisable.
The thing though is that Deolalikar does not actually seem to prove 1. Instead he proves
1′. If k-SAT was in P, then _after embedding the original space of variables into a polynomially larger space of variables_ (see page 92), the measure mu lifts to a polylog parameterisable measure mu’ in the larger space. In other words, mu is the _projection_ of a polylog parameterisable measure.
For instance, if mu was computed using the final output x_N of a straight line program, one would have to add back in all the intermediate literals of that program before one obtained the polylog parameterisable structure. I give some examples of this on the wiki at
http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability
But what is completely missing from the paper is any indication why the property of polylog parameterisability would be preserved under projections. Thus, for instance, the uniform measure on the space of k-SAT solutions may well be so complicated that it defies a polylog parameterisation, but that if one lifts this space into a polynomially larger space by adding in a lot of additional variables, then polylog parameterisability might well be restored (imagine for instance if SAT could be solved by a polynomial length straight line program, then by adding all the intermediate stages of computation of that program we would get a polylog parameterisation).
Now I may be wrong in that the stability of polylog parameterisation with respect to embedding the problem in a polynomially larger space has been addressed somewhere in Deolalikar’s paper, but I have read through it a few times now and not seen anything of the sort.
• August 14, 2010 12:58 pm
Terry, you indicated why problems in P leads to polylog param. distribution mu but isnt it the case also for problems in NP?
• August 14, 2010 1:31 pm
Sorry, my earlier formulation of polylog parameterisation was misleading in this regard. It’s not enough that mu has a product formulation prod_C p_C(x_C) into local factors (which would already allow the uniform distribution of SAT to qualify, regardless of whether SAT is in P); the p_C(x_C) have to be organised in a “directed acyclic manner”, which means that they take the form p_i(x_i; x_{pa(i)}) where for each choice of parent literals x_{pa(i)}, p_i is a probability distribution in x_i.
Basically, polylog parameterisability of (say) k-SAT provides an efficient way to generate a random solution of that k-SAT problem (though not necessarily with the uniform distribution, see comment below, but merely a distribution that gives each such solution a nonzero probability).
• August 14, 2010 1:51 pm
What I was asking is this. Problems in P lead to a measure mu on the set of satisfying inputs which are “polylog parameterisable”. My question is: Isnt it also the case (perhaps by a similar argument) for all problems in NP?
If this is the case case then the missing argument “that k-SAT does not lead to such a parametrization” will also prove that k-SAT is not in NP which, while a surprising twist of events, much too good to be true.
• August 14, 2010 2:08 pm
Gil, I guess it may clarify things if we rename “polylog parameterisable” as “polylog recursively factorisable” (see Definition 2.12 of the paper). The uniform distribution on the k-SAT solution can be factorised as the product of local factors (one for each clause), but it is not _recursively_ factorisable as the product of local conditional probability distributions. On the other hand, if a problem and all of its partial completions are in P, then this seems to give a polylog recursively factorisable distribution on the solution space _after_ one expands the space to a polynomially larger one by throwing in additional literals in order to perform the P computations.
August 14, 2010 2:13 pm
Isn’t there a trivial way to show that, if we allow projections, every problem i k-SAT is polylog parametrizable. For instance, we might make a graph that has many copies for each variable x_i, so that it is forced to be equal to original x_i, and then somehow to make a graph that directly gives distribution of solutions of k-SAT formula, using clauses as conditions.
I am also a bit puzzled why number of p_i in the product is n, equal to number of variables. Also, why are sums not in the definition. It seems rather restrictive to have just n factors.
August 14, 2010 2:29 pm
I see now that there is probably no trivial way to do this, as we can sample efficiently even when we extend the space. Yet clearly, sampling solutions is as hard as getting them.
• August 15, 2010 5:31 am
I got it, thanks. With this definition I am not even sure that the claim in D’s paper is stronger than NP=!P and amounts to a statement that there is no polynomial algorithm for k-SAT for some nice (albeit non uniform) distribution.
August 14, 2010 12:59 pm
So, if somehow one can construct an argument that polylog parametrizability is preserved under projection, will that fix the proof ? or that itself is impossible?
• August 14, 2010 1:33 pm
Yes, but the task of proving that the projection of a simple computation is again a simple computation is essentially the claim that P = NP. Given what Deolalikar is trying to prove, this looks unlikely, to say the least…
• August 14, 2010 1:39 pm
Sorry, I should have said “is analogous to the claim that P=NP” rather than “is essentially the claim that P=NP”. The computational model here is quite different (polylog recursive factorisability rather than polynomial time computation).
• August 14, 2010 2:00 pm
To put it another way, I now think that what Deolalikar really proves in the paper is not that P != NP (i.e. the projections of polynomially computable sets need not be polynomially computable), but rather the much weaker statement that the projections of polylog recursively factorisable measures are not necessarily polylog recursively factorisable.
August 14, 2010 1:22 pm
If this is the definition of polylog parameterizability, then isn’t it true that every k-SAT formula is polylog parametrizable? Namely, if N is number of satisfying assignments, then uniform distribution mu on the space of solutions can be expressed as
1/N prod_{i=1}^{m} p_i
Where p_i corresponds to the clause c_i, has the same k arguments, and is 1 if clause is satisfied, 0 if it is not satisfied. But clearly he wants number of parameters that is exponential.
What am I missing here?
What do you mean by “organised according to some acyclic graph”? I guess this would not allow my simple expression, but what expressions are allowed?
Perhaps you want uniform distribution to satisfy Markov network property (conditional independece) according to this graph, but how do you know such a graph exists?
• August 14, 2010 1:27 pm
I’ve written up a more accurate definition of the concept on the wiki at
http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability
and see also Definition 2.12 of the third draft of the paper (an earlier version of this wiki page was not precise enough). Basically, to be polylog parameterisable, one has to have an algorithm to sample from the solution space, in which the conditional distribution of each literal x_i is determined by the values of only polylog(n) parent literals.
In other words, each p_i must take the form p_i(x_i; x_{pa}(i)), where x_{pa(i)} are a polylog number of parent literals of x_i, and for each fixed instance of x_{pa}(i), p_i is a probability distribution in i.
Also, I realised that the distributions here were not quite uniform as claimed above; see my comment below.
August 14, 2010 1:40 pm
Thanks. That makes some sense. So basically, you want to express uniform (or something like that) distribution with a distribution which is coming from some Markov network, satisfying conditional independence properties.
I can understand that having his model of computation, he can find such a directed graph (this is what he did in chapter 7, now 8), and sample a uniform distribution.
However, there should be also a computational model-independent way to determine this number of parameters, as he claims this number is exponential for k-SAT in hard phase. We could perhaps say number of parameters is minimum over all possible such graphs, but I don’t see why a uniform distribution can have such a directed graph at all.
So, a priori, given a distribution mu corresponding to solutions of k-SAT formula f, why is there a directed graph and expression in the corresponding way at all?
• August 14, 2010 2:13 pm
Yes, I do find it quite unlikely that the uniform distribution on the k-SAT solution space (or any other distribution with this solution space as its support) admits a recursive factorisation along a directed graph with only polylog degree. And it may well be that Deolalikar actually proves this using what is known about the clusters of k-SAT etc.
But I don’t see the paper address the real question, which is whether a measure on the k-SAT solution space can be the _projection_ of a polylog recursively factorisable measure in a polynomially larger space.
August 14, 2010 2:27 pm
I meant expression in that form without polylog restriction.
But I see now there is one for any distribution, namely
p(x_1)*p(x_2;x_1)*….p(x_n;x_2,x_3,…x_n)
which takes 2^n-1 parameters, as you defined them.
As for the projection issue, perhaps there is some trivial way to get solutions to k-SAT with polylog parameters if we are allowed to extend number of variables, and then take projections.
However, this given that you can sample solutions efficiently if you have polylog parametrization in that sense (with additional variables, projection), perhaps what he showed is that P=NP implies that you can do this sampling easily, which is of course not hard to see.
So, the missing part would be: in the hard phase of k-SAT, sampling is hard.
• August 14, 2010 2:44 pm
So, the missing part would be: in the hard phase of k-SAT, sampling is hard.
Yes, this appears to indeed be the missing part. Deolalikar has apparently shown that sampling is hard if one is not allowed to introduce some intermediate variables for the sampling process, due to the geometry of the SAT clusters, but that is like saying that solving an n-bit SAT problem is hard if one is only allowed to use n steps in the computation. Once the intermediate variables are thrown in (as they are done in Section 8.3), this lifts the solution space to another space in which the cluster geometry could be completely different.
• August 15, 2010 10:30 am
Isn’t that the result from statistical physics for k=9, that he uses? In the critical regime it takes an exp number of these parameters to specify the joint distribution. Hence, the contradiction.
• August 15, 2010 11:09 am
It may well require an exponential number of parameters to specify a distribution on the original n-dimensional k-SAT solution space in the precise recursively factorisable form that Deolalikar needs. However, this does not seem to preclude the existence of a much shorter parameterisation of a distribution on some _lift_ of the k-SAT solution space formed by creating more variables. Since variables are certainly being created in Deolalikar’s argument, this creates a serious gap in the argument.
Let me quote the paragraph immediately preceding Remark 8.8 on page 92 of the paper, which I think is very revealing with regards to the entire strategy, and its inherent flaw:
We have embedded our original set of variates into a polynomially larger
product space, and obtained a directed graphical model on this larger space.
This product space has a nice factorization due to the directed graph structure. This is what we will exploit. (emphasis mine)
• August 14, 2010 1:22 pm
A slight correction: the measure mu considered in the paper is not quite the uniform measure on the solution space k-SAT, but a slightly biased measure formed by selecting the literals x_1,…,x_n in the following slightly correlated manner. Assume inductively that one has a partial solution x_1,….,x_i to k-SAT, then at least one of x_1,…,x_i,0 or x_1,…,x_i,1 is also a partial solution. If only the former is a partial solution, we set x_{i+1} := 0 (we have no choice); similarly, if only the latter is a partial solution, we set x_{i+1} := 1. If they are both 1, we choose x_{i+1} to be 0 or 1 uniformly at random.
Note that if k-SAT was in P, this gives a polynomial time algorithm to generate a random instance of a solution to k-SAT, which gives each such solution a nonzero probability, but the probability is not quite uniform. As stated, it is not obviously polylog parameterisable because the distribution of each literal x_i depends on all of the preceding literals x_1,..,x_{i-1}. But if SAT was solvable by (say) a straight line program, then one could lift this distribution to one involving polynomially more literals in such a way that it was polylog parameterisable (each literal is selected in terms of a conditional probability distribution that involves only polylog many ancestors).
• V Vinay permalink
August 14, 2010 2:14 pm
Terry, Several characterizations of P are known and let us say we prove one more that says problems in P are “polylog parametrizable.” All of these are resource shifting characterizations that shift polynomial time into another resource. Adding one more to the list will not be a surprise.
To me, the real play is on the other side, showing 3SAT/9SAT does not have this “polylog parametrizable” property. This is still a hard unconditional lower bound argument. How do we know that the k-Sat soln space cannot be “factorized” etc? Because there is no algorithm … cannnot be the answer without being circular.
Here is my intuition. Just as all NP-complete problems are isomorphic to each other (at least under AC^0 reductions) and are just disguises of each other, all characterizations are also just resource shifting which are basically the same in different disguises. Occasionally, we get a surprising characterization like IP=PSPACE or the PCP thm. They surprise because computation seem more powerful than we had anticipated or expected. The same is true with Primality as well. Given that the power of computation continues to surprise us, we really do not understand well what cannot be done. Which is why results such as derandomizing identity testing implying lower bounds excites us because they seem to give us hope.
In any case, there is nothing new that I have said that the community does not already know. The nub being that (1) and (2) in your comment points to a traditional resource shifting strategy in D’s paper. Your point that (1) is difficult to prove is likely true and well taken; but even if it is proved, so what? Can we really show (2)? I believe this is what holds the key. I am asking for a relative change in emphasis. Yet another characterization of P is not necessarily useful unless we make some simultaneous progress on (2).
August 14, 2010 6:24 pm
Many thought V Vinay has taken permanent recluse from complexity theory. Great to see him back, at least in a commentary mode, and sharing sharp insights.
• Jun Tarui permalink
August 14, 2010 11:07 pm
I think that Terry’s definition of polylog parameterizability above and at Polymathwiki is a fairly strong (severely limiting) one; in particular, the following two examples will not be polylog-parametrizable. Maybe this is just part of Terry’s point, but I thought I’d point out easy concrete things in case this piece of info is epsilon-useful.
(1) uniform (or any positive) measure on Even={x in the n-cube : x1+…+xn =0 (mod 2)}
(2) uniform (or any positive) measure on the n-cube – {(0,0,…,0)}
Claim: Let S be a subset of the n-cube, let Chi_S be the characteristic function of S on the n-cube, and let mu be a measure on S such that mu(x) > 0 for each x in S. Assume that mu is polylog parametriable. Then, there is some permutation (reordering) of [n] such that Boolean function Chi_S(x1,..,xn) can be expressed/computed as follows:
Chi_S(x1,…,xn) = AND_{i=1,…, n} g(xi; xj’s), where each g is a Boolean function of xi and polylog-many xj’s with j < i.
We can see the claim by focusing on measure being zero or nonzero in the definition of polylog paramateizability. Now consider (1). With respect to any ordering (permutation) of [n], the membership in Even cannot be expressed by such an AND as in the claim because the last factor g(xn; xj's) only depend on *fixed* polylog many xj's. By the same reason, (2) is not polylog parametrizable. (To express (1) and (2) in such a form, the last g(xn; xj's) have to depend on all n variables.)
August 15, 2010 2:03 am
Indeed, though your example is not in k-XORSAT, it clearly shows his point:
there are distributions polylog parametrizable in projections, but not polylog parametrizable without projections.
EVEN can be polylog parametrized if we add a number of auxilary variables, placeholders for partial sums, so that we never add more than polylog elements (we can always add a fixed number).
However, as you pointed out, without these variables, EVEN is not polylog parametrizable according to Tao’s definition.
It is also easy to see that k-XORSAT can be polylog parametrized with projections, but it is not clear to me if it can be polylog parametrized without projection according to Tao’s definition.
• August 15, 2010 4:18 am
Jun, earlier you were saying that Deolalikar’s simplicity definition applied only to problems in a subset of P — problems with all their projections in P. But Terry’s interpretation of polylog-parametrizability (with projections) applies to all functions in P. This appears to give us mildly contradictory accounts of what Deolalikar is claiming. Do you have a view about which is correct? (Perhaps they are both correct and Deolalikar’s strategy is different in different parts of the paper …)
• August 15, 2010 11:12 am
I like these examples, particularly the second one, because the uniform distribution on $\{0,1\}^n \backslash (0,\ldots,0)$ has no non-trivial conditional independence properties whatsoever, which reveals that the “link between polynomial time computation and conditional independence” to which Deolalikar devotes an entire chapter is completely absent if one does not allow the ability to add variables (and thus change the geometry of the solution space drastically).
• August 15, 2010 1:46 pm
Case 2 is interesting because clearly the last factor only needs 2 numbers to specify it’s distribution, one for when all the previous variables are 0 and one for all other cases, in other words the last variable becomes independent if any of the previous are non-zero.
If we can’t capture these kind of relationships in the solution space then the representation seems too crude. How would we distinguish XOR-SAT from 3-SAT if this was the case?
• Ørjan Johansen permalink
August 15, 2010 6:53 pm
I guess this is obvious to you experts, but I’d just like to point out explicitly that both (1) EVEN and (2) have the property that all their projections are in P, and that there are obvious polynomial algorithms to extend any partial assignment.
So from what I think I understand of this discussion, they seem to be in the class of problems that Deolalikar needs to have “easy” structure, and so the version of parametrizability that doesn’t use projections cannot be used.
• Jun Tarui permalink
August 15, 2010 9:20 am
Tim, my point was only that Deolalikar is using, in his reasoning, the simple fact that if k-SAT is in P, one can decide, given a partial assignment s, whether a satisfying solution y extending s exists. I think the same thing was also mentioned by Russell Impagliazzo: https://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/#comment-5429
I think D’s reasoning above certainly is *not* a big deal; it is not a possible way around the natural proof’s barrier or anything. On the other hand, I found your post at your blog (= Tim Gowers’ blog) interesting, where you talk about a class of poly-time functions with the property that for any projection P, existence of a solution in P can be poly-time computable (the same thing as the n-dim 0-1-colored hybercube induced by a poly-time Boolean function with the property that determining if there is 1 in any given subcube is also in P).
As for polylog parametrizalibility, I think what we may consider include the following. They are related but seem to be nonequivalent. I will just give a list; I don’t have anything useful to say about them at present.
(1) Terry’s definition with projections (with poly-many auxiliary variables) for the distribution on the *graph* of f:{0,1}^n –> {0,1}, i.e., the distribution on {(x1,…,xn, f(x1,…,xn) ): x in the n-cube} induced by the uniform distribution on the n-cube.
(2) parametrizability of the uniform, or some, distribution on the solution space, i.e,. on f^{-1} (1) (as opposed to the graph space above)
(3) sampling (= generating) a point in the solution space with *some* distribution with support equal to the solution space (This is explained in the last part of Terry’s wiki entry for polylog parametrizability)
(4) poly-time samplability of the solution space, i.e., sampling a solution uniformly (or nearly uniformly): This is in some sense a reverse view of Terry’s definition with projections: Start with the uniform distribution the r-cube, the set of random r-bits that a sampling algorithm can use; use auxiliary variables to represent intermediate states of computation, which will be projected out (thrown out); the distribution on the *outputs* x1,,,,,xn must be (nearly) equal to the uniform distribution on the solution space.
(5) In my previous comment, I pointed out that Terry’s definition without projection is fairly limiting. I guess one can consider extending his definition (without projection) in some ways.
For example, we can consider distributions D(x1,…,xn) expressible as a *sum* over poly-many distributions L(x1,.,,,,xn) each of which is expressible by Terry’s form (Ordering of variables may be depend on each L).
August 15, 2010 12:06 pm
“But what is completely missing from the paper is any indication why the property of polylog parameterisability would be preserved under projections. Thus, for instance, the uniform measure on the space of k-SAT solutions may well be so complicated that it defies a polylog parameterisation, but that if one lifts this space into a polynomially larger space by adding in a lot of additional variables, then polylog parameterisability might well be restored (imagine for instance if SAT could be solved by a polynomial length straight line program, then by adding all the intermediate stages of computation of that program we would get a polylog parameterisation).”
Does D actually need to show this? He is assuming that P=NP and deriving a contraction. So if D has lemma of the form:
“Suppose P=NP. Then the projection of a polylog paramaterisation is polylog,”
then the actual status of projections of polylog paramaterisation is irrelevant. (Sorry if I’ve missed the point, or if this is dealt with below.)
• August 15, 2010 2:37 pm
This is a fair point; P=NP is certainly a very big tool available to Deolalikar. On the other hand, it does seem now that we have some basic counterexamples (see e.g. the wiki) that show that projections of polylog parameterizable distributions need not be polylog parameterizable. It would still be that P=NP implies the negation of this statement, but then we could prove P!=NP more efficiently by a contradiction argument than by using any of Deolalikar’s machinery at all.
• Jun Tarui permalink
August 15, 2010 1:23 pm
First on notations: (i) quasipoly(n) = exp(polylog(n)) = 2^polylog(n) = n^polylog(n); (ii) I’ll use the abbreviation ‘ppp’ for ‘projection-polylog-parameterisable’ as defined by Terry above and at the wiki *except* that I’ll allow quasipoly-many auxiliary variables; (iii) f will denote a function mapping {0,1}^n to {0,1}; (iv) the *solution space* of f means the set f^{-1} (1) ; we’ll assume this set is nonempty; (v) the *graph* of f, graph(f), is the following subset of the (n+1)-cube with size 2^n: graph(f) = {(x1,…,xn, f(x)): (x1,…,xn) in the n-cube}; we’ll consider the distribution on graph(f) with support size 2^n induced by the uniform distribution on the n-cube.
I think that the following holds.
Prop. The following are equivalent.
(a) f:{0,1}^n –> {0,1} is in nonuniform-quasiP.
(b) The distribution on graph(f) is projection-polylog-parameterisable (ppp).
(c) The uniform distribution on the solution space of f is ppp.
(d) Some distribution whose support equals the solution space of f is ppp.
Proof Sketch: (a) –> (b): Explained by Terry.
(b) –> (a): The same thing as what I said above about the two examples (also explained at the wiki) applies; that is, ppp for the distribution directly yields a factorization representation by 0/1 *Boolean* factors. If each Boolean factor depends only on polylog many parents xj’s, this can be expressed by a quaipoly-size Boolean circuit, thus f can be computed by a quasipoly-size Boolean circuit.
(a) –> (c) (hence also (d)): Explained by Terry; express the uniform probability weight of each solution in the solution space by dividing Terry’s factorization expression by the size of the solution space (as opposed to by 2^n in Terry’s formula).
(d) –> (a): By the same reasoning as in the part for (b) –> (a) above: All that matters is nonzero (=positive) vs zero.
Remarks: (1) I’m talking in terms of quasipoly because that helps stating things in a simple way; the same reasoning above works for P.
(2) In Terry’s expression for ppp, consider posing the further limitation that, istead of p_i(x_i; polylog many x_j’s), each p_i has the form p_i(x_i; x_j), where *single* x_j is one of input variables x_1, …, x_n, i.e., j is an integer between 1 and n. This definition leads the complexity class L (= deterministic logspace): Think in terms of deterministic polynomial-size branching programs; use poly-many auxiliary variables to represent nodes in the branching program.
Caveat: My confidence for what I’ve said is not exactly high… ; I’m writing from Tokyo, and it is getting fairly late…
August 15, 2010 1:55 pm
I see a potential problem with (c)(d) when you use projections. Without projections, this is clear: you divide by number of solutions N and each factor is 0 or 1. But when you take projections, the extended space has much more than N solutions, its “geometry” is completely different. Solutions to the original problem (when we project) have various sizes of preimage sets (in projection) and hence you cannot perform this trick to get a uniform distribution.
Or in other word, using equivalence of ppp to distributions which can be computed using probabilistic TM (see my simple proof below), if we can get some distribution using probabilistic TM, your result would imply that we can get a uniform distribution, which I am not sure is true.
• August 15, 2010 2:34 pm
Yes, it may be that a problem needs to be in something like #P in order to be able to easily sample the _uniform_ distribution in a ppp manner, in order to get all the normalisations working properly. (Without #P but just non-uniform quasiP, I think one can still write the uniform distribution as the product of local factors in quasipolynomially many variables, but these factors will not be organised along a directed acyclic graph and so cannot be used to easily sample the solution space uniformly.)
• Jun Tarui permalink
August 15, 2010 4:01 pm
I am assuming that, for projection poly parametrizability, auxiliary variables (those which will be projected out) are *binary*. With this assumption/qualification, I think what I’m claiming holds. For the direction (a) –> (b) or (c) or (d), one automatically gets binary binary auxiliary variables. For the other direction, I need the assumption that each auxiliary variable takes only quasipoly many values with positive probability; but to make a story simple, I want to assume these are binary. If we allow auxiliary random variables to take exponentially many values, then these auxiliary random variables can “encode” input n-bit vectors as their values, and any function can be parameterized by O(n) such random variables.
With the assumption above that auxiliary variables are binary, I think what I’m claiming holds.
As for the comment by Terry: Please note that I am *not* saying anything about poly-time sampling, and that since we are talking about a *fixed* function f, the number of its solutions is *fixed*, and *not* something we have to count (by a #P computation, etc).
As for the comment by vloodin: Suppose that we are given a projection-polylog-parameterisation for a Boolean function f on the n-cube with m solutions, i.e, |f^{-1}(1)| = m. When we transform this ppp into a circuit, it is true that the number of bit vectors (x_1, …, x_n, x_{n+1}, …, x_q) with nonzero probability can be much larger than m, where xi’s other than x1, …, xn are auxiliary variables. But the sum of the probability weights of all such bit vectors with a *common* n-bit initial prefix (x1,…,xn) is 1/m, and thus our transformation does work through (I think).
• August 15, 2010 6:00 pm
It’s not just the number of solutions of {x: f(x)=1} that one has to count in order to factorise the uniform distribution; one also has to count the number of solutions to partial problems such as { (x_{i+1},…,x_n): f(x_1,…,x_n) = 1}. To do this in a quasipoly manner basically requires the various projections of f to be in (quasi-)#P.
• Jun Tarui permalink
August 16, 2010 2:02 am
I retract my “proposition” above (in my comment at 1:23pm) about (a)–(d) being equivalent; my “proof” was wrong. (vloodin and Terry, thank you for pointing out and explaining) For convenience, let me repeat conditions (a) — (d):
(a) f : {0,1}^n –> {0,1} is in nonuniform-quasiP.
(b) The distribution on graph(f) is projection-polylog-parameterisable (ppp).
(c) The uniform distribution on the solution space of f is ppp.
(d) Some distribution whose support equals the solution space of f is ppp.
The implication (a) –> (b) does hold as explained by Terry at the wiki; for all the other implications that I claimed, my proof sketches were wrong. But I think that there are some things we can say about (b) — (d); I’ll continue.
I think that (c) is basically equivalent to the condition that there is a poly-time randomized algorithm that samples a solution with respect to the uniform distribution on the solution space. I say “basically” because we have to allow an error of 2^{-quasipoly(n)} for the statistical distance.
For the direction (c) –> sampling, we can give a sampling algorithm A as follows. Let y1,…,yq be binary random variables in a projection-polylog-parameterisation for f, where (q-n) variables among y1,…,yq are auxiliary ones. For y1,…,y_polylog, i.e., for the first polylog-may yj’s, algorithm A computes their joint probability weights for all the possible 2^polylog binary values within precision 2^{-quasipoly}. Using its own random bits, algorithm A “picks” one particular polylog bit vector as the value for (y1,…,y_polylog), and continue. Note that the factorization expression naturally yields an expression for the distribution conditioned upon the first polylog yj’s being a particular vector. So indeed algorithm A can continue till it obtains a particular sample for (y1,…,yq); A can then return the appropriate n-dimensional projection of this sample.
In general, up to within 2^{-quasipoly} accuracy, for any measure mu on the n-cube, the following two are equivalent: (1) measure mu admits ppp with quasipoly-many auxiliary variables; (2) sampling according to mu can be done by a quasipoly-size probabilistic circuit. So condition (b) and (d) respectively imply corresponding sampling.
As for (b) vs (a): For some function g, it may well be the case that sampling from graph(g) is easy, but even the average complexity of g is high: Consider G:(x, r) –> (f^{-1}(x), r, x*r), where f is a one-way permutation on the n-cube, both of x and r are n-bit vectors, and * is the inner product mod 2, i.e., the setting of Goldreich-Levin Theorem. Computing G is hard, but sampling on graph(G) is easy.
August 14, 2010 1:04 pm
I do not like the structure of comments at wordpress, many of the recent comments are hidden in the middle of the page, so I would like to recall Lenka Zdeborova’s comment as I am very curious to know the answer:
Could Deolalikar’s approach prove that P != #P?
August 14, 2010 1:24 pm
No, that comment had to do with speculation about Z(T), and I think the answer is clear no, in the context it was given.
• An Update from Deolalikar's homepage permalink
August 14, 2010 1:37 pm
Vinay Deolalikar. P is not equal to NP. 6th August, 2010 (66 pages 10pt, 102 pages 12pt). Manuscript sent on 6th August to several leading researchers in various areas. You can find an older version here. Many researchers advised me to prepare a concise synopsis of the proof. I have done so, you may obtain it here. The 3 issues that were raised were (a) the use of locality and the techniques used to deal with complex fixed points (b) the difference between 9-SAT and XORSAT/2-SAT and (c) clarification on the method of parametrization. I will address all these in the version being prepared, which will be sent for peer and journal review.
August 14, 2010 1:45 pm
This is excellent news. If he does clarify this, then we will be able to see clearly if this proof works after all.
49. John Goodwin permalink
August 14, 2010 1:15 pm
For any physics types in the low-rent ‘crowdsource’ section watching, like me, this wonderful conversation with a bag of popcorn in hand:
The author has a notably good taste in textbooks and having a bunch of us comb them for perspective is a contribution well within our reach, or in any event a fine way to spend a bit of spare time in August.
My copy of Mezard and Montanari, Information, Physics, and Computation just arrived and it looks like a page turner. I’ll quote the first paragraph:
Over the last few years, several research areas have witnessed important progress through the unexpected collaboration of statistical physicists, computer scientists, and information theorists. This dialogue between scientific disciplines has not been without difficulties, as each field has its own objectives and rules of behaviour. Nonetheless, there is increasing consensus that a common ground exists and that it can be fruitful.
(emphasis mine).
Indeed.
Now that Statistical Physics, Information Theory, and the Theory of Computation has had it Woodstock, perhaps we will be seeing more of one another. The hot topic driving this in Industry is, by the way, message passing algorithms in a Cloud Computing context, and their promise of scale. It surely comes as a shock to us Physicists that Belief Propagation and neural nets obey Hamilton-Jacobi formalism! And even recently schooled grad students of Neural Nets seem never to have heard of Spin Glasses or Glauber dynamics. Perhaps this event will finally break down that barrier — and I suspect that will be its enduring value to both the Physics and the Industrial Mathematics communities.
In any event the book has a high text to formula ratio and looks to be a good read. Its ToC is available at Amazon link, and it covers the 1RSB cavity method. I see contributers above among its readers. I am unrelated to the authors, their institutions, and live on the other side of the planet from them….
August 15, 2010 3:46 am
The term “Industrial Mathematics” is cute ;-)
• August 15, 2010 12:49 pm
Can you give a reference on this Belief Propagation – neural nets – Hamilton-Jacobi connection, I would be interested to learn. Thanks.
• August 15, 2010 4:26 pm
This is definitely a sidebar, however there seems to be a deep connection between Belief Propagation (and its generalisation Expectation Propagation) and a sort of theory that comes out of optimal control theory for stochastic signals.
Here is a good page on Expectation Propagation:
Anyone reading this not already familiar with Loopy BP, EP, and factor graphs (probably a minority of the contributors to this conversation but certainly including me since I am just learning!) will really enjoy Minka’s 1-hour lecture on the topic, the first link on the page cited. This may give you a feel for where ‘Industrial Maths’ is at these days.
The connection to signal theory is profound: in that lecture, Minka shows the connection to Kalman filters, for example, as a natural consequence of running forwards then backwards over a certain sort of factor graph.
In any event, it seems that Hamilton-Jacobi has a stochastic cousin:
http://en.wikipedia.org/wiki/Hamilton-Jacobi-Bellman_equation
the discrete version of which is Bellman’s equation (this is the same topic that Economists call ‘Recursive Methods’ — see the famous text by Sargeant and Lundkvist for an example) and which OR types call ‘Dynamic Programming’ – the reason why one is ‘working backwards’ is obvious to anyone who has seen these techniques in action in those respective fields).
and the claim is that that *backwards* belief propagation just recovers HJB the same way that forwards and backwards combined, running over the factor graph, gives you Kalman filtering. I don’t claim to understand the result — just to be surprised and amazed by it as a physicist who is now aware computer science is a lot more interesting than it looked at first glance. ;)
Again, it looks like physics and comp sci (and signal theory and ‘the stochastic calculus’) are coming together in unexpected ways. However, this is so far off topic that anyone who wished to discuss it may contact me via my (dated but still functional) LinkedIn page, linked to my name.
Best Regards and good luck to you all in your quest!
August 14, 2010 2:46 pm
On the k-XORSAT problem:
Using definition of number of parameters that Terrence Tao has suggested, we can analyze uniform distribution on solutions of k-XORSAT.
Since k-XORSAT is a linear problem, and space of solutions can be expressed in terms of parameters (like in the linear algebra), which we can take to be some of the variables. Without loss of generality (permuting variables if necessary), for easy notation, assume x_1,… x_l are independent parameters for this linear space.
Then let p_i for i<=l be uniform distribution i.e. (1/2, 1/2), then we can factor uniform distribution on the solution space:
p_1(x_1)p_2(x_2)*..*p_l(x_l)*p(x_{l+1};x_1,…x_l)* …*p(x_n;x_1,…x_l)
However, this is not polylog parametrization if l is not of order polylog(n). The following question is then interesting: can we do a polylog parametrization of this form?
Note that all equations are of the fixed size (k variables, it is k-XORSAT). So, can we always get a polylog parametrization? This seems like a fun thing to consider
August 14, 2010 2:58 pm
It is easy to see that if we allow additional parameters, we can indeed do a parametrization which is polylog. All we have to do is to introduce intermediate results, say if
x_{l+1} is expressed as sum of many of the basic variables x_1…x_n, we can introduce intermediate variables, holding sums of polylog factors. We could even do sum two by two. We are allowed to do O(n) such sums, since we add number of parameters for each factor, and for each factor, we have 2^arity parameters. So we easily get 2^polylog parametrization, but only at the cost of extending number of variables.
Thus, given what D. is assuming (he does this extension of number of parameters in chapter 8), for him, this would be enough.
But if we do not allow this extension of parameters, is then polylog parametrization possible?
August 14, 2010 3:03 pm
I meant variables (parameters in the linear algebra problem) above, when I said parameters at the beginning. This should not be confused with parameters with respect to Tao’s definition, which are also mentioned; sorry if this caused confusion.
51. H. Tuuri permalink
August 14, 2010 4:08 pm
What is the the experts’ opinion? Will P != NP ever be proved? I have been sceptical on the grounds that the collection of all programs that can be computed in a polynomial time is so immensely complex.
I see that Razborov and Rudich have been able to prove the non-existence of a ‘natural proof’ for P != NP. In this blog discussion someone said that P != NP might be independent in some axiom systems.
52. August 14, 2010 7:05 pm
One thing puzzling me with his synopsis is the comment there is empirical evidence that 3-SAT does not enter a d1RSB phase, which would imply no contradiction if it was polylog parameterizable.
Isn’t this different behavior between 3-SAT and 9-SAT a fundamental contradiction since both are in NP and his machinery seems to suggest that 3-SAT is polylog parameterizable and 9-SAT isn’t?
• August 15, 2010 12:00 pm
Jeff, the trouble with 3-SAT is that the clustering was not proven rigorously, because it is technically more difficult. In the statistical physics terminology (which is different from the one of Deolalikar) 3-SAT does not have this d1RSB phase. But it has a frozen phase very nearby to the satisfiability threshold, which is so far empirically unpenetrable for polynomial algorithms.
53. Cristopher Moore permalink
August 14, 2010 9:23 pm
I have been blissfully away from the blogosphere. I just wanted to mention a few things. First of all, the goal of sampling uniformly random satisfying assignments is, of course, a much harder goal than finding a single one.
1. we believe, but have no proof, that 3-SAT enters a clustered phase, but not with frozen variables.
2. we believe that in this phase we can find a satisfying assignment in polynomial time — but that we _cannot_ sample uniformly random satisfying assignments in polynomial time.
This is partly why physicists call this the “dynamical” transition, since local search algorithms will get stuck in particular clusters and fail to sample the entire space.
3. we have a proof for sufficiently large k that there is a clustered phase with frozen variables. We believe that in this phase, even finding a single satisfying assignment is hard.
4. we have a proof that XORSAT enters a clustered phase with frozen variables. However, because of its linearity, even in this phase we can sample uniformly random satisfying assignments.
I’m sure this is redundant with some other people’s comments (including my own :-)
• August 15, 2010 12:08 pm
I think very little is missing (if anything) to prove Cris’s statement n. 2 in the 3-coloring problem if one restricts to local Markov chain sampling. The 1RSB threshold is at average degree 4, there is a rigorous link between necessary local Markov chain sampling time and reconstruction on trees. And in 3-COL and old lower bound on reconstruction tells us that above 4 sampling with Glauber dynamics takes exponential time. At the same time Cris himself proved that simple algorithms work up to 4.03.
54. August 14, 2010 10:25 pm
As far as I can tell, any SAT instance can be reduced to a SAT instance with large (and deterministic) separation of solutions. Attempts to use arguments about “separation of solutions” would then need to explain an $n^{1-\epsilon}$-separation, for any $\epsilon > 0$. Apologies for the diversion, but editing complex comments is quite tough, so have posted details over at A simple reduction.
(This seems like a colossal cheat so it may well be wrong. Comments would be appreciated.)
55. August 15, 2010 12:42 am
I will look directly to the ACKNOWLEDGMENTS in the revised version of Deolalikar’s paper.
August 15, 2010 2:45 am
On the Tao’s definition of number of parameters:
With this definition, the following problem, solvable in P, has exponential number of parameters (see comment above by Jun Tarui about EVEN):
n-XORSAT (which is just system of linear equations, say m equations of n variables; we ask if there solution to this system, system not assumed homogenous)
On the other side, it (i.e. uniform distribution on solution space) is a projection of a c^polylog parametrizable distribution (obtained by adding variables, placeholders for partial sums) in the Tao’s sense.
I don’t know if the same holds for k-XORSAT, but for purpose of a counterexample, n-XORSAT is just as good. It is in P, yet not 2^polylog parametrizable.
Hence, if Tao’s definition is what D. had in mind proof does not stand. He needs to prove he does not have a polylog parametrization with projections in hard phase (as Tao suggested, and this is just a concrete illustration/counterexample for this).
If D. had definition of the more general form in mind, namely this: solution space uniform distribution (or the like) is polylog parametrizable if it is a PROJECTION of polylog parametrizable distribution in the Tao’s sense, then the following would be true:
1. In this more general sense k-XORSAT is c^polylog parametrizable, but possibly not k-SAT.
2. His arguments based on computational model could as well show that if P=NP, then k-SAT solution space has a (quasi uniform) distribution that is c^polylog parametrizable in this more general sense. His promise to fix the bug in this part, which seems plausible enough to give it benefit of a doubt, could as well turn out to be fulfilled.
3. With this more general definition, the shift is to the part that claims, from statistical physics, that in hard phase in this sense distribution a-priori not 2^polylog parametrizable.
The point 3. is not even considered in D.’s paper at this point, other than referring to the physics results.
But even if he has some proof in mind, it is more likely that he proved it for more restrictive Tao’s definition. As example above clearly shows, this definition cannot work.
It seems to be much more easy to come to ends with a paper, when definitions are given. It also shows that good starting point for analysis is knowing what is EXACTLY meant by number of parameters. Hand waving there can easily obscure fundamental errors.
August 15, 2010 2:57 am
Nevertheless, if he does fix bug in the part 2. D will have a potentially useful reformulation of P=NP problem. It would be enough to prove that in the extended sense (which includes projections) k-SAT is not polylog parametrizable.
It is unclear if this leads to anywhere, but it is still interesting (provided 2. is fixed). It reduces P=NP question to a form that may be more tractable, certainly a partial result that can stand on its own. It also gives (all that if 2 is fixable) a new way, other than circuits, finite models etc. to study complexity problems. Number of parameters as a measure of complexity (providing the definition is clear, like the one Tao suggested or rather a bit stronger version) would be this new approach, and essential contribution coming from D’s work.
August 15, 2010 4:49 am
In the view of my comment below, that seems to give an easy argument for 2. it seems that it is a bit of an overstatement to say this is a valuable new approach. It just shifts the question of P vs NP to impossibility of polynomial time circuit sampling of k-SAT. So it shifts P vs NP to a harder question, and thus, it is useless.
57. Micki St James permalink
August 15, 2010 4:18 am
Ryan Williams has a map which takes any infinite collection C
of satisfiable formulas into a subset C’ of SAT0, the set of
formulas satisfied by the all-zeroes assignment.
He then claims that this set C’ has “analogous solution structure”
to C.
But I don’t know what such a claim is supposed to signify.
Solution structures are attached to problem definitions, not to
sets of formulas, right? What exactly are the problem definitions?
If the original problem was [here’s a formula, is it in set C],
then maybe he’s considering the transformed problem to be
[here’s a formula, is it in set C’].
But the trivial decidability of SAT0 doesn’t seem to solve this
problem; it can sometimes give quick “no” answers but can’t give quick “yes”
answers because the map from C to C’ is non-uniform. In other words,
the hard part isn’t the variable assignment, it’s the [is it in set C’] part.
August 15, 2010 4:42 am
I believe his remark does not show what he claims. For there is no “solution space” for SAT0 (which corresponds to a closed formula) to speak of, i.e. SAT0 is not analogous to k-SAT in a way that it can replace it in D’s proof.
• Ryan Williams permalink
August 15, 2010 11:40 am
I will explain what I wrote once more. I can see that the phrase “solution space of SAT0″ causes a nasty knee-jerk reaction. So I will state the problem completely in terms of k-SAT:
There is an infinite collection C of satisfiable k-SAT formulas with two properties:
(a) C is decided by a single, simple algorithm
(b) For any other collection C’ of satisfiable k-SAT formulas, the collection C contains a subcollection of formulas with solution distributions that require at least as many parameters to describe as C’ (provided I understand “parameters” correctly, which I think I do now, thanks to Tao and vloodin).
You can guess what C is. So the idea that we are capturing the complexity of a problem by showing how hard it is to generate satisfying assignments at random, doesn’t work. The statement that random k-SAT requires too many parameters should have a hole. I apologize if I didn’t make that clear.
If the current draft of the paper no longer is subject to this objection, that’s OK with me. I am planning for the future.
August 15, 2010 4:39 am
Let us consider the following claim: Suppose we can get a certain probability distribution on a set of binary variables (x_1,…,x_n) using a polynomial time run on a probabilistic Turing machine. Then our distribution is a projection of a c^polylog parametrizable (in the sense of Tao) distribution.
This claim seems to be pretty much what is done in the last chapter of D’s draft, using finite model theory, as it seems. But there appears to be a much easier direct proof of this.
It involves looking at a Cook-like table for the probabilistic Turing machine.
So, if this is correct, then there is a trivial proof that polynomial sampling gives rise to a projection of polynomial-parametrizable distribution, which is content of chapter 8.
So, it appears that step 2. can be fixed, in fact it is something easy, no need to prove this using finite model theory. Also, if we have a distribution with polynomial number of parameters, then we can sample it in polynomial number of steps (though this is on a circuit level). Thus, polynomial parametrizibility is just the same as polynomial random circuit sampling, and c^polylog parametrizibility is c^polylog random circuit sampling.
The non-trivial part is to show that k-SAT distribution does not have polynomial number of parameters (this is weaker than O(2^polylog), and of course than the claim that we need exponential number of parameters, in the Tao’s sense but allowing for projections). But as we see, this is just as hard as showing that k-SAT cannot be sampled in polynomial time (using circuits), and harder than showing k-SAT cannot be sampled in polynomial time (using algorithms); note that distribution is not uniform, as Terrence pointed out.
So does this follow from statistical physics results? Of course not, this would be an entirely new result. Yet not a word about how condensing implies exponential number of parameters is in D’s draft. And clearly, any proof of this would be proof that k-SAT does not have easy sampling, which is no easier than P!=NP in the first place.
Thus, as Terrence said, it seems that this paper does not even touch the P!=NP question.
August 15, 2010 5:08 am
Let’s not overstate and say that part 2 is true and has an easy proof and part 1 is the only missing part. Deolalikar’s claim was that poly-time algorithms have simple solution spaces and this is not what part 2 shows. We all know that projections of simple objects need not be simple themselves. This is the P vs NP question itself and has provable manifestations in many other domains.
August 15, 2010 7:25 am
There is a small correction – to “add” T numbers (using dummies) we need O(T) nodes, not O(log T) – we construct a tree of depth log(T), but on first level there are T/2 nodes, then T/4 etc.
On another note, about the question of a-priori estimating number of parameters (with projection), there seems to be some point to D’s approach. He essentially defined hardness of solution space by difficulty of sampling a quasi-uniform distribution on it (distribution is not uniform, but is easy enough to define, see Terrence’s comments). Thus, while in circuit complexity we had to deal with circuits that solved various instances of k-SAT, here we have probabilistic circuits that sample solutions of a single instance k-SAT formula. Now perhaps someone can answer this: how does the natural proofs barrier apply to this setting? I do not know enough about natural proofs barrier to answer this, but it appears that these probabilistic circuit samplings are subject to the same barrier as ordinary circuit complexity. But maybe, just maybe, there is a difference.
August 15, 2010 7:32 am
Also, D’s approach does give a way to separate easy from hard instances of k-SAT formulas. For each formula, his approach defines (difficult to compute) number of parameters constant, that is minimal number of parameters needed for distribution in Tao’s sense, but allowing projections. It is similar to another constant which can be defined to measure complexity of a k-SAT formula, that is complexity of a circuit determining if a partial assignment can be extended, which is just an application of circuit complexity, known to be subject to the natural proofs barrier.
August 15, 2010 7:38 am
Note that non-uniform distribution that appears seems to depend on the order of variables. So a complexity invariant is up to this order, but one would guess these should not vary that much (or one can take order that has minimal number of parameters). But I am skeptical how useful this invariant, a number attached to each single formula f from k-SAT, really is.
• August 15, 2010 9:35 am
vloodin: A measure of “instance hardness” would have many practical applications, but I don’t see how one could efficiently compute this measure. This seems at first glance as hard as establishing the Kolmogorov complexity of an associated circuit (as you also seem to suggest).
Also the variable ordering is critical: for algorithms based on exhaustive search, this is the main challenge. By arranging the search tree with the solution leaves clustered together, a deterministic depth-first search can find the first solution in logarithmic time, and list the rest in near-constant time per solution.
August 15, 2010 3:03 pm
Yes, it is as hard as that. This definition seems to have same problems as circuit complexity approach, which is natural proofs. So it seems rather useless when we come to the bottom of it.
59. Janos Simon permalink
August 15, 2010 6:20 am
vloodin’s proof can be made even simpler: there is no need for a complicated encoding of the head position. One can just use oblivious machines, at the cost of a multiplicative logT factor, which is OK for polynomial T.
August 15, 2010 12:20 pm
“But what is completely missing from the paper is any indication why the property of polylog parameterisability would be preserved under projections. Thus, for instance, the uniform measure on the space of k-SAT solutions may well be so complicated that it defies a polylog parameterisation, but that if one lifts this space into a polynomially larger space by adding in a lot of additional variables, then polylog parameterisability might well be restored (imagine for instance if SAT could be solved by a polynomial length straight line program, then by adding all the intermediate stages of computation of that program we would get a polylog parameterisation).”
Does D actually need to show this? He is assuming that P=NP and deriving a contraction. So if D has lemma of the form:
“Suppose P=NP. Then the projection of a polylog paramaterisation is polylog.”
then the actual status of projections of polylog paramaterisation is irrelevant. (Sorry if I’ve missed the point, or if this is dealt with elsewhere.)
61. Micki St James permalink
August 22, 2010 2:39 am
What is the arity of Rp ?
August 23, 2010 10:00 am
If a researcher were to prove that P=NP, presumably the proof would be easily checked because he or she would suggest an algorithm and we’d all type it up in our favorite programming language and watch it solve hard problems fast.
A proof of the non-existence of such an algorithm is obviously much harder, which is why (I assume) most of the attempted proofs are long sequences of math statements instead of algorithms we could check by running. I’m curious if there are any ideas for a (possibly very long running) algorithm that could test all the cases of 3SAT and prove that for each possible configuration, no simple solution exists, essentially a proof of NP!=P that is similar to the 4-color theorem proof. Anyone know of any work in that direction?
• Bruce Smith permalink
September 2, 2010 7:37 pm
(A proof that P = NP might not lead to a practical algorithm — it might be a non-constructive proof, or it might provide an algorithm whose complexity was a “very large” polynomial, e.g. of exponent 1000 or even 10^(10^1000), which would be useless in practice. In real life, even an exponent of 4 or 5 is usually hard to make practical use of.)
September 3, 2010 2:27 pm
Ok, you’re right about the non-constructive case, where all the person does is prove “there exists…”. But for the high power polynomials, we could still confirm time empirically on small values of N, even if it couldn’t handle anything large enough to be of real world value.
That’s still a tangent off of my main question — is there any brute force “division of the 3SAT into testable cases” algorithm that anyone has proposed? Is that a direction from which researchers have ever tried to attack P=NP ?
63. August 30, 2010 4:37 am
I am a professor of Computer Science in University of Hyderabad located in Hyderabad, India. Ours is a post-graduate and research institution and is recognized as one of the best universities in India. On Saturday, 28 August 2010, we heard a talk by Prof. Narendra S. Chaudhari (currently a visiting professor here in our department but in real-life, Professor at Dept. of Computer Science and Math at IIT, Indore). He said that the work is a result of more than 4 years of his efforts and also it is the first time that he is going public with it. The talk, titled Computationally Difficult Problems: Some Investigations gave a constructive procedure for solving 3-SAT problem in polynomial time (O(n^13)). Of course, the implications, if the procedure is true and holds in general are enormous as it also implies that P = NP. I found the procedure fascinating as it is fundamentally different from all existing ones that I have seen. The main trick lies in representation that allows Chaudhari to cast the 3-SAT problem in terms of the CLAUSES and not the literals. The complexity is fundamentally different because the mapping from literals to clauses is exponential while from clauses to 3-SAT is linear. There are space-time trade-offs and also aspects of dynamic programming involved in his constructive procedure. One the whole, the attempt appeared very serious and worth looking into more closely. I cannot still recover from the talk and there is a feeling of being a part of something momentous even if someone later finds a flaw in his procedure. The procedure appears to ring true on many critical aspects and I am speaking as a true-blue CS academic:-) A full account of the work is published in the Journal of the Indian Academy of Mathematics (Vol 31(2), 2009 pp 407,444) and a copy is available at this link. Prof. Chaudhari said that he has communicated the paper to some of the experts in our field and is awaiting comments, but thought that your blog is a great place to post given the people that seem to read it.
• August 30, 2010 9:19 pm
The paper does not report on actual runs of the algorithm, which seems like it should be easy to program. Neither do the slides. There are competitions on heuristics for solving SAT going on all the time—see section 7 of Wikipedia’s SAT page and references from there, including http://www.satcompetition.org
Most to the point, these competitions have lists of hard instances on which to test the algorithm. Of course I expect that carefully programming it would reveal errors. The paper needs to compare the algorithm to the well-known technique of resolution. The slides should also reference the mid-1980’s attack on P=NP by Ted Swart of Guelph University, whose error though blunt was tough to refute concretely, and hence led to some interesting work.
• Narendra S Chaudhari permalink
September 20, 2010 6:29 am
Thanks for the suggestion that careful programming is expected to reveal the errors: a few independent groups are in my contact and they have initiated the careful programming you have suggested. Testing on a few datasets in fact gave encouraging results.
Ted Swart’s work, if I remember correctly, was on Hamiltonian Circuit problem: I had seen the work earlier sometime in mid 1980’s but I am unable to trace it now. I dont think my approach has some similarity with that work. I would highly appreciate a copy on my gmail account ([email protected]) to explore the possibility of refering to the same.
• Micki St James permalink
September 1, 2010 12:07 am
3-SAT-Satisfiability Algorithm step V suggests possible invariants for the necessary
algorithm correctness proof, namely that T contains a list of top-level restricted
literals, Wx,x contains such a list when x = true and Wx,y contains such a list when
the pair of literals (x,y) = (true, true). Naturally one wonders why there is no
need for a set Wx,y,z to contain such a list when the
triple of literals (x,y,z) = (true, true, true).
I’d want to try the algorithm on \$(\notp \or \notq \or t) \and (\notr \or s \or \not t)
to observe how the Truth-Value Assignments Algorithm avoids serving up the
non-solution p=T q=T r=T s=F.
• Narendra S Chaudhari permalink
September 20, 2010 6:11 am
Part 1: (why there is no need for Wx,y,z): The justification is based on how the approach is developed using “literal-pair” as antecedent (for 3-Clause representation).
Part 2: (explanation of how the non-satisfying truth-value assignment is avoided) : explanation based on C-matrix (Clause representation) takes care of avoiding the non-satisfying truth-value assignment. (i have prepared the detailed steps for the example you have suggested, and I would email you the file containing the same: I could not attach that file with the present comment).
September 1, 2010 2:02 pm
Vinay gave a talk at HP Labs today – it was very high level and had no details. He claimed that some of the experts in the field have “okayed” his revised version.
65. September 6, 2010 9:19 am
A proof of the non-existence of such an algorithm is obviously much harder, which is why (I assume) most of the attempted proofs are long sequences of math statements instead of algorithms we could check by running. I’m curious if there are any ideas for a (possibly very long running) algorithm that could test all the cases of 3SAT and prove that for each possible configuration, no simple solution exists, essentially a proof of NP!=P that is similar to the 4-color theorem proof. Anyone know of any work in that direction?
66. September 6, 2010 9:52 am
That is not the problem. The clique size is k, which is independent on n. In the expression for potential, we have m clauses, and if each is given by this number of parameters, number of parameters is not exponential. So he must mean something else. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 24, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8038583993911743, "perplexity": 969.3552761489383}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928965.67/warc/CC-MAIN-20150521113208-00341-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/229690-identity-sum-print.html | # Identity of a sum
• Jun 17th 2014, 10:37 AM
Peppo
Identity of a sum
Let $x_{1},\dots,x_{N}$ real numbers such that $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{i+ j}=\frac{1}{2j+1}$$ for each natural number $j\in\left[1,N\right]$. Find (in function of $N$) the value of $$\underset{i=1}{\overset{N}{\sum}}\frac{x_{i}}{2i +1}.$$
• Jun 19th 2014, 12:26 PM
Peppo
Re: Identity of a sum
I know that I have to use the inverse of the Hilbert matrix, but I can understand how.
• Jun 20th 2014, 02:11 AM
Idea
Re: Identity of a sum
Well, yes but...
First
$X = H^{-1} B$
where H is (like) the Hilbert matrix and
B is the matrix with entries $\frac{1}{2j+1}$
Then evaluate
$B^t H^{-1} B=\frac{n(n+1)}{(2n+1)^2}$
• Jun 21st 2014, 12:34 AM
Peppo
Re: Identity of a sum
Quote:
Originally Posted by Idea
Well, yes but...
First
$X = H^{-1} B$
where H is (like) the Hilbert matrix and
B is the matrix with entries $\frac{1}{2j+1}$
Then evaluate
$B^t H^{-1} B=\frac{n(n+1)}{(2n+1)^2}$
Yes I see what you say... but how I can prove that $B^{t}H^{-1}B=\frac{N\left(N+1\right)}{\left(2N+1\right)^{2} }?$ Have I to use the explicit expression of $H^{-1}?$
• Jun 21st 2014, 04:02 AM
Idea
Re: Identity of a sum
To start, let me say that I don't know how to solve this problem.
I arrived at the answer by checking the results for some small values of n and then making a wild guess.
If you have an explicit expression for the i-j entry of the inverse of H then that would be one way.
There is such a formula for the inverse of the Hilbert matrix in terms of binomial coefficients and it is quite complicated.
Another way would be to use induction on the size of H.
• Jul 8th 2014, 03:55 AM
Peppo
Re: Identity of a sum
I'm not able to use the explicit inverse of the Cauchy (Hlibert) matrix. This is a new "hint" for this problem: if you consider the Shifted Legendre polynomials $$P_{n}\left(x\right)=\frac{1}{n!}\frac{d^{n}}{dx^ {n}}\left(x-x^{2}\right)^{n}$$ you have $$\int_{0}^{1} P_{n}\left(x\right)P_{m}\left(x\right)dx=\frac{ \delta_{mn} }{2n+1}$$ where $\delta_{mn}$ is the Kronecker delta. How can use it? Can someone help me? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9699133634567261, "perplexity": 580.2543996574856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560282937.55/warc/CC-MAIN-20170116095122-00423-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/159669/how-to-print-a-warning-sign-triangle-with-exclamation-point/159701 | How to print a warning sign (triangle with exclamation point)?
I am interested in including a triangle with an exclamation point (unicode #9888: ⚠) in a tex document. I've looked in the usual places (detexify, etc.) and not found this symbol. The document should be compilable with pdflatex and the source should have ASCII encoding. I already have a tikz version that I like reasonably well, given by
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}
\newcommand{\warningsign}{\tikz[baseline=-.75ex] \node[shape=regular polygon, regular polygon sides=3, inner sep=0pt, draw, thick] {\textbf{!}};}
\begin{document}
\warningsign Warning
\end{document}
so I am also requiring that the answer be an actual character, or at least not done using graphics.
• I usually use {\fontencoding{U}\fontfamily{futs}\selectfont\char 66\relax} for that symbol. Feb 10 '14 at 17:38
• @Nicola Talbot: Works like a charm! Please add that as an answer so that I can upvote it (and accept it unless someone comes along with a deeper explanation). Feb 10 '14 at 17:53
• You also have \usepackage{bclogo} and \bcattention. Feb 11 '14 at 1:51
• this command is incompatible with \usepackage[french]{babel} Sep 7 '15 at 13:28
The fourier package provides \danger:
\documentclass{article}
\usepackage{fourier}
\begin{document}
\danger
\end{document}
I sometimes find that some of the fourier commands conflict with other packages I use, so if I only want this symbol I do:
\documentclass{article}
\begin{document}
{\fontencoding{U}\fontfamily{futs}\selectfont\char 66\relax}
\end{document}
which is essentially what \danger does. This requires the futs font family which is provided with the fourier package, so the package must still be installed even though it's not actually being loaded.
If the futs font isn't available, the transcript will show the message:
LaTeX Font Warning: Font shape U/futs/m/n' undefined
(Font) using U/cmr/m/n' instead on input line 5.
This means that the cmr font is being used instead, which has the letter B in the \char 66 slot.
• In the second example, note that you need to have the fourier package installed in your TeX Live distribution even though it is not explicitly loaded. Otherwise, you will just get a "B" instead of a warning triangle. May 10 '17 at 3:08
• Thanks, @RadonRosborough. I was initially getting a "B". [at]NicolaTalbot could you add that to the answer as well? Jun 8 '17 at 10:13
• I get this error in a Beamer document on Arch Linux: "mktexpk: don't know how to create bitmap font for futr8r. mktexpk: perhaps futr8r is missing from the map file." Is there another package I have to install? Sep 25 '19 at 8:49
A \warning symbol (previously \danger) is provided by the fourier package (see The Comprehensive LATEX Symbol List, table 475, page 177).
If you don't have the need for the complete fourier package, but you want to use that symbol, you can extract it and use in your document:
\newcommand*{\TakeFourierOrnament}[1]{{%
\fontencoding{U}\fontfamily{futs}\selectfont\char#1}}
\newcommand*{\danger}{\TakeFourierOrnament{66}}
MWE
\documentclass{article}
\newcommand*{\TakeFourierOrnament}[1]{{%
\fontencoding{U}\fontfamily{futs}\selectfont\char#1}}
\newcommand*{\danger}{\TakeFourierOrnament{66}}
\begin{document}
\danger
\end{document}
Output
A simple build assigned to the unicode symbol, so you can use ⚠ or \Warning:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{xcolor}
\usepackage{newunicodechar}
\newcommand\Warning{%
\makebox[1.4em][c]{%
\makebox[0pt][c]{\raisebox{.1em}{\small!}}%
\makebox[0pt][c]{\color{red}\Large$\bigtriangleup$}}}%
\newunicodechar{⚠}{\Warning}
\begin{document}
on the power plug.
\end{document}
Here, I choose to build my own by overlaying a black, tiny ! atop a red $\triangle$, and then scaling the result to a desired [optional] size
\documentclass{article}
\usepackage{stackengine}
\usepackage{scalerel}
\usepackage{xcolor}
\newcommand\dangersign[1][2ex]{%
\renewcommand\stacktype{L}%
\scaleto{\stackon[1.3pt]{\color{red}$\triangle$}{\tiny !}}{#1}%
}
\begin{document}
This is a danger sign 5ex tall: \dangersign[5ex]\par
Here is the default (2ex) size: \dangersign
\end{document}
Zoom of result, to clarify my response to Charles' comment.
If one prefers a bolder !, just make it \bfseries:
\documentclass{article}
\usepackage{stackengine}
\usepackage{scalerel}
\usepackage{xcolor}
\newcommand\dangersign[1][2ex]{%
\renewcommand\stacktype{L}%
\scaleto{\stackon[1.3pt]{\color{red}$\triangle$}{\tiny\bfseries !}}{#1}%
}
\begin{document}
This is a danger sign 5ex tall: \dangersign[5ex]\par
Here is the default (2ex) size: \dangersign
\end{document}
• Perhaps it's my imagination, but the exclamation point in the small version looks slightly off-center. Feb 10 '14 at 20:03
• @CharlesStaats Sometimes, when glyphs are overlaid, inexact rendering will make them appear off center. Zooming in reveals a better approximation, and the printed result is accurate. Zooming the small symbol to 6400% reveals aligned glyphs. Feb 10 '14 at 20:05
• A question about the stackrel package: why does it render the \danger symbol from the fourier package as a letter B? Feb 11 '14 at 2:26
• @CharlesStaats Do you mean scalerel package? I would note that scalerel macros process their argument in math mode, by default, and one must actually delimit them with $...$ to process in text mode. So my guess is that the symbol is being processed in the wrong mode. Feb 11 '14 at 2:33
• Your guess seems to have been entirely correct. Thanks! Feb 11 '14 at 4:29
When you use XeLaTeX or LuaLaTeX you can simply use the character as is:
\documentclass{article}
\usepackage{fontspec}
\setmainfont[Ligatures=TeX]{STIXGeneral}
\newcommand\warningsign{⚠}
\begin{document}
\warningsign Warning
\end{document}
Of course the font must contain it. (STIX does.)
It looks like this:
Notice that it is more common to give unicode code points in hexadecimal, here U+26A0.
If you don’t want to have unusual characters in your source code, you can specify the warning sign by its code point:
\newcommand\warningsign{\symbol{"26A0}}
• With the most recent version of the STIX fonts, the call should be \setmainfont[Ligatures=TeX]{STIX} Feb 10 '14 at 18:36
• I appreciate the information, but prefer that my source code contain no characters that do not show up on my (American) keyboard. Feb 10 '14 at 18:48
• @CharlesStaats: You can use ASCII characters: ^^^^26a0, see The ^^ notation in various engines Feb 10 '14 at 18:54
The package fontawesome provides the commands \faWarning (equiv., \faExclamationTriangle for the warning icon, which looks like this:
A minimal example would be:
\documentclass{article}
\usepackage{fontawesome}
\begin{document}
\faWarning\, Warning: This product contains peanuts, %
which might not be very suitable for certain individuals.
\end{document}
The fontawesome package might be new, but it provides a wide range of high-quality web icons, and that makes it pretty relevant tool in terms of modern typesetting and design. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8308762311935425, "perplexity": 2970.7943909746623}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358233.7/warc/CC-MAIN-20211127193525-20211127223525-00259.warc.gz"} |
http://lt-jds.jinr.ru/record/61865?ln=en | / hep-ex CMS-SMP-12-020
Measurement of the production cross section for $Z\gamma \to \nu\bar{\nu}\gamma$ in pp collisions at $\sqrt{s}$ = 7 TeV and limits on $ZZ\gamma$ and $Z\gamma\gamma$ triple gauge boson couplings
Pages: 29
Abstract: A measurement of the $Z\gamma \to \nu\bar{\nu}\gamma$ cross section in pp collisions at $\sqrt{s}$ = 7 TeV is presented, using data corresponding to an integrated luminosity of 5.0 inverse femtobarns collected with the CMS detector. This measurement is based on the observation of events with an imbalance of transverse energy in excess of 130 GeV and a single photon in the absolute pseudorapidity range abs(eta) < 1.4 with transverse energy above 145 GeV. The $Z\gamma \to \nu\bar{\nu}\gamma$ production cross section is measured to be 21.1 +/- 4.2 (stat.) +/- 4.3 (syst.) +/- 0.5 (lum.) fb, which agrees with the standard model prediction of 21.9 +/- 1.1 fb. The results are combined with the CMS measurement of Z$\gamma$ production in the $l^+ l^- \gamma$ final state (where l is an electron or a muon) to yield the most stringent limits to date on triple gauge boson couplings: abs(h[3,Z]) < 2.7E-3, abs(h[4,Z]) < 1.3E-5 for $ZZ\gamma$ and abs(h[3,gamma]) < 2.9E-3, abs(h[4,gamma]) < 1.5E-5 for $Z\gamma\gamma$ couplings.
Note: Submitted to JHEP
Total numbers of views: 5457
Numbers of unique views: 947 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9936675429344177, "perplexity": 1433.223133600658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526401.41/warc/CC-MAIN-20190720004131-20190720030131-00125.warc.gz"} |
http://nrich.maths.org/1408&part= | ### Calendar Capers
Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
### Rotating Triangle
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
# Volume of a Pyramid and a Cone
##### Stage: 3
Published October 2001,October 2004,February 2011.
These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.
(The ideas used in this article are also used here to find the volume of a sphere and the, impressively named, mouhefanggai shape.)
### What you need to know.
You need to be familiar with the formulae for the area of a square, the area of a circle, and the volume of a cuboid. We will also use ideas of enlargement and ratio.
### The yangma.
When we think of a pyramid, we usually think of one where the vertex (the top point) is above the centre of the base, a right pyramid . However in this article, we will begin with a yangma. Yangma is an ancient Chinese name for a rectangular-based pyramid whose vertex is vertically above one of the vertices of the base. We will take a yangma with a square base of side length $a$, and height also length $a$. Three of these can be fitted together to form a cube.
Since we know that the volume of the cube is $a \times a \times a = a^3$, it follows that the volume of each of these yangmas is $\frac{a^3}{3}$.
### Enlargement
Consider, for a moment, a cube with edges 1 unit long. You know that the volume is $1 \times 1 \times 1$. Now stretch that cube in a horizontal direction so that it measures a by 1 by 1. Its volume is now $a \times 1 \times 1$. If we imagine the cube sliced horizontally, there will be the same number of slices, but each will be a times as long.
If we stretch the cube in a perpendicular direction, so that it measures a by b by 1, the volume will be $a \times b \times 1$, or $b$ times larger. If we stretch the cube in the third perpendicular direction, by a scale factor of $c$, the volume will be $a \times b \times c$, which we know as the formula for the volume of a cuboid.
There are three independent directions in which we could enlarge a 3D shape. If we enlarge it by a scale factor $k$ in one of these directions, the volume will be $k$ times as large.
#### Back to our yangma:
Suppose we want the volume of a yangma whose height is different to the base lengths, perhaps height $h$ instead of a. Well, this is just an enlargement in the vertical direction.
The scale factor? If we are turning $a$ into $h$, we have multiplied by $\frac{h}{a}$.
So the volume of our new yangma is $\frac{h}{a} \times \frac{a^3}{3}$, or $\frac{a^2h}{3}$.
### Sliding the slices
We now have a formula for the volume of any square-based pyramid whose vertex is above one of the vertices of the base. What about if the vertex is somewhere else - the middle, for instance? What we are going to do is to imagine the pyramid cut into lots of slices horizontally. We are going to slide these slices across, so that the top of the pyramid is above the middle of the base.
(Please click here for an animation of this process, although be aware that it might take a very long time to load on some computers).
If we had an infinite number of slices, our pyramid would have nice straight edges. You can probably imagine that with more slices, it would look smoother than in this illustration. Has the area of any of the slices changed? So has the volume of the shape changed?
We can now see that the volume of any square-based pyramid is $\frac{a^2h}{3}$.
### Comparing a cone with a pyramid
We will now look at a cone. We'll start with a right cone , whose vertex is above the centre of the base. In fact, by slicing it as in the previous section, we can show that the same formula applies for any cone.
Imagine a cone whose base is a circle radius $r$, and height is $h$. This cone will fit exactly inside a square-based pyramid with base length $2r$ and height $h$.
Suppose we take a slice of the pyramid with the cone inside, from some way up the pyramid. This will look like a square with a circle fitting inside. We don't know the radius of the cone at this point, so we'll call it $x$.
The area of the circle is $\pi x^2$.
The area of the square is $2x \times 2x=4x^2$.
The ratio of the circle to the square is $\pi : 4$.
The same is true for every slice we take: the area of the circle is $\frac{\pi}{4}$ of the area of the square.
So, if each slice is $\frac{\pi}{4}$ the size, the volume of the cone will be $\frac{\pi}{4}$ the volume of the pyramid.
The pyramid's volume is $\frac{(2r)^2h}{3} = \frac{4r^2h}{3}$
So the cone's volume is $\frac{\pi}{4} \times \frac{4r^2h}{3} = \frac{\pi r^2 h}{3}$.
### Non-square-based pyramids
We can use the same principles to find the volume of any pyramid.
#### Rectangular-based pyramid
If we have a pyramid with rectangular base measuring $a$ by $b$, and height $h$, then this can be obtained by stretching our square-based pyramid by scale factor $\frac{b}{a}$. The new volume will be $\frac{b}{a} \times \frac{a^2 h}{3} = \frac{a b h}{3}$.
#### Triangular-based pyramid
If the base of the pyramid is a triangle with base $a$ and perpendicular height $b$, it will fit exactly in the rectangular pyramid above.
Any slice will look like this:
Although we don't know the measurements of the rectangle in this slice, the sides will still be in the ratio $a:b$ (this may take some thinking about). Let's call them $k{a}$ and $k{b}$ (where $k$ is less than 1).
Area of rectangle = $k{a} \times k{b} = k^2 a{b}$
Area of triangle = $\frac{1}{2}{k}{a} \times {k}{b} = \frac{1}{2}k^2{a}{b} = \frac{1}{2} \times \mbox{area of rectangle}$.
If each triangle is half the size of the rectangle, the volume of the triangular-based pyramid will be half the volume of the rectangular-based pyramid, or $a{b}h/6$.
### Generalisation
The formula for the volume of any pyramid is $\frac{1}{3}\mbox{base area} \times \mbox{height}$.
Verify that this works for the pyramids above (and indeed for the cone). Can you convince yourself that this is always true? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8618983030319214, "perplexity": 262.44763680549903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861812410.28/warc/CC-MAIN-20160428164332-00220-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.mrmath.com/misfit/algebra-stuff/binomial-expansions/ | # Binomial Expansions
## Binomial Expansions
By algebraic definition, binomials are the sum of two objects. Let $(x+y)$ be any binomial (that is to say, whatever the two objects being added together are, let $x$ be the first and $y$ be the second).Binomial expansion refers to the result you get when you raise a binomial to a power. As much as we may wish it so,$$(x+y)^n \ne x^n + y^n$$This can be instantly proven for $n=2$:$$(x+y)^2 = x^2 + 2xy + y^2 \ne x^2 + y^2$$The hard truth is that, in order to multiply binomials, you need to distribute (FOIL, in the case of $n=2$). That is, for example,$$(x+y)^2 = (x+y)(x+y)$$This quickly becomes tedious and annoying - even multiplying three binomials together seems like a drag.$$(x+y)^3 = (x+y)(x+y)(x+y)$$However, Pascal's Triangle gives a magnificent shortcut for this process! If you want to multiply $(x+y)^n$, follow these guidelines:
• The variable part of the first term is $x^n$
• The variable part of the next term is $x^{n-1} y$
• Each sequential term's variable piece decreases the exponent on $x$ and increases the exponent on $y$. i.e. the third term is of the form $x^{n-2} y^2$
• Keep going until you have decreased $x$ to nothing, at which point your last term will be $y^n$
• If you have $(x+y)$ then all terms are positive, and if you have $(x-y)$ then the first term is positive, the second term is negative, and each term after that alternates
• Finally, and significantly - the coefficients of each term will come from the $n+1$ row in Pascal's Triangle
Recall that Pascal's Triangle » looks like this:Here are three examples. See if you can line up the guidelines above with the results. When you know the pattern, you don't actually have to do the tedious multiplication to get the answer!$$(x+y)^5$$
$$(x-y)^7$$
$$(2a + 3b)^4$$
• Popular Content
• Get Taught
• Other Stuff | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9118796586990356, "perplexity": 440.82226101992876}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385534.85/warc/CC-MAIN-20210308235748-20210309025748-00234.warc.gz"} |
https://devopedia.org/hidden-markov-model | • HMM states (X), observations (O) and proabilities (A, B). Source: Stamp 2018, fig. 1.
• An example of Hidden Markov Model. Source: Wikipedia 2019.
• Typical notation used in HMM. Source: Kang 2017.
• Complex birdsong analyzed using HMM. Source: Adapted from Katahira et al. 2011, fig. 4.
• Some types of HMMs. Source: Rabiner 1989, fig. 7.
• Trellis diagrams showing forward and backward algorithms. Source: Adapted from Jana 2019b.
# Hidden Markov Model
arvindpdmn
1365 DevCoins
Last updated by arvindpdmn
on 2019-09-06 05:47:21
Created by arvindpdmn
on 2019-09-05 16:45:46
## Summary
Consider weather, stock prices, DNA sequence, human speech or words in a sentence. In all these cases, current state is influenced by one or more previous states. Moreover, often we can observe the effect but not the underlying cause that remains hidden from the observer. Hidden Markov Model (HMM) helps us figure out the most probable hidden state given an observation.
In practice, we use a sequence of observations to estimate the sequence of hidden states. In HMM, the next state depends only on the current state. As such, it's good for modelling time series data.
We can classify HMM as a generative probabilistic model since a sequence of observed variables is generated by a sequence of hidden states. HMM is also seen as a specific kind of Bayesian network.
## Milestones
1913
Russian mathematician A. A. Markov recognizes that in a sequence of random variables, one variable may not be independent of the previous variable. For example, two successive coin tosses are independent but today's weather might depend on yesterday's weather. He models this as a chain of linked events with probability assigned to each link. This technique later is named Markov Chain.
1966
Baum and Petrie at the Institute of Defense Analyses, Princeton, introduce the Hidden Markov Model (HMM), though this name is not used. They state the problem of estimating transition and emission probabilities from observations. They use maximum likelihood estimate.
1967
Andrew Viterbi publishes an algorithm to decode information at the receiver in a communication system. Later named Viterbi algorithm, it's directly applicable to the decoding problem in HMM. Vintsyuk first applies this algorithm to speech and language processing in 1968.
1970
The Baum-Welch algorithm is proposed to solve the learning problem in HMM. This algorithm is a special case of the Expectation-Maximization (EM) algorithm. However, the name HMM is not used in the paper and mathematicians refer to HMM as "probabilistic functions of Markov chains".
1975
James Baker at CMU applies HMM to speech recognition in the DRAGON speech understanding system. This is one of the earliest engineering applications of HMM. HMM is further applied to speech recognition through the 1970s and 1980s by Jelinek, Bahl and Mercer at IBM.
1989
Lawrence Rabiner publishes a tutorial on HMM covering theory, practice and applications. He notes that HMM originated in mathematics and was not widely read by engineers. Even when it was applied to speech processing in the 1970s, there were no tutorials to help translate theory into practice.
2003
HMM is typically used when the number of states is small but one research team applies it to large scale web traffic analysis. This involves hundreds of states and tens of millions of observations.
## Discussion
• Could you explain HMM with an example?
Suppose Bob tells his friend Alice what he did earlier today. Based on this information Alice guesses today's weather at Bob's location. In HMM, we model weather as states and Bob's activity as observations.
To solve this problem, Alice needs to know three things:
• Transition Probabilities: Probability of moving from one state to another. For example, "If today was sunny, what's the probability that it will rain tomorrow?" If there are N states, this is an NxN matrix.
• Emission Probabilities: Probability of a particular output given a particular state. For example, "What's the chance that Bob is walking if it's raining?" Given a choice of M possible observation symbols, this is an NxM matrix. This is also called output or observation probabilities.
• Initial Probabilities: Probability of being in a state at the start, say, yesterday or ten days ago.
Unlike a typical Markov chain, we can't see the states in HMM. However, we can observe the output and then predict the state. Thus, the states are hidden, giving rise to the term "hidden" in the name HMM.
• What types of problems can be solved by HMM?
Let A, B and π denote the transition matrix, observation matrix and initial state distribution respectively. HMM can be represented as λ = (A, B, π). Let observation sequence be O and state sequence be Q.
HMM can be used to solve three types of problems:
• Likelihood Problem: Given O and λ, find the likelihood P(O|λ). How likely is a particular sequence of observations? Forward algorithm solves this problem.
• Decoding Problem: Given O and λ, find the best possible Q that explains O. Given the observation sequence, what's the best possible state sequence? Viterbi algorithm solves this problem.
• Learning Problem: Given O and Q, learn λ, perhaps by maximizing P(O|λ). What model best maps states to observations? Baum-Welch algorithm, also called forward-backward algorithm, solves this problem. In the language of machine learning, we can say that O is training data and the number of states N is the model's hyperparameter.
• What are some applications where HMM is useful?
HMM has been applied in many areas including automatic speech recognition, handwriting recognition, gesture recognition, part-of-speech tagging, musical score following, partial discharges and bioinformatics.
In speech recognition, a spectral analysis of speech gives us suitable observations for HMM. States are modelled after phonemes or syllables, or after the average number of observations in a spoken word. Each word gets its own model.
To tag words with their parts of speech, the tags are modelled as hidden states and the words are the observations.
In computer networking, HMMs are used in intrusion detection systems. This has two flavours: anomaly detection in which normal behaviour is modelled; or misuse detection in which a predefined set of attacks is modelled.
In computer vision, HMM has been used to label human activities from skeleton output. Each activity is modelled with a HMM. By linking multiple HMMs on common states, a compound HMM is formed. The purpose is to allow robots to be aware of human activity.
• What are the different types of Hidden Markov Models?
In the typical model, called the ergodic HMM, the states of the HMM are fully connected so that we can transition to a state from any other state. Left-right HMM is a more constrained model in which state transitions are allowed only from lower indexed states to higher indexed ones. Variations and combinations of these two types are possible, such as having two parallel left-to-right state paths.
HMM started with observations of discrete symbols governed by discrete probabilities. If observations are continuous signals, then we would use continuous observation density.
There are also domain-specific variations of HMM. For example, in biological sequence analysis, there are at least three types including profile-HMMs, pair-HMMs, and context-sensitive HMMs.
• Could you explain forward algorithm and backward algorithm?
Every state sequence has a probability that it will lead to a given sequence of observations. Given T observations and N states, there are $$N^T$$ possible state sequences. Thus, the complexity of calculating the probability of a given sequence of observations is $$O(N^{T}T)$$. Both forward and backward algorithms bring down the complexity to $$O(N^{2}T)$$ through dynamic programming.
In the forward algorithm, we consider the probability of being in a state at the current time step. Then we consider the transition probabilities to calculate the state probabilities for the next step. Thus, at each time step we have considered all state sequences preceding it. The algorithm is more efficient since it reuses calculations from earlier steps. Instead of keeping all path sequences, paths are folded into a forward trellis.
Backward algorithm is similar except that we start from the last time step and calculate in reverse. We're finding the probability that from a given state, the model will generate the output sequence that follows.
A combination of both algorithms, called forward-backward algorithm, is used to solve the learning problem.
• What's the algorithm for solving HMM's decoding problem?
Viterbi algorithm solves HMM's decoding problem. It's similar to the forward algorithm except that instead of summing the probabilities of all paths leading to a state, we retain only one path that gives maximum probability. Thus, at every time step or iteration, given that we have N states, we retain only N paths, the most likely path for each state. For the next iteration, we use the most likely paths of current iteration and repeat the process.
When we reach the end of the sequence, we'll have N most likely paths, each ending in a unique state. We then select the most likely end state. Once this selection is made, we backtrack to read the state sequence, that is, how we got to the end state. This state sequence is now the most likely sequence given our sequence of observations.
• How can we solve the learning problem of HMM?
In HMM's learning problem, we are required to learn the transition (A) and observation (B) probabilities when given a sequence of observations and the vocabulary of hidden states. The forward-backward algorithm solves this problem. It's an iterative algorithm. It starts with an initial estimate of the probabilities and improves these estimates with each iteration.
The algorithm consists of two steps:
• Expectation or E-step: We compute the expected state occupancy count and the expected state transition count based on current probabilities A and B.
• Maximization or M-step: We use the expected counts from the E-step to recompute A and B.
While this algorithm is unsupervised, in practice, initial conditions are very important. For this reason, often extra information is given to the algorithm. For example, in speech recognition, the HMM structure is set manually and the model is trained to set the initial probabilities.
• Could you describe some tools for doing HMM?
In Python, hmmlearn package implements HMM. Three models are available: hmm.GaussianHMM, hmm.GMMHMM and hmm.MultinomialHMM. This package is also part of Scikit-learn but will be removed in v0.17. Stephen Marsland has shared Python code in NumPy and Pandas that implements many essential algorithms for HMM.
In R, HMM package implements HMM. It has functions for forward, backward, Viterbi and Baum-Welch algorithms. Another package depmixS4 implements dependent mixture models that can be used to fit HMM to observed data. R-bloggers has an example use of depmixS4.
## Sample Code
• # Source: https://stats.stackexchange.com/questions/31746/what-is-the-difference-between-the-forward-backward-and-viterbi-algorithms
# Accessed: 2019-09-05
library(HMM)
# in education setting,
# hidden state: Rainy and Sunny
# observation: Walk, Shop, Clean
# state transition
P <- as.matrix(rbind(c(0.7,0.3),
c(0.4,0.6)))
# emission prob
R <- as.matrix(rbind(c(0.1, 0.4, 0.5),
c(0.6,0.3, 0.1)))
hmm = initHMM(States=c("Rainy","Sunny"),
Symbols=c("Walk","Shop", "Clean"),
startProbs=c(0.6,0.4),
transProbs=P,
emissionProbs=R)
hmm
obs=c("Walk","Shop","Walk", "Clean")
print(posterior(hmm, obs)) # gives the state probabilities for each observation
print(viterbi(hmm, obs)) # gives the sequence of states
## Milestones
1913
Russian mathematician A. A. Markov recognizes that in a sequence of random variables, one variable may not be independent of the previous variable. For example, two successive coin tosses are independent but today's weather might depend on yesterday's weather. He models this as a chain of linked events with probability assigned to each link. This technique later is named Markov Chain.
1966
Baum and Petrie at the Institute of Defense Analyses, Princeton, introduce the Hidden Markov Model (HMM), though this name is not used. They state the problem of estimating transition and emission probabilities from observations. They use maximum likelihood estimate.
1967
Andrew Viterbi publishes an algorithm to decode information at the receiver in a communication system. Later named Viterbi algorithm, it's directly applicable to the decoding problem in HMM. Vintsyuk first applies this algorithm to speech and language processing in 1968.
1970
The Baum-Welch algorithm is proposed to solve the learning problem in HMM. This algorithm is a special case of the Expectation-Maximization (EM) algorithm. However, the name HMM is not used in the paper and mathematicians refer to HMM as "probabilistic functions of Markov chains".
1975
James Baker at CMU applies HMM to speech recognition in the DRAGON speech understanding system. This is one of the earliest engineering applications of HMM. HMM is further applied to speech recognition through the 1970s and 1980s by Jelinek, Bahl and Mercer at IBM.
1989
Lawrence Rabiner publishes a tutorial on HMM covering theory, practice and applications. He notes that HMM originated in mathematics and was not widely read by engineers. Even when it was applied to speech processing in the 1970s, there were no tutorials to help translate theory into practice.
2003
HMM is typically used when the number of states is small but one research team applies it to large scale web traffic analysis. This involves hundreds of states and tens of millions of observations.
Author
No. of Edits
No. of Chats
DevCoins
4
0
1365
1736
Words
0
Chats
4
Edits
3
Likes
1234
Hits
## Cite As
Devopedia. 2019. "Hidden Markov Model." Version 4, September 6. Accessed 2020-01-26. https://devopedia.org/hidden-markov-model
• Site Map | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8983246684074402, "perplexity": 1288.4261627642136}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251690095.81/warc/CC-MAIN-20200126165718-20200126195718-00295.warc.gz"} |
http://tex.stackexchange.com/questions/121605/macro-for-typesetting-semantic-denotations-linguistics | # Macro for Typesetting Semantic Denotations (Linguistics)
I tried to write a macro for typesetting semantic denotations, using the stmaryrd package, but there are a few issues that I would like to resolve. My first attempt at the macro was the following:
\documentclass{article}
\usepackage{stmaryrd}
\newcommand{\den}[2][]{$\llbracket$#2$\rrbracket^{#1}$}
\begin{document}
\den[w,g]{example} = \ldots
\end{document}
The issue that I would like to resolve is that, in some instances, the text between the denotation brackets gets lengthy and is hard to look at since it runs across multiple lines. (It seems that I can't insert a picture yet, but it looks something like the following.)
[[This is a purposefully long sentence whose denotation I wish to specify, but because it is so long it runs across multiple lines and ends up being both really hard to read and pretty gross, aesthetically speaking]] = . . .
Now, the stmaryrd is written such that \llbracket and \rrbracket are delimiters, so then I tried this:
\documentclass{article}
\usepackage{stmaryrd}
\newcommand{\den}[2][]{
$\left\llbracket#2\right\rrbracket^{#1}$
}
\begin{document}
\den{example}
\end{document}
I'm not too familiar with delimiters, but it seems you cannot insert line breaks between the \left and \right commands, so there is no way of breaking the text inside of the brackets across multiple lines. This second attempt also centers the denotation and italicizes the text inside of the brackets, two things that I do not want to happen.
With regard to the first issue, is there some way to limit the horizontal space of the text inside of the brackets, perhaps with a \parbox command? The text would then have to wrap inside of the limited horizontal space, of course, but this is something I have no idea how to do myself. And, ideally, the size of the horizontal space would then be an optional argument of the new \den command, so that it could either be specified or scaled when necessary.
-
Welcome to TeX.SX! – Papiro Jun 28 '13 at 16:54
I would do this a slightly different way (taking on your suggestion of a \parbox). I've created a second command to wrap a long denotation. You can then use this as needed to wrap longer denotations to a specified width (I've set a default width of 1in). I've also used the ragged2e package to wrap the long texts with a ragged margin and hyphenation, and also put the entire argument of the denotation in the amsmath \text command so that the text is in roman and not math italic. Finally, by using inline mode instead of displaymath, we can left align the denotations for use with an example environment.
### Update
I've used the varwidth package to set the wrapped text in a box its natural width. Thanks to David Carlisle for telling me about this package. I've also put the examples with a gb4e example to show what they would look like.
\documentclass{article}
\usepackage{stmaryrd}
\usepackage{amsmath}
\usepackage{ragged2e}
\usepackage{varwidth}
\newcommand{\den}[2][]{
$$\left\llbracket\;\text{#2}\;\right\rrbracket^{#1}$$
}
\newcommand{\wraptext}[2][1in]{\begin{varwidth}{#1}{\RaggedRight#2}\end{varwidth}}
\begin{document}
\den[\alpha]{\wraptext[3in]{example of a really long denotation which will continue as long as we like}}
\den[\alpha]{\wraptext[2in]{example of a really long denotation which will continue as long as we like}}
\den[\alpha]{a smaller one}
\end{document}
-
This is more or less perfect; thanks. I messed around with it a little bit, and there only seems to be one minor issue. I'm not sure if it is something that can easily be resolved or not. But, if you use your exact MWE and set the optional \wraptext argument to 3in in the "example of a really long denotation which will continue as long as we like", the right bracket is offset because of how the text wraps. Any ideas? If not, it works well and the width can be manually manipulated until it works out. – Adam Liter Jun 28 '13 at 20:42
@Adam Thanks for pointing that out. See my updated answer. – Alan Munn Jun 28 '13 at 21:00
A simple implementation could be
\documentclass{article}
\usepackage{stmaryrd}
\newcommand\den[2][]{%
\ensuremath{%
\left\llbracket
\begin{tabular}{@{}l@{}}
#2
\end{tabular}
\right\rrbracket^{#1}}}
\begin{document}
Test \den{foo} and
$\den[\beta]{some text\\ others} = \Gamma$
\end{document}
A tabular environment provides the ability to (explicitly) break lines as you wish. So you don't need to specify the width of the text.
There are certainly more possibilities, but I believe this simple implementation is easy to understand. If you have more requirements, let me know.
-
Is there a way of implementing the marco so that you actually can specify the horizontal width of the text as an argument of the command and then have the text wrap? It would be ideal for my purposes, since these expressions are often embedded in nested list environments using the gb4e package, for example. Being able to specify such a parameter in the \den command would ultimately be more useful than having to insert line breaks by hand each time. Also, is there some way to prevent the output from being centered? Is that a result of embedding it inside $ and $? – Adam Liter Jun 28 '13 at 19:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9592157602310181, "perplexity": 772.8324653478145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500833461.95/warc/CC-MAIN-20140820021353-00221-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://www.ck12.org/geometry/Pythagorean-Theorem-and-Pythagorean-Triples/lesson/Pythagorean-Theorem-and-Pythagorean-Triples-Intermediate/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Pythagorean Theorem and Pythagorean Triples
%
Progress
Practice Pythagorean Theorem and Pythagorean Triples
Progress
%
Pythagorean Theorem and Pythagorean Triples
What if a friend of yours wanted to design a rectangular building with one wall 65 ft long and the other wall 72 ft long? How can he ensure the walls are going to be perpendicular? After completing this Concept, you'll be able to apply the Pythagorean Theorem in order to solve problems like these.
### Guidance
The sides of a right triangle are called legs (the sides of the right angle) and the side opposite the right angle is the hypotenuse. For the Pythagorean Theorem, the legs are “ $a$ ” and “ $b$ ” and the hypotenuse is “ $c$ ”.
Pythagorean Theorem: Given a right triangle with legs of lengths $a$ and $b$ and a hypotenuse of length $c$ , then $a^2 + b^2 = c^2$ .
Pythagorean Theorem Converse: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
There are several proofs of the Pythagorean Theorem, shown below.
##### Investigation: Proof of the Pythagorean Theorem
Tools Needed: pencil, 2 pieces of graph paper, ruler, scissors, colored pencils (optional)
1. On the graph paper, draw a 3 in. square, a 4 in. square, a 5 in square and a right triangle with legs of 3 and 4 inches.
2. Cut out the triangle and square and arrange them like the picture on the right.
3. This theorem relies on area. Recall from a previous math class, that the area of a square is length times width. But, because the sides are the same you can rewrite this formula as $A_{square} = length \times width = side \times side = side^2$ . So, the Pythagorean Theorem can be interpreted as $(square \ with \ side \ a)^2 + (square \ with \ side \ b)^2 = (square \ with \ side \ c)^2$ . In this Investigation, the sides are 3, 4 and 5 inches. What is the area of each square?
4. Now, we know that $9 + 16 = 25$ , or $3^2 + 4^2 = 5^2$ . Cut the smaller squares to fit into the larger square, thus proving the areas are equal.
##### Another Proof of the Pythagorean Theorem
This proof is “more formal,” meaning that we will use letters, $a, b,$ and $c$ to represent the sides of the right triangle. In this particular proof, we will take four right triangles, with legs $a$ and $b$ and hypotenuse $c$ and make the areas equal.
For two animated proofs, go to http://www.mathsisfun.com/pythagoras.html and scroll down to “And You Can Prove the Theorem Yourself.”
##### Pythagorean Triples
A Pythagorean Triple is a set of three whole numbers that makes the Pythagorean Theorem true. The most frequently used Pythagorean triple is 3, 4, 5, as in Investigation 8-1. Any multiple of a Pythagorean triple is also considered a triple because it would still be three whole numbers. Therefore, 6, 8, 10 and 9, 12, 15 are also sides of a right triangle. Other Pythagorean triples are:
$3, 4, 5 \qquad 5, 12, 13 \qquad 7, 24, 25 \qquad 8, 15, 17$
There are infinitely many Pythagorean triples. To see if a set of numbers makes a triple, plug them into the Pythagorean Theorem.
#### Example A
Do 6, 7, and 8 make the sides of a right triangle?
Plug in the three numbers into the Pythagorean Theorem. The largest length will always be the hypotenuse . $6^2 + 7^2 = 36 + 49 = 85 \neq 8^2$ . Therefore, these lengths do not make up the sides of a right triangle.
#### Example B
Find the length of the hypotenuse of the triangle below.
Let’s use the Pythagorean Theorem. Set $a$ and $b$ equal to 8 and 15 and solve for $c$ , the hypotenuse.
$8^2 + 15^2 & = c^2\\64 + 225 & = c^2\\289 & = c^2 \qquad \quad Take \ the \ square \ root \ of \ both \ sides.\\17 & = c$
When you take the square root of an equation, usually the answer is +17 or -17. Because we are looking for length, we only use the positive answer. Length is never negative .
#### Example C
Is 20, 21, 29 a Pythagorean triple?
If $20^2 + 21^2$ is equal to $29^2$ , then the set is a triple.
$20^2 + 21^2 & = 400 + 441 = 841\\29^2 & = 841$
Therefore, 20, 21, and 29 is a Pythagorean triple.
#### Example D
Determine if the triangle below is a right triangle.
Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent $c$ , in the equation.
$a^2 + b^2 &= c^2\\ 8^2 + 16^2 &= \left ( 8 \sqrt{5} \right )^2\\64 + 256&= 64 \cdot 5\\320 &= 320$
The triangle is a right triangle.
Watch this video for help with the Examples above.
#### Concept Problem Revisited
To make the walls perpendicular, find the length of the diagonal.
$65^2 + 72^2 & = c^2\\4225 + 5184 & = c^2\\9409 & = c^2\\97 & = c$
In order to make the building rectangular, both diagonals must be 97 feet.
### Guided Practice
1. Find the missing side of the right triangle below.
2. What is the diagonal of a rectangle with sides 10 and $16 \sqrt{5}$ ?
3. Determine if the triangle below is a right triangle.
1. Here, we are given the hypotenuse and a leg. Let’s solve for $b$ .
$7^2 + b^2 & = 14^2\\49 + b^2 & = 196\\b^2 & = 147\\b & = \sqrt{147} = \sqrt{7 \cdot 7 \cdot 3} =7 \sqrt{3}$
2. For any square and rectangle, you can use the Pythagorean Theorem to find the length of a diagonal. Plug in the sides to find $d$ .
$10^2 + \left ( 16 \sqrt{5} \right )^2 & = d^2\\100 + 1280 & = d^2\\1380 & = d^2\\d & = \sqrt{1380} = 2 \sqrt{345}$
3. Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest sides represent $c$ , in the equation.
$a^2 + b^2 &= c^2\\22^2 + 24^2&= 26^2\\484 + 576 &= 676\\1060 &\neq 676$
The triangle is not a right triangle.
### Explore More
Find the length of the missing side. Simplify all radicals.
1. If the legs of a right triangle are 10 and 24, then the hypotenuse is _____________.
2. If the sides of a rectangle are 12 and 15, then the diagonal is _____________.
3. If the legs of a right triangle are $x$ and $y$ , then the hypotenuse is ____________.
4. If the sides of a square are 9, then the diagonal is _____________.
Determine if the following sets of numbers are Pythagorean Triples.
1. 12, 35, 37
2. 9, 17, 18
3. 10, 15, 21
4. 11, 60, 61
5. 15, 20, 25
6. 18, 73, 75
Pythagorean Theorem Proofs
The first proof below is similar to the one done earlier in this Concept. Use the picture below to answer the following questions.
1. Find the area of the square with sides $(a + b)$ .
2. Find the sum of the areas of the square with sides $c$ and the right triangles with legs $a$ and $b$ .
3. The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.
Major General James A. Garfield (and former President of the U.S) is credited with deriving this next proof of the Pythagorean Theorem using a trapezoid.
1. Find the area of the trapezoid using the trapezoid area formula: $A = \frac{1}{2} (b_1 + b_2)h$
2. Find the sum of the areas of the three right triangles in the diagram.
3. The areas found in the previous two problems should be the same value. Set the expressions equal to each other and simplify to get the Pythagorean Theorem.
### Vocabulary Language: English
Circle
Circle
A circle is the set of all points at a specific distance from a given point in two dimensions.
Conic
Conic
Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
degenerate conic
degenerate conic
A degenerate conic is a conic that does not have the usual properties of a conic section. Since some of the coefficients of the general conic equation are zero, the basic shape of the conic is merely a point, a line or a pair of intersecting lines.
Ellipse
Ellipse
Ellipses are conic sections that look like elongated circles. An ellipse represents all locations in two dimensions that are the same distance from two specified points called foci.
hyperbola
hyperbola
A hyperbola is a conic section formed when the cutting plane intersects both sides of the cone, resulting in two infinite “U”-shaped curves.
Hypotenuse
Hypotenuse
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle
Legs of a Right Triangle
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Parabola
Parabola
A parabola is the characteristic shape of a quadratic function graph, resembling a "U".
Pythagorean number triple
Pythagorean number triple
A Pythagorean number triple is a set of three whole numbers $a,b$ and $c$ that satisfy the Pythagorean Theorem, $a^2 + b^2 = c^2$.
Pythagorean Theorem
Pythagorean Theorem
The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.
Right Triangle
Right Triangle
A right triangle is a triangle with one 90 degree angle. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 49, "texerror": 0, "math_score": 0.861575186252594, "perplexity": 314.7871878740586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246636104.0/warc/CC-MAIN-20150417045716-00213-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://ch.mathworks.com/help/stats/ksdensity.html | # ksdensity
Kernel smoothing function estimate for univariate and bivariate data
## Description
example
[f,xi] = ksdensity(x) returns a probability density estimate, f, for the sample data in the vector or two-column matrix x. The estimate is based on a normal kernel function, and is evaluated at equally-spaced points, xi, that cover the range of the data in x. ksdensity estimates the density at 100 points for univariate data, or 900 points for bivariate data.
ksdensity works best with continuously distributed samples.
example
[f,xi] = ksdensity(x,pts) specifies points (pts) to evaluate f. Here, xi and pts contain identical values.
example
[f,xi] = ksdensity(___,Name,Value) uses additional options specified by one or more name-value pair arguments in addition to any of the input arguments in the previous syntaxes. For example, you can define the function type ksdensity evaluates, such as probability density, cumulative probability, survivor function, and so on. Or you can specify the bandwidth of the smoothing window.
example
[f,xi,bw] = ksdensity(___) also returns the bandwidth of the kernel smoothing window, bw. The default bandwidth is the optimal for normal densities.
example
ksdensity(___) plots the kernel smoothing function estimate.
ksdensity(ax,___) plots the results using axes with the handle, ax, instead of the current axes returned by gca.
## Examples
collapse all
Generate a sample data set from a mixture of two normal distributions.
rng('default') % For reproducibility
x = [randn(30,1); 5+randn(30,1)];
Plot the estimated density.
[f,xi] = ksdensity(x);
figure
plot(xi,f);
The density estimate shows the bimodality of the sample.
Generate a nonnegative sample data set from the half-normal distribution.
rng('default') % For reproducibility
pd = makedist('HalfNormal','mu',0,'sigma',1);
x = random(pd,100,1);
Estimate pdfs with two different boundary correction methods, log transformation and reflection, by using the 'BoundaryCorrection' name-value pair argument.
pts = linspace(0,5,1000); % points to evaluate the estimator
[f1,xi1] = ksdensity(x,pts,'Support','positive');
[f2,xi2] = ksdensity(x,pts,'Support','positive','BoundaryCorrection','reflection');
Plot the two estimated pdfs.
plot(xi1,f1,xi2,f2)
lgd = legend('log','reflection');
title(lgd, 'Boundary Correction Method')
xl = xlim;
xlim([xl(1)-0.25 xl(2)])
ksdensity uses a boundary correction method when you specify either positive or bounded support. The default boundary correction method is log transformation. When ksdensity transforms the support back, it introduces the 1/x term in the kernel density estimator. Therefore, the estimate has a peak near x = 0. On the other hand, the reflection method does not cause undesirable peaks near the boundary.
Compute and plot the estimated cdf evaluated at a specified set of values.
pts = (min(hospital.Weight):2:max(hospital.Weight));
figure()
ecdf(hospital.Weight)
hold on
[f,xi,bw] = ksdensity(hospital.Weight,pts,'Support','positive',...
'Function','cdf');
plot(xi,f,'-g','LineWidth',2)
legend('empirical cdf','kernel-bw:default','Location','northwest')
xlabel('Patient weights')
ylabel('Estimated cdf')
ksdensity seems to smooth the cumulative distribution function estimate too much. An estimate with a smaller bandwidth might produce a closer estimate to the empirical cumulative distribution function.
Return the bandwidth of the smoothing window.
bw
bw = 0.1070
Plot the cumulative distribution function estimate using a smaller bandwidth.
[f,xi] = ksdensity(hospital.Weight,pts,'Support','positive',...
'Function','cdf','Bandwidth',0.05);
plot(xi,f,'--r','LineWidth',2)
legend('empirical cdf','kernel-bw:default','kernel-bw:0.05',...
'Location','northwest')
hold off
The ksdensity estimate with a smaller bandwidth matches the empirical cumulative distribution function better.
Plot the estimated cdf evaluated at 50 equally spaced points.
figure()
ksdensity(hospital.Weight,'Support','positive','Function','cdf',...
'NumPoints',50)
xlabel('Patient weights')
ylabel('Estimated cdf')
Generate sample data from an exponential distribution with mean 3.
rng('default') % For reproducibility
x = random('exp',3,100,1);
Create a logical vector that indicates censoring. Here, observations with lifetimes longer than 10 are censored.
T = 10;
cens = (x>T);
Compute and plot the estimated density function.
figure
ksdensity(x,'Support','positive','Censoring',cens);
Compute and plot the survivor function.
figure
ksdensity(x,'Support','positive','Censoring',cens,...
'Function','survivor');
Compute and plot the cumulative hazard function.
figure
ksdensity(x,'Support','positive','Censoring',cens,...
'Function','cumhazard');
Generate a mixture of two normal distributions, and plot the estimated inverse cumulative distribution function at a specified set of probability values.
rng('default') % For reproducibility
x = [randn(30,1); 5+randn(30,1)];
pi = linspace(.01,.99,99);
figure
ksdensity(x,pi,'Function','icdf');
Generate a mixture of two normal distributions.
rng('default') % For reproducibility
x = [randn(30,1); 5+randn(30,1)];
Return the bandwidth of the smoothing window for the probability density estimate.
[f,xi,bw] = ksdensity(x);
bw
bw = 1.5141
The default bandwidth is optimal for normal densities.
Plot the estimated density.
figure
plot(xi,f);
xlabel('xi')
ylabel('f')
hold on
Plot the density using an increased bandwidth value.
[f,xi] = ksdensity(x,'Bandwidth',1.8);
plot(xi,f,'--r','LineWidth',1.5)
A higher bandwidth further smooths the density estimate, which might mask some characteristics of the distribution.
Now, plot the density using a decreased bandwidth value.
[f,xi] = ksdensity(x,'Bandwidth',0.8);
plot(xi,f,'-.k','LineWidth',1.5)
legend('bw = default','bw = 1.8','bw = 0.8')
hold off
A smaller bandwidth smooths the density estimate less, which exaggerates some characteristics of the sample.
Create a two-column vector of points at which to evaluate the density.
gridx1 = -0.25:.05:1.25;
gridx2 = 0:.1:15;
[x1,x2] = meshgrid(gridx1, gridx2);
x1 = x1(:);
x2 = x2(:);
xi = [x1 x2];
Generate a 30-by-2 matrix containing random numbers from a mixture of bivariate normal distributions.
rng('default') % For reproducibility
x = [0+.5*rand(20,1) 5+2.5*rand(20,1);
.75+.25*rand(10,1) 8.75+1.25*rand(10,1)];
Plot the estimated density of the sample data.
figure
ksdensity(x,xi);
## Input Arguments
collapse all
Sample data for which ksdensity returns f values, specified as a column vector or two-column matrix. Use a column vector for univariate data, and a two-column matrix for bivariate data.
Example: [f,xi] = ksdensity(x)
Data Types: single | double
Points at which to evaluate f, specified as a vector or two-column matrix. For univariate data, pts can be a row or column vector. The length of the returned output f is equal to the number of points in pts.
Example: pts = (0:1:25); ksdensity(x,pts);
Data Types: single | double
Axes handle for the figure ksdensity plots to, specified as a handle.
For example, if h is a handle for a figure, then ksdensity can plot to that figure as follows.
Example: ksdensity(h,x)
### Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose Name in quotes.
Example: 'Censoring',cens,'Kernel','triangle','NumPoints',20,'Function','cdf' specifies that ksdensity estimates the cdf by evaluating at 20 equally spaced points that covers the range of data, using the triangle kernel smoothing function and accounting for the censored data information in vector cens.
The bandwidth of the kernel-smoothing window, which is a function of the number of points in x, specified as the comma-separated pair consisting of 'Bandwidth' and a scalar value. If the sample data is bivariate, Bandwidth can also be a two-element vector. The default is optimal for estimating normal densities [1], but you might want to choose a larger or smaller value to smooth more or less.
If you specify 'BoundaryCorrection' as 'log'(default) and 'Support' as either 'positive' or a vector [L U], ksdensity converts bounded data to be unbounded by using log transformation. The value of 'Bandwidth' is on the scale of the transformed values.
Example: 'Bandwidth',0.8
Data Types: single | double
Boundary correction method, specified as the comma-separated pair consisting of 'BoundaryCorrection' and 'log' or 'reflection'.
ValueDescription
'log'
ksdensity converts bounded data x to be unbounded by one of the following transformations. Then, it transforms back to the original bounded scale after density estimation.
• For univariate data, if you specify 'Support','positive', then ksdensity applies log(x).
• For univariate data, if you specify 'Support',[L U], where L and U are numeric scalars and L < U, then ksdensity applies log((x-L)/(U–x)).
• For bivariate data, ksdensity transforms each column of x in the same way with the univariate data.
The value of 'Bandwidth' and the bw output are on the scale of the transformed values.
'reflection'
ksdensity augments bounded data by adding reflected data near the boundaries, then it returns estimates corresponding to the original support. For details, see Reflection Method.
ksdensity applies boundary correction only when you specify 'Support' as a value other than 'unbounded'.
Example: 'BoundaryCorrection','reflection'
Logical vector indicating which entries are censored, specified as the comma-separated pair consisting of 'Censoring' and a vector of binary values. A value of 0 indicates there is no censoring, 1 indicates that observation is censored. Default is there is no censoring. This name-value pair is only valid for univariate data.
Example: 'Censoring',censdata
Data Types: logical
Function to estimate, specified as the comma-separated pair consisting of 'Function' and one of the following.
ValueDescription
'pdf'Probability density function.
'cdf'Cumulative distribution function.
'icdf'
Inverse cumulative distribution function. ksdensity computes the estimated inverse cdf of the values in x, and evaluates it at the probability values specified in pi.
This value is valid only for univariate data.
'survivor'Survivor function.
'cumhazard'
Cumulative hazard function.
This value is valid only for univariate data.
Example: 'Function','icdf'
Type of kernel smoother, specified as the comma-separated pair consisting of 'Kernel' and one of the following.
• 'normal' (default)
• 'box'
• 'triangle'
• 'epanechnikov'
• A kernel function that is a custom or built-in function. Specify the function as a function handle (for example, @myfunction or @normpdf) or as a character vector or string scalar (for example, 'myfunction' or 'normpdf'). The software calls the specified function with one argument that is an array of distances between data values and locations where the density is evaluated. The function must return an array of the same size containing corresponding values of the kernel function.
When 'Function' is 'pdf', the kernel function returns density values. Otherwise, it returns cumulative probability values.
Specifying a custom kernel when 'Function' is 'icdf' returns an error.
For bivariate data, ksdensity applies the same kernel to each dimension.
Example: 'Kernel','box'
Number of equally spaced points in xi, specified as the comma-separated pair consisting of 'NumPoints' and a scalar value. This name-value pair is only valid for univariate data.
For example, for a kernel smooth estimate of a specified function at 80 equally spaced points within the range of sample data, input:
Example: 'NumPoints',80
Data Types: single | double
Support for the density, specified as the comma-separated pair consisting of 'support' and one of the following.
ValueDescription
'unbounded'Default. Allow the density to extend over the whole real line.
'positive'Restrict the density to positive values.
Two-element vector, [L U]Give the finite lower and upper bounds for the support of the density. This option is only valid for univariate sample data.
Two-by-two matrix, [L1 L2; U1 U2]Give the finite lower and upper bounds for the support of the density. The first row contains the lower limits and the second row contains the upper limits. This option is only valid for bivariate sample data.
For bivariate data, 'Support' can be a combination of positive, unbounded, or bounded variables specified as [0 -Inf; Inf Inf] or [0 L; Inf U].
Example: 'Support','positive'
Example: 'Support',[0 10]
Data Types: single | double | char | string
Function used to create kernel density plot, specified as the comma-separated pair consisting of 'PlotFcn' and one of the following.
ValueDescription
'surf'3-D shaded surface plot, created using surf
'contour'Contour plot, created using contour
'plot3'3-D line plot, created using plot3
'surfc'Contour plot under a 3-D shaded surface plot, created using surfc
This name-value pair is only valid for bivariate sample data.
Example: 'PlotFcn','contour'
Weights for sample data, specified as the comma-separated pair consisting of 'Weights' and a vector of length size(x,1), where x is the sample data.
Example: 'Weights',xw
Data Types: single | double
## Output Arguments
collapse all
Estimated function values, returned as a vector whose length is equal to the number of points in xi or pts.
Evaluation points at which ksdensity calculates f, returned as a vector or a two-column matrix. For univariate data, the default is 100 equally-spaced points that cover the range of data in x. For bivariate data, the default is 900 equally-spaced points created using meshgrid from 30 equally-spaced points in each dimension.
Bandwidth of smoothing window, returned as a scalar value.
If you specify 'BoundaryCorrection' as 'log'(default) and 'Support' as either 'positive' or a vector [L U], ksdensity converts bounded data to be unbounded by using log transformation. The value of bw is on the scale of the transformed values.
collapse all
### Kernel Distribution
A kernel distribution is a nonparametric representation of the probability density function (pdf) of a random variable. You can use a kernel distribution when a parametric distribution cannot properly describe the data, or when you want to avoid making assumptions about the distribution of the data. A kernel distribution is defined by a smoothing function and a bandwidth value, which control the smoothness of the resulting density curve.
The kernel density estimator is the estimated pdf of a random variable. For any real values of x, the kernel density estimator's formula is given by
${\stackrel{^}{f}}_{h}\left(x\right)=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{x-{x}_{i}}{h}\right)\text{\hspace{0.17em}},$
where x1, x2, …, xn are random samples from an unknown distribution, n is the sample size, $K\left(·\right)$ is the kernel smoothing function, and h is the bandwidth.
The kernel estimator for the cumulative distribution function (cdf), for any real values of x, is given by
${\stackrel{^}{F}}_{h}\left(x\right)={\int }_{-\infty }^{x}{\stackrel{^}{f}}_{h}\left(t\right)dt=\frac{1}{n}\sum _{i=1}^{n}G\left(\frac{x-{x}_{i}}{h}\right)\text{\hspace{0.17em}},$
where $G\left(x\right)={\int }_{-\infty }^{x}K\left(t\right)dt$.
For more details, see Kernel Distribution.
### Reflection Method
The reflection method is a boundary correction method that accurately finds kernel density estimators when a random variable has bounded support. If you specify 'BoundaryCorrection','reflection', ksdensity uses the reflection method. This method augments bounded data by adding reflected data near the boundaries, and estimates the pdf. Then, ksdensity returns the estimated pdf corresponding to the original support with proper normalization, so that the estimated pdf's integral over the original support is equal to one.
If you additionally specify 'Support',[L U], then ksdensity finds the kernel estimator as follows.
• If 'Function' is 'pdf', then the kernel density estimator is
${\stackrel{^}{f}}_{h}\left(x\right)=\frac{1}{nh}\sum _{i=1}^{n}\left[K\left(\frac{x-{x}_{i}^{-}}{h}\right)+K\left(\frac{x-{x}_{i}}{h}\right)+K\left(\frac{x-{x}_{i}^{+}}{h}\right)\right]$ for L ≤ x ≤ U,
where ${x}_{i}^{-}=2L-{x}_{i}$, ${x}_{i}^{+}=2U-{x}_{i}$, and xi is the ith sample data.
• If 'Function' is 'cdf', then the kernel estimator for cdf is
${\stackrel{^}{F}}_{h}\left(x\right)=\frac{1}{n}\sum _{i=1}^{n}\left[G\left(\frac{x-{x}_{i}^{-}}{h}\right)+G\left(\frac{x-{x}_{i}}{h}\right)+G\left(\frac{x-{x}_{i}^{+}}{h}\right)\right]-\frac{1}{n}\sum _{i=1}^{n}\left[G\left(\frac{L-{x}_{i}^{-}}{h}\right)+G\left(\frac{L-{x}_{i}}{h}\right)+G\left(\frac{L-{x}_{i}^{+}}{h}\right)\right]$ for L ≤ x ≤ U.
• To obtain a kernel estimator for an inverse cdf, a survivor function, or a cumulative hazard function (when 'Function' is 'icdf', 'survivor', or 'cumhazrd'), ksdensity uses both ${\stackrel{^}{f}}_{h}\left(x\right)$ and ${\stackrel{^}{F}}_{h}\left(x\right)$.
If you additionally specify 'Support' as 'positive' or [0 inf], then ksdensity finds the kernel estimator by replacing [L U] with [0 inf] in the above equations.
## References
[1] Bowman, A. W., and A. Azzalini. Applied Smoothing Techniques for Data Analysis. New York: Oxford University Press Inc., 1997.
[2] Hill, P. D. “Kernel estimation of a distribution function.” Communications in Statistics - Theory and Methods. Vol 14, Issue. 3, 1985, pp. 605-620.
[3] Jones, M. C. “Simple boundary correction for kernel density estimation.” Statistics and Computing. Vol. 3, Issue 3, 1993, pp. 135-146.
[4] Silverman, B. W. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, 1986.
## Version History
Introduced before R2006a | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 10, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8396868109703064, "perplexity": 2298.6832461336467}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572304.13/warc/CC-MAIN-20220816120802-20220816150802-00700.warc.gz"} |
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_ChemPRIME_(Moore_et_al.)/19%3A_Nuclear_Chemistry/19.03%3A_Radioactive_Series | $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$
Naturally occurring uranium contains more than 99% $$\ce{_{92}^{238}U}$$ that decays to $$\ce{_{90}^{234}Th}$$ by $$α$$ emission:
$\ce{_{92}^{238}U -> _{90}^{234}Th + _{2}^{4}He} \nonumber$
The product of this reaction is also radioactive, however, and undergoes $$β$$ decay:
$\ce{ _{90}^{234}Th -> _{91}^{234}Pa + _{-1}^{0}e} \nonumber$
The $$\ce{_{91}^{234}Pa}$$ produced in this second reaction also emits a $$β$$ particle:
$\ce{ _{91}^{234}Pa -> _{92}^{234}U + _{-1}^{0}e} \nonumber$
These three reactions are only the first of 14 steps. After emission of eight $$α$$ particles and six $$β$$ particles, the isotope $$\ce{ _{82}^{206}Pb}$$ is produced. It has a stable nucleus which does not disintegrate further. The complete process may be written as follows:
$$\text{ }{}_{\text{92}}^{\text{238}}\text{U}\xrightarrow{\alpha }{}_{\text{90}}^{\text{234}}\text{Th}\xrightarrow{\beta }{}_{\text{91}}^{\text{234}}\text{Pa}\xrightarrow{\beta }{}_{\text{92}}^{\text{234}}\text{U}\xrightarrow{\alpha }{}_{\text{90}}^{\text{230}}\text{Th}\xrightarrow{\alpha }{}_{\text{88}}^{\text{226}}\text{Ra}\xrightarrow{\alpha }{}_{\text{88}}^{\text{222}}\text{Rn}$$
$$\downarrow ^{\alpha }$$ (2a)
$${}_{\text{82}}^{\text{206}}\text{Pb}\xleftarrow{\alpha }{}_{\text{84}}^{\text{210}}\text{Po}\xleftarrow{\beta }{}_{\text{83}}^{\text{210}}\text{Bi}\xleftarrow{\alpha }{}_{\text{82}}^{\text{210}}\text{Pb}\xleftarrow{\alpha }{}_{\text{84}}^{\text{214}}\text{Po}\xleftarrow{\beta }{}_{\text{83}}^{\text{214}}\text{Bi}\xleftarrow{\beta }{}_{\text{82}}^{\text{214}}\text{Pb}\xleftarrow{\alpha }{}_{\text{84}}^{\text{218}}\text{Po }$$
While the net reaction is
${}_{\text{92}}^{\text{238}}\text{U }\to \text{ }{}_{\text{82}}^{\text{206}}\text{Pb + 8}{}_{\text{2}}^{\text{4}}\text{He + 6}{}_{-\text{1}}^{\text{0}}e \nonumber$
Such a series of successive nuclear reactions is called a radioactive series. Two other radioactive series similar to the one just described occur in nature. One of these starts with the isotope $$\ce{ _{90}^{232}Th}$$ and involves 10 successive stages, while the other starts with $$\ce{_{92}^{235}U}$$ and involves 11 stages. Each of the three series produces a different stable isotope of lead.
##### Example $$\PageIndex{1}$$: Uranium-Actinium Series
The first four stages in the uranium-actinium series involve the emission of an α particle from a $$\ce{_{92}^{235}U}$$ nucleus, followed successively by the emission of a $$β$$ particle, a second $$α$$ particle, and then a second β particle. Write out equations to describe all four nuclear reactions.
Solution
The emission of an a particle lowers the atomic number by 2 (from 92 to 90). Since element 90 is thorium, we have
$\ce{ _{92}^{235}U -> _{90}^{231}Th + _{2}^{4}He} \nonumber$
The emission of a β particle now increases the atomic number by 1 to give an isotope of element 91, protactinium:
$\ce{_{90}^{231}Th -> _{91}^{231}Pa + _{-1}^{0}e} \nonumber$
The next two stages follow similarly:
$\ce{_{91}^{231}Pa -> _{89}^{227}Ac + _{2}^{4}He} \nonumber$
and
$\ce{_{89}^{227}Ac -> _{90}^{227}Th + _{-1}^{0}e} \nonumber$
##### Example $$\PageIndex{2}$$: Thorium Series
In the thorium series, $$\ce{_{90}^{232}Th}$$ loses a total of six α particles and four β particles in a 10-stage process. What isotope is finally produced in this series?
Solution
The loss of six α particles and four $$β$$ particles:
$\ce{6 _{2}^{4}He + 4 _{-1}^{0}e} \nonumber$
involves the total loss of 24 nucleons and 6 × 2 – 4 = 8 positive charges from the $$\ce{_{90}^{232}Th}$$ nucleus. The eventual result will be an isotope of mass number 232 – 24 = 208 and a nuclear charge of 90 – 8 = 82. Since element 82 is $$\ce{Pb}$$, we can write
$\ce{ _{90}^{232}Th -> _{82}^{208}Pb + 6 _{2}^{4}He + 4 _{-1}^{0}e} \nonumber$
This page titled 19.3: Radioactive Series is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Ed Vitz, John W. Moore, Justin Shorb, Xavier Prat-Resina, Tim Wendorff, & Adam Hahn. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9479125142097473, "perplexity": 1109.672849251453}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030336674.94/warc/CC-MAIN-20221001132802-20221001162802-00384.warc.gz"} |
http://math.stackexchange.com/questions/38787/existence-of-n-distinct-real-roots-of-an-orthogonal-polynomial | # Existence of n distinct (real) roots of an orthogonal polynomial
I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof http://en.wikipedia.org/wiki/Orthogonal_polynomials#Existence_of_real_roots so far is that (by contradiction):
• Assume we have $m \le n$ roots. We'll show $m=n$
• Let $\displaystyle S(x) = \prod_{j=1}^m (x-x_j)$
• Gives us that $S(x)$ is an nth degree polynomial
• $S(x)$ changes sign at each of the $x_j$
My problem is this statement:
$S(x)P_n(x)$ is therefore strictly positive, or strictly negative, everywhere except at the $x_j$.
The $x_j$? What $x_j$? The lecture notes I have also say "except at $x_i$" so I'm pretty confused.
If someone can help me out here I'd greatly appreciate it. Thank you!
-
The polynomial $P_n$ is of the form $P_n=S Q$ where $S$ is the product of the roots and $Q$ is a factor without roots that therefore cannot change its sign. – Phira May 12 '11 at 23:07
If you multiply by $S$, you have the factor $Q$ that is strictly positive or strictly negative and the factor $S^2$ that is only zero for $x=x_i$. – Phira May 12 '11 at 23:08
When you say "for $x=x_i$", is that $x_i$ for each $0 \leq i \leq m$? – Tim Green May 12 '11 at 23:30
Since your starting assumption is that $P_n$ might have multiple roots, what you do is to take all those roots and ignore multiplicity, form a polynomial from them, and call it $S(x)$; something like forming $(x-1)(x-3)$ as the $S$ if $P$ is $(x-1)^5(x-3)^2$... – J. M. May 13 '11 at 0:37
Did your lecture notes include: - "Let $\displaystyle S(x) = \prod_{i=1}^m (x-x_i)$ - Gives us that $S(x)$...changes sign at each of the $x_i$"? I'm simply asking if your confusion about the $x_j$ is simply due to the particular letter chosen for the index.? – amWhy May 13 '11 at 0:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.924124538898468, "perplexity": 335.5739023611663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783403502.46/warc/CC-MAIN-20160624155003-00128-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.johndcook.com/blog/tag/python/ | # Curvature and automatic differentiation
Curvature is tedious to calculate by hand because it involves calculating first and second order derivatives. Of course other applications require derivatives too, but curvature is the example we’ll look at in this post.
## Computing derivatives
It would be nice to write programs that only explicitly implement the original function and let software take care of finding the derivatives.
### Numerical differentiation
Finite difference approximations for derivatives are nothing new. For example, Euler (1707–1783) used finite differences to numerically solve differential equations. But numerical differentiation can be inaccurate or unstable, especially for higher order derivatives.
### Symbolic differentiation
Symbolic differentiation is another approach, having the computer manipulate expressions much as a person would do by hand. It works well for many problems, though it scales poorly for large problems. It also requires functions to be presented in traditional mathematical form, not in the form of source code.
### Automatic differentiation
Automatic differentiation is a third way. Like numerical differentiation, it works with floating point numbers, not symbolic expressions. But unlike numerical differentiation, the result does not have approximation error.
As someone put it, automatic differentiation applies the chain rule to floating point numbers rather than to symbolic expressions.
## Python implementation
I’ll use the Python library autograd to compute curvature and illustrate automatic differentiation. autograd is not the most powerful automatic differentiation library for Python, but it is the simplest I’ve seen.
We will compute the curvature of a logistic curve.
The curvature of the graph of a function is given by
Here’s Python code using autograd to compute the curvature.
def f(x):
return 1/(1 + np.exp(-x))
f1 = grad(f) # 1st derivative of f
f2 = grad(f1) # 2nd derivative of f
def curvature(x):
return abs(f2(x))*(1 + f1(x)**2)**-1.5
## Curvature plots
The graph is relatively flat in the middle and at the far ends. In between, the graph bends creating two regions of higher curvature.
import matplotlib.pyplot as plt
x = np.linspace(-5, 5, 300)
plt.plot(x, f(x))
plt.xlabel("$x$")
plt.ylabel("$y$")
plt.title("Logisitic curve")
plt.savefig("logistic_curve.svg")
Now let’s look at the curvature.
y = [curvature(t) for t in x]
plt.plot(x, y)
plt.xlabel("$x$")
plt.ylabel(r"$\kappa(x)$")
plt.title("Curvature")
plt.savefig("plot_logistic_curvature.svg")
As we should expect, the curvature is small at the ends and in the middle, with local maxima in between.
We can also look at the signed curvature, the same expression as curvature but without the absolute value.
We plot this with the following code.
def signed_curvature(x):
return f2(x)*(1 + f1(x)**2)**-1.5
y = [signed_curvature(t) for t in x]
plt.plot(x, y)
plt.xlabel("$x$")
plt.ylabel(r"$k(x)$")
plt.title("Signed curvature")
plt.savefig("graph_signed_curvature.svg")
The result looks more like a sine wave.
The positive values mean the curve is bending counterclockwise, and the negative values mean the curve is bending clockwise.
Related post: Squircles and curvature
# Summing random powers up to a threshold
Pick random numbers uniformly between 0 and 1, adding them as you go, and stop when you get a result bigger than 1. How many numbers would you expect to add together on average?
You need at least two samples, and often two are enough, but you might get any number, and those larger numbers will pull the expected value up.
Here’s a simulation program in Python.
from random import random
from collections import Counter
N = 1000000
c = Counter()
for _ in range(N):
x = 0
steps = 0
while x < 1:
x += random()
steps += 1
c[steps] += 1
print( sum([ k*c[k] for k in c.keys() ])/N )
When I ran it I got 2.718921. There’s a theoretical result first proved by W. Weissblum that the expected value is e = 2.71828… Our error was on the order of 1/√N, which is what we’d expect from the central limit theorem.
Now we can explore further in a couple directions. We could take a look at a the distribution of the number steps, not just its expected value. Printing c shows us the raw data.
Counter({
2: 499786,
3: 333175,
4: 125300,
5: 33466,
6: 6856,
7: 1213,
8: 172,
9: 29,
10: 3
})
And here’s a plot.
We could also generalize the problem by taking powers of the random numbers. Here’s what we get when we use exponents 1 through 20.
There’s a theoretical result that the expected number of steps is asymptotically equal to cn where
I computed c = 1.2494. The plot above shows that the dependence on the exponent n does look linear. The simulation results appear to be higher than the asymptotic prediction by a constant, but that’s consistent with the asymptotic prediction since relative to n, a constant goes away as n increases.
Reference for theoretical results: D. J. Newman and M. S. Klamkin. Expectations for Sums of Powers. The American Mathematical Monthly, Vol. 66, No. 1 (Jan., 1959), pp. 50-51
# Average fraction round up
Pick a large number n. Divide n by each of the positive integers up to n and round the results up to the nearest integer. On average, how far do you round up?
Or in terms of probability, what is the expected distance between a fraction n/r, where n is large and fixed and r is chosen randomly between 1 and n, and the nearest larger integer?
In symbols, the question above is asking for the approximate value of
for large n, i.e. in the limit as n goes to ∞. Here ⌈x⌉ denotes the ceiling of x, the smallest integer greater than or equal to x.
Let’s plot this as a function of n and see what it looks like. Here’s the Python code.
import matplotlib.pyplot as plt
from numpy import ceil, arange
def f(n):
return sum( [ceil(n/r) - n/r for r in range(1, n)] )/n
x = arange(1, 100)
y = [f(n) for n in x]
plt.plot(x, y)
plt.show()
And here’s the result.
It appears the graph may be converging to some value, and in fact it is. Charles de la Vallée Poussin proved in 1898 that the limiting value is the Euler–Mascheroni constant γ = 0.5772…. This constant is the limiting difference between the nth harmonic number and log n, i.e.
We can add a horizontal line to our plot to see how well the graph seems to match γ. To do this we need to import the constant euler_gamma from numpy and add the
plt.axhline(y=euler_gamma, linestyle=":")
after the plot command. When we do, this is what we see.
It looks like the plot is converging to a value slightly less than γ. Apparently the convergence is very slow. When we go out to 10,000 the plot is closer to being centered around γ but still maybe below γ more than above.
If we evaluate our function at n = 1,000,000, we get 0.577258… while γ = 0.577215….
At n = 10,000,000 we get 0.577218…. So taking 100 times as many terms in our sum gives us one extra correct decimal place, as we’d expect of a random processes since convergence usually goes like 1/√n.
# Equation for the Eiffel Tower
Robert Banks’s book Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics describes the Eiffel Tower’s shape as approximately the logarithmic curve
where y* and x0 are chosen to match the tower’s dimensions.
Here’s a plot of the curve:
And here’s the code that produced the plot:
from numpy import log, exp, linspace, vectorize
import matplotlib.pyplot as plt
# Taken from "Towing Icebergs, Falling Dominoes,
# and Other Adventures in Applied Mathematics"
# by Robert B. Banks
# Constants given in Banks in feet. Convert to meters.
feet_to_meter = 0.0254*12
ystar = 201*feet_to_meter
x0 = 207*feet_to_meter
height = 984*feet_to_meter
# Solve for where to cut off curve to match height of the tower.
# - ystar log xmin/x0 = height
xmin = x0 * exp(-height/ystar)
def f(x):
if -xmin < x < xmin:
return height
else:
return -ystar*log(abs(x/x0))
curve = vectorize(f)
x = linspace(-x0, x0, 400)
plt.plot(x, curve(x))
plt.xlim(-2*x0, 2*x0)
plt.xlabel("Meters")
plt.ylabel("Meters")
plt.title("Eiffel Tower")
plt.axes().set_aspect(1)
plt.savefig("eiffel_tower.svg")
Related post: When length equals area
The St. Louis arch is approximately a catenary, i.e. a hyperbolic cosine.
# Why is Kullback-Leibler divergence not a distance?
The Kullback-Leibler divergence between two probability distributions is a measure of how different the two distributions are. It is sometimes called a distance, but it’s not a distance in the usual sense because it’s not symmetric. At first this asymmetry may seem like a bug, but it’s a feature. We’ll explain why it’s useful to measure the difference between two probability distributions in an asymmetric way.
The Kullback-Leibler divergence between two random variables X and Y is defined as
This is pronounced/interpreted several ways:
• The divergence from Y to X
• The relative entropy of X with respect to Y
• How well Y approximates X
• The information gain going from the prior Y to the posterior X
• The average surprise in seeing Y when you expected X
A theorem of Gibbs proves that K-L divergence is non-negative. It’s clearly zero if X and Y have the same distribution.
The K-L divergence of two random variables is an expected value, and so it matters which distribution you’re taking the expectation with respect to. That’s why it’s asymmetric.
As an example, consider the probability densities below, one exponential and one gamma with a shape parameter of 2.
The two densities differ mostly on the left end. The exponential distribution believes this region is likely while the gamma does not. This means that an expectation with respect to the exponential distribution will weigh things in this region more heavily. In an information-theoretic sense, an exponential is a better approximation to a gamma than the other way around.
Here’s some Python code to compute the divergences.
from scipy.stats import expon, gamma
from scipy import inf
def KL(X, Y):
f = lambda x: -X.pdf(x)*(Y.logpdf(x) - X.logpdf(x))
e = expon
g = gamma(a = 2)
print( KL(e, g) )
print( KL(g, e) )
This returns
(0.5772156649008394, 1.3799968612282498e-08)
(0.4227843350984687, 2.7366807708872898e-09)
The first element of each pair is the integral and the second is the error estimate. So apparently both integrals have been computed accurately, and the first is clearly larger. This backs up our expectation that it’s more surprising to see a gamma when expecting an exponential than vice versa.
Although K-L divergence is asymmetric in general, it can be symmetric. For example, suppose X and Y are normal random variables with the same variance but different means. Then it would be equally surprising to see either one when expecting the other. You can verify this in the code above by changing the KL function to integrate over the whole real line
def KL(X, Y):
f = lambda x: -X.pdf(x)*(Y.logpdf(x) - X.logpdf(x))
and trying an example.
n1 = norm(1, 1)
n2 = norm(2, 1)
print( KL(n1, n2) )
print( KL(n2, n1) )
This returns
(0.4999999999999981, 1.2012834963423225e-08)
(0.5000000000000001, 8.106890774205374e-09)
and so both integrals are equal to within the error in the numerical integration.
# Negative correlation introduced by success
Suppose you measure people on two independent attributes, X and Y, and take those for whom X+Y is above some threshold. Then even though X and Y are uncorrelated in the full population, they will be negatively correlated in your sample.
This article gives the following example. Suppose beauty and acting ability were uncorrelated. Knowing how attractive someone is would give you no advantage in guessing their acting ability, and vice versa. Suppose further that successful actors have a combination of beauty and acting ability. Then among successful actors, the beautiful would tend to be poor actors, and the unattractive would tend to be good actors.
Here’s a little Python code to illustrate this. We take two independent attributes, distributed like IQs, i.e. normal with mean 100 and standard deviation 15. As the sum of the two attributes increases, the correlation between the two attributes becomes more negative.
from numpy import arange
from scipy.stats import norm, pearsonr
import matplotlib.pyplot as plt
# Correlation.
# The function pearsonr returns correlation and a p-value.
def corr(x, y):
return pearsonr(x, y)[0]
x = norm.rvs(100, 15, 10000)
y = norm.rvs(100, 15, 10000)
z = x + y
span = arange(80, 260, 10)
c = [ corr( x[z > low], y[z > low] ) for low in span ]
plt.plot( span, c )
plt.xlabel( "minimum sum" )
plt.ylabel( "correlation coefficient" )
plt.show()
# Random minimum spanning trees
I just ran across a post by John Baez pointing to an article by Alan Frieze on random minimum spanning trees.
Here’s the problem.
1. Create a complete graph with n nodes, i.e. connect every node to every other node.
2. Assign each edge a uniform random weight between 0 and 1.
3. Find the minimum spanning tree.
4. Add up the edges of the weights in the minimum spanning tree.
The surprise is that as n goes to infinity, the expected value of the process above converges to the Riemann zeta function at 3, i.e.
ζ(3) = 1/1³ + 1/2³ + 1/3³ + …
Incidentally, there are closed-form expressions for the Riemann zeta function at positive even integers. For example, ζ(2) = π² / 6. But no closed-form expressions have been found for odd integers.
## Simulation
Here’s a little Python code to play with this.
import networkx as nx
from random import random
N = 1000
G = nx.Graph()
for i in range(N):
for j in range(i+1, N):
T = nx.minimum_spanning_tree(G)
edges = T.edges(data=True)
print( sum([e[2]["weight"] for e in edges]) )
When I ran this, I got 1.2307, close to ζ(3) = 1.20205….
I ran this again, putting the code above inside a loop, averaging the results of 100 simulations, and got 1.19701. That is, the distance between my simulation result and ζ(3) went from 0.03 to 0.003.
There are two reasons I wouldn’t get exactly ζ(3). First, I’m only running a finite number of simulations (100) and so I’m not computing the expected value exactly, but only approximating it. (Probably. As in PAC: probably approximately correct.) Second, I’m using a finite graph, of size 1000, not taking a limit as graph size goes to infinity.
My limited results above suggest that the first reason accounts for most of the difference between simulation and theory. Running 100 replications cut the error down by a factor of 10. This is exactly what you’d expect from the central limit theorem. This suggests that for graphs as small as 1000 nodes, the expected value is close to the asymptotic value.
You could experiment with this, increasing the graph size and increasing the number of replications. But be patient. It takes a while for each replication to run.
## Generalization
The paper by Frieze considers more than the uniform distribution. You can use any non-negative distribution with finite variance and whose cumulative distribution function F is differentiable at zero. The more general result replaces ζ(3) with ζ(3) / F ‘(0). We could, for example, replace the uniform distribution on weights with an exponential distribution. In this case the distribution function is 1 – exp(-x), at its derivative at the origin is 1, so our simulation should still produce approximately ζ(3). And indeed it does. When I took the average of 100 runs with exponential weights I got a value of 1.2065.
There’s a little subtlety around using the derivative of the distribution at 0 rather than the density at 0. The derivative of the distribution (CDF) is the density (PDF), so why not just say density? One reason would be to allow the most general probability distributions, but a more immediate reason is that we’re up against a discontinuity at the origin. We’re looking at non-negative distributions, so the density has to be zero to the left of the origin.
When we say the derivative of the distribution at 0, we really mean the derivative at zero of a smooth extension of the distribution. For example, the exponential distribution has density 0 for negative x and density exp(-x) for non-negative x. Strictly speaking, the CDF of this distribution is 1 – exp(-x) for non-negative x and 0 for negative x. The left and right derivatives are different, so the derivative doesn’t exist. By saying the distribution function is simply exp(-x), we’ve used a smooth extension from the non-negative reals to all reals.
# Nearly all the area in a high-dimensional sphere is near the equator
Nearly all the area of a high-dimensional sphere is near the equator. And by symmetry, it doesn’t matter which equator you take. Draw any great circle and nearly all of the area will be near that circle. This is the canonical example of “concentration of measure.”
What exactly do we mean by “nearly all the area” and “near the equator”? You get to decide. Pick your standard of “nearly all the area,” say 99%, and your definition of “near the equator,” say within 5 degrees. Then it’s always possible to take the dimension high enough that your standards are met. The more demanding your standard, the higher the dimension will need to be, but it’s always possible to pick the dimension high enough.
This result is hard to imagine. Maybe a simulation will help make it more believable.
In the simulation below, we take as our “north pole” the point (1, 0, 0, 0, …, 0). We could pick any unit vector, but this choice is convenient. Our equator is the set of points orthogonal to the pole, i.e. that have first coordinate equal to zero. We draw points randomly from the sphere, compute their latitude (i.e. angle from the equator), and make a histogram of the results.
The area of our planet isn’t particularly concentrated near the equator.
But as we increase the dimension, we see more and more of the simulation points are near the equator.
Here’s the code that produced the graphs.
from scipy.stats import norm
from math import sqrt, pi, acos, degrees
import matplotlib.pyplot as plt
def pt_on_sphere(n):
# Return random point on unit sphere in R^n.
# Generate n standard normals and normalize length.
x = norm.rvs(0, 1, n)
length = sqrt(sum(x**2))
return x/length
def latitude(x):
# Latitude relative to plane with first coordinate zero.
angle_to_pole = acos(x[0]) # in radians
latitude_from_equator = 0.5*pi - angle_to_pole
return degrees( latitude_from_equator )
N = 1000 # number of samples
for n in [3, 30, 300, 3000]: # dimension of R^n
latitudes = [latitude(pt_on_sphere(n)) for _ in range(N)]
plt.hist(latitudes, bins=int(sqrt(N)))
plt.xlabel("Latitude in degrees from equator")
plt.title("Sphere in dimension {}".format(n))
plt.xlim((-90, 90))
plt.show()
Not only is most of the area near the equator, the amount of area outside a band around the equator decreases very rapidly as you move away from the band. You can see that from the histograms above. They look like a normal (Gaussian) distribution, and in fact we can make that more precise.
If A is a band around the equator containing at least half the area, then the proportion of the area a distance r or greater from A is bound by exp( -(n-1)r² ). And in fact, this holds for any set A containing at least half the area; it doesn’t have to be a band around the equator, just any set of large measure.
# Simple random number generator does surprisingly well
I was running the NIST statistical test suite recently. I wanted an example of a random number generator where the tests failed, and so I used a simple generator, a linear congruence generator. But to my surprise, the generator passed nearly all the tests, even though some more sophisticated generators failed some of the same tests.
This post will implement a couple of the simplest tests in Python and show that the generator does surprisingly well.
The linear congruential generator used here starts with an arbitrary seed, then at each step produces a new number by multiplying the previous number by a constant and taking the remainder by 231 – 1. The multiplier constant was chosen to be one of the multipliers recommended in [1].
We’ll need a couple math functions:
from math import sqrt, log
and we need to define the constants for our generator.
# Linear congruence generator (LCG) constants
z = 20170705 # seed
a = 742938285 # multiplier
e = 31 # will need this later
m = 2**e -1 # modulus
Next we form a long string of 0’s and 1’s using our generator
# Number of random numbers to generate
N = 100000
# Format to print bits, padding with 0's on the left if needed
formatstr = "0" + str(e) + "b"
bit_string = ""
for _ in range(N):
z = a*z % m # LCG
bit_string += format(z, formatstr)
Next we run a couple tests. First, we count the number of 1’s in our string of bits. We expect about half the bits to be 1’s. We can quantify “about” as within two standard deviations.
def count_ones(string):
ones = 0
for i in range(len(string)):
if string[i] == '1':
ones += 1
return ones
ones = count_ones(bit_string)
expected = e*N/2
sd = sqrt(0.25*N)
print( "Number of 1's: {}".format(ones) )
print( "Expected: {} to {}".format(expected - 2*sd, expected + 2*sd) )
The results are nothing unusual:
Number of 1's: 1550199
Expected: 1549683.8 to 1550316.2
Next we look at the length of the longest runs on 1’s. I’ve written before about the probability of long runs and the code below uses a couple results from that post.
def runs(string):
max_run = 0
current_run = 0
for i in range(len(string)):
if string[i] == '1':
current_run += 1
else:
current_run = 0
max_run = max(max_run, current_run)
return max_run
runlength = runs(bit_string)
expected = -log(0.5*e*N)/log(0.5)
sd = 1/log(2)
print( "Run length: {}".format(runlength) )
print( "Expected: {} to {}".format(expected - 2*sd, expected + 2*sd) )
Again the results are nothing unusual:
Run length: 19
Expected: 17.7 to 23.4
Simple random number generators are adequate for many uses. Some applications, such as high dimensional integration and cryptography, require more sophisticated generators, but sometimes its convenient and sufficient to use something simple. For example, code using the LCG generator above would be easier to debug than code using the Mersenne Twister. The entire state of the LCG is a single number, whereas the Mersenne Twister maintains an internal state of 312 numbers.
One obvious limitation of the LCG used here is that it couldn’t possibly produce more than 231 – 1 values before repeating itself. Since the state only depends on the last value, every time it comes to a given output, the next output will be whatever the next output was the previous time. In fact, [1] shows that it does produce 231 – 1 values before cycling. If the multiplier were not chosen carefully it could have a shorter period.
So our LCG has a period of about two billion values. That’s a lot if you’re writing a little game, for example. But it’s not enough for many scientific applications.
* * *
[1] George S. Fishman and Louis R. Moore III, An exhaustive analysis of multiplicative congruential random number generators with modulus 231 – 1, SIAM Journal of Scientific and Statistical Computing, Vol. 7, no. 1, January 1986.
# Polynomials evaluated at integers
Let p(x) = a0 + a1x + a2x2 + … + anxn and suppose at least one of the coefficients ai is irrational for some i ≥ 1. Then a theorem by Weyl says that the fractional parts of p(n) are equidistributed as n varies over the integers. That is, the proportion of values that land in some interval is equal to the length of that interval.
Clearly it’s necessary that one of the coefficients be irrational. What may be surprising is that it is sufficient.
If the coefficients are all rational with common denominator N, then the sequence would only contain multiples of 1/N. The interval [1/3N, 2/3N], for example, would never get a sample. If a0 were irrational but the rest of the coefficients were rational, we’d have the same situation, simply shifted by a0.
This is a theorem about what happens in the limit, but we can look at what happens for some large but finite set of terms. And we can use a χ2 test to see how evenly our sequence is compared to what one would expect from a random sequence.
In the Python code below, we use the polynomial p(x) = √2 x² + πx + 1 and evaluate p at 0, 1, 2, …, 99,999. We then count how many fall in the bins [0, 0.01), [0.01, 0.02), … [0.99, 1] and compute a chi-square statistic on the counts.
from math import pi, sqrt, floor
def p(x):
return 1 + pi*x + sqrt(2)*x*x
def chisq_stat(O, E):
return sum( [(o - e)**2/e for (o, e) in zip(O, E)] )
def frac(x):
return x - floor(x)
N = 100000
data = [frac(p(n)) for n in range(N)]
count = [0]*100
for d in data:
count[ int(floor(100*d)) ] += 1
expected = [N/100]*100
print(chisq_stat(count, expected))
We get a chi-square statistic of 95.4. Since we have 100 bins, there are 99 degrees of freedom, and we should compare our statistic to a chi-square distribution with 99 degrees of freedom. Such a distribution has mean 99 and standard deviation √(99*2) = 14.07, so a value of 95.4 is completely unremarkable.
If we had gotten a very large chi-square statistic, say 200, we’d have reason to suspect something was wrong. Maybe a misunderstanding on our part or a bug in our software. Or maybe the sequence was not as uniformly distributed as we expected.
If we had gotten a very small chi-square statistic, say 10, then again maybe we misunderstood something, or maybe our sequence is remarkably evenly distributed, more evenly than one would expect from a random sequence.
Related posts:
# Fractional parts, invariant measures, and simulation
A function f: XX is measure-preserving if for each iteration of f sends the same amount of stuff into a given set. To be more precise, given a measure μ and any μ-measurable set E with μ(E) > 0, we have
You can read the right side of the equation above as “the measure of the set of points that f maps into E.” You can apply this condition repeatedly to see that the measure of the set of points mapped into E after n iterations is still just the measure of E.
If X is a probability space, i.e. μ( ) = 1, then you could interpret the definition of measure-preserving to mean that the probability that a point ends up in E after n iterations is independent of n. We’ll illustrate this below with a simulation.
Let X be the half-open unit interval (0, 1] and let f be the Gauss map, i.e.
where [z] is the integer part of z. The function f is measure-preserving, though not for the usual Lebesgue measure. Instead it preserves the following measure:
Let’s take as our set E an interval [a, b] and test via simulation whether the probability of landing in E after n iterations really is just the measure of E, independent of n.
We can’t just generate points uniformly in the interval (0, 1]. We have to generate the points so that the probability of a point coming from a set E is μ(E). That means the PDF of the distribution must be p(x) = 1 / (log(2) (1 + x)). We use the inverse-CDF method to generate points with this density in the Python code below.
from math import log, floor
from random import random
def gauss_map(x):
y = 1.0/x
return y - floor(y)
# iterate gauss map n times
def iterated_gauss_map(x, n):
while n > 0:
x = gauss_map(x)
n = n - 1
return x
# measure mu( [a,b] )
def mu(a, b):
return (log(1.0+b) - log(1.0+a))/log(2.0)
# Generate samples with Prob(x in E) = mu( E )
def sample():
u = random()
return 2.0**u - 1.0
def simulate(num_points, num_iterations, a, b):
count = 0
for _ in range(num_points):
x = sample()
y = iterated_gauss_map(x, num_iterations)
if a < y < b:
count += 1
return count / num_points
# Change these parameters however you like
a, b = 0.1, 0.25
N = 1000000
exact = mu(a, b)
print("Exact probability:", exact)
print("Simulated probability after n iterations")
for n in range(1, 10):
simulated = simulate(N, n, a, b)
print("n =", n, ":", simulated)
Here’s the output I got:
Exact probability: 0.18442457113742736
Simulated probability after n iterations
n = 1 : 0.184329
n = 2 : 0.183969
n = 3 : 0.184233
n = 4 : 0.184322
n = 5 : 0.184439
n = 6 : 0.184059
n = 7 : 0.184602
n = 8 : 0.183877
n = 9 : 0.184834
With 1,000,000 samples, we expect the results to be the same to about 3 decimal places, which is what we see above.
Related post: Irrational rotations are ergodic.
(A transformation f is ergodic if it is measure preserving and the only sets E with f –1(E) = E are those with measure 0 or full measure. Rational rotations are measure-preserving but not ergodic. The Gauss map above is ergodic.)
# Irrational rotations are ergodic
In a blog post yesterday, I mentioned that the golden angle is an irrational portion of a circle, and so a sequence of rotations by the golden angle will not repeat itself. We can say more: rotations by an irrational portion of a circle are ergodic. Roughly speaking, this means that not only does the sequence not repeat itself, the sequence “mixes well” in a technical sense.
Ergodic functions have the property that “the time average equals the space average.” We’ll unpack what that means and illustrate it by simulation.
Suppose we pick a starting point x on the circle then repeatedly rotate it by a golden angle. Take an integrable function f on the circle and form the average of its values at the sequence of rotations. This is the time average. The space average is the integral of f over the circle, divided by the circumference of the circle. The ergodic theorem says that the time average equals the space average, except possibly for a setting of starting values of measure zero.
More generally, let X be a measure space (like the unit circle) with measure μ let T be an ergodic transformation (like rotating by a golden angle), Then for almost all starting values x we have the following:
Let’s do a simulation to see this in practice by running the following Python script.
from scipy import pi, cos
from scipy.constants import golden
golden_angle = 2*pi*golden**-2
def T(x):
return (x + golden_angle) % (2*pi)
def time_average(x, f, T, n):
s = 0
for k in range(n):
s += f(x)
x = T(x)
return s/n
def space_average(f):
return integral / (2*pi)
f = lambda x: cos(x)**2
N = 1000000
print( time_average(0, f, T, N) )
print( space_average(f) )
In this case we get 0.49999996 for the time average, and 0.5 for the space average. They’re not the same, but we only used a finite value of n; we didn’t take a limit. We should expect the two values to be close because n is large, but we shouldn’t expect them to be equal.
Update: The code and results were updated to fix a bug pointed out in the comments below. I had written ... % 2*pi when I should have written ... % (2*pi). I assumed the modulo operator was lower precedence than multiplication, but it’s not. It was a coincidence that the buggy code was fairly accurate.
A friend of mine, a programmer with decades of experience, recently made a similar error. He’s a Clojure fan but was writing in C or some similar language. He rightfully pointed out that this kind of error simply cannot happen in Clojure. Lisps, including Clojure, don’t have operator precedence because they don’t have operators. They only have functions, and the order in which functions are called is made explicit with parentheses. The Python code x % 2*pi corresponds to (* (mod x 2) pi) in Clojure, and the Python code x % (2*pi) corresponds to (mod x (* 2 pi)).
Related: Origin of the word “ergodic”
# Weibull distribution and Benford’s law
## Introduction to Benford’s law
In 1881, Simon Newcomb noticed that the edges of the first pages in a book of logarithms were dirty while the edges of the later pages were clean. From this he concluded that people were far more likely to look up the logarithms of numbers with leading digit 1 than of those with leading digit 9. Frank Benford studied the same phenomena later and now the phenomena is known as Benford’s law, or sometime the Newcomb-Benford law.
A data set follows Benford’s law if the proportion of elements with leading digit d is approximately
log10((d + 1)/d).
You could replace “10” with b if you look at the leading digits in base b.
Sets of physical constants often satisfy Benford’s law, as I showed here for the constants defined in SciPy.
Incidentally, factorials satisfy Benford’s law exactly in the limit.
## Weibull distributions
The Weibull distribution is a generalization of the exponential distribution. It’s a convenient distribution for survival analysis because it can have decreasing, constant, or increasing hazard, depending on whether the value of a shape parameter γ is less than, equal to, or greater than 1 respectively. The special case of constant hazard, shape γ = 1, corresponds to the exponential distribution.
## Weibull and Benford
If the shape parameter of a Weibull distributions is “not too large” then samples from that distribution approximately follow Benford’s law (source). We’ll explore this statement with a little Python code.
SciPy doesn’t contain a Weibull distribution per se, but it does have support for a generalization of the Weibull known as the exponential Weibull. The latter has two shape parameters. We set the first of these to 1 to get the ordinary Weibull distribution.
from math import log10, floor
from scipy.stats import exponweib
y = log10(x) % 1
return int(floor(10**y))
def weibull_stats(gamma):
distribution = exponweib(1, gamma)
N = 10000
samples = distribution.rvs(N)
counts = [0]*10
for s in samples:
print (counts)
Here’s how the leading digit distribution of a simulation of 10,000 samples from an exponential (Weibull with γ = 1) compares to the distribution predicted by Benford’s law.
|---------------+----------+-----------|
| Leading digit | Observed | Predicted |
|---------------+----------+-----------|
| 1 | 3286 | 3010 |
| 2 | 1792 | 1761 |
| 3 | 1158 | 1249 |
| 4 | 851 | 969 |
| 5 | 754 | 792 |
| 6 | 624 | 669 |
| 7 | 534 | 580 |
| 8 | 508 | 511 |
| 9 | 493 | 458 |
|---------------+----------+-----------|
Looks like a fairly good fit. How could we quantify the fit so we can compare how the fit varies with the shape parameter? The most common approach is to use the chi-square goodness of fit test.
def chisq_stat(O, E):
return sum( [(o - e)**2/e for (o, e) in zip(O, E)] )
Here “O” stands for “observed” and “E” stands for “expected.” The observed counts are the counts we actually saw. The expected values are the values Benford’s law would predict:
expected = [N*log10((i+1)/i) for i in range(1, 10)]
Note that we don’t want to pass counts to chisq_stat but counts[1:] instead. This is because counts starts with 0 index, but leading digits can’t be 0 for positive samples.
Here are the chi square goodness of fit statistics for a few values of γ. (Smaller is better.)
|-------+------------|
| Shape | Chi-square |
|-------+------------|
| 0.1 | 1.415 |
| 0.5 | 9.078 |
| 1.0 | 69.776 |
| 1.5 | 769.216 |
| 2.0 | 1873.242 |
|-------+------------|
This suggests that samples from a Weibull follow Benford’s law fairly well for shape γ < 1, i.e. for the case of decreasing hazard.
# Recreating the Vertigo poster
In his new book The Perfect Shape, Øyvind Hammer shows how to create a graph something like the poster for Alfred Hitchcock’s movie Vertigo.
Hammer’s code uses a statistical language called Past that I’d never heard of. Here’s my interpretation of his code using Python.
import matplotlib.pyplot as plt
from numpy import arange, sin, cos, exp
i = arange(5000)
x1 = 1.0*cos(i/10.0)*exp(-i/2500.0)
y1 = 1.4*sin(i/10.0)*exp(-i/2500.0)
d = 450.0
vx = cos(i/d)*x1 - sin(i/d)*y1
vy = sin(i/d)*x1 + cos(i/d)*y1
plt.plot(vx, vy, "k")
h = max(vy) - min(vy)
w = max(vx) - min(vx)
plt.axes().set_aspect(w/h)
plt.show()
This code produces what’s called a harmonograph, related to the motion of a pendulum free to move in x and y directions:
It’s not exactly the same as the movie poster, but it’s definitely similar. If you find a way to modify the code to make it closer to the poster, leave a comment below.
# Approximate inverse of the gamma function
The other day I ran across a blog post by Brian Hayes that linked to an article by David Cantrell on how to compute the inverse of the gamma function. Cantrell gives an approximation in terms of the Lambert W function.
In this post we’ll write a little Python code to kick the tires on Cantrell’s approximation. The post also illustrates how to do some common tasks using SciPy and matplotlib.
Here are the imports we’ll need.
import matplotlib.pyplot as plt
from scipy import pi, e, sqrt, log, linspace
from scipy.special import lambertw, gamma, psi
from scipy.optimize import root
First of all, the gamma function has a local minimum k somewhere between 1 and 2, and so it only makes sense to speak of its inverse to the left or right of this point. Gamma is strictly increasing for real values larger than k.
To find k we look for where the derivative of gamma is zero. It’s more common to work with the derivative of the logarithm of the gamma function than the derivative of the gamma function itself. That works just as well because gamma has a minimum where its log has a minimum. The derivative of the log of the gamma function is called ψ and is implemented in SciPy as scipy.special.psi. We use the function scipy.optimize.root to find where ψ is zero.
The root function returns more information than just the root we’re after. The root(s) are returned in the arrayx, and in our case there’s only one root, so we take the first element of the array:
k = root(psi, 1.46).x[0]
Now here is Cantrell’s algorithm:
c = sqrt(2*pi)/e - gamma(k)
def L(x):
return log((x+c)/sqrt(2*pi))
def W(x):
return lambertw(x)
def AIG(x):
return L(x) / W( L(x) / e) + 0.5
Cantrell uses AIG for Approximate Inverse Gamma.
How well goes this algorithm work? For starters, we’ll see how well it does when we do a round trip, following the exact gamma with the approximate inverse.
x = linspace(5, 30, 100)
plt.plot(x, AIG(gamma(x)))
plt.show()
This produces the following plot:
We get a straight line, as we should, so next we do a more demanding test. We’ll look at the absolute error in the approximate inverse. We’ll use a log scale on the x-axis since gamma values get large quickly.
y = gamma(x)
plt.plot(y, x- AIG(y))
plt.xscale("log")
plt.show()
This shows the approximation error is small, and gets smaller as its argument increases.
Cantrell’s algorithm is based on an asymptotic approximation, so it’s not surprising that it improves for large arguments.
Related posts: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8411609530448914, "perplexity": 1122.7029547964462}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945448.38/warc/CC-MAIN-20180421203546-20180421223546-00531.warc.gz"} |
https://brilliant.org/discussions/thread/mathfights/ | # Mathfights
Main post link -> http://mathfights.com/
There is a relatively new site for practicing math problems. You compete real-time against other mathfighters. If any of you have an Art of Problem Solving account, it is similar to FTW (although if you have an AoPS account then chances are you have heard of this before I post this discussion) If you want to join, then just go to mathfights.com and sign up. It's a great place to practice your speed and problem solving skills under pressure of time.
A note: the people you play in the early divisions may seem like robots but they are real. Just a bit slow for the likes of us.
Have fun!
EDIT: If you have a Google+ account, you can find the official community here:
Also, here is the official Facebook page. https://www.facebook.com/mathfights
EDIT2: Thank you everyone for the great feedback on this site! And thanks for joining!
Note by Daniel Liu
6 years, 4 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
Sort by:
It's soooo cool! Reminds me of pokemon battles a little bit!
- 6 years, 4 months ago
i think brilliant should also give similar feature
- 6 years, 4 months ago
thank you for posting this!
- 6 years, 4 months ago
coool
- 6 years, 4 months ago
I am currently in division 7 and can't find players for a fight. I hit the start button and sit waiting but the fight doesn't take place! Is this just with me or are others also facing a similar problem?
- 6 years, 4 months ago
Hmm... It may be because there aren't any division 7 players in your time zone at the moment. But don't worry, there always should be some. It may be just a coincidence that everyone happened to be playing each other or logged off.
- 6 years, 4 months ago
Have fun! And show the community why you belong in division 1.
- 6 years, 4 months ago
It's just soooo cooolll!!thnx 4 suggesting this website..itz awesome!!
- 6 years, 4 months ago
I'm not sure if its just me or not but it seems like when the clock gets to zero the match does not immediately stop and there is a delay of about 10-20 seconds.
- 6 years, 4 months ago
Yes, there is a bit of lag. Unfortunately, that probably won't be fixed anytime soon; the lag seems to be unpreventable. I've talked with the Founder a couple of times about this issue.
- 6 years, 4 months ago
Yay! Fixed!
- 5 years, 11 months ago
It is amazing! I love it!
- 6 years, 4 months ago
this is very amazing!! :D
- 6 years, 4 months ago
For some reason the site doesn't seem to be working :(
- 6 years, 4 months ago
Figured it out: Chrome was being annoying. Works on Firefox for some reason
- 6 years, 4 months ago
It's a good website but the problems in the early divisions are very easy
- 6 years, 4 months ago
Placement tests are out now!
- 5 years, 11 months ago
I love this it is amazing!
- 6 years, 4 months ago
Do the problems ever get harder?
- 6 years, 4 months ago
Yes they do. By the time you get into division 1, if you can get there, the problems will get a LOT harder.
- 6 years, 4 months ago
Ok thanks that's good to know
- 6 years, 4 months ago
Suuuupperb .....!!!!!!!! Thank you ....
- 6 years, 4 months ago
So damn awesome !
- 6 years, 4 months ago
Do we get solutions to the questions ?
- 6 years, 4 months ago
Not for now. This feature will come out soon so stay tuned.
- 6 years, 4 months ago
Yay, review buttons are out!
- 5 years, 11 months ago
A nice site kudos to you guy
- 6 years, 4 months ago
Wow, this is an awesome math practice tool!
- 6 years, 4 months ago
Brilliant website! ;)
- 6 years, 4 months ago
the site is very cool!! really
- 5 years, 8 months ago
lol, I am already using this. I am in division 3, my name is supermessi! Just math friend me and tell me you are from brilliant, and then we can play :)
- 5 years ago
when i press start it says "waiting for the opponent"and when i waite the never starts can anybody plz help me
- 6 years, 4 months ago
It may be a glitch that happened. It has happened before; it's because the server crashed I think. It should be fixed now; I didn't notice it.
- 6 years, 4 months ago
But the problems are really child like! Will it remain so?
- 5 years, 8 months ago
@Sagnik Saha after you get to division 3, the problems get way more serious!
- 5 years, 8 months ago
really Daniel it is awesome
- 4 years, 5 months ago
It's great! Division 3 is the hardest b/c the people are really competitive and had to redo it 2 times :/ But I am now division 1 and my rating is 2159! Thanks for introducing this Daniel.
- 4 years, 2 months ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9570003747940063, "perplexity": 4215.215571837464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371624083.66/warc/CC-MAIN-20200406102322-20200406132822-00469.warc.gz"} |
https://openstax.org/books/calculus-volume-1/pages/5-key-concepts | Calculus Volume 1
# Key Concepts
Calculus Volume 1Key Concepts
### 5.1Approximating Areas
• The use of sigma (summation) notation of the form $∑i=1nai∑i=1nai$ is useful for expressing long sums of values in compact form.
• For a continuous function defined over an interval $[a,b],[a,b],$ the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
• The width of each rectangle is $Δx=b−an.Δx=b−an.$
• Riemann sums are expressions of the form $∑i=1nf(xi*)Δx,∑i=1nf(xi*)Δx,$ and can be used to estimate the area under the curve $y=f(x).y=f(x).$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of ${xi*}{xi*}$ are chosen to be the left or right endpoints of the subintervals, respectively.
• Riemann sums allow for much flexibility in choosing the set of points ${xi*}{xi*}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.
### 5.2The Definite Integral
• The definite integral can be used to calculate net signed area, which is the area above the x-axis less the area below the x-axis. Net signed area can be positive, negative, or zero.
• The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
• Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
• The properties of definite integrals can be used to evaluate integrals.
• The area under the curve of many functions can be calculated using geometric formulas.
• The average value of a function can be calculated using definite integrals.
### 5.3The Fundamental Theorem of Calculus
• The Mean Value Theorem for Integrals states that for a continuous function over a closed interval, there is a value c such that $f(c)f(c)$ equals the average value of the function. See The Mean Value Theorem for Integrals.
• The Fundamental Theorem of Calculus, Part 1 shows the relationship between the derivative and the integral. See Fundamental Theorem of Calculus, Part 1.
• The Fundamental Theorem of Calculus, Part 2 is a formula for evaluating a definite integral in terms of an antiderivative of its integrand. The total area under a curve can be found using this formula. See The Fundamental Theorem of Calculus, Part 2.
### 5.4Integration Formulas and the Net Change Theorem
• The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero.
• The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
### 5.5Substitution
• Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable u and du for appropriate expressions in the integrand.
• When using substitution for a definite integral, we also have to change the limits of integration.
### 5.6Integrals Involving Exponential and Logarithmic Functions
• Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay.
• Substitution is often used to evaluate integrals involving exponential functions or logarithms.
### 5.7Integrals Resulting in Inverse Trigonometric Functions
• Formulas for derivatives of inverse trigonometric functions developed in Derivatives of Exponential and Logarithmic Functions lead directly to integration formulas involving inverse trigonometric functions.
• Use the formulas listed in the rule on integration formulas resulting in inverse trigonometric functions to match up the correct format and make alterations as necessary to solve the problem.
• Substitution is often required to put the integrand in the correct form. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9931431412696838, "perplexity": 246.63958501381416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147154.70/warc/CC-MAIN-20200228104413-20200228134413-00358.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=315668 | # Non-constructible number
by keiop3
Tags: nonconstructible, number
P: 1 why is the sin(2*pi/7) non-constructible?
Sci Advisor HW Helper P: 9,398 Becuase it doesn't lie in a quadratic extension of a quadratic extension of (etc) R: a number is constructible iff (using straight edge and compass) if it lies in an extension of degree 2^n for some n. The proof is elementary and a good exposition can be found in almost any Galois THeory book. To check this particular example find the minimal polynomial of sin2pi/7, which i imagine is the cycltomic x^5+x^4+x^3+x^2+x+1
Sci Advisor HW Helper P: 9,406 constructible means it is obtained by intersecting some lines and circles, hence given by quadratic equations. thus a sequence of extension fields of degree 2. since field extension degree is multiplicative, repeating them gives fields of degree 2^n. so any number satisfying an irreducible equation of degree not a power of 2 is not constructible.
Related Discussions Calculus & Beyond Homework 1 Calculus & Beyond Homework 0 Linear & Abstract Algebra 3 Calculus & Beyond Homework 1 Calculus & Beyond Homework 23 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9029589891433716, "perplexity": 1190.3865918541887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010216492/warc/CC-MAIN-20140305090336-00065-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/188491-criterion-closed-print.html | # Criterion for closed
• September 21st 2011, 09:09 AM
Sheld
Criterion for closed
I have this theorem.
If $X$ is a metric space and $Y \subset X$. $D \subset Y$ is closed in $Y$ iff $D = C \cap Y$ for some closed set $C$ in $X$.
(I have proved a very similar theorem for open sets.)
However I cant seem to make any progress for the above theorem. (In either direction)
Going forward, we have $D$ is closed.
So $Y-D$ is open in $Y$.
By using similar theorem for open sets.
$Y-D = E \cap Y$ for some open set $E$ in $X$.
But trying to get back to $D$ by taking complements, I get
$D = X-E \cup X-Y$
Which doesn't seem to help me at all.
Also using $D$ has all its limit points doesn't give me a lot to work with either.
This leads me to believe that the theorem is not true.
Is there something I am not considering? Even going backwards, I can't figure it out.
If $X$ is a metric space and $Y \subset X$. $D \subset Y$ is closed in $Y$ iff $D = C \cap Y$ for some closed set $C$ in $X$.
The closure of $Y$, $\overline{Y}~,$ in $X$ is the 'smallest' closed set containing $Y$.
If $D\subseteq Y$ then $\overline{D}\subseteq\overline{Y}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 29, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9592493176460266, "perplexity": 120.11676770921781}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701508530/warc/CC-MAIN-20130516105148-00039-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://chem.libretexts.org/Courses/Louisville_Collegiate_School/General_Chemistry/LibreTexts%2F%2FLouisville_Collegiate_School%2F%2FChapters%2F%2F12%3A_Kinetics/LibreTexts%2F%2FLouisville_Collegiate_School%2F%2FChapters%2F%2F12%3A_Kinetics%2F%2F12.2%3A_Factors_Affecting_Reaction_Rates | Skip to main content
# 12.2: Factors Affecting Reaction Rates
##### Learning Objectives
• Describe the effects of chemical nature, physical state, temperature, concentration, and catalysis on reaction rates
The rates at which reactants are consumed and products are formed during chemical reactions vary greatly. We can identify five factors that affect the rates of chemical reactions: the chemical nature of the reacting substances, the state of subdivision (one large lump versus many small particles) of the reactants, the temperature of the reactants, the concentration of the reactants, and the presence of a catalyst.
## The Chemical Nature of the Reacting Substances
The rate of a reaction depends on the nature of the participating substances. Reactions that appear similar may have different rates under the same conditions, depending on the identity of the reactants. For example, when small pieces of the metals iron and sodium are exposed to air, the sodium reacts completely with air overnight, whereas the iron is barely affected. The active metals calcium and sodium both react with water to form hydrogen gas and a base. Yet calcium reacts at a moderate rate, whereas sodium reacts so rapidly that the reaction is almost explosive.
## The State of Subdivision of the Reactants
Except for substances in the gaseous state or in solution, reactions occur at the boundary, or interface, between two phases. Hence, the rate of a reaction between two phases depends to a great extent on the surface contact between them. A finely divided solid has more surface area available for reaction than does one large piece of the same substance. Thus a liquid will react more rapidly with a finely divided solid than with a large piece of the same solid. For example, large pieces of iron react slowly with acids; finely divided iron reacts much more rapidly (Figure $$\PageIndex{1}$$). Large pieces of wood smolder, smaller pieces burn rapidly, and saw dust burns explosively.
Video $$\PageIndex{1}$$: The reaction of cesium with water in slow motion and a discussion of how the state of reactants and particle size affect reaction rates.
## Temperature of the Reactants
Chemical reactions typically occur faster at higher temperatures. Food can spoil quickly when left on the kitchen counter. However, the lower temperature inside of a refrigerator slows that process so that the same food remains fresh for days. We use a burner or a hot plate in the laboratory to increase the speed of reactions that proceed slowly at ordinary temperatures. In many cases, an increase in temperature of only 10 °C will approximately double the rate of a reaction in a homogeneous system.
## Concentrations of the Reactants
The rates of many reactions depend on the concentrations of the reactants. Rates usually increase when the concentration of one or more of the reactants increases. For example, calcium carbonate ($$\mathrm{CaCO_3}$$) deteriorates as a result of its reaction with the pollutant sulfur dioxide. The rate of this reaction depends on the amount of sulfur dioxide in the air (Figure $$\PageIndex{2}$$). As an acidic oxide, sulfur dioxide combines with water vapor in the air to produce sulfurous acid in the following reaction:
$\ce{SO}_{2(g)}+\ce{H_2O}_{(g)}⟶\ce{H_2SO}_{3(aq)} \label{12.3.1}$
Calcium carbonate reacts with sulfurous acid as follows:
$\ce{CaCO}_{3(s)}+\ce{H_2SO}_{3(aq)}⟶\ce{CaSO}_{3(aq)}+\ce{CO}_{2(g)}+\ce{H_2O}_{(l)} \label{12.3.2}$
In a polluted atmosphere where the concentration of sulfur dioxide is high, calcium carbonate deteriorates more rapidly than in less polluted air. Similarly, phosphorus burns much more rapidly in an atmosphere of pure oxygen than in air, which is only about 20% oxygen.
Video $$\PageIndex{2}$$: Phosphorous burns rapidly in air, but it will burn even more rapidly if the concentration of oxygen in is higher.
## The Presence of a Catalyst
Hydrogen peroxide solutions foam when poured onto an open wound because substances in the exposed tissues act as catalysts, increasing the rate of hydrogen peroxide’s decomposition. However, in the absence of these catalysts (for example, in the bottle in the medicine cabinet) complete decomposition can take months. A catalyst is a substance that increases the rate of a chemical reaction by lowering the activation energy without itself being consumed by the reaction. Activation energy is the minimum amount of energy required for a chemical reaction to proceed in the forward direction. A catalyst increases the reaction rate by providing an alternative pathway or mechanism for the reaction to follow (Figure $$\PageIndex{3}$$). Catalysis will be discussed in greater detail later in this chapter as it relates to mechanisms of reactions.
Chemical reactions occur when molecules collide with each other and undergo a chemical transformation. Before physically performing a reaction in a laboratory, scientists can use molecular modeling simulations to predict how the parameters discussed earlier will influence the rate of a reaction. Use the PhET Reactions & Rates interactive to explore how temperature, concentration, and the nature of the reactants affect reaction rates.
## Summary
The rate of a chemical reaction is affected by several parameters. Reactions involving two phases proceed more rapidly when there is greater surface area contact. If temperature or reactant concentration is increased, the rate of a given reaction generally increases as well. A catalyst can increase the rate of a reaction by providing an alternative pathway that causes the activation energy of the reaction to decrease.
## Glossary
catalyst
substance that increases the rate of a reaction without itself being consumed by the reaction
## Contributors and Attributions
• Was this article helpful? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8768300414085388, "perplexity": 1084.9883839125828}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300574.19/warc/CC-MAIN-20220117151834-20220117181834-00365.warc.gz"} |
http://physics.stackexchange.com/questions/73292/lorentz-group-and-classification-of-fields-by-their-transformation-under-lorentz | # Lorentz group and classification of fields by their transformation under Lorentz transformations
Let's have Lorentz group with generators of 3-rotations, $\hat {R}_{i}$, and Lorentz boosts, $\hat {L}_{i}$. By introducing operators
$\hat {J}_{i} = \frac{1}{2}\left(\hat {R}_{i} + i\hat {L}_{i}\right), \quad \hat {K}_{i} = \frac{1}{2}\left(\hat {R}_{i} - i\hat {L}_{i}\right)$
we makes algebra of the Lorentz group the same as SU(2) (or SO(3)) group. So each irreducible representation of the Lorentz group can be built as $$\hat {\mathbf S}^{(j_{1}, j_{2})} = \hat {\mathbf S}^{j_{1}}\times \hat {\mathbf S}^{j_{2}},$$ where $j_{1}, j_{2}$ are the max eigenvalues of $\hat {J}_{i}, \hat {K}_{i}$,
and it has dimention $(2j_{1} + 1)\times (2j_{2} + 1)$. The type of object, transforming via boosts and 3-rotations, is depend on $(j_{1}, j_{2})$: $$\Psi_{\alpha \beta} = S^{j_{1}}_{\alpha \mu}S^{j_{2}}_{\beta \nu}\Psi_{\mu \nu}.$$ For $(0, 0)$ we have scalar, for $\left(\frac{1}{2}, 0 \right), \left(0, \frac{1}{2}\right)$ we have spinor (left- and right-handled) etc. The value $j_{1} + j_{2}$ corresponds to the maximum value of $\hat {J}_{i} + \hat {K}_{i} = \hat {R}_{i}$, so it is an eigenvalue of irreducible rep of 3-rotation operator and corresponds to the spin number.
But the irreducible rep of Lorentz group isn't unitary.
So, the question: how can we classify the objects via transformations by using non-unitary reps?
-
Whether the representation of Lorentz group on space of fields is unitary or not is not of any physical significance. One requires that representation of Lorentz group on space of states be unitary. Space of states is a Fock space generated by Fourier modes of fields and even though the fields themselves are under finite dimensional (hence nonunitary) representation of Lorentz group the Fock space generated by their Fourier modes give a unitary representation of the Lorentz group. – user10001 Aug 5 '13 at 19:19
@user10001 . How exactly does Fock space give a unitary representation of the Lorentz group? – user8817 Aug 5 '13 at 23:06
@PhysiXxx That's a good question, but one you should ask in its own post, rather than buried in the comments here. – user1504 Aug 6 '13 at 11:49
Note that particles correspond to irreductible unitary representations of the Poincaré group (alias inhomogeneous Lorentz group), not the Lorentz group alone.
In these Poincaré representations, states are represented by $|p, \lambda \rangle$. $p$ is the momentum.
Let's consider positive massive representations ($p^2 = m^2, p^o >0$) Let $\pi=(m,\vec 0)$ . We see that we have a freedom to choose polarization, which corresponds to a $S0(3)$ symmetry. Looking at unitary representations of $SO(3)$ is the same thing that looking at representations of $SU(2)$
Here, $\lambda$ is a state basis for a little group $SU(2)$ representation $s$.
For a translation, we have :
$$U(a)|p, \lambda \rangle = e^{ iP.a}|p, \lambda \rangle$$
For a member $R$ of the little group $SU(2)$ , we have :
$$U(R)|\pi, \lambda \rangle = \sum_{\lambda'} D^{(s)}_{\lambda' \lambda}(R)|\pi, \lambda' \rangle$$
For any $SL(2,C)$ matrix $A$ , and for any $p$, it is possible to write an expression :
$$U(A)|p, \lambda \rangle = \sum_{\lambda'} D^{(s)}_{\lambda' \lambda}(W(p, A))| \Lambda_ap, \lambda' \rangle$$ where $W(p,A)$ is a $SU(2)$ little group element (see formula $18$ in the reference cited below for details)
With all this, you get an unitary representation of the Poincaré group.
The "Fock space" is the quantum version of these representations, that is it allows several-particles states.
See Reference pages 4 and 5
[EDIT] "For fields isn't important to have lorentz-invariant positive definite norm?"
No. Take for instance the Dirac equations for the bi-spinor field. The representation is $(1/2,0) + (0,1/2)$. This is not a unitary representation. There is a left and a right spinor. The transformation could be written :
$$\psi_{L,R} =\rightarrow e^{1/2(i\vec \sigma. \vec \theta \mp \vec \sigma. \vec \phi)}\psi_{L,R},$$
The parameters $\vec \theta$ correspond to rotations, the parameters $\vec \phi$ correspond to boosts.
Because the boost part is not unitary, we see clearly that the representation is not unitary.
So, this means that the bispinor bilinear expression $\psi^* \psi = \psi^*_{L}\psi_{L} + \psi^*_{R}\psi_{R}$ is not conserved in a Lorentz transformation [in fact, separarely, the spinor bilinear expressions $\psi^{*}_{L} \psi_{L}$ or $\psi^{*}_{R} \psi_{R}$ are not conserved too]. Remember here that the $\psi,\psi_{L}, \psi_{R}$ are fields, not "wave function".
Is this a problem ? No.
What is $\psi^*(x) \psi(x)$ ? It is just (mutliplied by $e$) the charge density of fields, that is $j^0(x)$
So, of corse, $j^0(x)$ is not an invariant for a Lorentz transformation, because it is the time component of a Lorentz vector.
The real Lorentz invariant is here : $\overline \psi(x) \psi(x)= \psi^*(x) \gamma^0 \psi(x)$
-
"...Note that particles correspond to irreductible unitary representations of the Poincaré group (alias inhomogeneous Lorentz group), not the Lorentz group alone...", - but we discussing about the fields, not about the wave-functions. For fields isn't important to have lorentz-invariant positive definite norm. – user8817 Aug 6 '13 at 16:05
@PhysiXxx : I have made an edit to the answer – Trimok Aug 7 '13 at 6:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9620550870895386, "perplexity": 440.7849083761405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701168998.49/warc/CC-MAIN-20160205193928-00205-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php/1977_IMO_Problems/Problem_6 | # 1977 IMO Problems/Problem 6
## Problem
Let be a function . Prove that if for each positive integer , then .
## Solution
We will prove this via induction. First we will prove there is a such that and then that is the only such solution.
Define the sequence with for and . By the given inequality we have that , this can be used to form a inequality chain of decreasing positive integers: By Infinite Descent, this sequence must terminate, and the only way it can terminate is if we input something into that is outside of its range. This can only happen if since the range and domain of are the positive integers. Since , there is a integer () such that .
Now if , then , which is impossible since by the range of , so we have is the only time when .
Now for the inductive step.
Assume that for all and these are the only times these values occur. We will prove that and that this is the only time this value occurs. Define the sequence similarly, except that , by the reasoning above, there is a such that , by the inductive assumption, this means that , we can repeat the inductive assumption to get that . This implies that . Thus, there is a such that .
Now for that , we have , which means that by the inductive assumption which implies since we must have , otherwise . So is the only time when
So the inductive step is complete. Therefore, by induction for all positive integers . | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9911176562309265, "perplexity": 198.69718579693168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038092961.47/warc/CC-MAIN-20210416221552-20210417011552-00137.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=156101 | Lorentz generators
by alphaone
Tags: generators, lorentz
Sci Advisor HW Helper P: 11,894 I don't think this is possible. $M_{\mu\nu}$ is different for every representation and the calculations are actually the other way around. The generators are computed by knowing how the spinors behave under restricted LT's. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9867537021636963, "perplexity": 1398.4485659895488}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894689.94/warc/CC-MAIN-20140722025814-00064-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://zims-en.kiwix.campusafrica.gos.orange.com/wikipedia_en_all_nopic/A/Irreducible_ideal | # Irreducible ideal
In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.[1]
## Examples
• Every prime ideal is irreducible.[2] Let two ideals ${\displaystyle J,K}$ be contained in some commutative ring ${\displaystyle R}$. If the intersection ${\displaystyle J\cap K}$ is a non-trivial ideal, then there exists some elements ${\displaystyle a\in J}$ and ${\displaystyle b\in K}$, where neither is in the intersection but the product is, which means a reducible ideal is not prime. A concrete example of this are the ideals ${\displaystyle 2\mathbb {Z} }$ and ${\displaystyle 3\mathbb {Z} }$ contained in ${\displaystyle \mathbb {Z} }$. The intersection is ${\displaystyle 6\mathbb {Z} }$, and ${\displaystyle 6\mathbb {Z} }$ is not a prime ideal.
• Every irreducible ideal of a Noetherian ring is a primary ideal,[1] and consequently for Noetherian rings an irreducible decomposition is a primary decomposition.[3]
• Every primary ideal of a principal ideal domain is an irreducible ideal.
• Every irreducible ideal is primal.[4]
## Properties
An element of an integral domain is prime if and only if the ideal generated by it is a nonzero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in ${\displaystyle \mathbb {Z} }$ for the ideal ${\displaystyle 4\mathbb {Z} }$ since it is not the intersection of two strictly greater ideals.
An ideal I of a ring R can be irreducible only if the algebraic set it defines is irreducible (that is, any open subset is dense) for the Zariski topology, or equivalently if the closed space of spec R consisting of prime ideals containing I is irreducible for the spectral topology. The converse does not hold; for example the ideal of polynomials in two variables with vanishing terms of first and second order is not irreducible.
If k is an algebraically closed field, choosing the radical of an irreducible ideal of a polynomial ring over k is exactly the same as choosing an embedding of the affine variety of its Nullstelle in the affine space. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 12, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9619389176368713, "perplexity": 130.70329098232975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710909.66/warc/CC-MAIN-20221202150823-20221202180823-00871.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/186279-time-series-theory-methods.html | # Math Help - Time Series: Theory and Methods
1. ## Time Series: Theory and Methods
Can someone help me with the follow textbook question from Time Series: Theory and Methods by Brockwell.
$Suppose\, that\, m_t = c_0 + c_1 + c_2t^2, \,t=0,\, \pm1,\, \pm2, ...$
a) $Show \, that\, m_t = \sum^2_{i=-2} a_im_{t+1} = \sum^3_{i=-3} b_im_{t+1}, \,t=0,\, \pm1,\, \pm2, ...$
$\, where \, a_2=a_{-2}=\frac{3}{35}, \, a_1=a_{-1}=\frac{7}{35}, a_0=\frac{17}{35}, \, b_3=b_{-3}=\frac{2}{21},\, b_2=b_{-2}=\frac{3}{21},\, b_1=b_{-1}=\frac{6}{21}, b_0=\frac{7}{21}$
I can do this with ease.
But i am stuck with this one :
b. $Suppose\, that\, X_t =m_t + Z_t \,where\, (Z_t, \, t=0,\, \pm1,\, \pm2, ...),$ $\, is\, a\, sequence\, of\, independent\, normal\, random\, variables,\, each\, with\, mean\, 0\, and\, variance\, \sigma^2.$
$\, Let\, U_t = \sum^2_{i=-2}{a_iX_{t+i}}\, and\, V_t=\sum^3_{i=-3}{a_iX_{t+i}}.$
$i Find\, the\, means\, and\, variances\, of\, U_t\, and\, V_t\,$
$ii. Find\, the\, Correlations\, between\, U_t\,and\,U_{t+1}\,and\,between\,V_t\,and\,V_{t+1}$
Thank=yoU!!
Thankyou!!
2. ## Re: Time Series: Theory and Methods
Can anyone help me with finding the mean and variances for bi?
3. ## Re: Time Series: Theory and Methods
Originally Posted by lpd
Can someone help me with the follow textbook question from Time Series: Theory and Methods by Brockwell.
$Suppose\, that\, m_t = c_0 + c_1 + c_2t^2, \,t=0,\, \pm1,\, \pm2, ...$
a) $Show \, that\, m_t = \sum^2_{i=-2} a_im_{t+1} = \sum^3_{i=-3} b_im_{t+1}, \,t=0,\, \pm1,\, \pm2, ...$
$\, where \, a_2=a_{-2}=\frac{3}{35}, \, a_1=a_{-1}=\frac{7}{35}, a_0=\frac{17}{35}, \, b_3=b_{-3}=\frac{2}{21},\, b_2=b_{-2}=\frac{3}{21},\, b_1=b_{-1}=\frac{6}{21}, b_0=\frac{7}{21}$
I can do this with ease.
But i am stuck with this one :
b. $Suppose\, that\, X_t =m_t + Z_t \,where\, (Z_t, \, t=0,\, \pm1,\, \pm2, ...),$ $\, is\, a\, sequence\, of\, independent\, normal\, random\, variables,\, each\, with\, mean\, 0\, and\, variance\, \sigma^2.$
$\, Let\, U_t = \sum^2_{i=-2}{a_iX_{t+i}}\, and\, V_t=\sum^3_{i=-3}{a_iX_{t+i}}.$
$i Find\, the\, means\, and\, variances\, of\, U_t\, and\, V_t\,$
$ii. Find\, the\, Correlations\, between\, U_t\,and\,U_{t+1}\,and\,between\,V_t\,and\,V_{t+1}$
Thank=yoU!!
Thankyou!!
$U_t=\sum_{i=-2}^2 X_{t+1}=\sum_{i=-2}^2\left( m_{t+i}+Z_{t+i}\right)=\sum_{i=-2}^2 m_{t+i}+ \sum_{i=-2}^2Z_{t+i}$
So the mean is:
$\overline{U_t}=\sum_{i=-2}^2 m_{t+i}$
and:
${\text{Var}}(U_t) = 5 \sigma^2$
CB
4. ## Re: Time Series: Theory and Methods
Originally Posted by CaptainBlack
$U_t=\sum_{i=-2}^2 X_{t+1}=\sum_{i=-2}^2\left( m_{t+i}+Z_{t+i}\right)=\sum_{i=-2}^2 m_{t+i}+ \sum_{i=-2}^2Z_{t+i}$
So the mean is:
$\overline{U_t}=\sum_{i=-2}^2 m_{t+i}$
and:
${\text{Var}}(U_t) = 5 \sigma^2$
CB
Thanks for that...
So let me try... could you check my working. I am a tad lost still...
$U_t=\sum_{i=-2}^2 a_i X_{t+1}=\sum_{i=-2}^2\left( a_i m_{t+i}+ a_iZ_{t+i}\right)=\sum_{i=-2}^2 a_i m_{t+i}+ \sum_{i=-2}^2 a_i Z_{t+i}$
So the mean is:
$\overline{U_t}=\sum_{i=-2}^2 a_i m_{t+i} = m_t$ from part a.
and:
$Var[U_t] = Var[m_t] + Var[\sum^2_{i=-2}a_iZ_{t+i}] = 0 + (\sum^2_{i=-2}a_i)^2Var[Z_{t+i}]$
$Var[U_t] = 0 + 1 \times \sigma^2 = \sigma^2$
I'm not sure how I can get 5. What am I doing wrong? ( ${\text{Var}}(U_t) = 5 \sigma^2$)
Or can I do something like...
$Var(U_t) = E(U_t^2) - (E(U_t))^2?$
$(E(U_t))^2 = m_t^2$
$E(U_t^2) = (m_t + 2\sum^2_{i=-2}a_iZ_{t+i})^2 = m_t^2 + m_t\sum^2_{i=-2}a_iZ_{t+i}) + (\sum^2_{i=-2}a_iZ_{t+i})^2$
$Var(U_t) = m_t^2 + 2m_t\sum^2_{i=-2}a_iZ_{t+i} + (\sum^2_{i=-2}a_iZ_{t+i})^2 - m_t^2$
$Var(U_t) = 2m_t\sum^2_{i=-2}a_iZ_{t+i} + (\sum^2_{i=-2}a_iZ_{t+i})^2$
$Var(U_t) = 2m_t\sigma^2 + \sigma^4$
$Var(U_t) = \sigma^2(2m_t + \sigma^2)=2m_t\sigma^2 + \sigma^4$
5. ## Re: Time Series: Theory and Methods
Originally Posted by lpd
Thanks for that...
So let me try... could you check my working. I am a tad lost still...
$U_t=\sum_{i=-2}^2 a_i X_{t+1}=\sum_{i=-2}^2\left( a_i m_{t+i}+ a_iZ_{t+i}\right)=\sum_{i=-2}^2 a_i m_{t+i}+ \sum_{i=-2}^2 a_i Z_{t+i}$
So the mean is:
$\overline{U_t}=\sum_{i=-2}^2 a_i m_{t+i} = m_t$ from part a.
and:
$Var[U_t] = Var[m_t] + Var[\sum^2_{i=-2}a_iZ_{t+i}] = 0 + (\sum^2_{i=-2}a_i)^2Var[Z_{t+i}]$
$Var[U_t] = 0 + 1 \times \sigma^2 = \sigma^2$
I'm not sure how I can get 5. What am I doing wrong? ( ${\text{Var}}(U_t) = 5 \sigma^2$)
the 5 was because I had used unit weights, what you have is nearly correct at the next to last line above
$Var[U_t] = Var[m_t] + Var\left[\sum^2_{i=-2}a_iZ_{t+i}\right] = 0 + \sum^2_{i=-2}a_i^2Var[Z_{t+i})]$
............. $=\sum^2_{i=-2}a_i^2 \sigma^2=\sigma^2 \sum^2_{i=-2}a_i^2$
We are using the independednce of the $Z$ s to replace the variance of the sum by the sum of the variances
Or can I do something like...
$Var(U_t) = E(U_t^2) - (E(U_t))^2?$
$(E(U_t))^2 = m_t^2$
$E(U_t^2) = (m_t + 2\sum^2_{i=-2}a_iZ_{t+i})^2 = m_t^2 + m_t\sum^2_{i=-2}a_iZ_{t+i}) + (\sum^2_{i=-2}a_iZ_{t+i})^2$
$Var(U_t) = m_t^2 + 2m_t\sum^2_{i=-2}a_iZ_{t+i} + (\sum^2_{i=-2}a_iZ_{t+i})^2 - m_t^2$
$Var(U_t) = 2m_t\sum^2_{i=-2}a_iZ_{t+i} + (\sum^2_{i=-2}a_iZ_{t+i})^2$
$Var(U_t) = 2m_t\sigma^2 + \sigma^4$
$Var(U_t) = \sigma^2(2m_t + \sigma^2)=2m_t\sigma^2 + \sigma^4$
You lose expectation operators part way through that should still be there, and have more than one operation that looks dubious to me.
CB
6. ## Re: Time Series: Theory and Methods
Originally Posted by CaptainBlack
the 5 was because I had used unit weights, what you have is nearly correct at the next to last line above
$Var[U_t] = Var[m_t] + Var\left[\sum^2_{i=-2}a_iZ_{t+i}\right] = 0 + \sum^2_{i=-2}a_i^2Var[Z_{t+i})]$
............. $=\sum^2_{i=-2}a_i^2 \sigma^2=\sigma^2 \sum^2_{i=-2}a_i^2$
We are using the independence of the $Z$ s to replace the variance of the sum by the sum of the variances
CB
Oh I See. So, $\sigma^2 \sum^2_{i=-2}a_i^2$ would look something like $\sigma^2 (2(\frac{3}{35})^2+ 2(\frac{12}{35})^2+ (\frac{17}{35})^2)$
How would I tackle the next part.
bii. $Find\,the\,correlations\, between\, U_{t}\, and\, U_{t+1}\,$
Do I do something like $Cov(U_t, U_{t+1})$?
$=E[(U_t-E(U_t))(U_{t+1}-E(U_{t+1}))]$
I should get something like...
$=E((\sum^2_{i=-2}a_i Z_{t+i})$ $(\sum^2_{i=-2}a_{i+1} Z_{t+1+i}))$
Then, is the correlation just equal to zero?
7. ## Re: Time Series: Theory and Methods
Originally Posted by lpd
Oh I See. So, $\sigma^2 \sum^2_{i=-2}a_i^2$ would look something like $\sigma^2 (2(\frac{3}{35})^2+ 2(\frac{12}{35})^2+ (\frac{17}{35})^2)$
How would I tackle the next part.
bii. $Find\,the\,correlations\, between\, U_{t}\, and\, U_{t+1}\,$
Do I do something like $Cov(U_t, U_{t+1})$?
$=E[(U_t-E(U_t))(U_{t+1}-E(U_{t+1}))]$
I should get something like...
$=E((\sum^2_{i=-2}a_i Z_{t+i})$ $(\sum^2_{i=-2}a_{i+1} Z_{t+1+i}))$
Then, is the correlation just equal to zero?
I will assume that you have a definition of the correlation something like:
${\rm{Cor}}(U,V)=\frac{{\rm{E}}[(U-\overline{U})(V-\overline{V})]}{\sigma_U \sigma_V}$
essentially the normalised covariance.
CB
8. ## Re: Time Series: Theory and Methods
Originally Posted by CaptainBlack
I will assume that you have a definition of the correlation something like:
${\rm{Cor}}(U,V)=\frac{{\rm{E}}[(U-\overline{U})(V-\overline{V})]}{\sigma_U \sigma_V}$
essentially the normalised covariance.
CB
I'm stuck on bii now.
Basically I got up to here... after a few cancellations in expanding the covariance equation out. $Cov(U_t, U_{t+1}) = ... = E[(\sum_{i=-2}^2 a_i Z_{t+i})(\sum_{i=-2}^2 a_{i+1} Z_{t+i+1})]$
How do I expand this out : $E((\sum_{i=-2}^2 a_i Z_{t+i})(\sum_{i=-2}^2 a_{i+1} Z_{t+i+1}))$
Can i say something like... $\sum_{i=-2}^2 a_i \sum_{i=-2}^2 a_{j+1}E[Z_{t+i}]E[Z_{t+j}]=0\,if\, i\neq j?$ $where\, j=i+1$
And then get something like...
$E[\sum_{i=-2}^1 a_i a_{i+1} Z^2_{t+i+j}] = (\sum_{i=-2}^1 a_i a_{i+1}) \sigma^2$?
9. ## Re: Time Series: Theory and Methods
Originally Posted by lpd
I'm stuck on bii now.
Basically I got up to here... after a few cancellations in expanding the covariance equation out. $Cov(U_t, U_{t+1}) = ... = E[(\sum_{i=-2}^2 a_i Z_{t+i})(\sum_{i=-2}^2 a_{i+1} Z_{t+i+1})]$
How do I expand this out : $E((\sum_{i=-2}^2 a_i Z_{t+i})(\sum_{i=-2}^2 a_{i+1} Z_{t+i+1}))$
Because the $Z$s are zero mean and independent $E(Z_m Z_n)=0$ if $m\ne n$ and $=\sigma^2$ if $m=n$
$E\left[\left(\sum_{i=-2}^2 a_i Z_{t+i}\right)\left(\sum_{i=-2}^2 a_{i+1} Z_{t+i+1}\right)\right]=E\left[(a_{-2}Z_{t-2}+..+a_2Z_2)(a_{-1}Z_{t-1}+..+a_2Z_3) \right]$
.......... $=\sigma^2\sum_{i=-1}^2a_i$
CB | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 82, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8946763277053833, "perplexity": 1056.0973659906529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119661285.56/warc/CC-MAIN-20141024030101-00160-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://numberworld.info/17054 | # Number 17054
### Properties of number 17054
Cross Sum:
Factorization:
Divisors:
1, 2, 8527, 17054
Count of divisors:
Sum of divisors:
Prime number?
No
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 2 (Binary):
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
429e
Base 32:
gku
sin(17054)
0.99080416769768
cos(17054)
0.13530373709884
tan(17054)
7.3228144982713
ln(17054)
9.744140059302
lg(17054)
4.2318262586473
sqrt(17054)
130.59096446539
Square(17054)
### Number Look Up
Look Up
17054 (seventeen thousand fifty-four) is a unique number. The cross sum of 17054 is 17. If you factorisate the figure 17054 you will get these result 2 * 8527. 17054 has 4 divisors ( 1, 2, 8527, 17054 ) whith a sum of 25584. The figure 17054 is not a prime number. The figure 17054 is not a fibonacci number. The number 17054 is not a Bell Number. The figure 17054 is not a Catalan Number. The convertion of 17054 to base 2 (Binary) is 100001010011110. The convertion of 17054 to base 3 (Ternary) is 212101122. The convertion of 17054 to base 4 (Quaternary) is 10022132. The convertion of 17054 to base 5 (Quintal) is 1021204. The convertion of 17054 to base 8 (Octal) is 41236. The convertion of 17054 to base 16 (Hexadecimal) is 429e. The convertion of 17054 to base 32 is gku. The sine of the number 17054 is 0.99080416769768. The cosine of the figure 17054 is 0.13530373709884. The tangent of the figure 17054 is 7.3228144982713. The square root of 17054 is 130.59096446539.
If you square 17054 you will get the following result 290838916. The natural logarithm of 17054 is 9.744140059302 and the decimal logarithm is 4.2318262586473. I hope that you now know that 17054 is impressive number! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8360105156898499, "perplexity": 2860.121471170963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573118.26/warc/CC-MAIN-20220817213446-20220818003446-00029.warc.gz"} |
https://dsp.stackexchange.com/questions/49879/auto-correlation-function-an-inverse-problem | # Auto-correlation function, an inverse problem
$x[n]$ is a complex function $n=0,1,2,\cdots,L-1$
we assume $x[n]$ is periodic in its index: $x[n+L]=x[n]$
Its auto-correlation function $C[n]$ is uniquely defined as: $$C[n]=\sum_{i=0}^{L-1} x[i+n]x^*[i]$$ $C[n]$ also has the periodic property: $$C[n+L]=C[n]\tag{1}$$
And ''conjugate-symmetry'' property: $C[-n]=C^*[n] \tag{2}$
### Now my question is:
For given $C[n]$, which satisfies property (1) and (2):
Can we find the corresponding $x[n]$ ?
If yes, is it unique? and what is the method to find $x[n]$?
If no, what other constraint properties should we add to $C[n]$, in order to make it yes?
• By the way, if $L$ is even, $(2) \wedge (1)$ implies that $C[-\frac L2]=C[\frac L2] = C[\frac z2 L] \in \mathbb R$ for all $z\in \mathbb Z$. That has a consequence $x$! Namely, every $x[i+\frac L2]= x^*[i]$. – Marcus Müller Jun 13 '18 at 21:03
• Hi: don't know about the complex case but, in statistics, if the covariance matrix is symmetric and positive definite, it is possible. see theorem 7.5 of this link for the details. I don't know if or how that would translate to the complex case ? fepress.org/wp-content/uploads/2014/06/…. – mark leeds Jun 13 '18 at 21:13
• also, I'm not sure if it helps or hurts but the periodicity that you describe is not necessary in the proof at the link because the concept of periodicity is already kind of embedded in the covariance matrix so it's not really dealt with in statistics ( atleast as far as my experience goes. ). – mark leeds Jun 13 '18 at 21:20
• matrix seems to be on the right track, but I don't understand what's the relation between covariance matrix and auto-correlation function here. What are $x[n]$ and $C[n]$ in the language of matrix? – wwwjjj Jun 13 '18 at 21:38
• In the convariance matrix theory, we have random variables $X[n]$, we can add a second dimension $t$ meaning the random time series, $X[n,t]$ the summation is over $t$ instead of the periodic $n$. I believe it must have some interesting relations with the non-random problem in my case. – wwwjjj Jun 13 '18 at 21:49
Let's look at the case $x[n] \in \mathbb{R}$, where $x[n]$ is real.
Autocorrelation is basically convolution of the signal with it's time inverse. This can be easily expressed in the frequency domain.
$$\mathscr{F}\Big\{ r_{xx}[n] \Big\} = \mathscr{F}\Big\{ x[n] \Big\} \cdot \mathscr{F}\Big\{ x[-n] \Big\}$$
$$R_{xx}(\omega) = X(\omega)\cdot X^*(\omega) = \Big| X(\omega) \Big|^2$$
So it's easy to see that the Fourier Transform of the auto correlation is simply the magnitude squared of the Fourier Transform of the input signal. That's sometimes referred to as the Power Spectrum.
It's also easy to see that information gets lost in the process. There are $N$ unique values going in but because of the symmetry properties of the auto correlation there are only $\frac{N}{2}$ unique (independent) values coming out. Looking in the frequency domain, we can see that the phase is lost.
If yes, is it unique? and what is the method to find $x[n]$?
No, it's not unique
Can we find the corresponding $x[n]$ ?
There is an infinite number of $x[n]$.
1. Take the Fourier Transform of the autocorrelation
2. Take the square root
3. Add an aribtiraty (but odd-symmetric) phase function
4. Do an inverse Fourier Transform
Any signal derived this way will have the same original auto correlation function.
if no, what other constraint properties should we add to $C[n]$, in order to make it yes?
You can't make it a yes, since it's not unique. No matter what auto correlation you choose, there will be infinite $x[n]$ that will have it as an autocorrelation.
• I’m not sure but the Weiner Filter might be a unique spectral factorization – Stanley Pawlukiewicz Jun 14 '18 at 1:31
• sorry Hil, but i can't resist wallowing in a little bit of OCD. – robert bristow-johnson Jun 14 '18 at 1:53
• actually Hilmar, if there are $N$ unique and non-zero values going into the autocorrelation, and it's linear correlation so it's just like convolving $x[n]$ with a time-reversed copy of the same: \begin{align} r_{xx}[n] &= x[n] \circledast x[-n] \\ &= \sum_i x[i]x[i+n] \\ \end{align} then $N$ values go in, $2N-1$ values come out, but there is symmetry and $N-1$ of the values of the $2N-1$ are redundant. that still leaves $N$ values. but i agree (from the frequency-domain argument) that the mapping of $x[n]$ to $r_{xx}[n]$ is not one-to-one. – robert bristow-johnson Jun 14 '18 at 1:59
• can someone explain why we can create the unique x in the time domain ( see link in my comment earlier ) ? does it have something to do with the periodicity requirement ? – mark leeds Jun 14 '18 at 6:24
• @markleeds : by "not unique" I mean the following: all signals that have the same power spectrum have the same auto correlation. Very different signals can have the same power spectrum. Example: A delta impulse and white noise have the same auto correlation. It's not just a bias. Another Example: the impulse response of every all pass filter has the same auto correlation, there many, many different allpass filters but they all have the same auto correlation – Hilmar Jun 14 '18 at 11:25
There is in general, as @Hilmar's answer points out, no unique solution to the question of a sequence that has the given perodic autocorrelation function. In the simplest case, that a shifted version $y$ of any sequence $x$ (e.g. $y[n] = x[n-3]$ for all $n$) has the same autocorrelation function as $x$. Similarly, $y[n] = x[-n]$ for all $n$ has the same autocorrelation function as $x$. If you feel that such $y$'s are really no different from $x$, then consider that all binary PN sequences of period $L = 2^m-1$ have the same periodic autocorrelation function $$C[n] = \begin{cases}L, & n \equiv 0\bmod L,\\-1,& \text{otherwise,} \end{cases}$$ and the PN sequences are very distinguishable from one another. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9054052233695984, "perplexity": 343.45196962607196}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999482.38/warc/CC-MAIN-20190624104413-20190624130413-00424.warc.gz"} |
http://mathhelpforum.com/trigonometry/197262-equation-axis-symmetry-graph-equation-y-x2-6x-2-a.html | # Thread: which is an equation of the axis of symmetry of the graph of the equation y=x2-6x+2
1. ## which is an equation of the axis of symmetry of the graph of the equation y=x2-6x+2
Castle learning question
which is an equation of the axis of symmetry of the graph of the equation y=x2-6x+2?
2. ## Re: which is an equation of the axis of symmetry of the graph of the equation y=x2-6x
Originally Posted by strdatmage
which is an equation of the axis of symmetry of the graph of the equation y=x2-6x+2?
Did you draw the graph and look at it?
Is it true that $y=x^2-6x+2=(x-3)^2-7~?$
3. ## Re: which is an equation of the axis of symmetry of the graph of the equation y=x2-6x
Originally Posted by strdatmage
Castle learning question
which is an equation of the axis of symmetry of the graph of the equation y=x2-6x+2?
Your equation describes a parabola opening up. Thus the symmetry axis passes through the vertex of the parabola.
Complete the square and determine the x-coordinate of the vertex:
$y=x^2-6x+2 = (x^2-6x \color{red}+ 9) - 9\color{black}+2= (x-3)^2-7$
4. ## Re: which is an equation of the axis of symmetry of the graph of the equation y=x2-6x
Hello, strdatmage!
Which is an equation of the axis of symmetry of the graph of: . $y\:=\:x^2-6x+2\,?$
It is worthwhile to learn this formula . . .
For the parabola: $y \:=\:ax^2 + bx + c$
. . the axis of symmetry is: . $x \:=\:\frac{\text{-}b}{2a}$
[Think of the "front half" of the Quadratic Formula.]
In this problem: . $a = 1,\;b = \text{-}6,\;c = 2$
We have: . $x \:=\:\frac{\text{-}(\text{-}6)}{2(1)} \:=\:3$
The axis of symmetry is the vertical line: $x \,=\,3.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.909804105758667, "perplexity": 256.12297467899947}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982937576.65/warc/CC-MAIN-20160823200857-00042-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://www.earthdoc.org/content/papers/10.3997/2214-4609.20147809 | 1887
### Abstract
Structural Uncertainties have a direct impact in exploration, production and drilling decisions. We suggest an approach to generate realisations of reservoir grid conditioned by structural uncertainties. In this paper we show how structural uncertainty can be estimated from geophysical data and used to constrain the shape of reservoir grid realisations. Firstly we present a practical and global approach for estimation of structural uncertainty (i.e. not limited to vertical displacements) which results in a 3D vectorial field attached to the structural model. Secondly we show how this vectorial field can be used in the process of constructing reservoir grids to produce realisations constrained by structural uncertainties. The presented technique enables the propagation of structural uncertainty in the computation of reservoir realisations, in such a way that a new dimension is added to probabilistic reserve calculation. Some examples have shown that including structural uncertainties in the generation of reservoir grid realisations is a major improvement for history matching.
/content/papers/10.3997/2214-4609.20147809
2008-06-09
2020-09-27 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8558424115180969, "perplexity": 1019.0087974143382}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400283990.75/warc/CC-MAIN-20200927152349-20200927182349-00366.warc.gz"} |
https://www.physicsforums.com/threads/complexifying-su-2-to-get-sl-2-c-group-thread-footnote.4671/ | # Complexifying su(2) to get sl(2,C)-group thread footnote
1. Aug 10, 2003
### marcus
Complexifying su(2) to get sl(2,C)---group thread footnote
On the group thread midterm exam (which we never had to take!) it says what is the LA of the matrix group SL(2, C)
and the answer is the TRACE ZERO 2x2 matrices.
So that is what sl(2,C) is.
When you exponentiate one of the little critters, det = exp trace,
so the determinant is one which is what SL means.
Any X in sl(2,C) has a unique decomposition into skew hermitians that goes like this
X = (X - X*)/2 + i(X + X*)/2i
and these two skew hermitians
(X - X*)/2 and (X + X*)/2i
are trace zero, because trace is linear
check the skew hermitiandom of them:
(X - X*)* = (X* - X) = - (X - X*)
the other one checks because (1/2i)* = - (1/2i)
since conjugation does not change (X + X*)* = (X + X*)
so the upshot is that any X in sl(2,C) is composed
X = A + iB
of two matrices A and B in su(2)
Also on the midterm was the fact that su(2) is the skew hermitian ones: A* = - A.
There was this footnote on complexification of LAs and the above suffices to show, without much further ado, that su(2)C the complexification of su(2) is isomorphic to sl(2, C)
2. Aug 10, 2003
### Tyger
SL(2,C) is a representation of the group of boosts and turns, so why doesn't it show up in our descriptions instead of the 4×4 Dirac spinors?
3. Aug 12, 2003
### r637h
Well, there you go: Topology/Non-Euclidian Geomerty, like poverty and ignorance: We will always have them with us.
Rudy
"Go Figure." - Archimedes
Last edited by a moderator: Aug 12, 2003 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9110056161880493, "perplexity": 4506.904756078849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501173866.98/warc/CC-MAIN-20170219104613-00567-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/distance-covered-by-a-runner.185834/ | # Distance covered by a runner
1. Sep 20, 2007
### rgluckin
1. The problem statement, all variables and given/known data
A runner accelerates from 0 to 15 ft/sec in 2 seconds.
She goes at a constant v from 2 sec to 10 sec.
She deccelerates to 0 at the 20 sec mark
Q: How much distance does she cover?
2. Relevant equations
3. The attempt at a solution
Here is my work:
From 0 to 2 sec, she accelerates from 0 ft/sec to 15 ft/sec.
Her average velocity is 7.5 ft/sec, so in 2 sec she covers 7.5 ft. (round to 8)
From 2 sec to 10 sec, she goes at a constant velocity of 15 ft/sec, so the distance covered is 15ft/sec*8sec, which is 120 feet
From 10 sec to 20 sec, she decelerates from 15 ft/sec to 0 ft/sec
Avg velocity of 15ft/sec / 10 sec = 1.5 ft/sec * 10 sec = 15 ft
8 ft + 120ft + 15ft = 143 ft in 20 sec.
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Sep 20, 2007
### darklide
D =(0.5*2*15)+(8*15)+(0.5*10*15)
=15+120+75
=180ft in 20 sec assuming that there is constant acc and deceleration
3. Sep 20, 2007
### rgluckin
OK, I get the first two figures -- but for the decceleration, she goes from 15ft/sec to 0, for a difference of 15 in 10 seconds, so that's an avg velocity of 1.5ft/sec *10 sec is 15 ft.
Where did I go wrong?
4. Sep 20, 2007
### darklide
You are assuming that the velocity is constant(average) which is not the case when accelerating or decelerating.
I did the question by using a velocity-time graph.
U could try to do so...
(i may be wrong...havent done these kind of questions for nearly one year) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9160864353179932, "perplexity": 2831.844647871484}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718285.69/warc/CC-MAIN-20161020183838-00145-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://worldwidescience.org/topicpages/p/period+doubling+bifurcations.html | #### Sample records for period doubling bifurcations
1. Perturbed period-doubling bifurcation. I. Theory
DEFF Research Database (Denmark)
Svensmark, Henrik; Samuelsen, Mogens Rugholm
1990-01-01
-defined way that is a function of the amplitude and the frequency of the signal. New scaling laws between the amplitude of the signal and the detuning δ are found; these scaling laws apply to a variety of quantities, e.g., to the shift of the bifurcation point. It is also found that the stability...... of a microwave-driven Josephson junction confirm the theory. Results should be of interest in parametric-amplification studies....
2. Fully developed turbulence via Feigenbaum's period-doubling bifurcations
International Nuclear Information System (INIS)
Duong-van, M.
1987-08-01
Since its publication in 1978, Feigenbaum's predictions of the onset of turbulence via period-doubling bifurcations have been thoroughly borne out experimentally. In this paper, Feigenbaum's theory is extended into the regime in which we expect to see fully developed turbulence. We develop a method of averaging that imposes correlations in the fluctuating system generated by this map. With this averaging method, the field variable is obtained by coarse-graining, while microscopic fluctuations are preserved in all averaging scales. Fully developed turbulence will be shown to be a result of microscopic fluctuations with proper averaging. Furthermore, this model preserves Feigenbaum's results on the physics of bifurcations at the onset of turbulence while yielding additional physics both at the onset of turbulence and in the fully developed turbulence regime
3. Period-doubling bifurcation and chaos control in a discrete-time mosquito model
Directory of Open Access Journals (Sweden)
Qamar Din
2017-12-01
Full Text Available This article deals with the study of some qualitative properties of a discrete-time mosquito Model. It is shown that there exists period-doubling bifurcation for wide range of bifurcation parameter for the unique positive steady-state of given system. In order to control the bifurcation we introduced a feedback strategy. For further confirmation of complexity and chaotic behavior largest Lyapunov exponents are plotted.
4. Amplitude calculation near a period-doubling bifurcation: An example
DEFF Research Database (Denmark)
Wiesenfeld, K.; Pedersen, Niels Falsig
1987-01-01
For the rf-driven Josephson junction, the dynamical behavior is studied near a period-doubling transition. The center-manifold theorem simplifies the problem and enables us to study only a first-order system, the parameters of which are expressed in terms of the Josephson-junction parameters....
5. On period doubling bifurcations of cycles and the harmonic balance method
International Nuclear Information System (INIS)
Itovich, Griselda R.; Moiola, Jorge L.
2006-01-01
This works attempts to give quasi-analytical expressions for subharmonic solutions appearing in the vicinity of a Hopf bifurcation. Starting with well-known tools as the graphical Hopf method for recovering the periodic branch emerging from classical Hopf bifurcation, precise frequency and amplitude estimations of the limit cycle can be obtained. These results allow to attain approximations for period doubling orbits by means of harmonic balance techniques, whose accuracy is established by comparison of Floquet multipliers with continuation software packages. Setting up a few coefficients, the proposed methodology yields to approximate solutions that result from a second period doubling bifurcation of cycles and to extend the validity limits of the graphical Hopf method
6. Spatial interaction creates period-doubling bifurcation and chaos of urbanization
International Nuclear Information System (INIS)
Chen Yanguang
2009-01-01
This paper provides a new way of looking at complicated dynamics of simple mathematical models. The complicated behavior of simple equations is one of the headstreams of chaos theory. However, a recent study based on dynamical equations of urbanization shows that there are still some undiscovered secrets behind the simple mathematical models such as logistic equation. The rural-urban interaction model can also display varied kinds of complicated dynamics, including period-doubling bifurcation and chaos. The two-dimension map of urbanization presents the same dynamics as that from the one-dimension logistic map. In theory, the logistic equation can be derived from the two-population interaction model. This seems to suggest that the complicated behavior of simple models results from interaction rather than pure intrinsic randomicity. In light of this idea, the classical predator-prey interaction model can be revised to explain the complex dynamics of logistic equation in physical and social sciences.
International Nuclear Information System (INIS)
2011-01-01
In this Letter, it is shown that from a two region partition of the phase space of a one-dimensional dynamical system, a p-region partition can be obtained for the CRL...LR...R orbits. That is, permutations associated with symbolic sequences are obtained. As a consequence, the trajectory in phase space is directly deduced from permutation. From this permutation other permutations associated with period-doubling and saddle-node bifurcation cascades are derived, as well as other composite permutations. - Research highlights: → Symbolic sequences are the usual topological approach to dynamical systems. → Permutations bear more physical information than symbolic sequences. → Period-doubling cascade permutations associated with original sequences are obtained. → Saddle-node cascade permutations associated with original sequences are obtained. → Composite permutations are derived.
8. Hamiltonian bifurcation perspective on two interacting vortex pairs: From symmetric to asymmetric leapfrogging, period doubling, and chaos
Science.gov (United States)
Whitchurch, Brandon; Kevrekidis, Panayotis G.; Koukouloyannis, Vassilis
2018-01-01
In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the two-dimensional plane. The vortices are considered as effective particles and their interaction can be described in classical mechanics terms. We first construct a Poincaré section, for a typical value of the energy, in order to acquire a picture of the structure of the phase space of the system. We divide the phase space in different regions which correspond to qualitatively distinct motions and we demonstrate its different temporal evolution in the "real" vortex space. Our main emphasis is on the leapfrogging periodic orbit, around which we identify a region that we term the "leapfrogging envelope" which involves mostly regular motions, such as higher order periodic and quasiperiodic solutions. We also identify the chaotic region of the phase plane surrounding the leapfrogging envelope as well as the so-called walkabout and braiding motions. Varying the energy as our control parameter, we construct a bifurcation tree of the main leapfrogging solution and its instabilities, as well as the instabilities of its daughter branches. We identify the symmetry-breaking instability of the leapfrogging solution (in line with earlier works), and also obtain the corresponding asymmetric branches of periodic solutions. We then characterize their own instabilities (including period doubling ones) and bifurcations in an effort to provide a more systematic perspective towards the types of motions available to this dynamical system.
9. Period-doubling bifurcation cascade observed in a ferromagnetic nanoparticle under the action of a spin-polarized current
Energy Technology Data Exchange (ETDEWEB)
Horley, Paul P., E-mail: [email protected] [Centro de Investigación en Materiales Avanzados, S.C. (CIMAV), Chihuahua/Monterrey, 120 Avenida Miguel de Cervantes, 31109 Chihuahua (Mexico); Kushnir, Mykola Ya. [Yuri Fedkovych Chernivtsi National University, 2 Kotsyubynsky str., 58012 Chernivtsi (Ukraine); Morales-Meza, Mishel [Centro de Investigación en Materiales Avanzados, S.C. (CIMAV), Chihuahua/Monterrey, 120 Avenida Miguel de Cervantes, 31109 Chihuahua (Mexico); Sukhov, Alexander [Institut für Physik, Martin-Luther Universität Halle-Wittenberg, 06120 Halle (Saale) (Germany); Rusyn, Volodymyr [Yuri Fedkovych Chernivtsi National University, 2 Kotsyubynsky str., 58012 Chernivtsi (Ukraine)
2016-04-01
We report on complex magnetization dynamics in a forced spin valve oscillator subjected to a varying magnetic field and a constant spin-polarized current. The transition from periodic to chaotic magnetic motion was illustrated with bifurcation diagrams and Hausdorff dimension – the methods developed for dissipative self-organizing systems. It was shown that bifurcation cascades can be obtained either by tuning the injected spin-polarized current or by changing the magnitude of applied magnetic field. The order–chaos transition in magnetization dynamics can be also directly observed from the hysteresis curves. The resulting complex oscillations are useful for development of spin-valve devices operating in harmonic and chaotic modes.
10. Multiple bifurcations and periodic 'bubbling' in a delay population model
International Nuclear Information System (INIS)
Peng Mingshu
2005-01-01
In this paper, the flip bifurcation and periodic doubling bifurcations of a discrete population model without delay influence is firstly studied and the phenomenon of Feigenbaum's cascade of periodic doublings is also observed. Secondly, we explored the Neimark-Sacker bifurcation in the delay population model (two-dimension discrete dynamical systems) and the unique stable closed invariant curve which bifurcates from the nontrivial fixed point. Finally, a computer-assisted study for the delay population model is also delved into. Our computer simulation shows that the introduction of delay effect in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node → stable focus → an lower-dimensional closed invariant curve (quasi-periodic solution, limit cycle) or/and stable periodic solutions → chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling) and the sudden change between two different attractors, etc
11. Double Hopf bifurcation in delay differential equations
Directory of Open Access Journals (Sweden)
Redouane Qesmi
2014-07-01
Full Text Available The paper addresses the computation of elements of double Hopf bifurcation for retarded functional differential equations (FDEs with parameters. We present an efficient method for computing, simultaneously, the coefficients of center manifolds and normal forms, in terms of the original FDEs, associated with the double Hopf singularity up to an arbitrary order. Finally, we apply our results to a nonlinear model with periodic delay. This shows the applicability of the methodology in the study of delay models arising in either natural or technological problems.
12. Bifurcation structure of successive torus doubling
International Nuclear Information System (INIS)
Sekikawa, Munehisa; Inaba, Naohiko; Yoshinaga, Tetsuya; Tsubouchi, Takashi
2006-01-01
The authors discuss the 'embryology' of successive torus doubling via the bifurcation theory, and assert that the coupled map of a logistic map and a circle map has a structure capable of generating infinite number of torus doublings
13. Two-Dimensional Simulation of Spatial-Temporal Behaviors About Period Doubling Bifurcation in an Atmospheric-Pressure Dielectric Barrier Discharge
International Nuclear Information System (INIS)
Zhang Jiao; Wang Yanhui; Wang Dezhen; Zhuang Juan
2014-01-01
As a spatially extended dissipated system, atmospheric-pressure dielectric barrier discharges (DBDs) could in principle possess complex nonlinear behaviors. In order to improve the stability and uniformity of atmospheric-pressure dielectric barrier discharges, studies on temporal behaviors and radial structure of discharges with strong nonlinear behaviors under different controlling parameters are much desirable. In this paper, a two-dimensional fluid model is developed to simulate the radial discharge structure of period-doubling bifurcation, chaos, and inverse period-doubling bifurcation in an atmospheric-pressure DBD. The results show that the period-2n (n = 1, 2…) and chaotic discharges exhibit nonuniform discharge structure. In period-2n or chaos, not only the shape of current pulses doesn't remains exactly the same from one cycle to another, but also the radial structures, such as discharge spatial evolution process and the strongest breakdown region, are different in each neighboring discharge event. Current-voltage characteristics of the discharge system are studied for further understanding of the radial structure. (low temperature plasma)
14. Chaos and bifurcations in periodic windows observed in plasmas
International Nuclear Information System (INIS)
Qin, J.; Wang, L.; Yuan, D.P.; Gao, P.; Zhang, B.Z.
1989-01-01
We report the experimental observations of deterministic chaos in a steady-state plasma which is not driven by any extra periodic forces. Two routes to chaos have been found, period-doubling and intermittent chaos. The fine structures in chaos such as periodic windows and bifurcations in windows have also been observed
15. Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate
Science.gov (United States)
Ren, Jingli; Yuan, Qigang
2017-08-01
A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincaré map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by Neimark-Sacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincaré map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last.
16. Bifurcation analysis of the logistic map via two periodic impulsive forces
International Nuclear Information System (INIS)
Jiang Hai-Bo; Li Tao; Zeng Xiao-Liang; Zhang Li-Ping
2014-01-01
The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincaré map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map. (general)
17. Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay
International Nuclear Information System (INIS)
Ding Yuting; Jiang Weihua; Wang Hongbin
2012-01-01
Highlights: ► We derive the unfolding of a financial system with Hopf-pitchfork bifurcation. ► We show the coexistence of a pair of stable small amplitudes periodic solutions. ► At the same time, also there is a pair of stable large amplitudes periodic solutions. ► Chaos can appear by period-doubling bifurcation far away from Hopf-pitchfork value. ► The study will be useful for interpreting economics phenomena in theory. - Abstract: In this paper, we identify the critical point for a Hopf-pitchfork bifurcation in a nonlinear financial system with delay, and derive the normal form up to third order with their unfolding in original system parameters near the bifurcation point by normal form method and center manifold theory. Furthermore, we analyze its local dynamical behaviors, and show the coexistence of a pair of stable periodic solutions. We also show that there coexist a pair of stable small-amplitude periodic solutions and a pair of stable large-amplitude periodic solutions for different initial values. Finally, we give the bifurcation diagram with numerical illustration, showing that the pair of stable small-amplitude periodic solutions can also exist in a large region of unfolding parameters, and the financial system with delay can exhibit chaos via period-doubling bifurcations as the unfolding parameter values are far away from the critical point of the Hopf-pitchfork bifurcation.
18. Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow
International Nuclear Information System (INIS)
Smith, L. D.; Rudman, M.; Lester, D. R.; Metcalfe, G.
2016-01-01
Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversal in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle–centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.
19. Nonresonant Double Hopf Bifurcation in Toxic Phytoplankton-Zooplankton Model with Delay
Science.gov (United States)
Yuan, Rui; Jiang, Weihua; Wang, Yong
This paper investigates a toxic phytoplankton-zooplankton model with Michaelis-Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.
20. Suppression of period-doubling and nonlinear parametric effects in periodically perturbed systems
International Nuclear Information System (INIS)
Bryant, P.; Wiesenfeld, K.
1986-01-01
We consider the effect on a generic period-doubling bifurcation of a periodic perturbation, whose frequency ω 1 is near the period-doubled frequency ω 0 /2. The perturbation is shown to always suppress the bifurcation, shifting the bifurcation point and stabilizing the behavior at the original bifurcation point. We derive an equation characterizing the response of the system to the perturbation, analysis of which reveals many interesting features of the perturbed bifurcation, including (1) the scaling law relating the shift of the bifurcation point and the amplitude of the perturbation, (2) the characteristics of the system's response as a function of bifurcation parameter, (3) parametric amplification of the perturbation signal including nonlinear effects such as gain saturation and a discontinuity in the response at a critical perturbation amplitude, (4) the effect of the detuning (ω 1 -ω 0 /2) on the bifurcation, and (5) the emergence of a closely spaced set of peaks in the response spectrum. An important application is the use of period-doubling systems as small-signal amplifiers, e.g., the superconducting Josephson parametric amplifier
1. Experimental observation of parametric effects near period doubling in a loss-modulated CO2 laser
OpenAIRE
Chizhevsky, V. N.
1996-01-01
A number of parametric effects, such as suppression of period doubling, shift of the bifurcation point, scaling law relating the shift and the perturbation amplitude, influence of the detuning on the suppression, reaching of the maximum gain between the original and shifted bifurcation points, and scaling law for idler power are experimentally observed near period doubling bifurcation in a loss-driven CO2 laser that is subjected to periodic loss perturbations at a frequency that is close to a...
2. On complex periodic motions and bifurcations in a periodically forced, damped, hardening Duffing oscillator
International Nuclear Information System (INIS)
Guo, Yu; Luo, Albert C.J.
2015-01-01
In this paper, analytically predicted are complex periodic motions in the periodically forced, damped, hardening Duffing oscillator through discrete implicit maps of the corresponding differential equations. Bifurcation trees of periodic motions to chaos in such a hardening Duffing oscillator are obtained. The stability and bifurcation analysis of periodic motion in the bifurcation trees is carried out by eigenvalue analysis. The solutions of all discrete nodes of periodic motions are computed by the mapping structures of discrete implicit mapping. The frequency-amplitude characteristics of periodic motions are computed that are based on the discrete Fourier series. Thus, the bifurcation trees of periodic motions are also presented through frequency-amplitude curves. Finally, based on the analytical predictions, the initial conditions of periodic motions are selected, and numerical simulations of periodic motions are carried out for comparison of numerical and analytical predictions. The harmonic amplitude spectrums are also given for the approximate analytical expressions of periodic motions, which can also be used for comparison with experimental measurement. This study will give a better understanding of complex periodic motions in the hardening Duffing oscillator.
3. Hydrodynamic bifurcation in electro-osmotically driven periodic flows
Science.gov (United States)
Morozov, Alexander; Marenduzzo, Davide; Larson, Ronald G.
2018-06-01
In this paper, we report an inertial instability that occurs in electro-osmotically driven channel flows. We assume that the charge motion under the influence of an externally applied electric field is confined to a small vicinity of the channel walls that, effectively, drives a bulk flow through a prescribed slip velocity at the boundaries. Here, we study spatially periodic wall velocity modulations in a two-dimensional straight channel numerically. At low slip velocities, the bulk flow consists of a set of vortices along each wall that are left-right symmetric, while at sufficiently high slip velocities, this flow loses its stability through a supercritical bifurcation. Surprisingly, the flow state that bifurcates from a left-right symmetric base flow has a rather strong mean component along the channel, which is similar to pressure-driven velocity profiles. The instability sets in at rather small Reynolds numbers of about 20-30, and we discuss its potential applications in microfluidic devices.
4. Periodic solutions and bifurcations of delay-differential equations
International Nuclear Information System (INIS)
He Jihuan
2005-01-01
In this Letter a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of delay-differential equations. An example is given to illustrate its convenience and effectiveness
5. Complex bifurcation patterns in a discrete predator-prey model with periodic environmental modulation
Science.gov (United States)
Harikrishnan, K. P.
2018-02-01
We consider the simplest model in the family of discrete predator-prey system and introduce for the first time an environmental factor in the evolution of the system by periodically modulating the natural death rate of the predator. We show that with the introduction of environmental modulation, the bifurcation structure becomes much more complex with bubble structure and inverse period doubling bifurcation. The model also displays the peculiar phenomenon of coexistence of multiple limit cycles in the domain of attraction for a given parameter value that combine and finally gets transformed into a single strange attractor as the control parameter is increased. To identify the chaotic regime in the parameter plane of the model, we apply the recently proposed scheme based on the correlation dimension analysis. We show that the environmental modulation is more favourable for the stable coexistence of the predator and the prey as the regions of fixed point and limit cycle in the parameter plane increase at the expense of chaotic domain.
6. Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional maps
International Nuclear Information System (INIS)
Avrutin, V; Granados, A; Schanz, M
2011-01-01
Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map becomes continuous. Depending on the properties of the map near the codimension-two bifurcation point, there exist different scenarios regarding how the infinite number of periodic orbits are born, mainly the so-called period adding and period incrementing. In our work we prove that, in order to undergo a big bang bifurcation of the period incrementing type, it is sufficient for a piecewise-defined one-dimensional map that the colliding fixed points are attractive and with associated eigenvalues of different signs
7. Sufficient conditions for a period incrementing big bang bifurcation in one-dimensional maps
Science.gov (United States)
Avrutin, V.; Granados, A.; Schanz, M.
2011-09-01
Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined discontinuous systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with one boundary in state space and become virtual, and, in the one-dimensional case, the map becomes continuous. Depending on the properties of the map near the codimension-two bifurcation point, there exist different scenarios regarding how the infinite number of periodic orbits are born, mainly the so-called period adding and period incrementing. In our work we prove that, in order to undergo a big bang bifurcation of the period incrementing type, it is sufficient for a piecewise-defined one-dimensional map that the colliding fixed points are attractive and with associated eigenvalues of different signs.
8. Period doubling phenomenon in a class of time delay equations
International Nuclear Information System (INIS)
Oliveira, C.R. de; Malta, C.P.
1985-01-01
The properties of the solution of a nonlinear time delayed differential equation (infinite dimension) as function of two parameters: the time delay tau and another parameter A (nonlinearity) are investigated. After a Hopf bifurcation period doubling may occur and is characterized by Feigenbaum's delta. A strange atractor is obtained after the period doubling cascade and the largest Lyapunov exponent is calculated indicating that the attractor has low dimension. The behaviour of this Liapunov exponent as function of tau is different from its behaviour as function of A. (Author) [pt
9. Analytical determination of bifurcations of periodic solution in three-degree-of-freedom vibro-impact systems with clearance
International Nuclear Information System (INIS)
Liu, Yongbao; Wang, Qiang; Xu, Huidong
2017-01-01
The smooth bifurcation and non-smooth grazing bifurcation of periodic solution of three-degree-of-freedom vibro-impact systems with clearance are studied in this paper. Firstly, six-dimensional Poincaré maps are established through choosing suitable Poincaré section and solving periodic solutions of vibro-impact system. Then, as the analytic expressions of all eigenvalues of Jacobi matrix of six-dimensional map are unavailable, the numerical calculations to search for the critical bifurcation values point by point is a laborious job based on the classical critical criterion described by the properties of eigenvalues. To overcome the difficulty from the classical bifurcation criteria, the explicit critical criterion without using eigenvalues calculation of high-dimensional map is applied to determine bifurcation points of Co-dimension-one bifurcations and Co-dimension-two bifurcations, and then local dynamical behaviors of these bifurcations are further analyzed. Finally, the existence of the grazing periodic solution of the vibro-impact system and grazing bifurcation point are analyzed, the discontinuous grazing bifurcation behavior is studied based on the compound normal form map near the grazing point, the discontinuous jumping phenomenon and the co-existing multiple solutions near the grazing bifurcation point are revealed.
10. Bifurcation of forced periodic oscillations for equations with Preisach hysteresis
International Nuclear Information System (INIS)
Krasnosel'skii, A; Rachinskii, D
2005-01-01
We study oscillations in resonant systems under periodic forcing. The systems depend on a scalar parameter and have the form of simple pendulum type equations with ferromagnetic friction represented by the Preisach hysteresis nonlinearity. If for some parameter value the period of free oscillations of the principal linear part of the system coincides with the period of the forcing term, then one may expect the existence of unbounded branches of periodic solutions for nearby parameter values. We present conditions for the existence and nonexistence of such branches and estimates of their number
11. Bifurcation structure of a model of bursting pancreatic cells
DEFF Research Database (Denmark)
Mosekilde, Erik; Lading, B.; Yanchuk, S.
2001-01-01
One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic P-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other. The transit......One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic P-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other....... The transition from this structure to the so-called period-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-III intermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which (n + 1)-spike bursting...
12. Limit cycles bifurcating from the periodic annulus of cubic homogeneous polynomial centers
Directory of Open Access Journals (Sweden)
Jaume Llibre
2015-10-01
Full Text Available We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of any cubic homogeneous polynomial center when it is perturbed inside the class of all polynomial differential systems of degree n.
13. Bifurcation analysis of delay-induced periodic oscillations
NARCIS (Netherlands)
Green, K.
2010-01-01
In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude
14. C-type period-doubling transition in nephron autoregulation
DEFF Research Database (Denmark)
Laugesen, Jakob Lund; Mosekilde, Erik; Holstein-Rathlou, Niels-Henrik
2011-01-01
The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bif......The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period......-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated...
15. Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system
Science.gov (United States)
Yu, Yue; Zhang, Zhengdi; Han, Xiujing
2018-03-01
In this work, we aim to demonstrate the novel routes to periodic and chaotic bursting, i.e., the different bursting dynamics via delayed pitchfork bifurcations around stable attractors, in the classical controlled Lü system. First, by computing the corresponding characteristic polynomial, we determine where some critical values about bifurcation behaviors appear in the Lü system. Moreover, the transition mechanism among different stable attractors has been introduced including homoclinic-type connections or chaotic attractors. Secondly, taking advantage of the above analytical results, we carry out a study of the mechanism for bursting dynamics in the Lü system with slowly periodic variation of certain control parameter. A distinct delayed supercritical pitchfork bifurcation behavior can be discussed when the control item passes through bifurcation points periodically. This delayed dynamical behavior may terminate at different parameter areas, which leads to different spiking modes around different stable attractors (equilibriums, limit cycles, or chaotic attractors). In particular, the chaotic attractor may appear by Shilnikov connections or chaos boundary crisis, which leads to the occurrence of impressive chaotic bursting oscillations. Our findings enrich the study of bursting dynamics and deepen the understanding of some similar sorts of delayed bursting phenomena. Finally, some numerical simulations are included to illustrate the validity of our study.
16. How universal is the period doubling phenomenon in equations with quadratic nonlinearity
International Nuclear Information System (INIS)
Malta, C.P.; Oliveira, C.R. de.
1983-09-01
Varying one parameter, the solution of nonlinear 1 sup(st) order differential equation with time delay tau is Fourier analysed. After the Hopf bifurcation, period-doubling phenomenon always occurs when tau is one of the fixed parameters (both for small and large tau). Varying tau, there are values of the fixed parameters for which no period-doubling occurs. 'Chaos' follows the period-doubling sequence and the rate at which 'chaos' is approached is very close to the universal delta = 4.6692016... characterising the period-doubling sequence to chaos in nonlinear difference equations. (Author) [pt
17. Genesis and bifurcations of unstable periodic orbits in a jet flow
International Nuclear Information System (INIS)
Uleysky, M Yu; Budyansky, M V; Prants, S V
2008-01-01
We study the origin and bifurcations of typical classes of unstable periodic orbits in a jet flow that was introduced before as a kinematic model of chaotic advection, transport and mixing of passive scalars in meandering oceanic and atmospheric currents. A method to detect and locate the unstable periodic orbits and classify them by the origin and bifurcations is developed. We consider in detail period-1 and period-4 orbits playing an important role in chaotic advection. We introduce five classes of period-4 orbits: western and eastern ballistic ones, whose origin is associated with ballistic resonances of the fourth-order, rotational ones, associated with rotational resonances of the second and fourth orders and rotational-ballistic ones associated with a rotational-ballistic resonance. It is a new kind of unstable periodic orbits that may appear in a chaotic flow with jets and/or circulation cells. Varying the perturbation amplitude, we track out the origin and bifurcations of the orbits for each class
18. Period-doubling cascades and strange attractors in the triple-well Φ6-Van der Pol oscillator
International Nuclear Information System (INIS)
Yu Jun; Zhang Rongbo; Pan Weizhen; Schimansky-Geier, L
2008-01-01
Duffing-Van der Pol equation with the fifth nonlinear-restoring force is investigated. The bifurcation structure and chaotic motion under the periodic perturbation are obtained by numerical simulations. Numerical simulations, including bifurcation diagrams, Lyapunov exponents, phase portraits and Poincare maps, exhibit some new complex dynamical behaviors of the system. Different routes to chaos, such as period doubling and quasi-periodic routes, and various kinds of strange attractors are also demonstrated
19. Stochastic response and bifurcation of periodically driven nonlinear oscillators by the generalized cell mapping method
Science.gov (United States)
Han, Qun; Xu, Wei; Sun, Jian-Qiao
2016-09-01
The stochastic response of nonlinear oscillators under periodic and Gaussian white noise excitations is studied with the generalized cell mapping based on short-time Gaussian approximation (GCM/STGA) method. The solutions of the transition probability density functions over a small fraction of the period are constructed by the STGA scheme in order to construct the GCM over one complete period. Both the transient and steady-state probability density functions (PDFs) of a smooth and discontinuous (SD) oscillator are computed to illustrate the application of the method. The accuracy of the results is verified by direct Monte Carlo simulations. The transient responses show the evolution of the PDFs from being Gaussian to non-Gaussian. The effect of a chaotic saddle on the stochastic response is also studied. The stochastic P-bifurcation in terms of the steady-state PDFs occurs with the decrease of the smoothness parameter, which corresponds to the deterministic pitchfork bifurcation.
20. Complex Dynamics of Droplet Traffic in a Bifurcating Microfluidic Channel: Periodicity, Multistability, and Selection Rules
Science.gov (United States)
Sessoms, D. A.; Amon, A.; Courbin, L.; Panizza, P.
2010-10-01
The binary path selection of droplets reaching a T junction is regulated by time-delayed feedback and nonlinear couplings. Such mechanisms result in complex dynamics of droplet partitioning: numerous discrete bifurcations between periodic regimes are observed. We introduce a model based on an approximation that makes this problem tractable. This allows us to derive analytical formulae that predict the occurrence of the bifurcations between consecutive regimes, establish selection rules for the period of a regime, and describe the evolutions of the period and complexity of droplet pattern in a cycle with the key parameters of the system. We discuss the validity and limitations of our model which describes semiquantitatively both numerical simulations and microfluidic experiments.
1. Bifurcations and Periodic Solutions for an Algae-Fish Semicontinuous System
Directory of Open Access Journals (Sweden)
Chuanjun Dai
2013-01-01
Full Text Available We propose an algae-fish semicontinuous system for the Zeya Reservoir to study the control of algae, including biological and chemical controls. The bifurcation and periodic solutions of the system were studied using a Poincaré map and a geometric method. The existence of order-1 periodic solution of the system is discussed. Based on previous analysis, we investigated the change in the location of the order-1 periodic solution with variable parameters and we described the transcritical bifurcation of the system. Finally, we provided a series of numerical results to illustrate the feasibility of the theoretical results. These results may help to facilitate a better understanding of algal control in the Zeya Reservoir.
2. Bifurcation with memory
International Nuclear Information System (INIS)
Olmstead, W.E.; Davis, S.H.; Rosenblat, S.; Kath, W.L.
1986-01-01
A model equation containing a memory integral is posed. The extent of the memory, the relaxation time lambda, controls the bifurcation behavior as the control parameter R is increased. Small (large) lambda gives steady (periodic) bifurcation. There is a double eigenvalue at lambda = lambda 1 , separating purely steady (lambda 1 ) from combined steady/T-periodic (lambda > lambda 1 ) states with T → infinity as lambda → lambda + 1 . Analysis leads to the co-existence of stable steady/periodic states and as R is increased, the periodic states give way to the steady states. Numerical solutions show that this behavior persists away from lambda = lambda 1
3. Resonant Homoclinic Flips Bifurcation in Principal Eigendirections
Directory of Open Access Journals (Sweden)
Tiansi Zhang
2013-01-01
Full Text Available A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.
4. Bifurcation and chaos in neural excitable system
International Nuclear Information System (INIS)
Jing Zhujun; Yang Jianping; Feng Wei
2006-01-01
In this paper, we investigate the dynamical behaviors of neural excitable system without periodic external current (proposed by Chialvo [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons and Fractals 1995;5(3-4):461-79] and with periodic external current as system's parameters vary. The existence and stability of three fixed points, bifurcation of fixed points, the conditions of existences of fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using bifurcation theory and center manifold theorem. The chaotic existence in the sense of Marotto's definition of chaos is proved. We then give the numerical simulated results (using bifurcation diagrams, computations of Maximum Lyapunov exponent and phase portraits), which not only show the consistence with the analytic results but also display new and interesting dynamical behaviors, including the complete period-doubling and inverse period-doubling bifurcation, symmetry period-doubling bifurcations of period-3 orbit, simultaneous occurrence of two different routes (invariant cycle and period-doubling bifurcations) to chaos for a given bifurcation parameter, sudden disappearance of chaos at one critical point, a great abundance of period windows (period 2 to 10, 12, 19, 20 orbits, and so on) in transient chaotic regions with interior crises, strange chaotic attractors and strange non-chaotic attractor. In particular, the parameter k plays a important role in the system, which can leave the chaotic behavior or the quasi-periodic behavior to period-1 orbit as k varies, and it can be considered as an control strategy of chaos by adjusting the parameter k. Combining the existing results in [Generic excitable dynamics on a two-dimensional map. Chaos, Solitons and Fractals 1995;5(3-4):461-79] with the new results reported in this paper, a more complete description of the system is now obtained
5. Bifurcation and Nonlinear Dynamic Analysis of Externally Pressurized Double Air Films Bearing System
Directory of Open Access Journals (Sweden)
Cheng-Chi Wang
2014-01-01
Full Text Available This paper studies the chaotic and nonlinear dynamic behaviors of a rigid rotor supported by externally pressurized double air films (EPDAF bearing system. A hybrid numerical method combining the differential transformation method and the finite difference method is used to calculate pressure distribution of EPDAF bearing system and bifurcation phenomenon of rotor center orbits. The results obtained for the orbits of the rotor center are in good agreement with those obtained using the traditional finite difference approach. The results presented summarize the changes which take place in the dynamic behavior of the EPDAF bearing system as the rotor mass and bearing number are increased and therefore provide a useful guideline for the bearing system.
6. Bifurcation structure of a model of bursting pancreatic cells
DEFF Research Database (Denmark)
Mosekilde, Erik; Lading, B.; Yanchuk, S.
2001-01-01
. The transition from this structure to the so-called period-adding structure is found to involve a subcritical period-doubling bifurcation and the emergence of type-III intermittency. The period-adding transition itself is not smooth but consists of a saddle-node bifurcation in which (n + 1)-spike bursting...... behavior is born, slightly overlapping with a subcritical period-doubling bifurcation in which n-spike bursting behavior loses its stability.......One- and two-dimensional bifurcation studies of a prototypic model of bursting oscillations in pancreatic P-cells reveal a squid-formed area of chaotic dynamics in the parameter plane, with period-doubling bifurcations on one side of the arms and saddle-node bifurcations on the other...
7. Quasi-periodic bifurcations and “amplitude death” in low-dimensional ensemble of van der Pol oscillators
Energy Technology Data Exchange (ETDEWEB)
Emelianova, Yu.P., E-mail: [email protected] [Department of Electronics and Instrumentation, Saratov State Technical University, Polytechnicheskaya 77, Saratov 410054 (Russian Federation); Kuznetsov, A.P., E-mail: [email protected] [Kotel' nikov' s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelyenaya 38, Saratov 410019 (Russian Federation); Turukina, L.V., E-mail: [email protected] [Kotel' nikov' s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelyenaya 38, Saratov 410019 (Russian Federation); Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24-25, 14476 Potsdam (Germany)
2014-01-10
The dynamics of the four dissipatively coupled van der Pol oscillators is considered. Lyapunov chart is presented in the parameter plane. Its arrangement is discussed. We discuss the bifurcations of tori in the system at large frequency detuning of the oscillators. Here are quasi-periodic saddle-node, Hopf and Neimark–Sacker bifurcations. The effect of increase of the threshold for the “amplitude death” regime and the possibilities of complete and partial broadband synchronization are revealed.
8. Further results on periods and period doubling for iterates of the trapezoic function
International Nuclear Information System (INIS)
Beyer, W.A.; Stein, P.R.
1982-01-01
The trapezoidal function lambda f/sub e/(x), is defined for fixed e element of (0,1] and for lambda element of [1,2] by lambda f/sub e/ (x) = lambda for /x-1/< 1-e and lambda f/sub e/(x) = lambda(1-/x-1/)/(1-e) for 1 greater than or equal to /x-1/greater than or equal to 1-e. For a fixed e, this is a one parameter family of endomorphisms of the interval [0,2]. The structure of the periods (or cycles) of these mappings is studied. In addition, the metric properties of the corresponding bifurcation diagrams are considered; in particular, the rate of convergence of a sequence of bifurcation points in the (x,lambda) plane is studied. It is shown to be different from that found by Feigenbaum and others for mappings which are not flat at the top. The limiting case e = 1 is of special interest. For cycles and containing a point x element of[e,2-e], the period quadruplicates instead of doubling as it does in the usual case
9. Stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks
Science.gov (United States)
Ferruzzo Correa, Diego P.; Bueno, Átila M.; Castilho Piqueira, José R.
2017-04-01
In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results.
10. The period adding and incrementing bifurcations: from rotation theory to applications
DEFF Research Database (Denmark)
Granados, Albert; Alseda, Lluis; Krupa, Maciej
2017-01-01
This survey article is concerned with the study of bifurcations of piecewise-smooth maps. We review the literature in circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich...
11. A period-doubling cascade precedes chaos for planar maps.
Science.gov (United States)
Sander, Evelyn; Yorke, James A
2013-09-01
A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic. Our method of proof in showing there is one cascade is to show there must be infinitely many cascades. We discuss in detail two-dimensional families like those which arise as a time-2π maps for the Duffing equation and the forced damped pendulum equation.
12. Period doubling cascades of prey-predator model with nonlinear harvesting and control of over exploitation through taxation
Science.gov (United States)
Gupta, R. P.; Banerjee, Malay; Chandra, Peeyush
2014-07-01
The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey-predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin's Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.
13. The transition mechanism from a symmetric single period discharge to a period-doubling discharge in atmospheric helium dielectric-barrier discharge
International Nuclear Information System (INIS)
Zhang, Dingzong; Wang, Yanhui; Wang, Dezhen
2013-01-01
Period-doubling and chaos phenomenon have been frequently observed in atmospheric-pressure dielectric-barrier discharges. However, how a normal single period discharge bifurcates into period-doubling state is still unclear. In this paper, by changing the driving frequency, we study numerically the transition mechanisms from a normal single period discharge to a period-doubling state using a one-dimensional self-consistent fluid model. The results show that before a discharge bifurcates into a period-doubling state, it first deviates from its normal operation and transforms into an asymmetric single period discharge mode. Then the weaker discharge in this asymmetric discharge will be enhanced gradually with increasing of the frequency until it makes the subsequent discharge weaken and results in the discharge entering a period-doubling state. In the whole transition process, the spatial distribution of the charged particle density and the electric field plays a definitive role. The conclusions are further confirmed by changing the gap width and the amplitude of the applied voltage
14. The transition mechanism from a symmetric single period discharge to a period-doubling discharge in atmospheric helium dielectric-barrier discharge
Energy Technology Data Exchange (ETDEWEB)
Zhang, Dingzong; Wang, Yanhui; Wang, Dezhen [School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 (China)
2013-06-15
Period-doubling and chaos phenomenon have been frequently observed in atmospheric-pressure dielectric-barrier discharges. However, how a normal single period discharge bifurcates into period-doubling state is still unclear. In this paper, by changing the driving frequency, we study numerically the transition mechanisms from a normal single period discharge to a period-doubling state using a one-dimensional self-consistent fluid model. The results show that before a discharge bifurcates into a period-doubling state, it first deviates from its normal operation and transforms into an asymmetric single period discharge mode. Then the weaker discharge in this asymmetric discharge will be enhanced gradually with increasing of the frequency until it makes the subsequent discharge weaken and results in the discharge entering a period-doubling state. In the whole transition process, the spatial distribution of the charged particle density and the electric field plays a definitive role. The conclusions are further confirmed by changing the gap width and the amplitude of the applied voltage.
15. Period doubling cascades of limit cycles in cardiac action potential models as precursors to chaotic early Afterdepolarizations.
Science.gov (United States)
Kügler, Philipp; Bulelzai, M A K; Erhardt, André H
2017-04-04
Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials (APs). EADs are caused by drugs, oxidative stress or ion channel disease, and they are considered as potential precursors to cardiac arrhythmias in recent attempts to redefine the cardiac drug safety paradigm. The irregular behaviour of EADs observed in experiments has been previously attributed to chaotic EAD dynamics under periodic pacing, made possible by a homoclinic bifurcation in the fast subsystem of the deterministic AP system of differential equations. In this article we demonstrate that a homoclinic bifurcation in the fast subsystem of the action potential model is neither a necessary nor a sufficient condition for the genesis of chaotic EADs. We rather argue that a cascade of period doubling (PD) bifurcations of limit cycles in the full AP system paves the way to chaotic EAD dynamics across a variety of models including a) periodically paced and spontaneously active cardiomyocytes, b) periodically paced and non-active cardiomyocytes as well as c) unpaced and spontaneously active cardiomyocytes. Furthermore, our bifurcation analysis reveals that chaotic EAD dynamics may coexist in a stable manner with fully regular AP dynamics, where only the initial conditions decide which type of dynamics is displayed. EADs are a potential source of cardiac arrhythmias and hence are of relevance both from the viewpoint of drug cardiotoxicity testing and the treatment of cardiomyopathies. The model-independent association of chaotic EADs with period doubling cascades of limit cycles introduced in this article opens novel opportunities to study chaotic EADs by means of bifurcation control theory and inverse bifurcation analysis. Furthermore, our results may shed new light on the synchronization and propagation of chaotic EADs in homogeneous and heterogeneous multicellular and cardiac tissue preparations.
16. Bifurcation and spatial pattern formation in spreading of disease with incubation period in a phytoplankton dynamics
Directory of Open Access Journals (Sweden)
Randhir Singh Baghel
2012-02-01
Full Text Available In this article, we propose a three dimensional mathematical model of phytoplankton dynamics with the help of reaction-diffusion equations that studies the bifurcation and pattern formation mechanism. We provide an analytical explanation for understanding phytoplankton dynamics with three population classes: susceptible, incubated, and infected. This model has a Holling type II response function for the population transformation from susceptible to incubated class in an aquatic ecosystem. Our main goal is to provide a qualitative analysis of Hopf bifurcation mechanisms, taking death rate of infected phytoplankton as bifurcation parameter, and to study further spatial patterns formation due to spatial diffusion. Here analytical findings are supported by the results of numerical experiments. It is observed that the coexistence of all classes of population depends on the rate of diffusion. Also we obtained the time evaluation pattern formation of the spatial system.
17. Study comparing the double kissing (DK) crush with classical crush for the treatment of coronary bifurcation lesions: the DKCRUSH-1 Bifurcation Study with drug-eluting stents.
Science.gov (United States)
Chen, S L; Zhang, J J; Ye, F; Chen, Y D; Patel, T; Kawajiri, K; Lee, M; Kwan, T W; Mintz, G; Tan, H C
2008-06-01
Classical crush has a lower rate of final kissing balloon inflation (FKBI) immediately after percutaneous coronary intervention (PCI). The double kissing (DK) crush technique has the potential to increase the FKBI rate, and no prospective studies on the comparison of classical with DK crush techniques have been reported. Three hundred and eleven patients with true bifurcation lesions were randomly divided into classical (n = 156) and DK crush (n = 155) groups. Clinical and angiographic details at follow-up at 8 months were indexed. The primary end point was major adverse cardiac events (MACE) including myocardial infarction, cardiac death and target lesion revascularization (TLR) at 8 months. FKBI was 76% in the classical crush group and 100% in the DK group (P DK crush group. Cumulative 8 month MACE was 24.4% in the classical crush group and 11.4% in the DK crush group (P = 0.02). The TLR-free survival rate was 75.4% in the classical crush group and 89.5% in the DK crush group (P = 0.002). DK crush technique has the potential of increasing FKBI rate and reducing stent thrombosis, with a further reduction of TLR and cumulative MACE rate at 8 months.
18. Bifurcation of the spin-wave equations
International Nuclear Information System (INIS)
Cascon, A.; Koiller, J.; Rezende, S.M.
1990-01-01
We study the bifurcations of the spin-wave equations that describe the parametric pumping of collective modes in magnetic media. Mechanisms describing the following dynamical phenomena are proposed: (i) sequential excitation of modes via zero eigenvalue bifurcations; (ii) Hopf bifurcations followed (or not) by Feingenbaum cascades of period doubling; (iii) local and global homoclinic phenomena. Two new organizing center for routes to chaos are identified; in the classification given by Guckenheimer and Holmes [GH], one is a codimension-two local bifurcation, with one pair of imaginary eigenvalues and a zero eigenvalue, to which many dynamical consequences are known; secondly, global homoclinic bifurcations associated to splitting of separatrices, in the limit where the system can be considered a Hamiltonian subjected to weak dissipation and forcing. We outline what further numerical and algebraic work is necessary for the detailed study following this program. (author)
19. DK crush (double-kissing and double-crush) technique for treatment of true coronary bifurcation lesions: illustration and comparison with classic crush.
Science.gov (United States)
Chen, Shaoliang; Zhang, Junjie; Ye, Fei; Zhu, Zhongsheng; Lin, Song; Shan, Shoujie; Kwan, Tak W
2007-04-01
Classic crush has a lower success rate compared to final kissing balloon inflation (FKBI). We previously reported the double-kissing (DK) crush technique that involves double-kissing along with double-crushing for the treatment of true bifurcation coronary lesions in 2005. This is a consecutive, nonrandomized, open-label study. Eighty-eight consecutive patients with single, true coronary bifurcation lesions according to Lefevre Classification2 and side branch diameter >2.0 mm were enrolled. The first 44 patients (from October 2004 to January 2005) were assigned to the classic crush treatment arm and the next 44 patients (from February 2005 to June 2005) were assigned to the DK crush technique arm, respectively. Data within 30 days were analyzed. Patients in the DK crush group, compared to those in classic crush group, were characterized by longer lesion length in the side branch (13.5 +/- 3.4 mm vs 7.8 +/- 3.1 mm; p DK crush group, as well as longer lesion length in the main vessel (24.3 +/- 8.6 mm vs 21.1 +/- 7.3 mm), though without significant differences (p >0.05). Subacute stent thrombosis was detected in 2 patients with failure of FKBI in the classic crush group (4.3%). In addition, patients in the classic crush group were characterized by a smaller minimum lumen diameter (MLD) at the side branch ostium (2.74 +/- 0.12 mm vs 3.01 +/- 0.13 mm; p DK crush has the potential to improve the clinical outcome in patients with coronary bifurcation lesions. Further randomized, prospective, multicenter studies are required to confirm these differences between the classic crush and DK crush techniques.
20. The acute changes of fractional flow reserve in DK (double kissing), crush, and 1-stent technique for true bifurcation lesions.
Science.gov (United States)
Ye, Fei; Zhang, Jun-Jie; Tian, Nai-Liang; Lin, Song; Liu, Zhi-Zhong; Kan, Jing; Xu, Hai-Mei; Zhu, Zhongsheng; Chen, Shao-Liang
2010-08-01
While many studies confirmed the importance of fractional flow reserve (FFR) in guiding complex percutaneous coronary interventions (PCI), data regarding the significance of FFR for bifurcation lesions are still lacking. Between October 2008 and October 2009, 51 patients with true bifurcation lesions were consecutively enrolled and randomized into double kissing (DK) crush (n = 25), and provisional 1-stent (n = 26) groups. FFR measurements at baseline and hyperemia were measured at pre-PCI, post-PCI, and at 8-month follow-up. Clinical follow-ups were available in 100% of patients while only 33% of patients underwent angiographic follow-up. Baseline clinical and angiographic characteristics were matched between the 2 groups. Pre-PCI FFR of the main branch (MB) in the DK group was 0.76 +/- 0.15, which was significantly lower than in the provisional 1-stent group (0.83 +/- 0.10, P = 0.029). This difference disappeared after the PCI procedure (0.92 +/- 0.04 vs. 0.92 +/- 0.05, P = 0.58). There were no significant differences in terms of baseline, angiographic, procedural indexes, and FFR of side branch (SB) between the 2 treatment arms. However, immediately after PCI, the patient with DK crush had higher FFR in the SB as compared to the provisional 1-stent group (0.94 +/- 0.03 vs. 0.90 +/- 0.08, P = 0.028, respectively) and also they had lower diameter stenosis (8.59 +/- 6.41% vs. 15.62 +/- 11.69%, P = 0.015, respectively). In the acute phase, immediately after PCI for bifurcation lesion, DK crush stenting was associated with higher FFR and lower residual diameter stenosis in the SB, as compared with the provisional 1-stent group.
1. Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider
International Nuclear Information System (INIS)
Luo, G.W.; Lv, X.H.; Ma, L.
2008-01-01
A two-degree-of-freedom plastic impact oscillator with a frictional slider is considered. Dynamics of the plastic impact oscillator are analyzed by a three-dimensional map, which describes free flight and sticking solutions of two masses of the system, between impacts, supplemented by transition conditions at the instants of impacts. Piecewise property and singularity are found to exist in the impact Poincare map. The piecewise property of the map is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of the map is generated via the grazing contact of two masses immediately before the impact. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. The influence of piecewise property, grazing singularity and parameter variation on dynamics of the vibro-impact system is analyzed. The global bifurcation diagrams of before-impact velocity as a function of the excitation frequency are plotted to predict much of the qualitative behavior of the system. The global bifurcations of period-N single-impact motions of the plastic impact oscillator are found to exhibit extensive and systematic characteristics. Dynamics of the impact oscillator, in the elastic impact case, is also analyzed. This type of impact is modelled by using the conditions of conservation of momentum and an instantaneous coefficient of restitution rule. The differences in periodic-impact motions and bifurcations are found by making a comparison between dynamic behaviors of the plastic and elastic impact oscillators with a frictional slider. The best progression of the plastic impact oscillator is found to occur in period-1 single-impact sticking motion with large impact velocity. The largest progression of the elastic impact oscillator occurs in period-1 multi-impact motion. The simulative results show that the plastic impact
2. Bifurcation and chaos in a Tessiet type food chain chemostat with pulsed input and washout
International Nuclear Information System (INIS)
Wang Fengyan; Hao Chunping; Chen Lansun
2007-01-01
In this paper, we introduce and study a model of a Tessiet type food chain chemostat with pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period doubling and period halving
3. Bifurcation scenarios for bubbling transition.
Science.gov (United States)
Zimin, Aleksey V; Hunt, Brian R; Ott, Edward
2003-01-01
Dynamical systems with chaos on an invariant submanifold can exhibit a type of behavior called bubbling, whereby a small random or fixed perturbation to the system induces intermittent bursting. The bifurcation to bubbling occurs when a periodic orbit embedded in the chaotic attractor in the invariant manifold becomes unstable to perturbations transverse to the invariant manifold. Generically the periodic orbit can become transversely unstable through a pitchfork, transcritical, period-doubling, or Hopf bifurcation. In this paper a unified treatment of the four types of bubbling bifurcation is presented. Conditions are obtained determining whether the transition to bubbling is soft or hard; that is, whether the maximum burst amplitude varies continuously or discontinuously with variation of the parameter through its critical value. For soft bubbling transitions, the scaling of the maximum burst amplitude with the parameter is derived. For both hard and soft transitions the scaling of the average interburst time with the bifurcation parameter is deduced. Both random (noise) and fixed (mismatch) perturbations are considered. Results of numerical experiments testing our theoretical predictions are presented.
4. Double Kissing Crush Versus Provisional Stenting for Left Main Distal Bifurcation Lesions: DKCRUSH-V Randomized Trial.
Science.gov (United States)
Chen, Shao-Liang; Zhang, Jue-Jie; Han, Yaling; Kan, Jing; Chen, Lianglong; Qiu, Chunguang; Jiang, Tiemin; Tao, Ling; Zeng, Hesong; Li, Li; Xia, Yong; Gao, Chuanyu; Santoso, Teguh; Paiboon, Chootopol; Wang, Yan; Kwan, Tak W; Ye, Fei; Tian, Nailiang; Liu, Zhizhong; Lin, Song; Lu, Chengzhi; Wen, Shangyu; Hong, Lang; Zhang, Qi; Sheiban, Imad; Xu, Yawei; Wang, Lefeng; Rab, Tanveer S; Li, Zhanquan; Cheng, Guanchang; Cui, Lianqun; Leon, Martin B; Stone, Gregg W
2017-11-28
Provisional stenting (PS) is the most common technique used to treat distal left main (LM) bifurcation lesions in patients with unprotected LM coronary artery disease undergoing percutaneous coronary intervention. The double kissing (DK) crush planned 2-stent technique has been shown to improve clinical outcomes in non-LM bifurcations compared with PS, and in LM bifurcations compared with culotte stenting, but has never been compared with PS in LM bifurcation lesions. The authors sought to determine whether a planned DK crush 2-stent technique is superior to PS for patients with true distal LM bifurcation lesions. The authors randomized 482 patients from 26 centers in 5 countries with true distal LM bifurcation lesions (Medina 1,1,1 or 0,1,1) to PS (n = 242) or DK crush stenting (n = 240). The primary endpoint was the 1-year composite rate of target lesion failure (TLF): cardiac death, target vessel myocardial infarction, or clinically driven target lesion revascularization. Routine 13-month angiographic follow-up was scheduled after ascertainment of the primary endpoint. TLF within 1 year occurred in 26 patients (10.7%) assigned to PS, and in 12 patients (5.0%) assigned to DK crush (hazard ratio: 0.42; 95% confidence interval: 0.21 to 0.85; p = 0.02). Compared with PS, DK crush also resulted in lower rates of target vessel myocardial infarction I (2.9% vs. 0.4%; p = 0.03) and definite or probable stent thrombosis (3.3% vs. 0.4%; p = 0.02). Clinically driven target lesion revascularization (7.9% vs. 3.8%; p = 0.06) and angiographic restenosis within the LM complex (14.6% vs. 7.1%; p = 0.10) also tended to be less frequent with DK crush compared with PS. There was no significant difference in cardiac death between the groups. In the present multicenter randomized trial, percutaneous coronary intervention of true distal LM bifurcation lesions using a planned DK crush 2-stent strategy resulted in a lower rate of TLF at 1 year than a PS strategy. (Double
5. Fibonacci order in the period-doubling cascade to chaos
International Nuclear Information System (INIS)
Linage, G.; Montoya, Fernando; Sarmiento, A.; Showalter, K.; Parmananda, P.
2006-01-01
In this contribution, we describe how the Fibonacci sequence appears within the Feigenbaum scaling of the period-doubling cascade to chaos. An important consequence of this discovery is that the ratio of successive Fibonacci numbers converges to the golden mean in every period-doubling sequence and therefore the convergence to φ, the most irrational number, occurs in concert with the onset of deterministic chaos
6. Fibonacci order in the period-doubling cascade to chaos
Energy Technology Data Exchange (ETDEWEB)
Linage, G. [Facultad de Ciencias UAEM, Avenida Universidad 1001, Colonia Chamilpa, C.P. 62210 Cuernavaca, Morelos (Mexico); Montoya, Fernando [Facultad de Ciencias UAEM, Avenida Universidad 1001, Colonia Chamilpa, C.P. 62210 Cuernavaca, Morelos (Mexico); Sarmiento, A. [Instituto de Matematicas, UNAM, Colonia Chamilpa, C.P. 62210 Cuernavaca, Morelos (Mexico); Showalter, K. [Department of Chemistry, West Virginia University, Morgantown, WV 26506-6045 (United States); Parmananda, P. [Facultad de Ciencias UAEM, Avenida Universidad 1001, Colonia Chamilpa, C.P. 62210 Cuernavaca, Morelos (Mexico)]. E-mail: [email protected]
2006-12-11
In this contribution, we describe how the Fibonacci sequence appears within the Feigenbaum scaling of the period-doubling cascade to chaos. An important consequence of this discovery is that the ratio of successive Fibonacci numbers converges to the golden mean in every period-doubling sequence and therefore the convergence to {phi}, the most irrational number, occurs in concert with the onset of deterministic chaos.
7. Bunch lengthening with bifurcation in electron storage rings
Energy Technology Data Exchange (ETDEWEB)
Kim, Eun-San; Hirata, Kohji [National Lab. for High Energy Physics, Tsukuba, Ibaraki (Japan)
1996-08-01
The mapping which shows equilibrium particle distribution in synchrotron phase space for electron storage rings is discussed with respect to some localized constant wake function based on the Gaussian approximation. This mapping shows multi-periodic states as well as double bifurcation in dynamical states of the equilibrium bunch length. When moving around parameter space, the system shows a transition/bifurcation which is not always reversible. These results derived by mapping are confirmed by multiparticle tracking. (author)
8. Reversing Period-Doubling Bifurcations in Models of Population Interactions Using Constant Stocking or Harvesting
Science.gov (United States)
James F. Selgrade; James H. Roberds
1998-01-01
This study considers a general class of two-dimensional, discrete population models where each per capita transition function (fitness) depends on a linear combination of the densities of the interacting populations. The fitness functions are either monotone decreasing functions (pioneer fitnesses) or one-humped functions (climax fitnesses). Conditions are derived...
9. Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates
Directory of Open Access Journals (Sweden)
2008-01-01
Full Text Available We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods market α, and the degree of capital mobility β. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values of β. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.
10. Universality of the topology of period doubling dynamical systems
International Nuclear Information System (INIS)
Beiersdorfer, P.
1983-10-01
The evolution of the topology of the invariant manifolds of the attractors of 3-D autonomous dynamical systems during period doubling is shown to be universal. The overall topology of the nth attractor is shown to depend only on the topology of the first attractor at birth
11. Period doubling in a model of magnetoconvection with Ohmic heating
International Nuclear Information System (INIS)
Osman, M. B. H.
2000-01-01
In this work it has been studied an idealized model of rotating nonlinear magneto convection to investigate the effects of Ohmic heating. In the over stable region it was found that Ohmic heating can lead to a period-doubling sequence
12. Relative Lyapunov Center Bifurcations
DEFF Research Database (Denmark)
Wulff, Claudia; Schilder, Frank
2014-01-01
Relative equilibria (REs) and relative periodic orbits (RPOs) are ubiquitous in symmetric Hamiltonian systems and occur, for example, in celestial mechanics, molecular dynamics, and rigid body motion. REs are equilibria, and RPOs are periodic orbits of the symmetry reduced system. Relative Lyapunov...... center bifurcations are bifurcations of RPOs from REs corresponding to Lyapunov center bifurcations of the symmetry reduced dynamics. In this paper we first prove a relative Lyapunov center theorem by combining recent results on the persistence of RPOs in Hamiltonian systems with a symmetric Lyapunov...... center theorem of Montaldi, Roberts, and Stewart. We then develop numerical methods for the detection of relative Lyapunov center bifurcations along branches of RPOs and for their computation. We apply our methods to Lagrangian REs of the N-body problem....
13. Numerical Exploration of Kaldorian Interregional Macrodynamics: Enhanced Stability and Predominance of Period Doubling under Flexible Exchange Rates
Directory of Open Access Journals (Sweden)
2010-01-01
Full Text Available We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movement β. By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is required together with high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.
14. Quasi-Periodicity and Border-Collision Bifurcations in a DC-DC Converter with Pulsewidth Modulation
DEFF Research Database (Denmark)
Zhusubalaliyev, Zh. T.; Soukhoterin, E.A.; Mosekilde, Erik
2003-01-01
border-collision bifurcations (BCB) on a two-dimensional torus. The arrangement of the resonance domains within the parameter plane is related to the Farey series, and their internal structure is described. It is shown that transitions to chaos mainly occur through finite sequences of BCB. Some other...
15. The genesis of period-adding bursting without bursting-chaos in the Chay model
International Nuclear Information System (INIS)
Yang Zhuoqin; Lu Qishao; Li Li
2006-01-01
According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding 'fold/homoclinic' bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to period-7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence
16. The genesis of period-adding bursting without bursting-chaos in the Chay model
International Nuclear Information System (INIS)
Yang Zhuoqin; Lu Qishao; Li Li
2006-01-01
According to the period-adding firing patterns without chaos observed in neuronal experiments, the genesis of the period-adding 'fold/homoclinic' bursting sequence without bursting-chaos is explored by numerical simulation, fast/slow dynamics and bifurcation analysis of limit cycle in the neuronal Chay model. It is found that each periodic bursting, from period-1 to 7, is separately generated by the corresponding periodic spiking pattern through two period-doubling bifurcations, except for the period-1 bursting occurring via a Hopf bifurcation. Consequently, it can be revealed that this period-adding bursting bifurcation without chaos has a compound bifurcation structure with transitions from spiking to bursting, which is closely related to period-doubling bifurcations of periodic spiking in essence
17. Bifurcation Analysis and Chaos Control in a Discrete Epidemic System
Directory of Open Access Journals (Sweden)
Wei Tan
2015-01-01
Full Text Available The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior R+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the system x→x+δ[rN(1-N/K-βxy/N-(μ+mx], y→y+δ[βxy/N-(μ+dy]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.
18. Bifurcations of propellant burning rate at oscillatory pressure
Energy Technology Data Exchange (ETDEWEB)
Novozhilov, Boris V. [N. N. Semenov Institute of Chemical Physics, Russian Academy of Science, 4 Kosygina St., Moscow 119991 (Russian Federation)
2006-06-15
A new phenomenon, the disparity between pressure and propellant burning rate frequencies, has revealed in numerical studies of propellant burning rate response to oscillatory pressure. As is clear from the linear approximation, under small pressure amplitudes, h, pressure and propellant burning rate oscillations occur with equal period T (T-solution). In the paper, however, it is shown that at a certain critical value of the parameter h the system in hand undergoes a bifurcation so that the T-solution converts to oscillations with period 2T (2T-solution). When the bifurcation parameter h increases, the subsequent behavior of the system becomes complicated. It is obtained a sequence of period doubling to 4T-solution and 8T-solution. Beyond a certain value of the bifurcation parameter h an apparently fully chaotic solution is found. These effects undoubtedly should be taken into account in studies of oscillatory processes in combustion chambers. (Abstract Copyright [2006], Wiley Periodicals, Inc.)
19. Bifurcation and Fractal of the Coupled Logistic Map
Science.gov (United States)
Wang, Xingyuan; Luo, Chao
The nature of the fixed points of the coupled Logistic map is researched, and the boundary equation of the first bifurcation of the coupled Logistic map in the parameter space is given out. Using the quantitative criterion and rule of system chaos, i.e., phase graph, bifurcation graph, power spectra, the computation of the fractal dimension, and the Lyapunov exponent, the paper reveals the general characteristics of the coupled Logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the coupled Logistic map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively; (2) during the process of double-period bifurcation, the system exhibits self-similarity and scale transform invariability in both the parameter space and the phase space. From the research of the attraction basin and Mandelbrot-Julia set of the coupled Logistic map, the following conclusions are indicated: (1) the boundary between periodic and quasiperiodic regions is fractal, and that indicates the impossibility to predict the moving result of the points in the phase plane; (2) the structures of the Mandelbrot-Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.
20. Calculation of coupling factor for double-period accelerating structure
International Nuclear Information System (INIS)
Bian Xiaohao; Chen Huaibi; Zheng Shuxin
2005-01-01
In the design of the linear accelerating structure, the coupling factor between cavities is a crucial parameter. The error of coupling factor accounts for the electric or magnetic field error mainly. To accurately design the coupling iris, the accurate calculation of coupling factor is essential. The numerical simulation is widely used to calculate the coupling factor now. By using MAFIA code, two methods have been applied to calculate the dispersion characteristics of the single-period structure, one method is to simulate the traveling wave mode by the period boundary condition; another method is to simulate the standing wave mode by the electrical boundary condition. In this work, the authors develop the two methods to calculate the coupling factor of double-period accelerating structure. Compared to experiment, the results for both methods are very similar, and in agreement with measurement within 15% deviation. (authors)
1. Bifurcations and Chaos of AN Immersed Cantilever Beam in a Fluid and Carrying AN Intermediate Mass
Science.gov (United States)
AL-QAISIA, A. A.; HAMDAN, M. N.
2002-06-01
The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.
2. Period doubling on a non-neutral magnetized electron beam
International Nuclear Information System (INIS)
Boswell, R.W.
1984-01-01
Low frequency oscillations on a non-neutral magnetized electron beam of very low density are investigated. A perturbation analysis of the slow mode of the rigid rotator equilibrium is developed to illustrate the nature of large amplitude fundamental mode oscillations. The results of this theoretical analysis show two important characteristics: firstly, as the perturbation amplitude is increasedthe waveform ceases to be purely sinusoidal and shows period doubling. Secondly, above a certain threshold, all harmonics of the wave grow and the wave breaks. The results of the former are compared with a simple electron beam experiment and are found to be in good qualitative agreement
3. Period doubling induced by thermal noise amplification in genetic circuits
KAUST Repository
Ruocco, G.
2014-11-18
Rhythms of life are dictated by oscillations, which take place in a wide rage of biological scales. In bacteria, for example, oscillations have been proven to control many fundamental processes, ranging from gene expression to cell divisions. In genetic circuits, oscillations originate from elemental block such as autorepressors and toggle switches, which produce robust and noise-free cycles with well defined frequency. In some circumstances, the oscillation period of biological functions may double, thus generating bistable behaviors whose ultimate origin is at the basis of intense investigations. Motivated by brain studies, we here study an “elemental” genetic circuit, where a simple nonlinear process interacts with a noisy environment. In the proposed system, nonlinearity naturally arises from the mechanism of cooperative stability, which regulates the concentration of a protein produced during a transcription process. In this elemental model, bistability results from the coherent amplification of environmental fluctuations due to a stochastic resonance of nonlinear origin. This suggests that the period doubling observed in many biological functions might result from the intrinsic interplay between nonlinearity and thermal noise.
4. Dynamic Bifurcations
CERN Document Server
1991-01-01
Dynamical Bifurcation Theory is concerned with the phenomena that occur in one parameter families of dynamical systems (usually ordinary differential equations), when the parameter is a slowly varying function of time. During the last decade these phenomena were observed and studied by many mathematicians, both pure and applied, from eastern and western countries, using classical and nonstandard analysis. It is the purpose of this book to give an account of these developments. The first paper, by C. Lobry, is an introduction: the reader will find here an explanation of the problems and some easy examples; this paper also explains the role of each of the other paper within the volume and their relationship to one another. CONTENTS: C. Lobry: Dynamic Bifurcations.- T. Erneux, E.L. Reiss, L.J. Holden, M. Georgiou: Slow Passage through Bifurcation and Limit Points. Asymptotic Theory and Applications.- M. Canalis-Durand: Formal Expansion of van der Pol Equation Canard Solutions are Gevrey.- V. Gautheron, E. Isambe...
5. Bifurcation and complex dynamics of a discrete-time predator-prey system involving group defense
Directory of Open Access Journals (Sweden)
S. M. Sohel Rana
2015-09-01
Full Text Available In this paper, we investigate the dynamics of a discrete-time predator-prey system involving group defense. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of R+2 by using bifurcation theory. Numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamical behaviors, including phase portraits, period-7, 20-orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance of the chaotic dynamics and attracting chaotic set. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors.
6. Regularization of the double period method for experimental data processing
Science.gov (United States)
Belov, A. A.; Kalitkin, N. N.
2017-11-01
In physical and technical applications, an important task is to process experimental curves measured with large errors. Such problems are solved by applying regularization methods, in which success depends on the mathematician's intuition. We propose an approximation based on the double period method developed for smooth nonperiodic functions. Tikhonov's stabilizer with a squared second derivative is used for regularization. As a result, the spurious oscillations are suppressed and the shape of an experimental curve is accurately represented. This approach offers a universal strategy for solving a broad class of problems. The method is illustrated by approximating cross sections of nuclear reactions important for controlled thermonuclear fusion. Tables recommended as reference data are obtained. These results are used to calculate the reaction rates, which are approximated in a way convenient for gasdynamic codes. These approximations are superior to previously known formulas in the covered temperature range and accuracy.
7. Shells, orbit bifurcations, and symmetry restorations in Fermi systems
Energy Technology Data Exchange (ETDEWEB)
Magner, A. G., E-mail: [email protected]; Koliesnik, M. V. [NASU, Institute for Nuclear Research (Ukraine); Arita, K. [Nagoya Institute of Technology, Department of Physics (Japan)
2016-11-15
The periodic-orbit theory based on the improved stationary-phase method within the phase-space path integral approach is presented for the semiclassical description of the nuclear shell structure, concerning themain topics of the fruitful activity ofV.G. Soloviev. We apply this theory to study bifurcations and symmetry breaking phenomena in a radial power-law potential which is close to the realistic Woods–Saxon one up to about the Fermi energy. Using the realistic parametrization of nuclear shapes we explain the origin of the double-humped fission barrier and the asymmetry in the fission isomer shapes by the bifurcations of periodic orbits. The semiclassical origin of the oblate–prolate shape asymmetry and tetrahedral shapes is also suggested within the improved periodic-orbit approach. The enhancement of shell structures at some surface diffuseness and deformation parameters of such shapes are explained by existence of the simple local bifurcations and new non-local bridge-orbit bifurcations in integrable and partially integrable Fermi-systems. We obtained good agreement between the semiclassical and quantum shell-structure components of the level density and energy for several surface diffuseness and deformation parameters of the potentials, including their symmetry breaking and bifurcation values.
8. Center manifolds, normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions
Science.gov (United States)
Holmes, Philip J.
1981-06-01
We study the instabilities known to aeronautical engineers as flutter and divergence. Mathematically, these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation. Making use of the center manifold and normal form theorems, we concentrate on the situation in which flutter and divergence become coupled, and show that there are essentially two ways in which this is likely to occur. In the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator. This system, which may be analyzed by conventional phase-plane techniques, captures all the qualitative features of the full system. We discuss the reduction and show how the nonlinear terms may be simplified and put into normal form. Invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics. Repeating the approach in the second case, we show that the essential model is now three dimensional and that far more complex behavior is possible, including nonperiodic and ‘chaotic’ motions. Throughout, we take a two degree of freedom system as an example, but the general methods are applicable to multi- and even infinite degree of freedom problems.
9. Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem. Pt. 1
International Nuclear Information System (INIS)
Kazantzis, P.G.
1979-01-01
New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the 'vertical-critical' orbits (αsub(ν) = -1, csub(ν) = 0) of the 'basic' plane families i. g 1 g 2 h, a, m and I. Further the numerical procedure employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections. (orig.)
10. Bifurcation and chaotic behavior in the Euler method for a Kaplan-Yorke prototype delay model
International Nuclear Information System (INIS)
Peng Mingshu
2004-01-01
A discrete model with a simple cubic nonlinearity term is treated in the study the rich dynamics of a prototype delayed dynamical system under Euler discretization. The effect of breaking the symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations
11. Codimension-2 bifurcations of the Kaldor model of business cycle
International Nuclear Information System (INIS)
Wu, Xiaoqin P.
2011-01-01
Research highlights: → The conditions are given such that the characteristic equation may have purely imaginary roots and double zero roots. → Purely imaginary roots lead us to study Hopf and Bautin bifurcations and to calculate the first and second Lyapunov coefficients. → Double zero roots lead us to study Bogdanov-Takens (BT) bifurcation. → Bifurcation diagrams for Bautin and BT bifurcations are obtained by using the normal form theory. - Abstract: In this paper, complete analysis is presented to study codimension-2 bifurcations for the nonlinear Kaldor model of business cycle. Sufficient conditions are given for the model to demonstrate Bautin and Bogdanov-Takens (BT) bifurcations. By computing the first and second Lyapunov coefficients and performing nonlinear transformation, the normal forms are derived to obtain the bifurcation diagrams such as Hopf, homoclinic and double limit cycle bifurcations. Some examples are given to confirm the theoretical results.
12. Building CX peanut-shaped disk galaxy profiles. The relative importance of the 3D families of periodic orbits bifurcating at the vertical 2:1 resonance
Science.gov (United States)
Patsis, P. A.; Harsoula, M.
2018-05-01
Context. We present and discuss the orbital content of a rather unusual rotating barred galaxy model, in which the three-dimensional (3D) family, bifurcating from x1 at the 2:1 vertical resonance with the known "frown-smile" side-on morphology, is unstable. Aims: Our goal is to study the differences that occur in the phase space structure at the vertical 2:1 resonance region in this case, with respect to the known, well studied, standard case, in which the families with the frown-smile profiles are stable and support an X-shaped morphology. Methods: The potential used in the study originates in a frozen snapshot of an N-body simulation in which a fast bar has evolved. We follow the evolution of the vertical stability of the central family of periodic orbits as a function of the energy (Jacobi constant) and we investigate the phase space content by means of spaces of section. Results: The two bifurcating families at the vertical 2:1 resonance region of the new model change their stability with respect to that of most studied analytic potentials. The structure in the side-on view that is directly supported by the trapping of quasi-periodic orbits around 3D stable periodic orbits has now an infinity symbol (i.e. ∞-type) profile. However, the available sticky orbits can reinforce other types of side-on morphologies as well. Conclusions: In the new model, the dynamical mechanism of trapping quasi-periodic orbits around the 3D stable periodic orbits that build the peanut, supports the ∞-type profile. The same mechanism in the standard case supports the X shape with the frown-smile orbits. Nevertheless, in both cases (i.e. in the new and in the standard model) a combination of 3D quasi-periodic orbits around the stable x1 family with sticky orbits can support a profile reminiscent of the shape of the orbits of the 3D unstable family existing in each model.
13. Does Kepler unveil the mystery of the Blazhko effect? First detection of period doubling in Kepler Blazhko RR Lyrae stars
DEFF Research Database (Denmark)
Szabó, R.; Kollath, Z.; Molnár, L.
2010-01-01
-doubling bifurcation in our non-linear RR Lyrae models computed by the Florida-Budapest hydrocode. This enabled us to trace the origin of this instability in RR Lyrae stars to a resonance, namely a 9:2 resonance between the fundamental mode and a high-order (ninth) radial overtone showing strange-mode characteristics...
14. Hopf bifurcation and chaos from torus breakdown in voltage-mode controlled DC drive systems
International Nuclear Information System (INIS)
Dai Dong; Ma Xikui; Zhang Bo; Tse, Chi K.
2009-01-01
Period-doubling bifurcation and its route to chaos have been thoroughly investigated in voltage-mode and current-mode controlled DC motor drives under simple proportional control. In this paper, the phenomena of Hopf bifurcation and chaos from torus breakdown in a voltage-mode controlled DC drive system is reported. It has been shown that Hopf bifurcation may occur when the DC drive system adopts a more practical proportional-integral control. The phenomena of period-adding and phase-locking are also observed after the Hopf bifurcation. Furthermore, it is shown that the stable torus can breakdown and chaos emerges afterwards. The work presented in this paper provides more complete information about the dynamical behaviors of DC drive systems.
15. Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays
NARCIS (Netherlands)
Visser, Sid; Meijer, Hil G.E.; van Putten, Michel J.A.M.; van Gils, Stephan A.
2012-01-01
A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential
16. Bifurcation and Nonlinear Oscillations.
Science.gov (United States)
1980-09-28
Structural stability and bifurcation theory. pp. 549-560 in Dinamical Systems (Ed. MI. Peixoto), Academic Press, 1973. [211 J. Sotomayor, Generic one...Dynamical Systems Brown University ELECTP" 71, Providence, R. I. 02912 1EC 2 4 1980j //C -*)’ Septabe-4., 1980 / -A + This research was supported in...problems are discussed. The first one deals with the characterization of the flow for a periodic planar system which is the perturbation of an autonomous
17. Hopf bifurcation in an Internet congestion control model
International Nuclear Information System (INIS)
Li Chunguang; Chen Guanrong; Liao Xiaofeng; Yu Juebang
2004-01-01
We consider an Internet model with a single link accessed by a single source, which responds to congestion signals from the network, and study bifurcation of such a system. By choosing the gain parameter as a bifurcation parameter, we prove that Hopf bifurcation occurs. The stability of bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical example is given to verify the theoretical analysis
18. Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment
International Nuclear Information System (INIS)
Li, Jinhui; Teng, Zhidong; Wang, Guangqing; Zhang, Long; Hu, Cheng
2017-01-01
In this paper, we introduce the saturated treatment and logistic growth rate into an SIR epidemic model with bilinear incidence. The treatment function is assumed to be a continuously differential function which describes the effect of delayed treatment when the medical condition is limited and the number of infected individuals is large enough. Sufficient conditions for the existence and local stability of the disease-free and positive equilibria are established. And the existence of the stable limit cycles also is obtained. Moreover, by using the theory of bifurcations, it is shown that the model exhibits backward bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcations. Finally, the numerical examples are given to illustrate the theoretical results and obtain some additional interesting phenomena, involving double stable periodic solutions and stable limit cycles.
19. Bifurcation and chaos in a dc-driven long annular Josephson junction
DEFF Research Database (Denmark)
Grnbech-Jensen, N.; Lomdahl, Peter S.; Samuelsen, Mogens Rugholm
1991-01-01
Simulations of long annular Josephson junctions in a static magnetic field show that in large regions of bias current the system can exhibit a period-doubling bifurcation route to chaos. This is in contrast to previously studied Josephson-junction systems where chaotic behavior has primarily been...
20. Bifurcations and Crises in a Shape Memory Oscillator
Directory of Open Access Journals (Sweden)
2004-01-01
Full Text Available The remarkable properties of shape memory alloys have been motivating the interest in applications in different areas varying from biomedical to aerospace hardware. The dynamical response of systems composed by shape memory actuators presents nonlinear characteristics and a very rich behavior, showing periodic, quasi-periodic and chaotic responses. This contribution analyses some aspects related to bifurcation phenomenon in a shape memory oscillator where the restitution force is described by a polynomial constitutive model. The term bifurcation is used to describe qualitative changes that occur in the orbit structure of a system, as a consequence of parameter changes, being related to chaos. Numerical simulations show that the response of the shape memory oscillator presents period doubling cascades, direct and reverse, and crises.
1. Walking dynamics of the passive compass-gait model under OGY-based state-feedback control: Analysis of local bifurcations via the hybrid Poincaré map
International Nuclear Information System (INIS)
Gritli, Hassène; Belghith, Safya
2017-01-01
Highlights: • We study the passive walking dynamics of the compass-gait model under OGY-based state-feedback control. • We analyze local bifurcations via a hybrid Poincaré map. • We show exhibition of the super(sub)-critical flip bifurcation, the saddle-node(saddle) bifurcation and a saddle-flip bifurcation. • An analysis via a two-parameter bifurcation diagram is presented. • Some new hidden attractors in the controlled passive walking dynamics are displayed. - Abstract: In our previous work, we have analyzed the passive dynamic walking of the compass-gait biped model under the OGY-based state-feedback control using the impulsive hybrid nonlinear dynamics. Such study was carried out through bifurcation diagrams. It was shown that the controlled bipedal gait exhibits attractive nonlinear phenomena such as the cyclic-fold (saddle-node) bifurcation, the period-doubling (flip) bifurcation and chaos. Moreover, we revealed that, using the controlled continuous-time dynamics, we encountered a problem in finding, identifying and hence following branches of (un)stable solutions in order to characterize local bifurcations. The present paper solves such problem and then provides a further investigation of the controlled bipedal walking dynamics using the developed analytical expression of the controlled hybrid Poincaré map. Thus, we show that analysis via such Poincaré map allows to follow branches of both stable and unstable fixed points in bifurcation diagrams and hence to explore the complete dynamics of the controlled compass-gait biped model. We demonstrate the generation, other than the conventional local bifurcations in bipedal walking, i.e. the flip bifurcation and the saddle-node bifurcation, of a saddle-saddle bifurcation, a subcritical flip bifurcation and a new type of a local bifurcation, the saddle-flip bifurcation. In addition, to further understand the occurrence of the local bifurcations, we present an analysis with a two-parameter bifurcation
2. Maximum entropy state of the quasi-geostrophic bi-disperse point vortex system: bifurcation phenomena under periodic boundary conditions
Energy Technology Data Exchange (ETDEWEB)
Funakoshi, Satoshi; Sato, Tomoyoshi; Miyazaki, Takeshi, E-mail: [email protected], E-mail: [email protected] [Department of Mechanical Engineering and Intelligent Systems, University of Electro-Communications, 1-5-1, Chofugaoka, Chofu, Tokyo 182-8585 (Japan)
2012-06-01
We investigate the statistical mechanics of quasi-geostrophic point vortices of mixed sign (bi-disperse system) numerically and theoretically. Direct numerical simulations under periodic boundary conditions are performed using a fast special-purpose computer for molecular dynamics (GRAPE-DR). Clustering of point vortices of like sign is observed and two-dimensional (2D) equilibrium states are formed. It is shown that they are the solutions of the 2D mean-field equation, i.e. the sinh-Poisson equation. The sinh-Poisson equation is generalized to study the 3D nature of the equilibrium states, and a new mean-field equation with the 3D Laplace operator is derived based on the maximum entropy theory. 3D solutions are obtained at very low energy level. These solution branches, however, cannot be traced up to the higher energy level at which the direct numerical simulations are performed, and transitions to 2D solution branches take place when the energy is increased. (paper)
3. Bifurcations of Tumor-Immune Competition Systems with Delay
Directory of Open Access Journals (Sweden)
Ping Bi
2014-01-01
Full Text Available A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.
4. Observation of bifurcation phenomena in an electron beam plasma system
International Nuclear Information System (INIS)
Hayashi, N.; Tanaka, M.; Shinohara, S.; Kawai, Y.
1995-01-01
When an electron beam is injected into a plasma, unstable waves are excited spontaneously near the electron plasma frequency f pe by the electron beam plasma instability. The experiment on subharmonics in an electron beam plasma system was performed with a glow discharge tube. The bifurcation of unstable waves with the electron plasma frequency f pe and 1/2 f pe was observed using a double-plasma device. Furthermore, the period doubling route to chaos around the ion plasma frequency in an electron beam plasma system was reported. However, the physical mechanism of bifurcation phenomena in an electron beam plasma system has not been clarified so far. We have studied nonlinear behaviors of the electron beam plasma instability. It was found that there are some cases: the fundamental unstable waves and subharmonics of 2 period are excited by the electron beam plasma instability, the fundamental unstable waves and subharmonics of 3 period are excited. In this paper, we measured the energy distribution functions of electrons and the dispersion relation of test waves in order to examine the physical mechanism of bifurcation phenomena in an electron beam plasma system
5. Bifurcation analysis of a delay differential equation model associated with the induction of long-term memory
International Nuclear Information System (INIS)
Hao, Lijie; Yang, Zhuoqin; Lei, Jinzhi
2015-01-01
Highlights: • A delay differentiation equation model for CREB regulation is developed. • Increasing the time delay can generate various bifurcations. • Increasing the time delay can induce chaos by two routes. - Abstract: The ability to form long-term memories is an important function for the nervous system, and the formation process is dynamically regulated through various transcription factors, including CREB proteins. In this paper, we investigate the dynamics of a delay differential equation model for CREB protein activities, which involves two positive and two negative feedbacks in the regulatory network. We discuss the dynamical mechanisms underlying the induction of long-term memory, in which bistability is essential for the formation of long-term memory, while long time delay can destabilize the high level steady state to inhibit the long-term memory formation. The model displays rich dynamical response to stimuli, including monostability, bistability, and oscillations, and can transit between different states by varying the negative feedback strength. Introduction of a time delay to the model can generate various bifurcations such as Hopf bifurcation, fold limit cycle bifurcation, Neimark–Sacker bifurcation of cycles, and period-doubling bifurcation, etc. Increasing the time delay can induce chaos by two routes: quasi-periodic route and period-doubling cascade.
6. Emergence of the bifurcation structure of a Langmuir–Blodgett transfer model
KAUST Repository
Köpf, Michael H
2014-10-07
© 2014 IOP Publishing Ltd & London Mathematical Society. We explore the bifurcation structure of a modified Cahn-Hilliard equation that describes a system that may undergo a first-order phase transition and is kept permanently out of equilibrium by a lateral driving. This forms a simple model, e.g., for the deposition of stripe patterns of different phases of surfactant molecules through Langmuir-Blodgett transfer. Employing continuation techniques the bifurcation structure is numerically investigated using the non-dimensional transfer velocity as the main control parameter. It is found that the snaking structure of steady front states is intertwined with a large number of branches of time-periodic solutions that emerge from Hopf or period-doubling bifurcations and end in global bifurcations (sniper and homoclinic). Overall the bifurcation diagram has a harp-like appearance. This is complemented by a two-parameter study in non-dimensional transfer velocity and domain size (as a measure of the distance to the phase transition threshold) that elucidates through which local and global codimension 2 bifurcations the entire harp-like structure emerges.
7. Heteroclinic Bifurcation Behaviors of a Duffing Oscillator with Delayed Feedback
Directory of Open Access Journals (Sweden)
Shao-Fang Wen
2018-01-01
Full Text Available The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.
8. Bifurcation and chaos in high-frequency peak current mode Buck converter
Science.gov (United States)
Chang-Yuan, Chang; Xin, Zhao; Fan, Yang; Cheng-En, Wu
2016-07-01
Bifurcation and chaos in high-frequency peak current mode Buck converter working in continuous conduction mode (CCM) are studied in this paper. First of all, the two-dimensional discrete mapping model is established. Next, reference current at the period-doubling point and the border of inductor current are derived. Then, the bifurcation diagrams are drawn with the aid of MATLAB. Meanwhile, circuit simulations are executed with PSIM, and time domain waveforms as well as phase portraits in i L-v C plane are plotted with MATLAB on the basis of simulation data. After that, we construct the Jacobian matrix and analyze the stability of the system based on the roots of characteristic equations. Finally, the validity of theoretical analysis has been verified by circuit testing. The simulation and experimental results show that, with the increase of reference current I ref, the corresponding switching frequency f is approaching to low-frequency stage continuously when the period-doubling bifurcation happens, leading to the converter tending to be unstable. With the increase of f, the corresponding I ref decreases when the period-doubling bifurcation occurs, indicating the stable working range of the system becomes smaller. Project supported by the National Natural Science Foundation of China (Grant No. 61376029), the Fundamental Research Funds for the Central Universities, China, and the College Graduate Research and Innovation Program of Jiangsu Province, China (Grant No. SJLX15_0092).
9. Period doubling for trapezoid function iteration: metric theory
International Nuclear Information System (INIS)
Beyer, W.A.; Stein, P.R.
1982-01-01
Iterations of a one-parameter family F(lambda,x) = lambda f(x) of endomorphisms of [0,2] having the form of a trapezoid f(x) = x/e for x belongs to [0,e], f(x) = 1 for x belongs to (e,2 - e) and f(x) = (2 - x)/e for x belongs to [2 - e,2], are investigated. Here lambda belongs to [1,2] and e belongs to (0,1). Let lambda/sub n/ be the smallest value of lambda > 1 for which x = 1 is a periodic point of period 2/sup n/. It is proved that for e < 0.99, lambda/sub n/ - lambda/sub n-1/ approx. = k(lambda/sub infinity//e)/sup -2n/, where k is some constant and lambda/sub infinity/ = lim/sub n→infinity/lambda/sub n/. The same conclusion probably holds for any e < 1. This behavior is substantially different from that found by Feigenbaum and others for the case where f(x) assumes its maximum value for a unique x. Numerical investigations are reported for functions related to the trapezoid function
10. Bifurcation and Chaos in a Pulse Width modulation controlled Buck Converter
DEFF Research Database (Denmark)
Kocewiak, Lukasz; Bak, Claus Leth; Munk-Nielsen, Stig
2007-01-01
by a system of piecewise-smooth nonautonomous differential equations. The research are focused on chaotic oscillations analysis and analytical search for bifurcations dependent on parameter. The most frequent route to chaos by the period doubling is observed in the second order DC-DC buck converter. Other...... bifurcations as a complex behaviour in power electronic system evidence are also described. In order to verify theoretical study the experimental DC-DC buck converter was build. The results obtained from three sources were presented and compared. A very good agreement between theory and experiment was observed....
11. Bifurcations in a discrete time model composed of Beverton-Holt function and Ricker function.
Science.gov (United States)
Shang, Jin; Li, Bingtuan; Barnard, Michael R
2015-05-01
We provide rigorous analysis for a discrete-time model composed of the Ricker function and Beverton-Holt function. This model was proposed by Lewis and Li [Bull. Math. Biol. 74 (2012) 2383-2402] in the study of a population in which reproduction occurs at a discrete instant of time whereas death and competition take place continuously during the season. We show analytically that there exists a period-doubling bifurcation curve in the model. The bifurcation curve divides the parameter space into the region of stability and the region of instability. We demonstrate through numerical bifurcation diagrams that the regions of periodic cycles are intermixed with the regions of chaos. We also study the global stability of the model. Copyright © 2015 Elsevier Inc. All rights reserved.
12. Homoclinic bifurcation in Chua's circuit
spiking and bursting behaviors of neurons. Recent experiments ... a limit cycle increases in a wiggle with alternate sequences of stable and unstable orbits via ... further changes in parameter, the system shows period-adding bifurcation when .... [21–23] transition from limit cycle to single scroll chaos via PD and then to alter-.
13. A double expansion method for the frequency response of finite-length beams with periodic parameters
Science.gov (United States)
Ying, Z. G.; Ni, Y. Q.
2017-03-01
A double expansion method for the frequency response of finite-length beams with periodic distribution parameters is proposed. The vibration response of the beam with spatial periodic parameters under harmonic excitations is studied. The frequency response of the periodic beam is the function of parametric period and then can be expressed by the series with the product of periodic and non-periodic functions. The procedure of the double expansion method includes the following two main steps: first, the frequency response function and periodic parameters are expanded by using identical periodic functions based on the extension of the Floquet-Bloch theorem, and the period-parametric differential equation for the frequency response is converted into a series of linear differential equations with constant coefficients; second, the solutions to the linear differential equations are expanded by using modal functions which satisfy the boundary conditions, and the linear differential equations are converted into algebraic equations according to the Galerkin method. The expansion coefficients are obtained by solving the algebraic equations and then the frequency response function is finally determined. The proposed double expansion method can uncouple the effects of the periodic expansion and modal expansion so that the expansion terms are determined respectively. The modal number considered in the second expansion can be reduced remarkably in comparison with the direct expansion method. The proposed double expansion method can be extended and applied to the other structures with periodic distribution parameters for dynamics analysis. Numerical results on the frequency response of the finite-length periodic beam with various parametric wave numbers and wave amplitude ratios are given to illustrate the effective application of the proposed method and the new frequency response characteristics, including the parameter-excited modal resonance, doubling-peak frequency response
14. Reverse bifurcation and fractal of the compound logistic map
Science.gov (United States)
Wang, Xingyuan; Liang, Qingyong
2008-07-01
The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot-Julia set of compound logistic map. We generalize the Welstead and Cromer's periodic scanning technology and using this technology construct a series of Mandelbrot-Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot-Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.
15. Bifurcations of a class of singular biological economic models
International Nuclear Information System (INIS)
Zhang Xue; Zhang Qingling; Zhang Yue
2009-01-01
This paper studies systematically a prey-predator singular biological economic model with time delay. It shows that this model exhibits two bifurcation phenomena when the economic profit is zero. One is transcritical bifurcation which changes the stability of the system, and the other is singular induced bifurcation which indicates that zero economic profit brings impulse, i.e., rapid expansion of the population in biological explanation. On the other hand, if the economic profit is positive, at a critical value of bifurcation parameter, the system undergoes a Hopf bifurcation, i.e., the increase of delay destabilizes the system and bifurcates into small amplitude periodic solution. Finally, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here. In addition, we study numerically that the system undergoes a saddle-node bifurcation when the bifurcation parameter goes through critical value of positive economic profit.
16. Walking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos
Science.gov (United States)
Gritli, Hassène; Belghith, Safya
2017-06-01
An analysis of the passive dynamic walking of a compass-gait biped model under the OGY-based control approach using the impulsive hybrid nonlinear dynamics is presented in this paper. We describe our strategy for the development of a simplified analytical expression of a controlled hybrid Poincaré map and then for the design of a state-feedback control. Our control methodology is based mainly on the linearization of the impulsive hybrid nonlinear dynamics around a desired nominal one-periodic hybrid limit cycle. Our analysis of the controlled walking dynamics is achieved by means of bifurcation diagrams. Some interesting nonlinear phenomena are displayed, such as the period-doubling bifurcation, the cyclic-fold bifurcation, the period remerging, the period bubbling and chaos. A comparison between the raised phenomena in the impulsive hybrid nonlinear dynamics and the hybrid Poincaré map under control was also presented.
17. Geometric pre-patterning based tuning of the period doubling onset strain during thin film wrinkling
Energy Technology Data Exchange (ETDEWEB)
Saha, Sourabh K. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2017-02-16
Wrinkling of supported thin films is an easy-to-implement and low-cost fabrication technique for generation of stretch-tunable periodic micro and nano-scale structures. However, the tunability of such structures is often limited by the emergence of an undesirable period doubled mode at high strains. Predictively tuning the onset strain for period doubling via existing techniques requires one to have extensive knowledge about the nonlinear pattern formation behavior. Herein, a geometric pre-patterning based technique is introduced to delay the onset of period doubling that can be implemented to predictively tune the onset strain even with limited system knowledge. The technique comprises pre-patterning the film/base bilayer with a sinusoidal pattern that has the same period as the natural wrinkle period of the system. The effectiveness of this technique has been verified via physical and computational experiments on the polydimethylsiloxane/glass bilayer system. It is observed that the period doubling onset strain can be increased from the typical value of 20% for flat films to greater than 30% with a modest pre-pattern aspect ratio (2∙amplitude/period) of 0.15. In addition, finite element simulations reveal that (i) the onset strain can be increased up to a limit by increasing the amplitude of the pre-patterns and (ii) the delaying effect can be captured entirely by the pre-pattern geometry. As a result, one can implement this technique even with limited system knowledge, such as material properties or film thickness, by simply replicating pre-existing wrinkled patterns to generate prepatterned bilayers. Thus, geometric pre-patterning is a practical scheme to suppress period doubling that can increase the operating range of stretch-tunable wrinkle-based devices by at least 50%.
18. Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays
OpenAIRE
Luo, Hongwei; Zhang, Jiangang; Du, Wenju; Lu, Jiarong; An, Xinlei
2017-01-01
A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the dire...
19. Bifurcation Observation of Combining Spiral Gear Transmission Based on Parameter Domain Structure Analysis
Directory of Open Access Journals (Sweden)
He Lin
2016-01-01
Full Text Available This study considers the bifurcation evolutions for a combining spiral gear transmission through parameter domain structure analysis. The system nonlinear vibration equations are created with piecewise backlash and general errors. Gill’s numerical integration algorithm is implemented in calculating the vibration equation sets. Based on cell-mapping method (CMM, two-dimensional dynamic domain planes have been developed and primarily focused on the parameters of backlash, transmission error, mesh frequency and damping ratio, and so forth. Solution demonstrates that Period-doubling bifurcation happens as the mesh frequency increases; moreover nonlinear discontinuous jump breaks the periodic orbit and also turns the periodic state into chaos suddenly. In transmission error planes, three cell groups which are Period-1, Period-4, and Chaos have been observed, and the boundary cells are the sensitive areas to dynamic response. Considering the parameter planes which consist of damping ratio associated with backlash, transmission error, mesh stiffness, and external load, the solution domain structure reveals that the system step into chaos undergoes Period-doubling cascade with Period-2m (m: integer periodic regions. Direct simulations to obtain the bifurcation diagram and largest Lyapunov exponent (LE match satisfactorily with the parameter domain solutions.
20. Renormalization of period doubling in symmetric four-dimensional volume-preserving maps
International Nuclear Information System (INIS)
Mao, J.; Greene, J.M.
1987-01-01
We have determined three maps (truncated at quadratic terms) that are fixed under the renormalization operator of pitchfork period doubling in symmetric four-dimensional volume-preserving maps. Each of these contains the previously known two-dimensional area-preserving map that is fixed under the period-doubling operator. One of these three fixed maps consists of two uncoupled two-dimensional (nonlinear) area-preserving fixed maps. The other two contain also the two-dimensional area-preserving fixed map coupled (in general) with a linear two-dimensional map. The renormalization calculation recovers all numerical results for the pitchfork period doubling in the symmetric four-dimensional volume-preserving maps, reported by Mao and Helleman [Phys. Rev. A 35, 1847 (1987)]. For a large class of nonsymmetric four-dimensional volume-preserving maps, we found that the fixed maps are the same as those for the symmetric maps
1. Bifurcations of heterodimensional cycles with two saddle points
Energy Technology Data Exchange (ETDEWEB)
Geng Fengjie [School of Information Technology, China University of Geosciences (Beijing), Beijing 100083 (China)], E-mail: [email protected]; Zhu Deming [Department of Mathematics, East China Normal University, Shanghai 200062 (China)], E-mail: [email protected]; Xu Yancong [Department of Mathematics, East China Normal University, Shanghai 200062 (China)], E-mail: [email protected]
2009-03-15
The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed.
2. Bifurcations of heterodimensional cycles with two saddle points
International Nuclear Information System (INIS)
Geng Fengjie; Zhu Deming; Xu Yancong
2009-01-01
The bifurcations of 2-point heterodimensional cycles are investigated in this paper. Under some generic conditions, we establish the existence of one homoclinic loop, one periodic orbit, two periodic orbits, one 2-fold periodic orbit, and the coexistence of one periodic orbit and heteroclinic loop. Some bifurcation patterns different to the case of non-heterodimensional heteroclinic cycles are revealed.
3. Incidence of double ovulation during the early postpartum period in lactating dairy cows.
Science.gov (United States)
Kusaka, Hiromi; Miura, Hiroshi; Kikuchi, Motohiro; Sakaguchi, Minoru
2017-03-15
In lactating cattle, the incidence of twin calving has many negative impacts on production and reproduction in dairy farming. In almost all cases, natural twinning in dairy cattle is the result of double ovulation. It has been suggested that the milk production level of cows influences the number of ovulatory follicles. The objective of the present study was to investigate the incidence of double ovulations during the early postpartum period in relation to the productive and reproductive performance of dairy cows. The ovaries of 43 Holstein cows (26 primiparous and 17 multiparous) were ultrasonographically scanned throughout the three postpartum ovulation sequences. The incidence of double ovulation in the unilateral ovaries was 66.7%, with a higher incidence in the right ovary than in the left, whereas that in bilateral ovaries was 33.3%. When double ovulations were counted dividing into each side ovary in which ovulations occurred, the total frequency of ovulations deviated from a 1:1 ratio (60.3% in the right side and 39.7% in the left side, P cows, double ovulation occurred more frequently than in primiparous cows (58.8% vs. 11.5% per cow and 30.0% vs. 3.8% per ovulation, respectively P cows, the double ovulators exhibited higher peak milk yield (P cows. Two multiparous cows that experienced double ovulation during the early postpartum period subsequently conceived twin fetuses. It can be speculated that the incidence of double ovulations during the early postpartum period partly contributes to the increased incidence of undesirable twin births in multiparous dairy cows. Copyright © 2016. Published by Elsevier Inc.
4. Unfolding the Riddling Bifurcation
DEFF Research Database (Denmark)
Maistrenko, Yu.; Popovych, O.; Mosekilde, Erik
1999-01-01
We present analytical conditions for the riddling bifurcation in a system of two symmetrically coupled, identical, smooth one-dimensional maps to be soft or hard and describe a generic scenario for the transformations of the basin of attraction following a soft riddling bifurcation.......We present analytical conditions for the riddling bifurcation in a system of two symmetrically coupled, identical, smooth one-dimensional maps to be soft or hard and describe a generic scenario for the transformations of the basin of attraction following a soft riddling bifurcation....
5. Viral infection model with periodic lytic immune response
International Nuclear Information System (INIS)
Wang Kaifa; Wang Wendi; Liu Xianning
2006-01-01
Dynamical behavior and bifurcation structure of a viral infection model are studied under the assumption that the lytic immune response is periodic in time. The infection-free equilibrium is globally asymptotically stable when the basic reproductive ratio of virus is less than or equal to one. There is a non-constant periodic solution if the basic reproductive ratio of the virus is greater than one. It is found that period doubling bifurcations occur as the amplitude of lytic component is increased. For intermediate birth rates, the period triplication occurs and then period doubling cascades proceed gradually toward chaotic cycles. For large birth rate, the period doubling cascade proceeds gradually toward chaotic cycles without the period triplication, and the inverse period doubling can be observed. These results can be used to explain the oscillation behaviors of virus population, which was observed in chronic HBV or HCV carriers
6. Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers
International Nuclear Information System (INIS)
Rahman, Aminur; Blackmore, Denis
2016-01-01
Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark–Sacker (N–S) bifurcations, and even chaos. For example, in [1], based on simulations Gilet conjectured the existence of a supercritical N-S bifurcation as the damping factor in his one- dimensional path model. We prove Gilet’s conjecture and more; in fact, both supercritical and subcritical (N-S) bifurcations are produced by separately varying the damping factor and wave-particle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation.
7. Invariants, Attractors and Bifurcation in Two Dimensional Maps with Polynomial Interaction
Science.gov (United States)
Hacinliyan, Avadis Simon; Aybar, Orhan Ozgur; Aybar, Ilknur Kusbeyzi
This work will present an extended discrete-time analysis on maps and their generalizations including iteration in order to better understand the resulting enrichment of the bifurcation properties. The standard concepts of stability analysis and bifurcation theory for maps will be used. Both iterated maps and flows are used as models for chaotic behavior. It is well known that when flows are converted to maps by discretization, the equilibrium points remain the same but a richer bifurcation scheme is observed. For example, the logistic map has a very simple behavior as a differential equation but as a map fold and period doubling bifurcations are observed. A way to gain information about the global structure of the state space of a dynamical system is investigating invariant manifolds of saddle equilibrium points. Studying the intersections of the stable and unstable manifolds are essential for understanding the structure of a dynamical system. It has been known that the Lotka-Volterra map and systems that can be reduced to it or its generalizations in special cases involving local and polynomial interactions admit invariant manifolds. Bifurcation analysis of this map and its higher iterates can be done to understand the global structure of the system and the artifacts of the discretization by comparing with the corresponding results from the differential equation on which they are based.
8. Bifurcations of Fibonacci generating functions
Energy Technology Data Exchange (ETDEWEB)
Ozer, Mehmet [Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul (Turkey) and Semiconductor Physics Institute, LT-01108 and Vilnius Gediminas Technical University, Sauletekio 11, LT-10223 (Lithuania)]. E-mail: [email protected]; Cenys, Antanas [Semiconductor Physics Institute, LT-01108 and Vilnius Gediminas Technical University, Sauletekio 11, LT-10223 (Lithuania); Polatoglu, Yasar [Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul (Turkey); Hacibekiroglu, Guersel [Istanbul Kultur University, E5 Karayolu Uzeri Sirinevler, 34191 Istanbul (Turkey); Akat, Ercument [Yeditepe University, 26 Agustos Campus Kayisdagi Street, Kayisdagi 81120, Istanbul (Turkey); Valaristos, A. [Aristotle University of Thessaloniki, GR-54124, Thessaloniki (Greece); Anagnostopoulos, A.N. [Aristotle University of Thessaloniki, GR-54124, Thessaloniki (Greece)
2007-08-15
In this work the dynamic behaviour of the one-dimensional family of maps F{sub p,q}(x) = 1/(1 - px - qx {sup 2}) is examined, for specific values of the control parameters p and q. Lyapunov exponents and bifurcation diagrams are numerically calculated. Consequently, a transition from periodic to chaotic regions is observed at values of p and q, where the related maps correspond to Fibonacci generating functions associated with the golden-, the silver- and the bronze mean.
9. Bifurcations of Fibonacci generating functions
International Nuclear Information System (INIS)
Ozer, Mehmet; Cenys, Antanas; Polatoglu, Yasar; Hacibekiroglu, Guersel; Akat, Ercument; Valaristos, A.; Anagnostopoulos, A.N.
2007-01-01
In this work the dynamic behaviour of the one-dimensional family of maps F p,q (x) = 1/(1 - px - qx 2 ) is examined, for specific values of the control parameters p and q. Lyapunov exponents and bifurcation diagrams are numerically calculated. Consequently, a transition from periodic to chaotic regions is observed at values of p and q, where the related maps correspond to Fibonacci generating functions associated with the golden-, the silver- and the bronze mean
10. Nonlinear stability control and λ-bifurcation
International Nuclear Information System (INIS)
Erneux, T.; Reiss, E.L.; Magnan, J.F.; Jayakumar, P.K.
1987-01-01
Passive techniques for nonlinear stability control are presented for a model of fluidelastic instability. They employ the phenomena of λ-bifurcation and a generalization of it. λ-bifurcation occurs when a branch of flutter solutions bifurcates supercritically from a basic solution and terminates with an infinite period orbit at a branch of divergence solutions which bifurcates subcritically from the basic solution. The shape of the bifurcation diagram then resembles the greek letter λ. When the system parameters are in the range where flutter occurs by λ-bifurcation, then as the flow velocity increase the flutter amplitude also increases, but the frequencies of the oscillations decrease to zero. This diminishes the damaging effects of structural fatigue by flutter, and permits the flow speed to exceed the critical flutter speed. If generalized λ-bifurcation occurs, then there is a jump transition from the flutter states to a divergence state with a substantially smaller amplitude, when the flow speed is sufficiently larger than the critical flutter speed
11. Bifurcation and complex dynamics of a discrete-time predator-prey system
Directory of Open Access Journals (Sweden)
S. M. Sohel Rana
2015-06-01
Full Text Available In this paper, we investigate the dynamics of a discrete-time predator-prey system of Holling-I type in the closed first quadrant R+2. The existence and local stability of positive fixed point of the discrete dynamical system is analyzed algebraically. It is shown that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation in the interior of R+2 by using bifurcation theory. It has been found that the dynamical behavior of the model is very sensitive to the parameter values and the initial conditions. Numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamic behaviors, including phase portraits, period-9, 10, 20-orbits, attracting invariant circle, cascade of period-doubling bifurcation from period-20 leading to chaos, quasi-periodic orbits, and sudden disappearance of the chaotic dynamics and attracting chaotic set. In particular, we observe that when the prey is in chaotic dynamic, the predator can tend to extinction or to a stable equilibrium. The Lyapunov exponents are numerically computed to characterize the complexity of the dynamical behaviors. The analysis and results in this paper are interesting in mathematics and biology.
12. Hopf bifurcation for tumor-immune competition systems with delay
Directory of Open Access Journals (Sweden)
Ping Bi
2014-01-01
Full Text Available In this article, a immune response system with delay is considered, which consists of two-dimensional nonlinear differential equations. The main purpose of this paper is to explore the Hopf bifurcation of a immune response system with delay. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solution are also given. Especially, the conditions of the global existence of periodic solutions bifurcating from Hopf bifurcations are given. Numerical simulations are carried out to illustrate the the theoretical analysis and the obtained results.
13. Bifurcations and chaos of a vibration isolation system with magneto-rheological damper
Energy Technology Data Exchange (ETDEWEB)
Zhang, Hailong [Magneto-electronics Lab, School of Physics and Technology, Nanjing Normal University, Nanjing 210046 (China); Vibration Control Lab, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042 (China); Zhang, Ning [Magneto-electronics Lab, School of Physics and Technology, Nanjing Normal University, Nanjing 210046 (China); Min, Fuhong; Yan, Wei; Wang, Enrong, E-mail: [email protected] [Vibration Control Lab, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042 (China)
2016-03-15
Magneto-rheological (MR) damper possesses inherent hysteretic characteristics. We investigate the resulting nonlinear behaviors of a two degree-of-freedom (2-DoF) MR vibration isolation system under harmonic external excitation. A MR damper is identified by employing the modified Bouc-wen hysteresis model. By numerical simulation, we characterize the nonlinear dynamic evolution of period-doubling, saddle node bifurcating and inverse period-doubling using bifurcation diagrams of variations in frequency with a fixed amplitude of the harmonic excitation. The strength of chaos is determined by the Lyapunov exponent (LE) spectrum. Semi-physical experiment on the 2-DoF MR vibration isolation system is proposed. We trace the time history and phase trajectory under certain values of frequency of the harmonic excitation to verify the nonlinear dynamical evolution of period-doubling bifurcations to chaos. The largest LEs computed with the experimental data are also presented, confirming the chaotic motion in the experiment. We validate the chaotic motion caused by the hysteresis of the MR damper, and show the transitions between distinct regimes of stable motion and chaotic motion of the 2-DoF MR vibration isolation system for variations in frequency of external excitation.
14. Bifurcations and chaos of a vibration isolation system with magneto-rheological damper
Directory of Open Access Journals (Sweden)
Hailong Zhang
2016-03-01
Full Text Available Magneto-rheological (MR damper possesses inherent hysteretic characteristics. We investigate the resulting nonlinear behaviors of a two degree-of-freedom (2-DoF MR vibration isolation system under harmonic external excitation. A MR damper is identified by employing the modified Bouc-wen hysteresis model. By numerical simulation, we characterize the nonlinear dynamic evolution of period-doubling, saddle node bifurcating and inverse period-doubling using bifurcation diagrams of variations in frequency with a fixed amplitude of the harmonic excitation. The strength of chaos is determined by the Lyapunov exponent (LE spectrum. Semi-physical experiment on the 2-DoF MR vibration isolation system is proposed. We trace the time history and phase trajectory under certain values of frequency of the harmonic excitation to verify the nonlinear dynamical evolution of period-doubling bifurcations to chaos. The largest LEs computed with the experimental data are also presented, confirming the chaotic motion in the experiment. We validate the chaotic motion caused by the hysteresis of the MR damper, and show the transitions between distinct regimes of stable motion and chaotic motion of the 2-DoF MR vibration isolation system for variations in frequency of external excitation.
15. Bifurcation and instability problems in vortex wakes
DEFF Research Database (Denmark)
Aref, Hassan; Brøns, Morten; Stremler, Mark A.
2007-01-01
A number of instability and bifurcation problems related to the dynamics of vortex wake flows are addressed using various analytical tools and approaches. We discuss the bifurcations of the streamline pattern behind a bluff body as a vortex wake is produced, a theory of the universal Strouhal......-Reynolds number relation for vortex wakes, the bifurcation diagram for "exotic" wake patterns behind an oscillating cylinder first determined experimentally by Williamson & Roshko, and the bifurcations in topology of the streamlines pattern in point vortex streets. The Hamiltonian dynamics of point vortices...... in a periodic strip is considered. The classical results of von Kármán concerning the structure of the vortex street follow from the two-vortices-in-a-strip problem, while the stability results follow largely from a four-vortices-in-a-strip analysis. The three-vortices-in-a-strip problem is argued...
16. Bifurcation of Jovian magnetotail current sheet
Directory of Open Access Journals (Sweden)
P. L. Israelevich
2006-07-01
Full Text Available Multiple crossings of the magnetotail current sheet by a single spacecraft give the possibility to distinguish between two types of electric current density distribution: single-peaked (Harris type current layer and double-peaked (bifurcated current sheet. Magnetic field measurements in the Jovian magnetic tail by Voyager-2 reveal bifurcation of the tail current sheet. The electric current density possesses a minimum at the point of the Bx-component reversal and two maxima at the distance where the magnetic field strength reaches 50% of its value in the tail lobe.
17. Bifurcation of Jovian magnetotail current sheet
Directory of Open Access Journals (Sweden)
P. L. Israelevich
2006-07-01
Full Text Available Multiple crossings of the magnetotail current sheet by a single spacecraft give the possibility to distinguish between two types of electric current density distribution: single-peaked (Harris type current layer and double-peaked (bifurcated current sheet. Magnetic field measurements in the Jovian magnetic tail by Voyager-2 reveal bifurcation of the tail current sheet. The electric current density possesses a minimum at the point of the Bx-component reversal and two maxima at the distance where the magnetic field strength reaches 50% of its value in the tail lobe.
18. V 1343 aquilae (SS 433) as a double-periodic star
International Nuclear Information System (INIS)
Goranskij, V.P.
1983-01-01
The new computer method of double periodicity search earlier tested on the Blazhko effect in RR Lyrae type variable stars is applied to define more precise periods of brightness variability of the binary V 1343 Aql (SS 433). Computer program was used in the two-parameter search regime. The obtained periods are P 1 = 13sup(d).074+-0sup(d).008 and P 2 = 163sup(d).8+-1sup(d).2. The periodically repeating brithness curve deeps treated as primary minima (the accretion disc eclipsed by the star) vary their shape with the phase of period P 2 . The expected eclipse at 1979 October 16 did nor occur
19. Attractors near grazing–sliding bifurcations
International Nuclear Information System (INIS)
Glendinning, P; Kowalczyk, P; Nordmark, A B
2012-01-01
In this paper we prove, for the first time, that multistability can occur in three-dimensional Fillipov type flows due to grazing–sliding bifurcations. We do this by reducing the study of the dynamics of Filippov type flows around a grazing–sliding bifurcation to the study of appropriately defined one-dimensional maps. In particular, we prove the presence of three qualitatively different types of multiple attractors born in grazing–sliding bifurcations. Namely, a period-two orbit with a sliding segment may coexist with a chaotic attractor, two stable, period-two and period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three orbit with two sliding segments may coexist
20. Stability and bifurcation analysis in a delayed SIR model
International Nuclear Information System (INIS)
Jiang Zhichao; Wei Junjie
2008-01-01
In this paper, a time-delayed SIR model with a nonlinear incidence rate is considered. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out
1. Linear theory period ratios for surface helium enhanced double-mode Cepheids
International Nuclear Information System (INIS)
Cox, A.N.; Hodson, S.W.; King, D.S.
1979-01-01
Linear nonadiabatic theory period ratios for models of double-mode Cepheids with their two periods between 1 and 7 days have been computed, assuming differing amounts and depths of surface helium enhancement. Evolution theory masses and luminosities are found to be consistent with the observed periods. All models give Pi 1 /Pi 0 approx. =0.70 as observed for the 11 known variables, contrary to previous theoretical conclusions. The composition structure that best fits the period ratios has the helium mass fraction in the outer 10 -3 of the stellar mass (T< or =250,000 K) as 0.65, similar to a previous model for the triple-mode pulsator AC And. This enrichment can be established by a Cepheid wind and downward inverted μ gradient instability mixing in the lifetime of these low-mass classical Cepheids
2. Bifurcation and chaos in the simple passive dynamic walking model with upper body.
Science.gov (United States)
Li, Qingdu; Guo, Jianli; Yang, Xiao-Song
2014-09-01
We present some rich new complex gaits in the simple walking model with upper body by Wisse et al. in [Robotica 22, 681 (2004)]. We first show that the stable gait found by Wisse et al. may become chaotic via period-doubling bifurcations. Such period-doubling routes to chaos exist for all parameters, such as foot mass, upper body mass, body length, hip spring stiffness, and slope angle. Then, we report three new gaits with period 3, 4, and 6; for each gait, there is also a period-doubling route to chaos. Finally, we show a practical method for finding a topological horseshoe in 3D Poincaré map, and present a rigorous verification of chaos from these gaits.
3. Bifurcation and chaos in the simple passive dynamic walking model with upper body
Energy Technology Data Exchange (ETDEWEB)
Li, Qingdu; Guo, Jianli [Key Laboratory of Industrial Internet of Things and Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065 (China); Yang, Xiao-Song, E-mail: [email protected] [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2014-09-01
We present some rich new complex gaits in the simple walking model with upper body by Wisse et al. in [Robotica 22, 681 (2004)]. We first show that the stable gait found by Wisse et al. may become chaotic via period-doubling bifurcations. Such period-doubling routes to chaos exist for all parameters, such as foot mass, upper body mass, body length, hip spring stiffness, and slope angle. Then, we report three new gaits with period 3, 4, and 6; for each gait, there is also a period-doubling route to chaos. Finally, we show a practical method for finding a topological horseshoe in 3D Poincaré map, and present a rigorous verification of chaos from these gaits.
4. Bifurcation and chaos in the simple passive dynamic walking model with upper body
International Nuclear Information System (INIS)
Li, Qingdu; Guo, Jianli; Yang, Xiao-Song
2014-01-01
We present some rich new complex gaits in the simple walking model with upper body by Wisse et al. in [Robotica 22, 681 (2004)]. We first show that the stable gait found by Wisse et al. may become chaotic via period-doubling bifurcations. Such period-doubling routes to chaos exist for all parameters, such as foot mass, upper body mass, body length, hip spring stiffness, and slope angle. Then, we report three new gaits with period 3, 4, and 6; for each gait, there is also a period-doubling route to chaos. Finally, we show a practical method for finding a topological horseshoe in 3D Poincaré map, and present a rigorous verification of chaos from these gaits
5. Study of periodically excited bubbly jets by PIV and double optical sensors
Energy Technology Data Exchange (ETDEWEB)
Milenkovic, Rade [Laboratorium fuer Thermalhydraulics PSI, Paul Scherrer Institut, OVGA 415, CH-5232 Villigen PSI (Switzerland)]. E-mail: [email protected]; Sigg, Beat [Laboratorium fuer Kerntechnik, ETHZ, ETH Zentrum CLT, CH-8092 Zurich (Switzerland); Yadigaroglu, George [Laboratorium fuer Kerntechnik, ETHZ, ETH Zentrum CLT, CH-8092 Zurich (Switzerland)
2005-12-15
Interactions between large coherent structures and bubbles in two-phase flow can be systematically observed in a periodically excited bubbly jet. Controlled excitation at fixed frequency causes large eddy structures to develop at regular intervals. Thus, interactions between large vortices and bubbles can be studied with PIV and double optical sensors (DOS) using phase-averaging techniques. A number of results on the time and space dependence of velocities and void fractions are presented revealing physical interactions between the liquid flow field and bubble movement as well as feedbacks from bubble agglomeration on the development of flow structures. A clear indication of bubble trapping inside the vortex ring is the generation of a bubble ring that travels with the same velocity as the vortex ring. The DOS results indicate clustering of the bubbles in coherent vortex structures, with a periodic variation of void fraction during the excitation period.
6. Study of periodically excited bubbly jets by PIV and double optical sensors
International Nuclear Information System (INIS)
2005-01-01
Interactions between large coherent structures and bubbles in two-phase flow can be systematically observed in a periodically excited bubbly jet. Controlled excitation at fixed frequency causes large eddy structures to develop at regular intervals. Thus, interactions between large vortices and bubbles can be studied with PIV and double optical sensors (DOS) using phase-averaging techniques. A number of results on the time and space dependence of velocities and void fractions are presented revealing physical interactions between the liquid flow field and bubble movement as well as feedbacks from bubble agglomeration on the development of flow structures. A clear indication of bubble trapping inside the vortex ring is the generation of a bubble ring that travels with the same velocity as the vortex ring. The DOS results indicate clustering of the bubbles in coherent vortex structures, with a periodic variation of void fraction during the excitation period
7. Bifurcation theory for finitely smooth planar autonomous differential systems
Science.gov (United States)
Han, Maoan; Sheng, Lijuan; Zhang, Xiang
2018-03-01
In this paper we establish bifurcation theory of limit cycles for planar Ck smooth autonomous differential systems, with k ∈ N. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and C∞ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.
8. Period doubling of azimuthal oscillations on a non-neutral magnetized electron column
International Nuclear Information System (INIS)
Boswell, R.W.
1985-01-01
The low-frequency azimuthal oscillations on a non-neutral magnetized electron column of very low density are investigated. A perturbation analysis of the slow mode of the rigid rotator equilibrium is developed to illustrate the nature of large-amplitude fundamental-mode oscillations. The results of this theoretical analysis show two important characteristics: firstly, as the perturbation amplitude is increased the wave form ceases to be purely sinusoidal and shows period doubling. Secondly, above a certain threshold, all harmonics of the wave grow and the wave breaks. The results of the former are compared with a simple electron beam experiment and are found to be in good qualitative agreement. (author)
9. Resonance spiking by periodic loss in the double-sided liquid cooling disk oscillator
Science.gov (United States)
Nie, Rongzhi; She, Jiangbo; Li, Dongdong; Li, Fuli; Peng, Bo
2017-03-01
A double-sided liquid cooling Nd:YAG disk oscillator working at a pump repetition rate of 20 Hz is demonstrated. The output energy of 376 mJ is realized, corresponding to the optical-optical efficiency of 12.8% and the slope efficiency of 14%. The pump pulse width is 300 µs and the laser pulse width is 260 µs. Instead of being a damped signal, the output of laser comprises undamped spikes. A periodic intra-cavity loss was found by numerical analysis, which has a frequency component near the eigen frequency of the relaxation oscillation. Resonance effect will induce amplified spikes even though the loss fluctuates in a small range. The Shark-Hartmann sensor was used to investigate the wavefront aberration induced by turbulent flow and temperature gradient. According to the wavefront and fluid mechanics analysis, it is considered that the periodic intra-cavity loss can be attributed to turbulent flow and temperature gradient.
10. Bifurcations of transition states: Morse bifurcations
International Nuclear Information System (INIS)
MacKay, R S; Strub, D C
2014-01-01
A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction. Transition states diffeomorphic to S 2m−3 are known to exist for energies just above any index-1 critical point of a Hamiltonian of m degrees of freedom, with dividing surfaces S 2m−2 . The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations. Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show how sequences of Morse bifurcations producing various interesting forms of transition state and dividing surface are present in reacting systems, by considering a hypothetical class of bimolecular reactions in gas phase. (paper)
11. Shell structure and orbit bifurcations in finite fermion systems
Science.gov (United States)
Magner, A. G.; Yatsyshyn, I. S.; Arita, K.; Brack, M.
2011-10-01
We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).
12. REM sleep complicates period adding bifurcations from monophasic to polyphasic sleep behavior in a sleep-wake regulatory network model for human sleep
OpenAIRE
Kalmbach, K.; Booth, V.; Behn, C. G. Diniz
2017-01-01
The structure of human sleep changes across development as it consolidates from the polyphasic sleep of infants to the single nighttime sleep period typical in adults. Across this same developmental period, time scales of the homeostatic sleep drive, the physiological drive to sleep that increases with time spent awake, also change and presumably govern the transition from polyphasic to monophasic sleep behavior. Using a physiologically-based, sleep-wake regulatory network model for human sle...
13. Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays
International Nuclear Information System (INIS)
Karaoglu, Esra; Merdan, Huseyin
2014-01-01
Highlights: • A ratio-dependent predator–prey system involving two discrete maturation time delays is studied. • Hopf bifurcations are analyzed by choosing delay parameters as bifurcation parameters. • When a delay parameter passes through a critical value, Hopf bifurcations occur. • The direction of bifurcation, the period and the stability of periodic solution are also obtained. - Abstract: In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations
14. Pitchfork bifurcation and vibrational resonance in a fractional-order ...
The fractional-order damping mainly determines the pattern of the vibrational resonance. There is a bifurcation point of the fractional order which, in the case of double-well potential, transforms vibrational resonance pattern from a single resonance to a double resonance, while in the case of single-well potential, transforms ...
15. Mode locking and spatiotemporal chaos in periodically driven Gunn diodes
DEFF Research Database (Denmark)
Mosekilde, Erik; Feldberg, Rasmus; Knudsen, Carsten
1990-01-01
oscillation entrains with the external signal. This produces a devil’s staircase of frequency-locked solutions. At higher microwave amplitudes, period doubling and other forms of mode-converting bifurcations can be seen. In this interval the diode also exhibits spatiotemporal chaos. At still higher microwave...
16. Bifurcation structure of an optical ring cavity
DEFF Research Database (Denmark)
Kubstrup, C.; Mosekilde, Erik
1996-01-01
One- and two-dimensional continuation techniques are applied to determine the basic bifurcation structure for an optical ring cavity with a nonlinear absorbing element (the Ikeda Map). By virtue of the periodic structure of the map, families of similar solutions develop in parameter space. Within...
17. Energetics and monsoon bifurcations
Science.gov (United States)
2017-01-01
Monsoons involve increases in dry static energy (DSE), with primary contributions from increased shortwave radiation and condensation of water vapor, compensated by DSE export via horizontal fluxes in monsoonal circulations. We introduce a simple box-model characterizing evolution of the DSE budget to study nonlinear dynamics of steady-state monsoons. Horizontal fluxes of DSE are stabilizing during monsoons, exporting DSE and hence weakening the monsoonal circulation. By contrast latent heat addition (LHA) due to condensation of water vapor destabilizes, by increasing the DSE budget. These two factors, horizontal DSE fluxes and LHA, are most strongly dependent on the contrast in tropospheric mean temperature between land and ocean. For the steady-state DSE in the box-model to be stable, the DSE flux should depend more strongly on the temperature contrast than LHA; stronger circulation then reduces DSE and thereby restores equilibrium. We present conditions for this to occur. The main focus of the paper is describing conditions for bifurcation behavior of simple models. Previous authors presented a minimal model of abrupt monsoon transitions and argued that such behavior can be related to a positive feedback called the moisture advection feedback'. However, by accounting for the effect of vertical lapse rate of temperature on the DSE flux, we show that bifurcations are not a generic property of such models despite these fluxes being nonlinear in the temperature contrast. We explain the origin of this behavior and describe conditions for a bifurcation to occur. This is illustrated for the case of the July-mean monsoon over India. The default model with mean parameter estimates does not contain a bifurcation, but the model admits bifurcation as parameters are varied.
18. Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays
Directory of Open Access Journals (Sweden)
Hongwei Luo
2017-01-01
Full Text Available A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.
19. Border-Collision Bifurcations and Chaotic Oscillations in a Piecewise-Smooth Dynamical System
DEFF Research Database (Denmark)
Zhusubaliyev, Z.T.; Soukhoterin, E.A.; Mosekilde, Erik
2002-01-01
Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject...... of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border......-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling...
20. TQ-bifurcations in discrete dynamical systems: Analysis of qualitative rearrangements of the oscillation mode
Energy Technology Data Exchange (ETDEWEB)
Makarenko, A. V., E-mail: [email protected] [Constructive Cybernetics Research Group (Russian Federation)
2016-10-15
A new class of bifurcations is defined in discrete dynamical systems, and methods for their diagnostics and the analysis of their properties are presented. The TQ-bifurcations considered are implemented in discrete mappings and are related to the qualitative rearrangement of the shape of trajectories in an extended space of states. Within the demonstration of the main capabilities of the toolkit, an analysis is carried out of a logistic mapping in a domain to the right of the period-doubling limit point. Five critical values of the parameter are found for which the geometric structure of the trajectories of the mapping experiences a qualitative rearrangement. In addition, an analysis is carried out of the so-called “trace map,” which arises in the problems of quantum-mechanical description of various properties of discrete crystalline and quasicrystalline lattices.
1. Travelling waves and their bifurcations in the Lorenz-96 model
Science.gov (United States)
van Kekem, Dirk L.; Sterk, Alef E.
2018-03-01
In this paper we study the dynamics of the monoscale Lorenz-96 model using both analytical and numerical means. The bifurcations for positive forcing parameter F are investigated. The main analytical result is the existence of Hopf or Hopf-Hopf bifurcations in any dimension n ≥ 4. Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub- or supercritical. The first Hopf bifurcation for F > 0 is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave. Furthermore, by unfolding the codimension two Hopf-Hopf bifurcation it is shown to act as an organising centre, explaining dynamics such as quasi-periodic attractors and multistability, which are observed in the original Lorenz-96 model. Finally, the region of parameter values beyond the first Hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and Lyapunov exponents. The observed routes to chaos are various but without clear pattern as n → ∞.
2. Application of Recurrence Analysis to the period doubling cascade of a confined buoyant flow
International Nuclear Information System (INIS)
Angeli, D; Corticelli, M A; Fichera, A; Pagano, A
2017-01-01
Recurrence Analysis (RA) is a promising and flexible tool to identify the behaviour of nonlinear dynamical systems. The potentialities of such a technique are explored in the present work, for the study of transitions to chaos of buoyant flow in enclosures. The case of a hot cylindrical source centred in a square enclosure, is considered here, for which an extensive database of results has been collected in recent years. For a specific value of the system aspect ratio, a sequence of period doublings has been identified, leading to the onset of chaos. RA is applied here to analyse the different flow regimes along the route to chaos. The qualitative visual identification of patterns and the statistics given by the quantitative analysis suggest that this kind of tool is well suited to the study of transitional flows in thermo-fluid dynamics. (paper)
3. Mechanism for boundary crises in quasiperiodically forced period-doubling systems
International Nuclear Information System (INIS)
Kim, Sang-Yoon; Lim, Woochang
2005-01-01
We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing ε, a chaotic attractor disappears suddenly via a 'standard' boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of ε, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant 'ring-shaped' unstable set which has no counterpart in the unforced case
4. Mechanism for boundary crises in quasiperiodically forced period-doubling systems
Energy Technology Data Exchange (ETDEWEB)
Kim, Sang-Yoon [Department of Physics, Kangwon National University, Chunchon, Kangwon-Do 200-701 (Korea, Republic of)]. E-mail: [email protected]; Lim, Woochang [Department of Physics, Kangwon National University, Chunchon, Kangwon-Do 200-701 (Korea, Republic of)]. E-mail: [email protected]
2005-01-10
We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing {epsilon}, a chaotic attractor disappears suddenly via a 'standard' boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of {epsilon}, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant 'ring-shaped' unstable set which has no counterpart in the unforced case.
5. Hopf bifurcation analysis of Chen circuit with direct time delay feedback
International Nuclear Information System (INIS)
Hai-Peng, Ren; Wen-Chao, Li; Ding, Liu
2010-01-01
Direct time delay feedback can make non-chaotic Chen circuit chaotic. The chaotic Chen circuit with direct time delay feedback possesses rich and complex dynamical behaviours. To reach a deep and clear understanding of the dynamics of such circuits described by delay differential equations, Hopf bifurcation in the circuit is analysed using the Hopf bifurcation theory and the central manifold theorem in this paper. Bifurcation points and bifurcation directions are derived in detail, which prove to be consistent with the previous bifurcation diagram. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodical orbit (oscillation) in Chen circuit with direct time delay feedback. Bifurcation boundaries are derived using the Hopf bifurcation analysis, which will be helpful for determining the parameters in the stabilisation of the originally chaotic circuit
6. Dedicated bifurcation stents
Directory of Open Access Journals (Sweden)
Ajith Ananthakrishna Pillai
2012-03-01
Full Text Available Bifurcation percutaneous coronary intervention (PCI is still a difficult call for the interventionist despite advancements in the instrumentation, technical skill and the imaging modalities. With major cardiac events relate to the side-branch (SB compromise, the concept and practice of dedicated bifurcation stents seems exciting. Several designs of such dedicated stents are currently undergoing trials. This novel concept and pristine technology offers new hope notwithstanding the fact that we need to go a long way in widespread acceptance and practice of these gadgets. Some of these designs even though looks enterprising, the mere complex delivering technique and the demanding knowledge of the exact coronary anatomy makes their routine use challenging.
7. Numerical analysis of bifurcations
International Nuclear Information System (INIS)
Guckenheimer, J.
1996-01-01
This paper is a brief survey of numerical methods for computing bifurcations of generic families of dynamical systems. Emphasis is placed upon algorithms that reflect the structure of the underlying mathematical theory while retaining numerical efficiency. Significant improvements in the computational analysis of dynamical systems are to be expected from more reliance of geometric insight coming from dynamical systems theory. copyright 1996 American Institute of Physics
8. Modeling, Dynamics, Bifurcation Behavior and Stability Analysis of a DC-DC Boost Converter in Photovoltaic Systems
Science.gov (United States)
Zhioua, M.; El Aroudi, A.; Belghith, S.; Bosque-Moncusí, J. M.; Giral, R.; Al Hosani, K.; Al-Numay, M.
A study of a DC-DC boost converter fed by a photovoltaic (PV) generator and supplying a constant voltage load is presented. The input port of the converter is controlled using fixed frequency pulse width modulation (PWM) based on the loss-free resistor (LFR) concept whose parameter is selected with the aim to force the PV generator to work at its maximum power point. Under this control strategy, it is shown that the system can exhibit complex nonlinear behaviors for certain ranges of parameter values. First, using the nonlinear models of the converter and the PV source, the dynamics of the system are explored in terms of some of its parameters such as the proportional gain of the controller and the output DC bus voltage. To present a comprehensive approach to the overall system behavior under parameter changes, a series of bifurcation diagrams are computed from the circuit-level switched model and from a simplified model both implemented in PSIM© software showing a remarkable agreement. These diagrams show that the first instability that takes place in the system period-1 orbit when a primary parameter is varied is a smooth period-doubling bifurcation and that the nonlinearity of the PV generator is irrelevant for predicting this phenomenon. Different bifurcation scenarios can take place for the resulting period-2 subharmonic regime depending on a secondary bifurcation parameter. The boundary between the desired period-1 orbit and subharmonic oscillation resulting from period-doubling in the parameter space is obtained by calculating the eigenvalues of the monodromy matrix of the simplified model. The results from this model have been validated with time-domain numerical simulation using the circuit-level switched model and also experimentally from a laboratory prototype. This study can help in selecting the parameter values of the circuit in order to delimit the region of period-1 operation of the converter which is of practical interest in PV systems.
9. Lamb wave band gaps in one-dimensional radial phononic crystal plates with periodic double-sided corrugations
Energy Technology Data Exchange (ETDEWEB)
Li, Yinggang [School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049 (China); School of Transportation, Wuhan University of Technology, Wuhan 430070 (China); Chen, Tianning [School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049 (China); Wang, Xiaopeng, E-mail: [email protected] [School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049 (China); Li, Suobin [School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, 710049 (China)
2015-11-01
In this paper, we present the theoretical investigation of Lamb wave propagation in one-dimensional radial phononic crystal (RPC) plates with periodic double-sided corrugations. The dispersion relations, the power transmission spectra, and the displacement fields of the eigenmodes are studied by using the finite element method based on two-dimensional axial symmetry models in cylindrical coordinates. Numerical results show that the proposed RPC plates with periodic double-sided corrugations can yield several band gaps with a variable bandwidth for Lamb waves. The formation mechanism of band gaps in the double-sided RPC plates is attributed to the coupling between the Lamb modes and the in-phase and out-phases resonant eigenmodes of the double-sided corrugations. We investigate the evolution of band gaps in the double-sided RPC plates with the corrugation heights on both sides arranged from an asymmetrical distribution to a symmetrical distribution gradually. Significantly, with the introduction of symmetric double-sided corrugations, the antisymmetric Lamb mode is suppressed by the in-phase resonant eigenmodes of the double-sided corrugations, resulting in the disappearance of the lowest band gap. Furthermore, the effects of the geometrical parameters on the band gaps are further explored numerically.
10. Evolution of the clock from yeast to man by period-doubling folds in the cellular oscillator.
Science.gov (United States)
Klevecz, R R; Li, C M
2007-01-01
Analysis of genome-wide oscillations in transcription reveals that the cell is an oscillator and an attractor and that the maintenance of a stable phenotype requires that maximums in expression in clusters of transcripts must be poised at antipodal phases around the steady state-this is the dynamic architecture of phenotype. Plots of the path through concentration phase space taken by all of the transcripts of Saccharomyces cerevisiae yield a simple three-dimensional surface. How this surface might change as period lengthens or as a cell differentiates is at the center of current work. We have shown that changes in gene expression in response to mutation or perturbation by drugs occur through a folding or unfolding of the surface described by this circle of transcripts and we suggest that the path from this 40-minute oscillation to the cell cycle and circadian rhythms takes place through a series of period-two or period-three bifurcations. These foldings in the surface of the putative attractor result in an increasingly dense set of nested trajectories in the concentrations of message and protein. Evolutionary advantage might accrue to an organism that could change period by changes in just one or a few genes as day length increased from 4 hours in the prebiotic Earth, through 8 hours during the expansion of photoautotrophs, to the present 24 hours.
11. Analysis of a Stochastic Chemical System Close to a SNIPER Bifurcation of Its Mean-Field Model
KAUST Repository
Erban, Radek; Chapman, S. Jonathan; Kevrekidis, Ioannis G.; Vejchodský , Tomá š
2009-01-01
A framework for the analysis of stochastic models of chemical systems for which the deterministic mean-field description is undergoing a saddle-node infinite period (SNIPER) bifurcation is presented. Such a bifurcation occurs, for example
12. Frequency locking, quasiperiodicity, subharmonic bifurcations and chaos in high frequency modulated stripe geometry DH semiconductor lasers
International Nuclear Information System (INIS)
Zhao Yiguang
1991-01-01
The method of obtaining self-consistent solutions of the field equation and the rate equations of photon density and carrier concentration has been used to study frequecny locking, quasiperiodicity, subharmonic bifurcations and chaos in high frequency modulated stripe geometry DH semiconductor lasers. The results show that the chaotic behavior arises in self-pulsing stripe geometry semiconductor lasers. The route to chaos is not period-double, but quasiperiodicity to chaos. All of the results agree with the experiments. Some obscure points in previous theory about chaos have been cleared up
13. Comparative study of optical properties of the one-dimensional multilayer Period-Doubling and Thue-Morse quasi-periodic photonic crystals
Directory of Open Access Journals (Sweden)
Y. Bouazzi
2012-10-01
Full Text Available The last decades have witnessed the growing interest in the use of photonic crystal as a new material that can be used to control electromagnetic wave. Actually, not only the periodic structures but also the quasi-periodic systems have become significant structures of photonic crystals. This work deals with optical properties of dielectric Thue-Morse multilayer and Period-Doubling multilayer. We use the so-called Transfer Matrix Method (TMM to determine the transmission spectra of the structures. Based on the representation of the transmittance spectra in the visible range a comparative analysis depending on the iteration number, number of layers and incidence angle is presented.
14. Bifurcation of solutions to Hamiltonian boundary value problems
Science.gov (United States)
McLachlan, R. I.; Offen, C.
2018-06-01
A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.
15. Modified jailed balloon technique for bifurcation lesions.
Science.gov (United States)
Saito, Shigeru; Shishido, Koki; Moriyama, Noriaki; Ochiai, Tomoki; Mizuno, Shingo; Yamanaka, Futoshi; Sugitatsu, Kazuya; Tobita, Kazuki; Matsumi, Junya; Tanaka, Yutaka; Murakami, Masato
2017-12-04
We propose a new systematic approach in bifurcation lesions, modified jailed balloon technique (M-JBT), and report the first clinical experience. Side branch occlusion brings with a serious complication and occurs in more than 7.0% of cases during bifurcation stenting. A jailed balloon (JB) is introduced into the side branch (SB), while a stent is placed in the main branch (MB) as crossing SB. The size of the JB is half of the MB stent size. While the proximal end of JB attaching to MB stent, both stent and JB are simultaneously inflated with same pressure. JB is removed and then guidewires are recrossed. Kissing balloon dilatation (KBD) and/or T and protrusion (TAP) stenting are applied as needed. Between February 2015 and February 2016, 233 patients (254 bifurcation lesions including 54 left main trunk disease) underwent percutaneous coronary intervention (PCI) using this technique. Procedure success was achieved in all cases. KBD was performed for 183 lesions and TAP stenting was employed for 31 lesions. Occlusion of SV was not observed in any of the patients. Bench test confirmed less deformity of MB stent in M-JBT compared with conventional-JBT. This is the first report for clinical experiences by using modified jailed balloon technique. This novel M-JBT is safe and effective in the preservation of SB patency during bifurcation stenting. © 2017 Wiley Periodicals, Inc.
16. Nonlinear response of a forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two
International Nuclear Information System (INIS)
Ji, J.C.; Zhang, N.
2009-01-01
Non-resonant bifurcations of codimension two may appear in the controlled van der Pol-Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation.
17. [Spectral analysis of fiber bragg grating modulated by double long period grating and its application in smart structure monitoring].
Science.gov (United States)
Lu, Ji-Yun; Liang, Da-Kai; Zhang, Xiao-Li; Zhu, Zhu
2009-12-01
Spectrum of fiber bragg grating (FBG) sensor modulated by double long period grating (LPFG) is proposed in the paper. Double LPFG consists of two LPFGS whose center wavelengths are the same and reflection spectrum of FBG sensor is located in linear range of double LPFG transmission spectrum. Based on spectral analysis of FBG and double LPFG, reflection spectrum of FBG modulated by double LPFG is obtained and studied by use of band-hider filter characteristics for double LPFG. An FBG sensor is attached on the surface of thin steel beam, which is strained by bending, and the center wavelength of FBG sensor will shift. The spectral peak of FBG sensor modulated by double LPFG is changed correspondingly, and the spectral change will lead to variation in exit light intensity from double LPFG. Experiment demonstrates that the relation of filtering light intensity from double LPFG monitored by optical power meter to center wavelength change of FBG sensor is linear and the minimum strain of material (steel beam) detected by the modulation and demodulation system is 1.05 microepsilon. This solution is used in impact monitoring of optical fibre smart structure, and FBG sensor is applied for impulse response signal monitoring induced by low-velocity impact, when impact pendulum is loaded to carbon fiber-reinforced plastics (CFP). The acquired impact response signal and fast Fourier transform of the signal detected by FBG sensor agree with the measurement results of eddy current displacement meter attached to the FBG sensor. From the results, the present method using FBG sensor is found to be effective for monitoring the impact. The research provides a practical reference in dynamic monitoring of optical fiber smart structure field.
18. Symmetry breaking bifurcations of a current sheet
International Nuclear Information System (INIS)
Parker, R.D.; Dewar, R.L.; Johnson, J.L.
1990-01-01
Using a time evolution code with periodic boundary conditions, the viscoresistive hydromagnetic equations describing an initially static, planar current sheet with large Lundquist number have been evolved for times long enough to reach a steady state. A cosh 2 x resistivity model was used. For long periodicity lengths L p , the resistivity gradient drives flows that cause forced reconnection at X point current sheets. Using L p as a bifurcation parameter, two new symmetry breaking bifurcations were found: a transition to an asymmetric island chain with nonzero, positive, or negative phase velocity, and a transition to a static state with alternating large and small islands. These states are reached after a complex transient behavior, which involves a competition between secondary current sheet instability and coalescence
19. Symmetry breaking bifurcations of a current sheet
International Nuclear Information System (INIS)
Parker, R.D.; Dewar, R.L.; Johnson, J.L.
1988-08-01
Using a time evolution code with periodic boundary conditions, the viscoresistive hydromagnetic equations describing an initially static, planar current sheet with large Lundquist number have been evolved for times long enough to reach a steady state. A cosh 2 x resistivity model was used. For long periodicity lengths, L p , the resistivity gradient drives flows which cause forced reconnection at X point current sheets. Using L p as a bifurcation parameter, two new symmetry breaking bifurcations were found - a transition to an asymmetric island chain with nonzero, positive or negative phase velocity, and a transition to a static state with alternating large and small islands. These states are reached after a complex transient behavior which involves a competition between secondary current sheet instability and coalescence. 31 refs., 6 figs
20. Bubble transport in bifurcations
Science.gov (United States)
2017-11-01
Motivated by a developmental gas embolotherapy technique for cancer treatment, we examine the transport of bubbles entrained in liquid. In gas embolotherapy, infarction of tumors is induced by selectively formed vascular gas bubbles that originate from acoustic vaporization of vascular droplets. In the case of non-functionalized droplets with the objective of vessel occlusion, the bubbles are transported by flow through vessel bifurcations, where they may split prior to eventually reach vessels small enough that they become lodged. This splitting behavior affects the distribution of bubbles and the efficacy of flow occlusion and the treatment. In these studies, we investigated bubble transport in bifurcations using computational and theoretical modeling. The model reproduces the variety of experimentally observed splitting behaviors. Splitting homogeneity and maximum shear stress along the vessel walls is predicted over a variety of physical parameters. Maximum shear stresses were found to decrease with increasing Reynolds number. The initial bubble length was found to affect the splitting behavior in the presence of gravitational asymmetry. This work was supported by NIH Grant R01EB006476.
1. A double-blind study of the efficacy of apomorphine and its assessment in "off-periods in Parkinson's disease
NARCIS (Netherlands)
van Laar, T.; Jansen, E.N.H.; Essink, A.W.G.; Neef, C.; Oosterloo, Sebe J.
1993-01-01
Five patients with idiopathic Parkinson's disease with severe response fluctuations were selected for a randomized double-blind placebo-controlled study, concerning the clinical effects of subcutaneous apomorphine and its assessment in off¿-periods. The study was designed as five n = 1 studies, in
2. Ayres' bifurcated solar model
International Nuclear Information System (INIS)
Kalkofen, W.
1985-01-01
The assumptions of Ayres' model of the upper solar atmosphere are examined. It is found that the bistable character of his model is postulated - through the assumptions concerning the opacity sources and the effect of mechanical waves, which are allowed to destroy the CO molecules but not to heat the gas. The neglect of cooling by metal lines is based on their reduced local cooling rate, but it ignores the increased depth over which this cooling occurs. Thus, the bifurcated model of the upper solar atmosphere consists of two models, one cold at the temperature minimum, with a kinetic temperature of 2900 K, and the other hot, with a temperature of 4900 K. 8 references
3. Bifurcations and chaos in convection taking non-Fourier heat-flux
Science.gov (United States)
Layek, G. C.; Pati, N. C.
2017-11-01
In this Letter, we report the influences of thermal time-lag on the onset of convection, its bifurcations and chaos of a horizontal layer of Boussinesq fluid heated underneath taking non-Fourier Cattaneo-Christov hyperbolic model for heat propagation. A five-dimensional nonlinear system is obtained for a low-order Galerkin expansion, and it reduces to Lorenz system for Cattaneo number tending to zero. The linear stability agreed with existing results that depend on Cattaneo number C. It also gives a threshold Cattaneo number, CT, above which only oscillatory solutions can persist. The oscillatory solutions branch terminates at the subcritical steady branch with a heteroclinic loop connecting a pair of saddle points for subcritical steady-state solutions. For subcritical onset of convection two stable solutions coexist, that is, hysteresis phenomenon occurs at this stage. The steady solution undergoes a Hopf bifurcation and is of subcritical type for small value of C, while it becomes supercritical for moderate Cattaneo number. The system goes through period-doubling/noisy period-doubling transition to chaos depending on the control parameters. There after the system exhibits Shil'nikov chaos via homoclinic explosion. The complexity of spiral strange attractor is analyzed using fractal dimension and return map.
4. Bifurcations sights, sounds, and mathematics
CERN Document Server
Matsumoto, Takashi; Kokubu, Hiroshi; Tokunaga, Ryuji
1993-01-01
Bifurcation originally meant "splitting into two parts. " Namely, a system under goes a bifurcation when there is a qualitative change in the behavior of the sys tem. Bifurcation in the context of dynamical systems, where the time evolution of systems are involved, has been the subject of research for many scientists and engineers for the past hundred years simply because bifurcations are interesting. A very good way of understanding bifurcations would be to see them first and study theories second. Another way would be to first comprehend the basic concepts and theories and then see what they look like. In any event, it is best to both observe experiments and understand the theories of bifurcations. This book attempts to provide a general audience with both avenues toward understanding bifurcations. Specifically, (1) A variety of concrete experimental results obtained from electronic circuits are given in Chapter 1. All the circuits are very simple, which is crucial in any experiment. The circuits, howev...
5. Influence of time-periodic potentials on electronic transport in double-well structure
International Nuclear Information System (INIS)
Chun-Lei, Li; Yan, Xu
2010-01-01
Within the framework of the Floquet theorem, we have investigated single-electron photon-assisted tunneling in a double-well system using the transfer matrix technique. The transmission probability displays satellite peaks on both sides of the main resonance peaks and these satellite peaks originate from emission or absorption photons. The single-electron resonance tunneling can be controlled through changing the applied harmonically potential positions, such as driven potential in wells, in barriers, or in whole double-well systems. This advantage should be useful in the optimization of the parameters of a transmission device. (condensed matter: electronic structure, electrical, magnetic, and optical properties)
6. Nonequilibrium ferroelectric-ferroelastic 10 nm nanodomains: wrinkles, period-doubling, and power-law relaxation.
Science.gov (United States)
Scott, James F; Evans, Donald M; Katiyar, Ram S; McQuaid, Raymond G P; Gregg, J Marty
2017-08-02
Since the 1935 work of Landau-Lifshitz and of Kittel in 1946 all ferromagnetic, ferroelectric, and ferroelastic domains have been thought to be straight-sided with domain widths proportional to the square root of the sample thickness. We show in the present work that this is not true. We also discover period doubling domains predicted by Metaxas et al (2008 Phys. Rev. Lett. 99 217208) and modeled by Wang and Zhao (2015 Sci. Rep. 5 8887). We examine non-equilibrium ferroic domain structures in perovskite oxides with respect to folding, wrinkling, and relaxation and suggest that structures are kinetically limited and in the viscous flow regime predicted by Metaxas et al in 2008 but never observed experimentally. Comparisons are made with liquid crystals and hydrodynamic instabilities, including chevrons, and fractional power-law relaxation. As Shin et al (2016 Soft Matter 12 3502) recently emphasized: 'An understanding of how these folds initiate, propagate, and interact with each other is still lacking'. Inside each ferroelastic domain are ferroelectric 90° nano-domains with 10 nm widths and periodicity in agreement with the 10 nm theoretical minima predicted by Feigl et al (2014 Nat. Commun. 5 4677). Evidence is presented for domain-width period doubling, which is common in polymer films but unknown in ferroic domains. A discussion of the folding-to-period doubling phase transition model of Wang and Zhao is included.
7. Analysis of stability and Hopf bifurcation for a viral infectious model with delay
International Nuclear Information System (INIS)
Sun Chengjun; Cao Zhijie; Lin Yiping
2007-01-01
In this paper, a four-dimensional viral infectious model with delay is considered. The stability of the two equilibria and the existence of Hopf bifurcation are investigated. It is found that there are stability switches and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. Using the normal form theory and center manifold argument [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981], the explicit formulaes which determine the stability, the direction and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to illustrate the validity of the main results
8. Stability and bifurcation analysis in a kind of business cycle model with delay
International Nuclear Information System (INIS)
Zhang Chunrui; Wei Junjie
2004-01-01
A kind of business cycle model with delay is considered. Firstly, the linear stability of the model is studied and bifurcation set is drawn in the appropriate parameter plane. It is found that there exist Hopf bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the normal form method and center manifold theorem. Finally, a group conditions to guarantee the global existence of periodic solutions is given, and numerical simulations are performed to illustrate the analytical results found
9. Geometrically Induced Interactions and Bifurcations
Science.gov (United States)
Binder, Bernd
2010-01-01
In order to evaluate the proper boundary conditions in spin dynamics eventually leading to the emergence of natural and artificial solitons providing for strong interactions and potentials with monopole charges, the paper outlines a new concept referring to a curvature-invariant formalism, where superintegrability is given by a special isometric condition. Instead of referring to the spin operators and Casimir/Euler invariants as the generator of rotations, a curvature-invariant description is introduced utilizing a double Gudermann mapping function (generator of sine Gordon solitons and Mercator projection) cross-relating two angular variables, where geometric phases and rotations arise between surfaces of different curvature. Applying this stereographic projection to a superintegrable Hamiltonian can directly map linear oscillators to Kepler/Coulomb potentials and/or monopoles with Pöschl-Teller potentials and vice versa. In this sense a large scale Kepler/Coulomb (gravitational, electro-magnetic) wave dynamics with a hyperbolic metric could be mapped as a geodesic vertex flow to a local oscillator singularity (Dirac monopole) with spherical metrics and vice versa. Attracting fixed points and dynamic constraints are given by special isometries with magic precession angles. The nonlinear angular encoding directly provides for a Shannon mutual information entropy measure of the geodesic phase space flow. The emerging monopole patterns show relations to spiral Fresnel holography and Berry/Aharonov-Bohm geometric phases subject to bifurcation instabilities and singularities from phase ambiguities due to a local (entropy) overload. Neutral solitons and virtual patterns emerging and mediating in the overlap region between charged or twisted holographic patterns are visualized and directly assigned to the Berry geometric phase revealing the role of photons, neutrons, and neutrinos binding repulsive charges in Coulomb, strong and weak interaction.
10. Riddling bifurcation and interstellar journeys
International Nuclear Information System (INIS)
Kapitaniak, Tomasz
2005-01-01
We show that riddling bifurcation which is characteristic for low-dimensional attractors embedded in higher-dimensional phase space can give physical mechanism explaining interstellar journeys described in science-fiction literature
11. Stability and Hopf Bifurcation in a Delayed SEIRS Worm Model in Computer Network
Directory of Open Access Journals (Sweden)
Zizhen Zhang
2013-01-01
Full Text Available A delayed SEIRS epidemic model with vertical transmission in computer network is considered. Sufficient conditions for local stability of the positive equilibrium and existence of local Hopf bifurcation are obtained by analyzing distribution of the roots of the associated characteristic equation. Furthermore, the direction of the local Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by using the normal form theory and center manifold theorem. Finally, a numerical example is presented to verify the theoretical analysis.
12. Dynamic bifurcations on financial markets
International Nuclear Information System (INIS)
Kozłowska, M.; Denys, M.; Wiliński, M.; Link, G.; Gubiec, T.; Werner, T.R.; Kutner, R.; Struzik, Z.R.
2016-01-01
We provide evidence that catastrophic bifurcation breakdowns or transitions, preceded by early warning signs such as flickering phenomena, are present on notoriously unpredictable financial markets. For this we construct robust indicators of catastrophic dynamical slowing down and apply these to identify hallmarks of dynamical catastrophic bifurcation transitions. This is done using daily closing index records for the representative examples of financial markets of small and mid to large capitalisations experiencing a speculative bubble induced by the worldwide financial crisis of 2007-08.
13. Threshold for strong thermal dephasing in periodically poled KTP in external cavity frequency doubling
DEFF Research Database (Denmark)
Lundeman, Jesper Holm; Jensen, Ole Bjarlin; Andersen, Peter E.
2009-01-01
We present a measurement series of the efficiency of periodically poled KTP used for second-harmonic generation in an external phase-locked cavity. Due to the high absorption (0.01 cm^−1) in the PPKTP crystal at the pump wavelength a strong thermal dephasing of the periodically poled grating...
14. Enhanced long-distance transport of periodic electron beams in an advanced double layer cone-channel target
Science.gov (United States)
Ji, Yanling; Duan, Tao; Zhou, Weimin; Li, Boyuan; Wu, Fengjuan; Zhang, Zhimeng; Ye, Bin; Wang, Rong; Wu, Chunrong; Tang, Yongjian
2018-02-01
An enhanced long-distance transport of periodic electron beams in an advanced double layer cone-channel target is investigated using two-dimensional particle-in-cell simulations. The target consists of a cone attached to a double-layer hollow channel with a near-critical-density inner layer. The periodic electron beams are generated by the combination of ponderomotive force and longitudinal laser electric field. Then a stable electron propagation is achieved in the double-layer channel over a much longer distance without evident divergency, compared with a normal cone-channel target. Detailed simulations show that the much better long-distance collimation and guidance of energetic electrons is attributed to the much stronger electromagnetic fields at the inner wall surfaces. Furthermore, a continuous electron acceleration is obtained by the more intense laser electric fields and extended electron acceleration length in the channel. Our investigation shows that by employing this advanced target, both the forward-going electron energy flux in the channel and the energy coupling efficiency from laser to electrons are about threefold increased in comparison with the normal case.
15. Enhanced long-distance transport of periodic electron beams in an advanced double layer cone-channel target
Directory of Open Access Journals (Sweden)
Yanling Ji
2018-02-01
Full Text Available An enhanced long-distance transport of periodic electron beams in an advanced double layer cone-channel target is investigated using two-dimensional particle-in-cell simulations. The target consists of a cone attached to a double-layer hollow channel with a near-critical-density inner layer. The periodic electron beams are generated by the combination of ponderomotive force and longitudinal laser electric field. Then a stable electron propagation is achieved in the double-layer channel over a much longer distance without evident divergency, compared with a normal cone-channel target. Detailed simulations show that the much better long-distance collimation and guidance of energetic electrons is attributed to the much stronger electromagnetic fields at the inner wall surfaces. Furthermore, a continuous electron acceleration is obtained by the more intense laser electric fields and extended electron acceleration length in the channel. Our investigation shows that by employing this advanced target, both the forward-going electron energy flux in the channel and the energy coupling efficiency from laser to electrons are about threefold increased in comparison with the normal case.
16. Global Hopf Bifurcation for a Predator-Prey System with Three Delays
Science.gov (United States)
Jiang, Zhichao; Wang, Lin
2017-06-01
In this paper, a delayed predator-prey model is considered. The existence and stability of the positive equilibrium are investigated by choosing the delay τ = τ1 + τ2 as a bifurcation parameter. We see that Hopf bifurcation can occur as τ crosses some critical values. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are also determined by using the center manifold and normal form theory. Furthermore, based on the global Hopf bifurcation theorem for general function differential equations, which was established by J. Wu using fixed point theorem and degree theory methods, the existence of global Hopf bifurcation is investigated. Finally, numerical simulations to support the analytical conclusions are carried out.
17. Stability, bifurcation and a new chaos in the logistic differential equation with delay
International Nuclear Information System (INIS)
Jiang Minghui; Shen Yi; Jian Jigui; Liao Xiaoxin
2006-01-01
This Letter is concerned with bifurcation and chaos in the logistic delay differential equation with a parameter r. The linear stability of the logistic equation is investigated by analyzing the associated characteristic transcendental equation. Based on the normal form approach and the center manifold theory, the formula for determining the direction of Hopf bifurcation and the stability of bifurcation periodic solution in the first bifurcation values is obtained. By theoretical analysis and numerical simulation, we found a new chaos in the logistic delay differential equation
18. Quantitative angiography methods for bifurcation lesions
DEFF Research Database (Denmark)
Collet, Carlos; Onuma, Yoshinobu; Cavalcante, Rafael
2017-01-01
Bifurcation lesions represent one of the most challenging lesion subsets in interventional cardiology. The European Bifurcation Club (EBC) is an academic consortium whose goal has been to assess and recommend the appropriate strategies to manage bifurcation lesions. The quantitative coronary...... angiography (QCA) methods for the evaluation of bifurcation lesions have been subject to extensive research. Single-vessel QCA has been shown to be inaccurate for the assessment of bifurcation lesion dimensions. For this reason, dedicated bifurcation software has been developed and validated. These software...
19. Magneto-elastic dynamics and bifurcation of rotating annular plate*
International Nuclear Information System (INIS)
Hu Yu-Da; Piao Jiang-Min; Li Wen-Qiang
2017-01-01
In this paper, magneto-elastic dynamic behavior, bifurcation, and chaos of a rotating annular thin plate with various boundary conditions are investigated. Based on the thin plate theory and the Maxwell equations, the magneto-elastic dynamic equations of rotating annular plate are derived by means of Hamilton’s principle. Bessel function as a mode shape function and the Galerkin method are used to achieve the transverse vibration differential equation of the rotating annular plate with different boundary conditions. By numerical analysis, the bifurcation diagrams with magnetic induction, amplitude and frequency of transverse excitation force as the control parameters are respectively plotted under different boundary conditions such as clamped supported sides, simply supported sides, and clamped-one-side combined with simply-anotherside. Poincaré maps, time history charts, power spectrum charts, and phase diagrams are obtained under certain conditions, and the influence of the bifurcation parameters on the bifurcation and chaos of the system is discussed. The results show that the motion of the system is a complicated and repeated process from multi-periodic motion to quasi-period motion to chaotic motion, which is accompanied by intermittent chaos, when the bifurcation parameters change. If the amplitude of transverse excitation force is bigger or magnetic induction intensity is smaller or boundary constraints level is lower, the system can be more prone to chaos. (paper)
20. Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays
International Nuclear Information System (INIS)
Song Yongli; Han Maoan; Peng Yahong
2004-01-01
We consider a Lotka-Volterra competition system with two delays. We first investigate the stability of the positive equilibrium and the existence of Hopf bifurcations, and then using the normal form theory and center manifold argument, derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions
1. Hopf bifurcation in a dynamic IS-LM model with time delay
International Nuclear Information System (INIS)
Neamtu, Mihaela; Opris, Dumitru; Chilarescu, Constantin
2007-01-01
The paper investigates the impact of delayed tax revenues on the fiscal policy out-comes. Choosing the delay as a bifurcation parameter we study the direction and the stability of the bifurcating periodic solutions. We show when the system is stable with respect to the delay. Some numerical examples are given to confirm the theoretical results
2. Effects of gas periodic stimulation on key enzyme activity in gas double-dynamic solid state fermentation (GDD-SSF).
Science.gov (United States)
Chen, Hongzhang; Shao, Meixue; Li, Hongqiang
2014-03-05
The heat and mass transfer have been proved to be the important factors in air pressure pulsation for cellulase production. However, as process of enzyme secretion, the cellulase formation has not been studied in the view of microorganism metabolism and metabolic key enzyme activity under air pressure pulsation condition. Two fermentation methods in ATPase activity, cellulase productivity, weight lose rate and membrane permeability were systematically compared. Results indicated that gas double-dynamic solid state fermentation had no obviously effect on cell membrane permeability. However, the relation between ATPase activity and weight loss rate was linearly dependent with r=0.9784. Meanwhile, the results also implied that gas periodic stimulation had apparently strengthened microbial metabolism through increasing ATPase activity during gas double-dynamic solid state fermentation, resulting in motivating the production of cellulase by Trichoderma reesei YG3. Therefore, the increase of ATPase activity would be another crucial factor to strengthen fermentation process for cellulase production under gas double-dynamic solid state fermentation. Copyright © 2013 Elsevier Inc. All rights reserved.
3. Pierce instability and bifurcating equilibria
International Nuclear Information System (INIS)
Godfrey, B.B.
1981-01-01
The report investigates the connection between equilibrium bifurcations and occurrence of the Pierce instability. Electrons flowing from one ground plane to a second through an ion background possess a countable infinity of static equilibria, of which only one is uniform and force-free. Degeneracy of the uniform and simplest non-uniform equilibria at a certain ground plan separation marks the onset of the Pierce instability, based on a newly derived dispersion relation appropriate to all the equilibria. For large ground plane separations the uniform equilibrium is unstable and the non-uniform equilibrium is stable, the reverse of their stability properties at small separations. Onset of the Pierce instability at the first bifurcation of equilibria persists in more complicated geometries, providing a general criterion for marginal stability. It seems probable that bifurcation analysis can be a useful tool in the overall study of stable beam generation in diodes and transport in finite cavities
4. Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations
International Nuclear Information System (INIS)
Zhang Shuwen; Tan Dejun; Chen Lansun
2006-01-01
The effect of periodic forcing and impulsive perturbations on predator-prey model with Holling type II functional response is investigated. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of prey. The impulsive perturbation is affected by introducing periodic constant impulsive immigration of predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including (1) quasi-periodic oscillating, (2) period doubling cascade, (3) chaos, (4) period halfing cascade, (5) non-unique dynamics
5. Chaos in periodically forced Holling type IV predator-prey system with impulsive perturbations
International Nuclear Information System (INIS)
Zhang Shuwen; Tan Dejun; Chen Lansun
2006-01-01
The effect of periodic forcing and impulsive perturbations on predator-prey model with Holling type IV functional response is investigated. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of the prey. The impulsive perturbations are affected by introducing periodic constant impulsive immigration of predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can easily give rise to complex dynamics, including (1) quasi-periodic oscillating, (2) period doubling cascade, (3) chaos, (4) period halfing cascade
6. Bifurcation Analysis and Spatiotemporal Patterns in Unidirectionally Delay-Coupled Vibratory Gyroscopes
Science.gov (United States)
Li, Li; Xu, Jian
Time delay is inevitable in unidirectionally coupled drive-free vibratory gyroscope system. The effect of time delay on the gyroscope system is studied in this paper. To this end, amplitude death and Hopf bifurcation induced by small time delay are first investigated by analyzing the related characteristic equation. Then, the direction of Hopf bifurcations and stability of Hopf-bifurcating periodic oscillations are determined by calculating the normal form on the center manifold. Next, spatiotemporal patterns of these Hopf-bifurcating periodic oscillations are analyzed by using the symmetric bifurcation theory of delay differential equations. Finally, it is found that numerical simulations agree with the associated analytic results. These phenomena could be induced although time delay is very small. Therefore, it is shown that time delay is an important factor which influences the sensitivity and accuracy of the gyroscope system and cannot be neglected during the design and manufacture.
7. Analysis of stability and Hopf bifurcation for a delayed logistic equation
International Nuclear Information System (INIS)
Sun Chengjun; Han Maoan; Lin Yiping
2007-01-01
The dynamics of a logistic equation with discrete delay are investigated, together with the local and global stability of the equilibria. In particular, the conditions under which a sequence of Hopf bifurcations occur at the positive equilibrium are obtained. Explicit algorithm for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [Hassard B, Kazarino D, Wan Y. Theory and applications of Hopf bifurcation. Cambridge: Cambridge University Press; 1981.]. Global existence of periodic solutions is also established by using a global Hopf bifurcation result of Wu [Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 350:1998;4799-38.
8. Bifurcation theory of ac electric arcing
International Nuclear Information System (INIS)
Christen, Thomas; Peinke, Emanuel
2012-01-01
The performance of alternating current (ac) electric arcing devices is related to arc extinction or its re-ignition at zero crossings of the current (so-called ‘current zero’, CZ). Theoretical investigations thus usually focus on the transient behaviour of arcs near CZ, e.g. by solving the modelling differential equations in the vicinity of CZ. This paper proposes as an alternative approach to investigate global mathematical properties of the underlying periodically driven dynamic system describing the electric circuit containing the arcing device. For instance, the uniqueness of the trivial solution associated with the insulating state indicates the extinction of any arc. The existence of non-trivial attractors (typically a time-periodic state) points to a re-ignition of certain arcs. The performance regions of arcing devices, such as circuit breakers and arc torches, can thus be identified with the regions of absence and existence, respectively, of non-trivial attractors. Most important for applications, the boundary of a performance region in the model parameter space is then associated with the bifurcation of the non-trivial attractors. The concept is illustrated for simple black-box arc models, such as the Mayr and the Cassie model, by calculating for various cases the performance boundaries associated with the bifurcation of ac arcs. (paper)
9. Secondary Channel Bifurcation Geometry: A Multi-dimensional Problem
Science.gov (United States)
Gaeuman, D.; Stewart, R. L.
2017-12-01
The construction of secondary channels (or side channels) is a popular strategy for increasing aquatic habitat complexity in managed rivers. Such channels, however, frequently experience aggradation that prevents surface water from entering the side channels near their bifurcation points during periods of relatively low discharge. This failure to maintain an uninterrupted surface water connection with the main channel can reduce the habitat value of side channels for fish species that prefer lotic conditions. Various factors have been proposed as potential controls on the fate of side channels, including water surface slope differences between the main and secondary channels, the presence of main channel secondary circulation, transverse bed slopes, and bifurcation angle. A quantitative assessment of more than 50 natural and constructed secondary channels in the Trinity River of northern California indicates that bifurcations can assume a variety of configurations that are formed by different processes and whose longevity is governed by different sets of factors. Moreover, factors such as bifurcation angle and water surface slope vary with discharge level and are continuously distributed in space, such that they must be viewed as a multi-dimensional field rather than a single-valued attribute that can be assigned to a particular bifurcation.
10. Global bifurcations in a piecewise-smooth Cournot duopoly game
International Nuclear Information System (INIS)
Tramontana, Fabio; Gardini, Laura; Puu, Toenu
2010-01-01
The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu . The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark-Sacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties differ significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.
11. Communication: Mode bifurcation of droplet motion under stationary laser irradiation
Energy Technology Data Exchange (ETDEWEB)
Takabatake, Fumi [Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502 (Japan); Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University, Sendai, Miyagi 980-8579 (Japan); Yoshikawa, Kenichi [Faculty of Life and Medical Sciences, Doshisha University, Kyotanabe, Kyoto 610-0394 (Japan); Ichikawa, Masatoshi, E-mail: [email protected] [Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502 (Japan)
2014-08-07
The self-propelled motion of a mm-sized oil droplet floating on water, induced by a local temperature gradient generated by CW laser irradiation is reported. The circular droplet exhibits two types of regular periodic motion, reciprocal and circular, around the laser spot under suitable laser power. With an increase in laser power, a mode bifurcation from rectilinear reciprocal motion to circular motion is caused. The essential aspects of this mode bifurcation are discussed in terms of spontaneous symmetry-breaking under temperature-induced interfacial instability, and are theoretically reproduced with simple coupled differential equations.
12. Bifurcation analysis of nephron pressure and flow regulation
DEFF Research Database (Denmark)
Barfred, Mikael; Mosekilde, Erik; Holstein-Rathlou, N.-H.
1996-01-01
One- and two-dimensional continuation techniques are applied to study the bifurcation structure of a model of renal flow and pressure control. Integrating the main physiological mechanisms by which the individual nephron regulates the incoming blood flow, the model describes the interaction between...... the tubuloglomerular feedback and the response of the afferent arteriole. It is shown how a Hopf bifurcation leads the system to perform self-sustained oscillations if the feedback gain becomes sufficiently strong, and how a further increase of this parameter produces a folded structure of overlapping period...
13. Bifurcation of steady tearing states
International Nuclear Information System (INIS)
Saramito, B.; Maschke, E.K.
1985-10-01
We apply the bifurcation theory for compact operators to the problem of the nonlinear solutions of the 3-dimensional incompressible visco-resistive MHD equations. For the plane plasma slab model we compute branches of nonlinear tearing modes, which are stationary for the range of parameters investigated up to now
14. Bifurcation of limit cycles for cubic reversible systems
Directory of Open Access Journals (Sweden)
Yi Shao
2014-04-01
Full Text Available This article is concerned with the bifurcation of limit cycles of a class of cubic reversible system having a center at the origin. We prove that this system has at least four limit cycles produced by the period annulus around the center under cubic perturbations
15. A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation
International Nuclear Information System (INIS)
Yong Chen; Qi Wang
2005-01-01
In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons and Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained
16. Hopf bifurcation and chaos in macroeconomic models with policy lag
International Nuclear Information System (INIS)
Liao Xiaofeng; Li Chuandong; Zhou Shangbo
2005-01-01
In this paper, we consider the macroeconomic models with policy lag, and study how lags in policy response affect the macroeconomic stability. The local stability of the nonzero equilibrium of this equation is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation. Some general stability criteria involving the policy lag and the system parameter are derived. By choosing the policy lag as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Moreover, we show that the government can stabilize the intrinsically unstable economy if the policy lag is sufficiently short, but the system become locally unstable when the policy lag is too long. We also find the chaotic behavior in some range of the policy lag
17. Bifurcation analysis of a delayed mathematical model for tumor growth
International Nuclear Information System (INIS)
Khajanchi, Subhas
2015-01-01
In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings
18. PERIODIC ORBIT FAMILIES IN THE GRAVITATIONAL FIELD OF IRREGULAR-SHAPED BODIES
Energy Technology Data Exchange (ETDEWEB)
Jiang, Yu [State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043 (China); Baoyin, Hexi, E-mail: [email protected] [School of Aerospace Engineering, Tsinghua University, Beijing 100084 (China)
2016-11-01
The discovery of binary and triple asteroids in addition to the execution of space missions to minor celestial bodies in the past several years have focused increasing attention on periodic orbits around irregular-shaped celestial bodies. In the present work, we adopt a polyhedron shape model for providing an accurate representation of irregular-shaped bodies and employ the model to calculate their corresponding gravitational and effective potentials. We also investigate the characteristics of periodic orbit families and the continuation of periodic orbits. We prove a fact, which provides a conserved quantity that permits restricting the number of periodic orbits in a fixed energy curved surface about an irregular-shaped body. The collisions of Floquet multipliers are maintained during the continuation of periodic orbits around the comet 1P/Halley. Multiple bifurcations in the periodic orbit families about irregular-shaped bodies are also discussed. Three bifurcations in the periodic orbit family have been found around the asteroid 216 Kleopatra, which include two real saddle bifurcations and one period-doubling bifurcation.
19. PERIODIC ORBIT FAMILIES IN THE GRAVITATIONAL FIELD OF IRREGULAR-SHAPED BODIES
International Nuclear Information System (INIS)
Jiang, Yu; Baoyin, Hexi
2016-01-01
The discovery of binary and triple asteroids in addition to the execution of space missions to minor celestial bodies in the past several years have focused increasing attention on periodic orbits around irregular-shaped celestial bodies. In the present work, we adopt a polyhedron shape model for providing an accurate representation of irregular-shaped bodies and employ the model to calculate their corresponding gravitational and effective potentials. We also investigate the characteristics of periodic orbit families and the continuation of periodic orbits. We prove a fact, which provides a conserved quantity that permits restricting the number of periodic orbits in a fixed energy curved surface about an irregular-shaped body. The collisions of Floquet multipliers are maintained during the continuation of periodic orbits around the comet 1P/Halley. Multiple bifurcations in the periodic orbit families about irregular-shaped bodies are also discussed. Three bifurcations in the periodic orbit family have been found around the asteroid 216 Kleopatra, which include two real saddle bifurcations and one period-doubling bifurcation.
20. Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring–mass resonators
Energy Technology Data Exchange (ETDEWEB)
Qian, Denghui, E-mail: [email protected]; Shi, Zhiyu, E-mail: [email protected]
2016-10-07
Bandgap properties of the locally resonant phononic crystal double panel structure made of a two-dimensional periodic array of a spring–mass resonator surrounded by n springs (n equals to zero at the beginning of the study) connected between the upper and lower plates are investigated in this paper. The finite element method is applied to calculate the band structure, of which the accuracy is confirmed in comparison with the one calculated by the extended plane wave expansion (PWE) method and the transmission spectrum. Numerical results and further analysis demonstrate that two bands corresponding to the antisymmetric vibration mode open a wide band gap but is cut narrower by a band corresponding to the symmetric mode. One of the regulation rules shows that the lowest frequency on the symmetric mode band is proportional to the spring stiffness. Then, a new design idea of adding springs around the resonator in a unit cell (n is not equal to zero now) is proposed in the need of widening the bandwidth and lowering the starting frequency. Results show that the bandwidth of the band gap increases from 50 Hz to nearly 200 Hz. By introducing the quality factor, the regulation rules with the comprehensive consideration of the whole structure quality limitation, the wide band gap and the low starting frequency are also discussed. - Highlights: • The locally resonant double panel structure opens a band gap in the low frequency region. • The band gap is the coupling between the symmetric and antisymmetric vibration modes. • The band structure of the double panel is the evolution of that of the single plate. • By adding springs around the resonator in a unit cell, the bandwidth gets wider. • The band gap can be controlled by tuning the parameters.
1. Complex oscillatory behaviour in a delayed protein cross talk model with periodic forcing
International Nuclear Information System (INIS)
Nikolov, Svetoslav
2009-01-01
The purpose of this paper is to examine the effects of periodic forcing on the time delay protein cross talk model behaviour. We assume periodic variation for the plasma membrane permeability. The dynamic behaviour of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing can very easily give rise to complex dynamics, including a period-doubling cascade, chaos, quasi-periodic oscillating, and periodic windows. Finally, we calculate the maximal Lyapunov exponent in the regions of the parameter space where chaotic motion of delayed protein cross talk model with periodic forcing exists.
2. Bifurcation in a buoyant horizontal laminar jet
Science.gov (United States)
Arakeri, Jaywant H.; Das, Debopam; Srinivasan, J.
2000-06-01
The trajectory of a laminar buoyant jet discharged horizontally has been studied. The experimental observations were based on the injection of pure water into a brine solution. Under certain conditions the jet has been found to undergo bifurcation. The bifurcation of the jet occurs in a limited domain of Grashof number and Reynolds number. The regions in which the bifurcation occurs has been mapped in the Reynolds number Grashof number plane. There are three regions where bifurcation does not occur. The various mechanisms that prevent bifurcation have been proposed.
3. Stability and Hopf bifurcation in a simplified BAM neural network with two time delays.
Science.gov (United States)
Cao, Jinde; Xiao, Min
2007-03-01
Various local periodic solutions may represent different classes of storage patterns or memory patterns, and arise from the different equilibrium points of neural networks (NNs) by applying Hopf bifurcation technique. In this paper, a bidirectional associative memory NN with four neurons and multiple delays is considered. By applying the normal form theory and the center manifold theorem, analysis of its linear stability and Hopf bifurcation is performed. An algorithm is worked out for determining the direction and stability of the bifurcated periodic solutions. Numerical simulation results supporting the theoretical analysis are also given.
4. Bifurcation structures of a cobweb model with memory and competing technologies
Science.gov (United States)
2018-05-01
In this paper we study a simple model based on the cobweb demand-supply framework with costly innovators and free imitators. The evolutionary selection between technologies depends on a performance measure which is related to the degree of memory. The resulting dynamics is described by a two-dimensional map. The map has a fixed point which may lose stability either via supercritical Neimark-Sacker bifurcation or flip bifurcation and several multistability situations exist. We describe some sequences of global bifurcations involving attracting and repelling closed invariant curves. These bifurcations, characterized by the creation of homoclinic connections or homoclinic tangles, are described through several numerical simulations. Particular bifurcation phenomena are also observed when the parameters are selected inside a periodicity region.
5. Bifurcation routes and economic stability
Czech Academy of Sciences Publication Activity Database
Vošvrda, Miloslav
2001-01-01
Roč. 8, č. 14 (2001), s. 43-59 ISSN 1212-074X R&D Projects: GA ČR GA402/00/0439; GA ČR GA402/01/0034; GA ČR GA402/01/0539 Institutional research plan: AV0Z1075907 Keywords : macroeconomic stability * foreign investment phenomenon * the Hopf bifurcation Subject RIV: AH - Economics
6. Periodically poled self-frequency-doubling green laser fabricated from Nd:Mg:LiNbO₃ single crystal.
Science.gov (United States)
Wang, Dong Zhou; Sun, De Hui; Kang, Xue Liang; Sang, Yuan Hua; Yan, Bo Xia; Liu, Hong; Bi, Yong
2015-07-13
Although a breakthrough in the fabrication of green laser diodes has occurred, the high costs associated with the difficulty of manufacture still present a great obstacle for its practical application. Another approach for producing a green laser, by combining a laser device and a nonlinear crystal, entails the fabrication of complex structures and exhibits unstable performance due to interface contact defects, thus limiting its application. In this work, we report the fabrication by domain engineering of high quality periodically poled LiNbO₃, co-doped with Nd³⁺ and Mg²⁺, which combines a laser medium and a high efficiency second harmonic conversion crystal into a single system that is designed to overcome the above problems. An 80 mW self-frequency doubling green laser was constructed for the first time from a periodically poled Nd:Mg:LiNbO₃ crystal of 16 mm in length. This crystal can be used for developing compact, stable, highly efficient mini-solid-state-lasers, which promise to have many applications in portable laser-based spectroscopy, photo-communications, terahertz wave generation, and laser displays.
7. Bifurcation analysis on a delayed SIS epidemic model with stage structure
Directory of Open Access Journals (Sweden)
Kejun Zhuang
2007-05-01
Full Text Available In this paper, a delayed SIS (Susceptible Infectious Susceptible model with stage structure is investigated. We study the Hopf bifurcations and stability of the model. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. The conditions to guarantee the global existence of periodic solutions are established. Also some numerical simulations for supporting the theoretical are given.
8. A case of complete double aortic arch visualized by transthoracic echocardiography.
Science.gov (United States)
Saito, Naka; Kato, Shingo; Saito, Noritaka; Nakachi, Tatsuya; Fukui, Kazuki; Iwasawa, Tae; Kosuge, Masami; Kimura, Kazuo
2017-08-01
A case of double aortic arch that was well visualized using transthoracic echocardiography is reported. A 38-year-old man underwent transthoracic echocardiography for the evaluation of dyspnea. A suprasternal view of transthoracic echocardiography showed the ascending aorta bifurcate to left and right aortic arches, with blood flow from the ascending aorta to bilateral aortic arches. The diagnosis of right side-dominant double aortic arch was made, and the patient's symptom was conceivably related to compression of the trachea due to a vascular ring. This report indicates the potential usefulness of transthoracic echocardiography for noninvasive detection of double aortic arch in adults. © 2017, Wiley Periodicals, Inc.
9. Equilibrium-torus bifurcation in nonsmooth systems
DEFF Research Database (Denmark)
Zhusubahyev, Z.T.; Mosekilde, Erik
2008-01-01
Considering a set of two coupled nonautonomous differential equations with discontinuous right-hand sides describing the behavior of a DC/DC power converter, we discuss a border-collision bifurcation that can lead to the birth of a two-dimensional invariant torus from a stable node equilibrium...... point. We obtain the chart of dynamic modes and show that there is a region of parameter space in which the system has a single stable node equilibrium point. Under variation of the parameters, this equilibrium may disappear as it collides with a discontinuity boundary between two smooth regions...... in the phase space. The disappearance of the equilibrium point is accompanied by the soft appearance of an unstable focus period-1 orbit surrounded by a resonant or ergodic torus. Detailed numerical calculations are supported by a theoretical investigation of the normal form map that represents the piecewise...
10. Some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers of finite extent
Energy Technology Data Exchange (ETDEWEB)
Li, K., E-mail: [email protected] [Key Laboratory of Microgravity, Chinese Academy of Sciences, Beijing 100190, China and National Microgravity Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190 (China); University of Chinese Academy of Sciences, Beijing 100190 (China); Xun, B.; Hu, W. R. [Key Laboratory of Microgravity, Chinese Academy of Sciences, Beijing 100190, China and National Microgravity Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190 (China)
2016-05-15
As a part of the preliminary studies for the future space experiment (Zona-K) in the Russian module of the International Space Station, some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers filled with 10 cSt silicone oil have been numerically studied in this paper. As the laterally applied temperature difference is raised, variations in the spatial structure and temporal evolution of the thermocapillary convection and a complex sequence of transitions are observed. The results show that the finite extent of the liquid layer significantly influences the tempo-spatial evolution of the thermocapillary convection. Moreover, the bifurcation route of the thermocapillary convection changes very sensitively by the aspect ratio of the liquid layer. With the increasing Reynolds number (applied temperature difference), the steady thermocapillary convection experiences two consecutive transitions from periodic oscillatory state to quasi-periodic oscillatory state with frequency-locking before emergence of chaotic convection in a liquid layer of aspect ratio 14.25, and the thermocapillary convection undergoes period-doubling cascades leading to chaotic convection in a liquid layer of aspect ratio 13.0.
11. Some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers of finite extent
International Nuclear Information System (INIS)
Li, K.; Xun, B.; Hu, W. R.
2016-01-01
As a part of the preliminary studies for the future space experiment (Zona-K) in the Russian module of the International Space Station, some bifurcation routes to chaos of thermocapillary convection in two-dimensional liquid layers filled with 10 cSt silicone oil have been numerically studied in this paper. As the laterally applied temperature difference is raised, variations in the spatial structure and temporal evolution of the thermocapillary convection and a complex sequence of transitions are observed. The results show that the finite extent of the liquid layer significantly influences the tempo-spatial evolution of the thermocapillary convection. Moreover, the bifurcation route of the thermocapillary convection changes very sensitively by the aspect ratio of the liquid layer. With the increasing Reynolds number (applied temperature difference), the steady thermocapillary convection experiences two consecutive transitions from periodic oscillatory state to quasi-periodic oscillatory state with frequency-locking before emergence of chaotic convection in a liquid layer of aspect ratio 14.25, and the thermocapillary convection undergoes period-doubling cascades leading to chaotic convection in a liquid layer of aspect ratio 13.0.
12. Stability and bifurcation of numerical discretization of a second-order delay differential equation with negative feedback
International Nuclear Information System (INIS)
Ding Xiaohua; Su Huan; Liu Mingzhu
2008-01-01
The paper analyzes a discrete second-order, nonlinear delay differential equation with negative feedback. The characteristic equation of linear stability is solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The existence of local Hopf bifurcations is investigated, and the direction and stability of periodic solutions bifurcating from the Hopf bifurcation of the discrete model are determined by the Hopf bifurcation theory of discrete system. Finally, some numerical simulations are performed to illustrate the analytical results found
13. The Effect of Ginger on Breast Milk Volume in the Early Postpartum Period: A Randomized, Double-Blind Controlled Trial.
Science.gov (United States)
Paritakul, Panwara; Ruangrongmorakot, Kasem; Laosooksathit, Wipada; Suksamarnwong, Maysita; Puapornpong, Pawin
2016-09-01
In Thailand, ginger is a popular natural galactagogue among breastfeeding women. However, there has never been evidence to support the effectiveness of ginger in increasing the breast milk volume. To compare breast milk volume on the third and seventh day postpartum between lactating mothers who receive 500 mg dried ginger capsules twice daily with those receiving placebo. A randomized, double-blind controlled trial was conducted. Women who deliver a term baby were randomly assigned to receive dried ginger or placebo for 7 days postpartum. Breast milk volume was measured on third day postpartum using test weight method for a period of 24 hours and on seventh day postpartum using 1 hour milk production. We also compared the third day serum prolactin level between the two groups. Data from 63 women were available for analysis, 30 from the ginger group and 33 from the placebo group. The two groups were similar regarding baseline characteristics. Women in the ginger group have higher milk volume than the placebo group (191.0 ± 71.2 mL/day versus 135.0 ± 61.5 mL/day, p ginger group does not differ from the placebo group (80.0 ± 58.5 mL versus 112.1 ± 91.6 mL, p = 0.24). The mean serum prolactin levels were similar in both groups (321.5 ± 131.8 ng/L in the ginger group, and 331.4 ± 100.7 ng/L in the placebo group, p = 0.74). No side effect was reported in this study. Ginger is a promising natural galactagogue to improve breast milk volume in the immediate postpartum period without any notable side effect.
14. Effects of End CAP and Aspect Ratio on Transmission of Sound across a Truss-Like Periodic Double Panel
Science.gov (United States)
EL-RAHEB, M.; WAGNER, P.
2002-02-01
Transmission of sound across 2-D truss-like periodic double panels separated by an air gap and in contact with an acoustic fluid on the external faces is analyzed. Each panel is made of repeated cells. Combining the transfer matrices of the unit cell forms a set of equations for the overall elastic frequency response. The acoustic pressure in the fluids is expressed using a source boundary element method. Adding rigid reflecting end caps confines the air in the gap between panels which influences sound transmission. Measured values of transmission loss differ from the 2-D model by the wide low-frequency dip of the mass-spring-mass or “msm” resonance also termed the “air gap resonance”. In this case, the panels act as rigid masses and the air gap acts as an adiabatic air spring. Results from the idealized 3-D and 2-D models, incorporating rigid cavities and elastic plates, reveal that the “msm” dip is absent in 2-D models radiating into a semi-infinite medium. The dip strengthens as aspect ratio approaches unity. Even when the dip disappears in 2-D, TL rises more steeply for frequencies above the “msm” frequency.
15. Multistability and gluing bifurcation to butterflies in coupled networks with non-monotonic feedback
International Nuclear Information System (INIS)
Ma Jianfu; Wu Jianhong
2009-01-01
Neural networks with a non-monotonic activation function have been proposed to increase their capacity for memory storage and retrieval, but there is still a lack of rigorous mathematical analysis and detailed discussions of the impact of time lag. Here we consider a two-neuron recurrent network. We first show how supercritical pitchfork bifurcations and a saddle-node bifurcation lead to the coexistence of multiple stable equilibria (multistability) in the instantaneous updating network. We then study the effect of time delay on the local stability of these equilibria and show that four equilibria lose their stability at a certain critical value of time delay, and Hopf bifurcations of these equilibria occur simultaneously, leading to multiple coexisting periodic orbits. We apply centre manifold theory and normal form theory to determine the direction of these Hopf bifurcations and the stability of bifurcated periodic orbits. Numerical simulations show very interesting global patterns of periodic solutions as the time delay is varied. In particular, we observe that these four periodic solutions are glued together along the stable and unstable manifolds of saddle points to develop a butterfly structure through a complicated process of gluing bifurcations of periodic solutions
16. Stability and Hopf Bifurcation for a Delayed SLBRS Computer Virus Model
Directory of Open Access Journals (Sweden)
Zizhen Zhang
2014-01-01
Full Text Available By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.
17. Stability and Hopf bifurcation for a delayed SLBRS computer virus model.
Science.gov (United States)
Zhang, Zizhen; Yang, Huizhong
2014-01-01
By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.
18. Stability and bifurcation analysis for a discrete-time bidirectional ring neural network model with delay
Directory of Open Access Journals (Sweden)
Yan-Ke Du
2013-09-01
Full Text Available We study a class of discrete-time bidirectional ring neural network model with delay. We discuss the asymptotic stability of the origin and the existence of Neimark-Sacker bifurcations, by analyzing the corresponding characteristic equation. Employing M-matrix theory and the Lyapunov functional method, global asymptotic stability of the origin is derived. Applying the normal form theory and the center manifold theorem, the direction of the Neimark-Sacker bifurcation and the stability of bifurcating periodic solutions are obtained. Numerical simulations are given to illustrate the main results.
19. Stability and Hopf bifurcation on a model for HIV infection of CD4{sup +} T cells with delay
Energy Technology Data Exchange (ETDEWEB)
Wang Xia [College of Mathematics and Information Science, Xinyang Normal University, Xinyang, Henan 464000 (China)], E-mail: [email protected]; Tao Youde [College of Mathematics and Information Science, Xinyang Normal University, Xinyang, Henan 464000 (China); Beijing Institute of Information Control, Beijing 100037 (China); Song Xinyu [College of Mathematics and Information Science, Xinyang Normal University, Xinyang, Henan 464000 (China) and Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091 (China)], E-mail: [email protected]
2009-11-15
In this paper, a delayed differential equation model that describes HIV infection of CD4{sup +} T cells is considered. The stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions.
20. Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus
Directory of Open Access Journals (Sweden)
Tao Dong
2012-01-01
Full Text Available By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.
1. Bifurcation and synchronization of synaptically coupled FHN models with time delay
International Nuclear Information System (INIS)
Wang Qingyun; Lu Qishao; Chen Guanrong; Feng Zhaosheng; Duan Lixia
2009-01-01
This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.
2. Bifurcation in the Lengyel–Epstein system for the coupled reactors with diffusion
Directory of Open Access Journals (Sweden)
Shaban Aly
2016-01-01
Full Text Available The main goal of this paper is to continue the investigations of the important system of Fengqi et al. (2008. The occurrence of Turing and Hopf bifurcations in small homogeneous arrays of two coupled reactors via diffusion-linked mass transfer which described by a system of ordinary differential equations is considered. I study the conditions of the existence as well as stability properties of the equilibrium solutions and derive the precise conditions on the parameters to show that the Hopf bifurcation occurs. Analytically I show that a diffusion driven instability occurs at a certain critical value, when the system undergoes a Turing bifurcation, patterns emerge. The spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. Numerically, at a certain critical value of diffusion the periodic solution gets destabilized and two new spatially nonconstant periodic solutions arise by Turing bifurcation.
3. Bifurcation and chaos of an axially accelerating viscoelastic beam
International Nuclear Information System (INIS)
Yang Xiaodong; Chen Liqun
2005-01-01
This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin-Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial-differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincare map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam
4. A bifurcation analysis for the Lugiato-Lefever equation
Science.gov (United States)
Godey, Cyril
2017-05-01
The Lugiato-Lefever equation is a cubic nonlinear Schrödinger equation, including damping, detuning and driving, which arises as a model in nonlinear optics. We study the existence of stationary waves which are found as solutions of a four-dimensional reversible dynamical system in which the evolutionary variable is the space variable. Relying upon tools from bifurcation theory and normal forms theory, we discuss the codimension 1 bifurcations. We prove the existence of various types of steady solutions, including spatially localized, periodic, or quasi-periodic solutions. Contribution to the Topical Issue: "Theory and Applications of the Lugiato-Lefever Equation", edited by Yanne K. Chembo, Damia Gomila, Mustapha Tlidi, Curtis R. Menyuk.
5. Overweight and obesity doubled over a 6-year period in young women living in poverty in Mexico.
Science.gov (United States)
Neufeld, Lynnette M; Hernández-Cordero, Sonia; Fernald, Lia C; Ramakrishnan, Usha
2008-03-01
To document the changes in BMI and the prevalence of overweight and obesity in young women living in poverty in a semi-urban community in Mexico. Women who had previously participated in a longitudinal research study (1997-2000) were re-assessed in 2005. Anthropometric measurements were obtained using standard procedures, and socio-demographic questionnaires were administered. Total and annual rate of change in BMI and change in the prevalence of overweight and obesity (BMI > or = 25.0 and > or =30.0) were estimated. Mean age in 2005 was 30.0 +/- 5.7 years (n = 683) and time between recruitment and follow-up was 6.4 +/- 1.0 years. Mean change in BMI was +3.6 +/- 2.7 (range -8.2 to +14.6). In 2005, 500 (73.2%) women were overweight, up from 263 (38.5%) in the original assessment. The prevalence of obesity tripled over the follow-up period (from 9.8% to 30.3%). The mean annual rate of change in BMI was +0.6 (+/-0.4). After adjustment for age and parity at baseline, an annual rate of change of BMI above the sample median (>0.5) was associated with lower levels of formal education. The annual increase in the prevalence of overweight and obesity in this sample is double that which was reported at a national level in Mexico. An understanding of the determinants of this rapid increase among the women living in poverty in Mexico is urgently needed.
6. Application of the bifurcation method to the modified Boussinesq equation
Directory of Open Access Journals (Sweden)
Shaoyong Li
2014-08-01
Firstly, we give a property of the solutions of the equation, that is, if $1+u(x, t$ is a solution, so is $1-u(x, t$. Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.
7. Bifurcations of optimal vector fields: an overview
NARCIS (Netherlands)
Kiseleva, T.; Wagener, F.; Rodellar, J.; Reithmeier, E.
2009-01-01
We develop a bifurcation theory for the solution structure of infinite horizon optimal control problems with one state variable. It turns out that qualitative changes of this structure are connected to local and global bifurcations in the state-costate system. We apply the theory to investigate an
8. Evidence for bifurcation and universal chaotic behavior in nonlinear semiconducting devices
International Nuclear Information System (INIS)
Testa, J.; Perez, J.; Jeffries, C.
1982-01-01
Bifurcations, chaos, and extensive periodic windows in the chaotic regime are observed for a driven LRC circuit, the capacitive element being a nonlinear varactor diode. Measurements include power spectral analysis; real time amplitude data; phase portraits; and a bifurcation diagram, obtained by sampling methods. The effects of added external noise are studied. These data yield experimental determinations of several of the universal numbers predicted to characterize nonlinear systems having this route to chaos
9. Stability and Hopf bifurcation in a delayed competitive web sites model
International Nuclear Information System (INIS)
Xiao Min; Cao Jinde
2006-01-01
The delayed differential equations modeling competitive web sites, based on the Lotka-Volterra competition equations, are considered. Firstly, the linear stability is investigated. It is found that there is a stability switch for time delay, and Hopf bifurcation occurs when time delay crosses through a critical value. Then the direction and stability of the bifurcated periodic solutions are determined, using the normal form theory and the center manifold reduction. Finally, some numerical simulations are carried out to illustrate the results found
10. Voltage stability, bifurcation parameters and continuation methods
Energy Technology Data Exchange (ETDEWEB)
1994-12-31
This paper considers the importance of the choice of bifurcation parameter in the determination of the voltage stability limit and the maximum power load ability of a system. When the bifurcation parameter is power demand, the two limits are equivalent. However, when other types of load models and bifurcation parameters are considered, the two concepts differ. The continuation method is considered as a method for determination of voltage stability margins. Three variants of the continuation method are described: the continuation parameter is the bifurcation parameter the continuation parameter is initially the bifurcation parameter, but is free to change, and the continuation parameter is a new arc length parameter. Implementations of voltage stability software using continuation methods are described. (author) 23 refs., 9 figs.
11. Forced phase-locked response of a nonlinear system with time delay after Hopf bifurcation
International Nuclear Information System (INIS)
Ji, J.C.; Hansen, Colin H.
2005-01-01
The trivial equilibrium of a nonlinear autonomous system with time delay may become unstable via a Hopf bifurcation of multiplicity two, as the time delay reaches a critical value. This loss of stability of the equilibrium is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. The resultant dynamic behaviour of the corresponding nonlinear non-autonomous system in the neighbourhood of the Hopf bifurcation is investigated based on the reduction of the infinite-dimensional problem to a four-dimensional centre manifold. As a result of the interaction between the Hopf bifurcating periodic solutions and the external periodic excitation, a primary resonance can occur in the forced response of the system when the forcing frequency is close to the Hopf bifurcating periodic frequency. The method of multiple scales is used to obtain four first-order ordinary differential equations that determine the amplitudes and phases of the phase-locked periodic solutions. The first-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration of the delay-differential equation. It is also found that the steady state solutions of the nonlinear non-autonomous system may lose their stability via either a pitchfork or Hopf bifurcation. It is shown that the primary resonance response may exhibit symmetric and asymmetric phase-locked periodic motions, quasi-periodic motions, chaotic motions, and coexistence of two stable motions
12. Local stability and Hopf bifurcation in small-world delayed networks
International Nuclear Information System (INIS)
Li Chunguang; Chen Guanrong
2004-01-01
The notion of small-world networks, recently introduced by Watts and Strogatz, has attracted increasing interest in studying the interesting properties of complex networks. Notice that, a signal or influence travelling on a small-world network often is associated with time-delay features, which are very common in biological and physical networks. Also, the interactions within nodes in a small-world network are often nonlinear. In this paper, we consider a small-world networks model with nonlinear interactions and time delays, which was recently considered by Yang. By choosing the nonlinear interaction strength as a bifurcation parameter, we prove that Hopf bifurcation occurs. We determine the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, we show a numerical example to verify the theoretical analysis
13. Local stability and Hopf bifurcation in small-world delayed networks
Energy Technology Data Exchange (ETDEWEB)
Li Chunguang E-mail: [email protected]; Chen Guanrong E-mail: [email protected]
2004-04-01
The notion of small-world networks, recently introduced by Watts and Strogatz, has attracted increasing interest in studying the interesting properties of complex networks. Notice that, a signal or influence travelling on a small-world network often is associated with time-delay features, which are very common in biological and physical networks. Also, the interactions within nodes in a small-world network are often nonlinear. In this paper, we consider a small-world networks model with nonlinear interactions and time delays, which was recently considered by Yang. By choosing the nonlinear interaction strength as a bifurcation parameter, we prove that Hopf bifurcation occurs. We determine the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, we show a numerical example to verify the theoretical analysis.
14. Hopf bifurcation in a environmental defensive expenditures model with time delay
International Nuclear Information System (INIS)
Russu, Paolo
2009-01-01
In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, τ 0 . It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.
15. Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses
Directory of Open Access Journals (Sweden)
Chuandong Li
2014-01-01
Full Text Available We extend the three-dimensional SIR model to four-dimensional case and then analyze its dynamical behavior including stability and bifurcation. It is shown that the new model makes a significant improvement to the epidemic model for computer viruses, which is more reasonable than the most existing SIR models. Furthermore, we investigate the stability of the possible equilibrium point and the existence of the Hopf bifurcation with respect to the delay. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. An analytical condition for determining the direction, stability, and other properties of bifurcating periodic solutions is obtained by using the normal form theory and center manifold argument. The obtained results may provide a theoretical foundation to understand the spread of computer viruses and then to minimize virus risks.
16. Local and global bifurcations at infinity in models of glycolytic oscillations
DEFF Research Database (Denmark)
Sturis, Jeppe; Brøns, Morten
1997-01-01
We investigate two models of glycolytic oscillations. Each model consists of two coupled nonlinear ordinary differential equations. Both models are found to have a saddle point at infinity and to exhibit a saddle-node bifurcation at infinity, giving rise to a second saddle and a stable node...... at infinity. Depending on model parameters, a stable limit cycle may blow up to infinite period and amplitude and disappear in the bifurcation, and after the bifurcation, the stable node at infinity then attracts all trajectories. Alternatively, the stable node at infinity may coexist with either a stable...... sink (not at infinity) or a stable limit cycle. This limit cycle may then disappear in a heteroclinic bifurcation at infinity in which the unstable manifold from one saddle at infinity joins the stable manifold of the other saddle at infinity. These results explain prior reports for one of the models...
17. Stability and Hopf Bifurcation of a Reaction-Diffusion Neutral Neuron System with Time Delay
Science.gov (United States)
Dong, Tao; Xia, Linmao
2017-12-01
In this paper, a type of reaction-diffusion neutral neuron system with time delay under homogeneous Neumann boundary conditions is considered. By constructing a basis of phase space based on the eigenvectors of the corresponding Laplace operator, the characteristic equation of this system is obtained. Then, by selecting time delay and self-feedback strength as the bifurcating parameters respectively, the dynamic behaviors including local stability and Hopf bifurcation near the zero equilibrium point are investigated when the time delay and self-feedback strength vary. Furthermore, the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form and the center manifold theorem for the corresponding partial differential equation. Finally, two simulation examples are given to verify the theory.
18. Renal denervation beyond the bifurcation: The effect of distal ablation placement on safety and blood pressure.
Science.gov (United States)
Beeftink, Martine M A; Spiering, Wilko; De Jong, Mark R; Doevendans, Pieter A; Blankestijn, Peter J; Elvan, Arif; Heeg, Jan-Evert; Bots, Michiel L; Voskuil, Michiel
2017-04-01
Renal denervation may be more effective if performed distal in the renal artery because of smaller distances between the lumen and perivascular nerves. The authors reviewed the angiographic results of 97 patients and compared blood pressure reduction in relation to the location of the denervation. No significant differences in blood pressure reduction or complications were found between patient groups divided according to their spatial distribution of the ablations (proximal to the bifurcation in both arteries, distal to the bifurcation in one artery and distal in the other artery, or distal to the bifurcation in both arteries), but systolic ambulatory blood pressure reduction was significantly related to the number of distal ablations. No differences in adverse events were observed. In conclusion, we found no reason to believe that renal denervation distal to the bifurcation poses additional risks over the currently advised approach of proximal denervation, but improved efficacy remains to be conclusively established. ©2017 Wiley Periodicals, Inc.
19. Stability and bifurcation in a simplified four-neuron BAM neural network with multiple delays
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We first study the distribution of the zeros of a fourth-degree exponential polynomial. Then we apply the obtained results to a simplified bidirectional associated memory (BAM neural network with four neurons and multiple time delays. By taking the sum of the delays as the bifurcation parameter, it is shown that under certain assumptions the steady state is absolutely stable. Under another set of conditions, there are some critical values of the delay, when the delay crosses these critical values, the Hopf bifurcation occurs. Furthermore, some explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and center manifold reduction. Numerical simulations supporting the theoretical analysis are also included.
20. Recent perspective on coronary artery bifurcation interventions.
Science.gov (United States)
Dash, Debabrata
2014-01-01
Coronary bifurcation lesions are frequent in routine practice, accounting for 15-20% of all lesions undergoing percutaneous coronary intervention (PCI). PCI of this subset of lesions is technically challenging and historically has been associated with lower procedural success rates and worse clinical outcomes compared with non-bifurcation lesions. The introduction of drug-eluting stents has dramatically improved the outcomes. The provisional technique of implanting one stent in the main branch remains the default approach in most bifurcation lesions. Selection of the most effective technique for an individual bifurcation is important. The use of two-stent techniques as an intention to treat is an acceptable approach in some bifurcation lesions. However, a large amount of metal is generally left unapposed in the lumen with complex two-stent techniques, which is particularly concerning for the risk of stent thrombosis. New technology and dedicated bifurcation stents may overcome some of the limitations of two-stent techniques and revolutionise the management of bifurcation PCI in the future.
1. Limit cycles bifurcating from a perturbed quartic center
Energy Technology Data Exchange (ETDEWEB)
Coll, Bartomeu, E-mail: [email protected] [Dept. de Matematiques i Informatica, Universitat de les Illes Balears, Facultat de ciencies, 07071 Palma de Mallorca (Spain); Llibre, Jaume, E-mail: [email protected] [Dept. de Matematiques, Universitat Autonoma de Barcelona, Edifici Cc 08193 Bellaterra, Barcelona, Catalonia (Spain); Prohens, Rafel, E-mail: [email protected] [Dept. de Matematiques i Informatica, Universitat de les Illes Balears, Facultat de ciencies, 07071 Palma de Mallorca (Spain)
2011-04-15
Highlights: We study polynomial perturbations of a quartic center. We get simultaneous upper and lower bounds for the bifurcating limit cycles. A higher lower bound for the maximum number of limit cycles is obtained. We obtain more limit cycles than the number obtained in the cubic case. - Abstract: We consider the quartic center x{sup .}=-yf(x,y),y{sup .}=xf(x,y), with f(x, y) = (x + a) (y + b) (x + c) and abc {ne} 0. Here we study the maximum number {sigma} of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n - 1)/2] + 4 {<=} {sigma} {<=} 5[(n - 1)/2] + 14, where [{eta}] denotes the integer part function of {eta}.
2. Bifurcation analysis on a generalized recurrent neural network with two interconnected three-neuron components
International Nuclear Information System (INIS)
Hajihosseini, Amirhossein; Maleki, Farzaneh; Rokni Lamooki, Gholam Reza
2011-01-01
Highlights: → We construct a recurrent neural network by generalizing a specific n-neuron network. → Several codimension 1 and 2 bifurcations take place in the newly constructed network. → The newly constructed network has higher capabilities to learn periodic signals. → The normal form theorem is applied to investigate dynamics of the network. → A series of bifurcation diagrams is given to support theoretical results. - Abstract: A class of recurrent neural networks is constructed by generalizing a specific class of n-neuron networks. It is shown that the newly constructed network experiences generic pitchfork and Hopf codimension one bifurcations. It is also proved that the emergence of generic Bogdanov-Takens, pitchfork-Hopf and Hopf-Hopf codimension two, and the degenerate Bogdanov-Takens bifurcation points in the parameter space is possible due to the intersections of codimension one bifurcation curves. The occurrence of bifurcations of higher codimensions significantly increases the capability of the newly constructed recurrent neural network to learn broader families of periodic signals.
3. Crossing Y-stent technique with dual open-cell stents for coiling of wide-necked bifurcation aneurysms.
Science.gov (United States)
Ko, Jun Kyeung; Han, In Ho; Cho, Won Ho; Choi, Byung Kwan; Cha, Seung Heon; Choi, Chang Hwa; Lee, Sang Weon; Lee, Tae Hong
2015-05-01
Double stenting in a Y-configuration is a promising therapeutic option for wide-necked cerebral aneurysms not amenable to reconstruction with a single stent. We retrospectively evaluated the efficacy and safety of the crossing Y-stent technique for coiling of wide-necked bifurcation aneurysms. By collecting clinical and radiological data we evaluated from January 2007 through December 2013, 20 wide-necked bifurcation aneurysms. Twelve unruptured and eight ruptured aneurysms in 20 patients were treated with crossing Y-stent-assisted coiling. Aneurysm size and neck size ranged from 3.2 to 28.2mm (mean 7.5mm) and from 1.9 to 9.1mm (mean 4.5mm). A Y-configuration was established successfully in all 20 patients. All aneurysms were treated with a pair of Neuroform stents. The immediate angiographic results were total occlusion in 17 aneurysms, residual neck in two, and residual sac in one. Peri-operative morbidity was only 5%. Fifteen of 18 surviving patients underwent follow-up conventional angiography (mean, 10.9 months). The result showed stable occlusion in all 15 aneurysms and asymptomatic in-stent occlusion in one branch artery. At the end of the observation period (mean, 33.5 months), all 12 patients without subarachnoid hemorrhage had excellent clinical outcomes (mRS 0), except one (mRS 2). Of eight patients with subarachnoid hemorrhage, four remained symptom free (mRS 0), while the other four had were dependent or dead (mRS score, 3-6). In this report on 20 patients, crossing Y-stent technique for coiling of wide-necked bifurcation aneurysms showed a good technical safety and favorable clinical and angiographic outcome. Copyright © 2015. Published by Elsevier B.V.
4. Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation
International Nuclear Information System (INIS)
Saiki, Yoshitaka; Yamada, Michio; Chian, Abraham C.-L.; Miranda, Rodrigo A.; Rempel, Erico L.
2015-01-01
The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs
5. Reconstruction of chaotic saddles by classification of unstable periodic orbits: Kuramoto-Sivashinsky equation
Energy Technology Data Exchange (ETDEWEB)
Saiki, Yoshitaka, E-mail: [email protected] [Graduate School of Commerce and Management, Hitotsubashi University, Tokyo 186-8601 (Japan); Yamada, Michio [Research Institute for Mathematical Sciences (RIMS), Kyoto University, Kyoto 606-8502 (Japan); Chian, Abraham C.-L. [Paris Observatory, LESIA, CNRS, 92195 Meudon (France); National Institute for Space Research (INPE), P.O. Box 515, São José dos Campos, São Paulo 12227-010 (Brazil); Institute of Aeronautical Technology (ITA) and World Institute for Space Environment Research (WISER), São José dos Campos, São Paulo 12228-900 (Brazil); School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005 (Australia); Department of Biomedical Engineering, George Washington University, Washington, DC 20052 (United States); Miranda, Rodrigo A. [Faculty UnB-Gama, and Plasma Physics Laboratory, Institute of Physics, University of Brasília (UnB), Brasília DF 70910-900 (Brazil); Rempel, Erico L. [Institute of Aeronautical Technology (ITA) and World Institute for Space Environment Research (WISER), São José dos Campos, São Paulo 12228-900 (Brazil)
2015-10-15
The unstable periodic orbits (UPOs) embedded in a chaotic attractor after an attractor merging crisis (MC) are classified into three subsets, and employed to reconstruct chaotic saddles in the Kuramoto-Sivashinsky equation. It is shown that in the post-MC regime, the two chaotic saddles evolved from the two coexisting chaotic attractors before crisis can be reconstructed from the UPOs embedded in the pre-MC chaotic attractors. The reconstruction also involves the detection of the mediating UPO responsible for the crisis, and the UPOs created after crisis that fill the gap regions of the chaotic saddles. We show that the gap UPOs originate from saddle-node, period-doubling, and pitchfork bifurcations inside the periodic windows in the post-MC chaotic region of the bifurcation diagram. The chaotic attractor in the post-MC regime is found to be the closure of gap UPOs.
6. Quantum entanglement and fixed-point bifurcations
International Nuclear Information System (INIS)
Hines, Andrew P.; McKenzie, Ross H.; Milburn, G.J.
2005-01-01
How does the classical phase-space structure for a composite system relate to the entanglement characteristics of the corresponding quantum system? We demonstrate how the entanglement in nonlinear bipartite systems can be associated with a fixed-point bifurcation in the classical dynamics. Using the example of coupled giant spins we show that when a fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state--the ground state--achieves its maximum amount of entanglement near the critical point. We conjecture that this will be a generic feature of systems whose classical limit exhibits such a bifurcation
7. A case study in bifurcation theory
Science.gov (United States)
Khmou, Youssef
This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.
8. Bifurcations of non-smooth systems
Science.gov (United States)
Angulo, Fabiola; Olivar, Gerard; Osorio, Gustavo A.; Escobar, Carlos M.; Ferreira, Jocirei D.; Redondo, Johan M.
2012-12-01
Non-smooth systems (namely piecewise-smooth systems) have received much attention in the last decade. Many contributions in this area show that theory and applications (to electronic circuits, mechanical systems, …) are relevant to problems in science and engineering. Specially, new bifurcations have been reported in the literature, and this was the topic of this minisymposium. Thus both bifurcation theory and its applications were included. Several contributions from different fields show that non-smooth bifurcations are a hot topic in research. Thus in this paper the reader can find contributions from electronics, energy markets and population dynamics. Also, a carefully-written specific algebraic software tool is presented.
9. Bifurcating Particle Swarms in Smooth-Walled Fractures
Science.gov (United States)
Pyrak-Nolte, L. J.; Sun, H.
2010-12-01
Particle swarms can occur naturally or from industrial processes where small liquid drops containing thousands to millions of micron-size to colloidal-size particles are released over time from seepage or leaks into fractured rock. The behavior of these particle swarms as they fall under gravity are affected by particle interactions as well as interactions with the walls of the fractures. In this paper, we present experimental results on the effect of fractures on the cohesiveness of the swarm and the formation of bifurcation structures as they fall under gravity and interact with the fracture walls. A transparent cubic sample (100 mm x 100 mm x 100 mm) containing a synthetic fracture with uniform aperture distributions was optically imaged to quantify the effect of confinement within fractures on particle swarm formation, swarm velocity, and swarm geometry. A fracture with a uniform aperture distribution was fabricated from two polished rectangular prisms of acrylic. A series of experiments were performed to determine how swarm movement and geometry are affected as the walls of the fracture are brought closer together from 50 mm to 1 mm. During the experiments, the fracture was fully saturated with water. We created the swarms using two different particle sizes in dilute suspension (~ 1.0% by mass). The particles were 3 micron diameter fluorescent polymer beads and 25 micron diameter soda-lime glass beads. Experiments were performed using swarms that ranged in size from 5 µl to 60 µl. The swarm behavior was imaged using an optical fluorescent imaging system composed of a CCD camera illuminated by a 100 mW diode-pumped doubled YAG laser. As a swarm falls in an open-tank of water, it forms a torroidal shape that is stable as long as no ambient or background currents exist in the water tank. When a swarm is released into a fracture with an aperture less than 5 mm, the swarm forms the torroidal shape but it is distorted because of the presence of the walls. The
10. Bifurcation Control of Chaotic Dynamical Systems
National Research Council Canada - National Science Library
Wang, Hua O; Abed, Eyad H
1992-01-01
A nonlinear system which exhibits bifurcations, transient chaos, and fully developed chaos is considered, with the goal of illustrating the role of two ideas in the control of chaotic dynamical systems...
11. Bifurcation and Control in a Singular Phytoplankton-Zooplankton-Fish Model with Nonlinear Fish Harvesting and Taxation
Science.gov (United States)
Meng, Xin-You; Wu, Yu-Qian
In this paper, a delayed differential algebraic phytoplankton-zooplankton-fish model with taxation and nonlinear fish harvesting is proposed. In the absence of time delay, the existence of singularity induced bifurcation is discussed by regarding economic interest as bifurcation parameter. A state feedback controller is designed to eliminate singularity induced bifurcation. Based on Liu’s criterion, Hopf bifurcation occurs at the interior equilibrium when taxation is taken as bifurcation parameter and is more than its corresponding critical value. In the presence of time delay, by analyzing the associated characteristic transcendental equation, the interior equilibrium loses local stability when time delay crosses its critical value. What’s more, the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are investigated based on normal form theory and center manifold theorem, and nonlinear state feedback controller is designed to eliminate Hopf bifurcation. Furthermore, Pontryagin’s maximum principle has been used to obtain optimal tax policy to maximize the benefit as well as the conservation of the ecosystem. Finally, some numerical simulations are given to demonstrate our theoretical analysis.
12. Bifurcation and category learning in network models of oscillating cortex
Science.gov (United States)
Baird, Bill
1990-06-01
A genetic model of oscillating cortex, which assumes “minimal” coupling justified by known anatomy, is shown to function as an associative memory, using previously developed theory. The network has explicit excitatory neurons with local inhibitory interneuron feedback that forms a set of nonlinear oscillators coupled only by long-range excitatory connections. Using a local Hebb-like learning rule for primary and higher-order synapses at the ends of the long-range connections, the system learns to store the kinds of oscillation amplitude patterns observed in olfactory and visual cortex. In olfaction, these patterns “emerge” during respiration by a pattern forming phase transition which we characterize in the model as a multiple Hopf bifurcation. We argue that these bifurcations play an important role in the operation of real digital computers and neural networks, and we use bifurcation theory to derive learning rules which analytically guarantee CAM storage of continuous periodic sequences-capacity: N/2 Fourier components for an N-node network-no “spurious” attractors.
13. Ternary choices in repeated games and border collision bifurcations
International Nuclear Information System (INIS)
Dal Forno, Arianna; Gardini, Laura; Merlone, Ugo
2012-01-01
Highlights: ► We extend a model of binary choices with externalities to include more alternatives. ► Introducing one more option affects the complexity of the dynamics. ► We find bifurcation structures which where impossible to observe in binary choices. ► A ternary choice cannot simply be considered as a binary choice plus one. - Abstract: Several recent contributions formalize and analyze binary choices games with externalities as those described by Schelling. Nevertheless, in the real world choices are not always binary, and players have often to decide among more than two alternatives. These kinds of interactions are examined in game theory where, starting from the well known rock-paper-scissor game, several other kinds of strategic interactions involving more than two choices are examined. In this paper we investigate how the dynamics evolve introducing one more option in binary choice games with externalities. The dynamics we obtain are always in a stable regime, that is, the structurally stable dynamics are only attracting cycles, but of any possible positive integer as period. We show that, depending on the structure of the game, the dynamics can be quite different from those existing when considering binary choices. The bifurcation structure, due to border collisions, is explained, showing the existence of so-called big-bang bifurcation points.
14. Discretization analysis of bifurcation based nonlinear amplifiers
Science.gov (United States)
Feldkord, Sven; Reit, Marco; Mathis, Wolfgang
2017-09-01
Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov-Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov-Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exemplarily that explicit Runge-Kutta methods transform the truncated normalform equation of the Andronov-Hopf bifurcation into the normalform equation of the Neimark-Sacker bifurcation. Dependent on the order of the integration method, higher order terms are added during this transformation.A rescaled normalform equation of the Neimark-Sacker bifurcation is introduced that allows a parametric design of a discrete-time system which corresponds to the rescaled Andronov-Hopf system. This system approximates the characteristics of the rescaled Hopf-type amplifier for a large range of parameters. The natural frequency and the peak amplitude are preserved for every set of parameters. The Neimark-Sacker bifurcation based systems avoid large computational effort that would be caused by applying higher order integration methods to the continuous-time normalform equations.
15. Homoclinic connections and subcritical Neimark bifurcation in a duopoly model with adaptively adjusted productions
International Nuclear Information System (INIS)
Agliari, Anna
2006-01-01
In this paper we study some global bifurcations arising in the Puu's oligopoly model when we assume that the producers do not adjust to the best reply but use an adaptive process to obtain at each step the new production. Such bifurcations cause the appearance of a pair of closed invariant curves, one attracting and one repelling, this latter being involved in the subcritical Neimark bifurcation of the Cournot equilibrium point. The aim of the paper is to highlight the relationship between the global bifurcations causing the appearance/disappearance of two invariant closed curves and the homoclinic connections of some saddle cycle, already conjectured in [Agliari A, Gardini L, Puu T. Some global bifurcations related to the appearance of closed invariant curves. Comput Math Simul 2005;68:201-19]. We refine the results obtained in such a paper, showing that the appearance/disappearance of closed invariant curves is not necessarily related to the existence of an attracting cycle. The characterization of the periodicity tongues (i.e. a region of the parameter space in which an attracting cycle exists) associated with a subcritical Neimark bifurcation is also discussed
16. Cutting Balloon Angioplasty in the Treatment of Short Infrapopliteal Bifurcation Disease.
Science.gov (United States)
Iezzi, Roberto; Posa, Alessandro; Santoro, Marco; Nestola, Massimiliano; Contegiacomo, Andrea; Tinelli, Giovanni; Paolini, Alessandra; Flex, Andrea; Pitocco, Dario; Snider, Francesco; Bonomo, Lorenzo
2015-08-01
To evaluate the safety, feasibility, and effectiveness of cutting balloon angioplasty in the management of infrapopliteal bifurcation disease. Between November 2010 and March 2013, 23 patients (mean age 69.6±9.01 years, range 56-89; 16 men) suffering from critical limb ischemia were treated using cutting balloon angioplasty (single cutting balloon, T-shaped double cutting balloon, or double kissing cutting balloon technique) for 47 infrapopliteal artery bifurcation lesions (16 popliteal bifurcation and 9 tibioperoneal bifurcation) in 25 limbs. Follow-up consisted of clinical examination and duplex ultrasonography at 1 month and every 3 months thereafter. All treatments were technically successful. No 30-day death or adverse events needing treatment were registered. No flow-limiting dissection was observed, so no stent implantation was necessary. The mean postprocedure minimum lumen diameter and acute gain were 0.28±0.04 and 0.20±0.06 cm, respectively, with a residual stenosis of 0.04±0.02 cm. Primary and secondary patency rates were estimated as 89.3% and 93.5% at 6 months and 77.7% and 88.8% at 12 months, respectively; 1-year primary and secondary patency rates of the treated bifurcation were 74.2% and 87.0%, respectively. The survival rate estimated by Kaplan-Meier analysis was 82.5% at 1 year. Cutting balloon angioplasty seems to be a safe and effective tool in the routine treatment of short/ostial infrapopliteal bifurcation lesions, avoiding procedure-related complications, overcoming the limitations of conventional angioplasty, and improving the outcome of catheter-based therapy. © The Author(s) 2015.
17. Bifurcation analysis for a discrete-time Hopfield neural network of two neurons with two delays and self-connections
International Nuclear Information System (INIS)
Kaslik, E.; Balint, St.
2009-01-01
In this paper, a bifurcation analysis is undertaken for a discrete-time Hopfield neural network of two neurons with two different delays and self-connections. Conditions ensuring the asymptotic stability of the null solution are found, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, Fold or Neimark-Sacker bifurcations occur, but Flip and codimension 2 (Fold-Neimark-Sacker, double Neimark-Sacker, resonance 1:1 and Flip-Neimark-Sacker) bifurcations may also be present. The direction and the stability of the Neimark-Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory
18. Stability and Bifurcation in Magnetic Flux Feedback Maglev Control System
Directory of Open Access Journals (Sweden)
Wen-Qing Zhang
2013-01-01
Full Text Available Nonlinear properties of magnetic flux feedback control system have been investigated mainly in this paper. We analyzed the influence of magnetic flux feedback control system on control property by time delay and interfering signal of acceleration. First of all, we have established maglev nonlinear model based on magnetic flux feedback and then discussed hopf bifurcation’s condition caused by the acceleration’s time delay. The critical value of delayed time is obtained. It is proved that the period solution exists in maglev control system and the stable condition has been got. We obtained the characteristic values by employing center manifold reduction theory and normal form method, which represent separately the direction of hopf bifurcation, the stability of the period solution, and the period of the period motion. Subsequently, we discussed the influence maglev system on stability of by acceleration’s interfering signal and obtained the stable domain of interfering signal. Some experiments have been done on CMS04 maglev vehicle of National University of Defense Technology (NUDT in Tangshan city. The results of experiments demonstrate that viewpoints of this paper are correct and scientific. When time lag reaches the critical value, maglev system will produce a supercritical hopf bifurcation which may cause unstable period motion.
19. A double-blind assessment of additive intolerance in children using a 12 day challenge period at home.
Science.gov (United States)
Wilson, N; Scott, A
1989-05-01
Alleged food-additive intolerance (respiratory, dermatological, behavioural or abdominal) was assessed in 19 children using daily challenge drinks of either, base product alone, base product plus sunset yellow/tartrazine, or base product plus sodium metabisulphite/sodium benzoate. The same type of drink was given for 12 days, double-blind and in random order. During the trial the children were maintained on an additive-free diet under supervision. Diary cards were used to record symptoms and medication usage. If there was an apparent association between symptoms and drink ingredient the trial was repeated, again double-blind. Additive intolerance was confirmed by a consistent deterioration of symptoms in only three children. In one, urticaria was induced by the colourings, in another extremely abnormal behaviour was induced by the preservatives and a third child was only free of asthma and abdominal pain on placebo. This form of individual trial, using 12 daily drinks, overcomes some of the objections to a single challenge study. Despite this, intolerance to the additives was only confirmed in 3/19 children in whom it had been believed to occur.
20. Simple or Complex Stenting for Bifurcation Coronary Lesions: A Patient-Level Pooled-Analysis of the Nordic Bifurcation Study and the British Bifurcation Coronary Study
DEFF Research Database (Denmark)
Behan, Miles W; Holm, Niels Ramsing; Curzen, Nicholas P
2011-01-01
Background— Controversy persists regarding the correct strategy for bifurcation lesions. Therefore, we combined the patient-level data from 2 large trials with similar methodology: the NORDIC Bifurcation Study (NORDIC I) and the British Bifurcation Coronary Study (BBC ONE). Methods and Results— B...
1. Stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable duffing oscillator and bifurcation of moment equation
International Nuclear Information System (INIS)
Zhang Guangjun; Xu Jianxue; Wang Jue; Yue Zhifeng; Zou Hailin
2009-01-01
In this paper stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator is analyzed by moment method. This kind of novel transition refers to the one among three potential well on two sides of bifurcation point of original system at the presence of internal noise. Several conclusions are drawn. First, the semi-analytical result of stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator can be obtained, and the semi-analytical result is qualitatively compatible with the one of Monte Carlo simulation. Second, a bifurcation of double-branch fixed point curves occurs in the moment equations with noise intensity as their bifurcation parameter. Third, the bifurcation of moment equations corresponds to stochastic resonance of original system. Finally, the mechanism of stochastic resonance is presented from another viewpoint through analyzing the energy transfer induced by the bifurcation of moment equation.
2. Bifurcation analysis of Rössler system with multiple delayed feedback
Directory of Open Access Journals (Sweden)
Meihong Xu
2010-10-01
Full Text Available In this paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of a Rössler system with multiple delayed feedback proposed by Ghosh and Chowdhury. At first we consider the stability of equilibrium and the existence of Hopf bifurcations. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, we give a numerical simulation example which indicates that chaotic oscillation is converted into a stable steady state or a stable periodic orbit when the delay passes through certain critical values.
3. Bifurcation and chaos response of a cracked rotor with random disturbance
Science.gov (United States)
Leng, Xiaolei; Meng, Guang; Zhang, Tao; Fang, Tong
2007-01-01
The Monte-Carlo method is used to investigate the bifurcation and chaos characteristics of a cracked rotor with a white noise process as its random disturbance. Special attention is paid to the influence of the stiffness change ratio and the rotating speed ratio on the bifurcation and chaos response of the system. Numerical simulations show that the affect of the random disturbance is significant as the undisturbed response of the cracked rotor system is a quasi-periodic or chaos one, and such affect is smaller as the undisturbed response is a periodic one.
4. Bifurcation analysis of a three dimensional system
Directory of Open Access Journals (Sweden)
Yongwen WANG
2018-04-01
Full Text Available In order to enrich the stability and bifurcation theory of the three dimensional chaotic systems, taking a quadratic truncate unfolding system with the triple singularity equilibrium as the research subject, the existence of the equilibrium, the stability and the bifurcation of the system near the equilibrium under different parametric conditions are studied. Using the method of mathematical analysis, the existence of the real roots of the corresponding characteristic equation under the different parametric conditions is analyzed, and the local manifolds of the equilibrium are gotten, then the possible bifurcations are guessed. The parametric conditions under which the equilibrium is saddle-focus are analyzed carefully by the Cardan formula. Moreover, the conditions of codimension-one Hopf bifucation and the prerequisites of the supercritical and subcritical Hopf bifurcation are found by computation. The results show that the system has abundant stability and bifurcation, and can also supply theorical support for the proof of the existence of the homoclinic or heteroclinic loop connecting saddle-focus and the Silnikov's chaos. This method can be extended to study the other higher nonlinear systems.
5. Prediction of fibre architecture and adaptation in diseased carotid bifurcations.
LENUS (Irish Health Repository)
Creane, Arthur
2011-12-01
Many studies have used patient-specific finite element models to estimate the stress environment in atherosclerotic plaques, attempting to correlate the magnitude of stress to plaque vulnerability. In complex geometries, few studies have incorporated the anisotropic material response of arterial tissue. This paper presents a fibre remodelling algorithm to predict the fibre architecture, and thus anisotropic material response in four patient-specific models of the carotid bifurcation. The change in fibre architecture during disease progression and its affect on the stress environment in the plaque were predicted. The mean fibre directions were assumed to lie at an angle between the two positive principal strain directions. The angle and the degree of dispersion were assumed to depend on the ratio of principal strain values. Results were compared with experimental observations and other numerical studies. In non-branching regions of each model, the typical double helix arterial fibre pattern was predicted while at the bifurcation and in regions of plaque burden, more complex fibre architectures were found. The predicted change in fibre architecture in the arterial tissue during plaque progression was found to alter the stress environment in the plaque. This suggests that the specimen-specific anisotropic response of the tissue should be taken into account to accurately predict stresses in the plaque. Since determination of the fibre architecture in vivo is a difficult task, the system presented here provides a useful method of estimating the fibre architecture in complex arterial geometries.
6. Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays
Directory of Open Access Journals (Sweden)
Huitao Zhao
2013-01-01
Full Text Available A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998 for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.
7. Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling
International Nuclear Information System (INIS)
Song Yongli; Tadé, Moses O; Zhang Tonghua
2009-01-01
In this paper, a delayed neural network with unidirectional coupling is considered which consists of two two-dimensional nonlinear differential equation systems with exponential decay where one system receives a delayed input from the other system. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the centre manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay-differential equations combined with representation theory of Lie groups. Then the global continuation of phase-locked periodic solutions is investigated. Numerical simulations are given to illustrate the results obtained
8. Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves
International Nuclear Information System (INIS)
Sushko, Iryna; Agliari, Anna; Gardini, Laura
2006-01-01
We study the structure of the 2D bifurcation diagram for a two-parameter family of piecewise smooth unimodal maps f with one break point. Analysing the parameters of the normal form for the border-collision bifurcation of an attracting n-cycle of the map f, we describe the possible kinds of dynamics associated with such a bifurcation. Emergence and role of border-collision bifurcation curves in the 2D bifurcation plane are studied. Particular attention is paid also to the curves of homoclinic bifurcations giving rise to the band merging of pieces of cyclic chaotic intervals
9. Anticontrol of Hopf bifurcation and control of chaos for a finance system through washout filters with time delay.
Science.gov (United States)
Zhao, Huitao; Lu, Mengxia; Zuo, Junmei
2014-01-01
A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system.
10. Bifurcation sets of the motion of a heavy rigid body around a fixed point in Goryatchev-Tchaplygin case
International Nuclear Information System (INIS)
Quazzani, T.H.A.; Dekkaki, S.; Kharbach, J.; Quazzani-Ja, M.
2000-01-01
In this paper, the topology of Hamiltonian flows is described on the real phase space for the Goryatchev-Tchaplygin top. By making use of Fomenko's theory of surgery on Liouville tori, it is given a complete description of the generic bifurcations of the common level sets of the first integrals. It is also given a numerical investigation of these bifurcations. Explicit periodic solutions for singular common level sets of the first integrals were determined
11. Discretizing the transcritical and pitchfork bifurcations – conjugacy results
KAUST Repository
Ló czi, Lajos
2015-01-01
© 2015 Taylor & Francis. We present two case studies in one-dimensional dynamics concerning the discretization of transcritical (TC) and pitchfork (PF) bifurcations. In the vicinity of a TC or PF bifurcation point and under some natural assumptions
12. The drug-eluting resorbable magnesium vascular scaffold in complex coronary bifurcations: insights from an in vivo multimodality imaging study.
Science.gov (United States)
Bennett, Johan; Vanhaverbeke, Maarten; Vanden Driessche, Nina; Hiltrop, Nick; Adriaenssens, Tom; Desmet, Walter; Sinnaeve, Peter; Dubois, Christophe
2018-04-20
This acute in vivo study sought to provide insights regarding the feasibility of performing complex bifurcation stenting with Magmaris magnesium alloy bioresorbable scaffolds (Biotronik, Bulach, Switzerland). Twenty-five New Zealand White rabbits underwent stenting of non-diseased aorto-iliac bifurcations with the Magmaris using provisional (PS; n=5), culotte (n=6), modified T (n=6), or T and protrusion (TAP, n=8) stenting techniques. Angiography, optical coherence tomography and micro-computed tomography were performed. Angiographic results were good without evidence of side branch (SB) compromise. In 9/25 procedures, strut fractures were identified with minimal luminal compromise in two cases. PS opened the SB optimally without evidence of scaffold compromise. Culotte resulted in complete bifurcation coverage and good scaffold expansion; single strut fractures were present in three out of six and double fractures in one out of six procedures. Modified T and TAP resulted in complete bifurcation coverage, minimal neocarina double-strut layers and good expansion. In two out of six modified T procedures, strut fractures were present with SB scaffold deformity present in an additional two out of six procedures. In three out of eight TAP procedures, strut fractures were present without compromising overall scaffold integrity. Bifurcation stenting using Magmaris appears feasible. PS with additional TAP whenever needed seems a reasonable approach. Whenever a two-stent technique is planned, TAP appears most favourable whilst modified T and culotte stenting should probably be avoided.
13. Comments on the Bifurcation Structure of 1D Maps
DEFF Research Database (Denmark)
Belykh, V.N.; Mosekilde, Erik
1997-01-01
-within-a-box structure of the total bifurcation set. This presents a picture in which the homoclinic orbit bifurcations act as a skeleton for the bifurcational set. At the same time, experimental results on continued subharmonic generation for piezoelectrically amplified sound waves, predating the Feigenbaum theory......, are called into attention....
14. NUMERICAL HOPF BIFURCATION OF DELAY-DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
2006-01-01
In this paper we consider the numerical solution of some delay differential equations undergoing a Hopf bifurcation. We prove that if the delay differential equations have a Hopf bifurcation point atλ=λ*, then the numerical solution of the equation also has a Hopf bifurcation point atλh =λ* + O(h).
15. Turing instability and bifurcation analysis in a diffusive bimolecular system with delayed feedback
Science.gov (United States)
Wei, Xin; Wei, Junjie
2017-09-01
A diffusive autocatalytic bimolecular model with delayed feedback subject to Neumann boundary conditions is considered. We mainly study the stability of the unique positive equilibrium and the existence of periodic solutions. Our study shows that diffusion can give rise to Turing instability, and the time delay can affect the stability of the positive equilibrium and result in the occurrence of Hopf bifurcations. By applying the normal form theory and center manifold reduction for partial functional differential equations, we investigate the stability and direction of the bifurcations. Finally, we give some simulations to illustrate our theoretical results.
16. Bifurcation of elastic solids with sliding interfaces
Science.gov (United States)
Bigoni, D.; Bordignon, N.; Piccolroaz, A.; Stupkiewicz, S.
2018-01-01
Lubricated sliding contact between soft solids is an interesting topic in biomechanics and for the design of small-scale engineering devices. As a model of this mechanical set-up, two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so this formulation is the objective of the present paper. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or spring-type' interfacial conditions are not able to predict bifurcations in tension, while experiments-one of which, ad hoc designed, is reported-show that these bifurcations are a reality and become possible when the correct sliding interface model is used. The presented results introduce a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact.
17. Climate bifurcation during the last deglaciation?
NARCIS (Netherlands)
Lenton, T.M.; Livina, V.N.; Dakos, V.; Scheffer, M.
2012-01-01
There were two abrupt warming events during the last deglaciation, at the start of the Bolling-Allerod and at the end of the Younger Dryas, but their underlying dynamics are unclear. Some abrupt climate changes may involve gradual forcing past a bifurcation point, in which a prevailing climate state
18. Resource competition: a bifurcation theory approach.
NARCIS (Netherlands)
Kooi, B.W.; Dutta, P.S.; Feudel, U.
2013-01-01
We develop a framework for analysing the outcome of resource competition based on bifurcation theory. We elaborate our methodology by readdressing the problem of competition of two species for two resources in a chemostat environment. In the case of perfect-essential resources it has been
19. Digital subtraction angiography of carotid bifurcation
International Nuclear Information System (INIS)
Vries, A.R. de.
1984-01-01
This study demonstrates the reliability of digital subtraction angiography (DSA) by means of intra- and interobserver investigations as well as indicating the possibility of substituting catheterangiography by DSA in the diagnosis of carotid bifurcation. Whenever insufficient information is obtained from the combination of non-invasive investigation and DSA, a catheterangiogram will be necessary. (Auth.)
20. Percutaneous coronary intervention for coronary bifurcation disease
DEFF Research Database (Denmark)
Lassen, Jens Flensted; Holm, Niels Ramsing; Banning, Adrian
2016-01-01
of combining the opinions of interventional cardiologists with the opinions of a large variety of other scientists on bifurcation management. The present 11th EBC consensus document represents the summary of the up-to-date EBC consensus and recommendations. It points to the fact that there is a multitude...
1. Bifurcation of self-folded polygonal bilayers
Science.gov (United States)
Abdullah, Arif M.; Braun, Paul V.; Hsia, K. Jimmy
2017-09-01
Motivated by the self-assembly of natural systems, researchers have investigated the stimulus-responsive curving of thin-shell structures, which is also known as self-folding. Self-folding strategies not only offer possibilities to realize complicated shapes but also promise actuation at small length scales. Biaxial mismatch strain driven self-folding bilayers demonstrate bifurcation of equilibrium shapes (from quasi-axisymmetric doubly curved to approximately singly curved) during their stimulus-responsive morphing behavior. Being a structurally instable, bifurcation could be used to tune the self-folding behavior, and hence, a detailed understanding of this phenomenon is appealing from both fundamental and practical perspectives. In this work, we investigated the bifurcation behavior of self-folding bilayer polygons. For the mechanistic understanding, we developed finite element models of planar bilayers (consisting of a stimulus-responsive and a passive layer of material) that transform into 3D curved configurations. Our experiments with cross-linked Polydimethylsiloxane samples that change shapes in organic solvents confirmed our model predictions. Finally, we explored a design scheme to generate gripper-like architectures by avoiding the bifurcation of stimulus-responsive bilayers. Our research contributes to the broad field of self-assembly as the findings could motivate functional devices across multiple disciplines such as robotics, artificial muscles, therapeutic cargos, and reconfigurable biomedical devices.
2. Complex bifurcation patterns in a discrete predator–prey model with ...
We consider the simplest model in the family of discrete predator–prey system and introduce for the first time an environmental factor in the evolution of the system by periodically modulating the natural death rateof the predator.We show that with the introduction of environmental modulation, the bifurcation structure ...
3. The bifurcation and peakons for the special C(3,2,2) equation
Keywords. C(3, 2, 2) equation; peakons; bell-shaped solitary waves; periodic cusp waves. .... In other words, the function φ is not well defined on ... Figure 2. The phase portrait bifurcation of system (22). 336. Pramana – J. Phys., Vol. 83, No.
4. Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation
International Nuclear Information System (INIS)
Ma Zhi-Min; Sun Yu-Huai; Liu Fu-Sheng
2013-01-01
In this paper, the generalized Boussinesq wave equation u tt — u xx + a(u m ) xx + bu xxxx = 0 is investigated by using the bifurcation theory and the method of phase portraits analysis. Under the different parameter conditions, the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained. (general)
5. Bifurcating Solutions to the Monodomain Model Equipped with FitzHugh-Nagumo Kinetics
Directory of Open Access Journals (Sweden)
Robert Artebrant
2009-01-01
cells surrounded by collections of normal cells. Thus, the cell model features a discontinuous coefficient. Analytical techniques are applied to approximate the time-periodic solution that arises at the Hopf bifurcation point. Accurate numerical experiments are employed to complement our findings.
6. Bifurcation analysis of a smoothed model of a forced impacting beam and comparison with an experiment
DEFF Research Database (Denmark)
Elmegård, Michael; Krauskopf, B.; Osinga, H.M.
2014-01-01
bifurca tions disappear when the transition of the switching is sufficiently and increasingly localized as the impact becomes harder. The bifurcation structure of the impact oscillator response is investigated via the one- and twoparameter continuation of periodic orbits in the driving frequency and....../or forcing amplitude. The results are in good agreement with experimental measurements....
7. Global Hopf bifurcation analysis on a BAM neural network with delays
Science.gov (United States)
Sun, Chengjun; Han, Maoan; Pang, Xiaoming
2007-01-01
A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.
8. Global Hopf bifurcation analysis on a BAM neural network with delays
International Nuclear Information System (INIS)
Sun Chengjun; Han Maoan; Pang Xiaoming
2007-01-01
A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large
9. Stability and Hopf Bifurcation Analysis on a Nonlinear Business Cycle Model
Directory of Open Access Journals (Sweden)
Liming Zhao
2016-01-01
Full Text Available This study begins with the establishment of a three-dimension business cycle model based on the condition of a fixed exchange rate. Using the established model, the reported study proceeds to describe and discuss the existence of the equilibrium and stability of the economic system near the equilibrium point as a function of the speed of market regulation and the degree of capital liquidity and a stable region is defined. In addition, the condition of Hopf bifurcation is discussed and the stability of a periodic solution, which is generated by the Hopf bifurcation and the direction of the Hopf bifurcation, is provided. Finally, a numerical simulation is provided to confirm the theoretical results. This study plays an important role in theoretical understanding of business cycle models and it is crucial for decision makers in formulating macroeconomic policies as detailed in the conclusions of this report.
10. Hopf bifurcation and chaos in a third-order phase-locked loop
Science.gov (United States)
Piqueira, José Roberto C.
2017-01-01
Phase-locked loops (PLLs) are devices able to recover time signals in several engineering applications. The literature regarding their dynamical behavior is vast, specifically considering that the process of synchronization between the input signal, coming from a remote source, and the PLL local oscillation is robust. For high-frequency applications it is usual to increase the PLL order by increasing the order of the internal filter, for guarantying good transient responses; however local parameter variations imply structural instability, thus provoking a Hopf bifurcation and a route to chaos for the phase error. Here, one usual architecture for a third-order PLL is studied and a range of permitted parameters is derived, providing a rule of thumb for designers. Out of this range, a Hopf bifurcation appears and, by increasing parameters, the periodic solution originated by the Hopf bifurcation degenerates into a chaotic attractor, therefore, preventing synchronization.
11. Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation
Science.gov (United States)
Renson, Ludovic; Barton, David A. W.; Neild, Simon A.
Control-based continuation (CBC) is a means of applying numerical continuation directly to a physical experiment for bifurcation analysis without the use of a mathematical model. CBC enables the detection and tracking of bifurcations directly, without the need for a post-processing stage as is often the case for more traditional experimental approaches. In this paper, we use CBC to directly locate limit-point bifurcations of a periodically forced oscillator and track them as forcing parameters are varied. Backbone curves, which capture the overall frequency-amplitude dependence of the system’s forced response, are also traced out directly. The proposed method is demonstrated on a single-degree-of-freedom mechanical system with a nonlinear stiffness characteristic. Results are presented for two configurations of the nonlinearity — one where it exhibits a hardening stiffness characteristic and one where it exhibits softening-hardening.
12. Double resonance long period fiber grating for detection of E. coli in trace concentration by choosing a proper bacteriophage
Science.gov (United States)
Chiniforooshan, Y.; Celebanska, A.; Janik, M.; Mikulic, P.; Haddad, F.; Perreault, J.; Bock, W. J.
2017-04-01
There is a critical need of a fast, specific and reliable assay for biological species. To address this need, long period fiber gratings (LPFG) among other fiber optic sensors can be used because of their high sensitivity to changes in surrounding medium. In this work we fabricated and used two over-etched LPFGs. One of them was covered with T4 Phage and the other was covered with MS2 phage that both specifically bind with Escherichia coli (E. coli) bacteria. This bacterium is a major cause of the food contaminations and outbreaks. We showed achieving a highest sensitivity region of the LPFG and the way to fine tune to that region by over-etching the grating. Finally, using the highly sensitive LPFG platform we could detect E. coli at concentrations as low as 100 colony forming units (CFU), by covering the LPFG with an optimized bio-functionalization of the fiber surface with MS2 bacteriophage.
13. Consistent gaussian basis sets of double- and triple-zeta valence with polarization quality of the fifth period for solid-state calculations.
Science.gov (United States)
Laun, Joachim; Vilela Oliveira, Daniel; Bredow, Thomas
2018-02-22
Consistent basis sets of double- and triple-zeta valence with polarization quality for the fifth period have been derived for periodic quantum-chemical solid-state calculations with the crystalline-orbital program CRYSTAL. They are an extension of the pob-TZVP basis sets, and are based on the full-relativistic effective core potentials (ECPs) of the Stuttgart/Cologne group and on the def2-SVP and def2-TZVP valence basis of the Ahlrichs group. We optimized orbital exponents and contraction coefficients to supply robust and stable self-consistent field (SCF) convergence for a wide range of different compounds. The computed crystal structures are compared to those obtained with standard basis sets available from the CRYSTAL basis set database. For the applied hybrid density functional PW1PW, the average deviations of calculated lattice constants from experimental references are smaller with pob-DZVP and pob-TZVP than with standard basis sets. © 2018 Wiley Periodicals, Inc. © 2018 Wiley Periodicals, Inc.
14. Simplest bifurcation diagrams for monotone families of vector fields on a torus
Science.gov (United States)
Baesens, C.; MacKay, R. S.
2018-06-01
In part 1, we prove that the bifurcation diagram for a monotone two-parameter family of vector fields on a torus has to be at least as complicated as the conjectured simplest one proposed in Baesens et al (1991 Physica D 49 387–475). To achieve this, we define ‘simplest’ by sequentially minimising the numbers of equilibria, Bogdanov–Takens points, closed curves of centre and of neutral saddle, intersections of curves of centre and neutral saddle, Reeb components, other invariant annuli, arcs of rotational homoclinic bifurcation of horizontal homotopy type, necklace points, contractible periodic orbits, points of neutral horizontal homoclinic bifurcation and half-plane fan points. We obtain two types of simplest case, including that initially proposed. In part 2, we analyse the bifurcation diagram for an explicit monotone family of vector fields on a torus and prove that it has at most two equilibria, precisely four Bogdanov–Takens points, no closed curves of centre nor closed curves of neutral saddle, at most two Reeb components, precisely four arcs of rotational homoclinic connection of ‘horizontal’ homotopy type, eight horizontal saddle-node loop points, two necklace points, four points of neutral horizontal homoclinic connection, and two half-plane fan points, and there is no simultaneous existence of centre and neutral saddle, nor contractible homoclinic connection to a neutral saddle. Furthermore, we prove that all saddle-nodes, Bogdanov–Takens points, non-neutral and neutral horizontal homoclinic bifurcations are non-degenerate and the Hopf condition is satisfied for all centres. We also find it has four points of degenerate Hopf bifurcation. It thus provides an example of a family satisfying all the assumptions of part 1 except the one of at most one contractible periodic orbit.
15. Hopf bifurcation in a reaction-diffusive two-species model with nonlocal delay effect and general functional response
International Nuclear Information System (INIS)
Han, Renji; Dai, Binxiang
2017-01-01
Highlights: • We model general two-dimensional reaction-diffusion with nonlocal delay. • The existence of unique positive steady state is studied. • The bilinear form for the proposed system is given. • The existence, direction of Hopf bifurcation are given by symmetry method. - Abstract: A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments.
16. Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling
Science.gov (United States)
Song, Yongli; Zhang, Tonghua; Tadé, Moses O.
2009-12-01
The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.
17. Oral hyaluronan relieves wrinkles: a double-blinded, placebo-controlled study over a 12-week period
Directory of Open Access Journals (Sweden)
Oe M
2017-07-01
Full Text Available Mariko Oe,1 Seigo Sakai,1 Hideto Yoshida,1 Nao Okado,1 Haruna Kaneda,1 Yasunobu Masuda,1 Osamu Urushibata2 1R&D Division, Kewpie Corporation, Sengawa-cho, Chofu-shi, 2Department of Dermatology, Toho University Ohashi Medical Center, Ohashi, Meguro-ku, Tokyo, Japan Background: Hyaluronan (HA has critical moisturizing property and high water retention capacity especially for human skin. This study aimed to evaluate the effect of oral intake of HA. Methods: The mean molecular weight (MW of HA is 2 k and 300 k. Sixty Japanese male and female subjects aged 22–59 years who presented with crow’s feet wrinkles were randomly assigned to the HA 2 k or HA 300 k at 120 mg/day or the placebo group. The subjects were administered HA at a rate of 120 mg/day or a placebo for 12 weeks. The skin wrinkles were evaluated by image analysis of skin wrinkle replicas, and their skin condition was evaluated using a questionnaire survey. Results: During the study period, the HA groups showed better level of the whole sulcus volume ratio, wrinkle area ratio, and wrinkle volume ratio than the placebo group. After 8 weeks of ingestion, the HA 300 k group showed significantly diminished wrinkles compared with the placebo group. Skin luster and suppleness significantly improved after 12 weeks in all groups compared with the baseline. Conclusion: The results suggest that oral HA (both HA 2 k and HA 300 k inhibits skin wrinkles and improves skin condition. Keywords: hyaluronic acid, dietary supplement, skin, wrinkle volume, molecular weight
18. Impact adding bifurcation in an autonomous hybrid dynamical model of church bell
Science.gov (United States)
Brzeski, P.; Chong, A. S. E.; Wiercigroch, M.; Perlikowski, P.
2018-05-01
In this paper we present the bifurcation analysis of the yoke-bell-clapper system which corresponds to the biggest bell "Serce Lodzi" mounted in the Cathedral Basilica of St Stanislaus Kostka, Lodz, Poland. The mathematical model of the system considered in this work has been derived and verified based on measurements of dynamics of the real bell. We perform numerical analysis both by direct numerical integration and path-following method using toolbox ABESPOL (Chong, 2016). By introducing the active yoke the position of the bell-clapper system with respect to the yoke axis of rotation can be easily changed and it can be used to probe the system dynamics. We found a wide variety of periodic and non-periodic solutions, and examined the ranges of coexistence of solutions and transitions between them via different types of bifurcations. Finally, a new type of bifurcation induced by a grazing event - an "impact adding bifurcation" has been proposed. When it occurs, the number of impacts between the bell and the clapper is increasing while the period of the system's motion stays the same.
19. Bone tunnel change develops within two weeks of double-bundle anterior cruciate ligament reconstruction using hamstring autograft: A comparison of different postoperative immobilization periods using computed tomography.
Science.gov (United States)
Shimizu, Ryo; Adachi, Nobuo; Ishifuro, Minoru; Nakamae, Atsuo; Ishikawa, Masakazu; Deie, Masataka; Ochi, Mitsuo
2017-10-01
The purpose of this study was to evaluate bone tunnel changes following anterior cruciate ligament (ACL) reconstruction during the early postoperative period using computed tomography (CT), and to understand the impact of postoperative immobilization on these changes. Twenty patients who underwent double-bundle ACL reconstruction using hamstring tendon autografts were included. We subcategorized patients into two groups: patients who underwent isolated ACL reconstruction and had three days of knee immobilization (Group A, n=10); and patients with concomitant meniscus injuries who underwent ACL reconstruction and meniscus repair simultaneously (Group B, n=10) had their knees immobilized for two weeks after surgery. Bone tunnel enlargement was evaluated using CT imaging at one to three days, two weeks, one month, three months and six months after surgery. The cross-sectional area of the femoral and tibial tunnels was measured, and enlargement rate was calculated. The tunnel center location at two weeks after surgery was also evaluated. The mean cross-sectional area adjacent to the joint space of the femoral and tibial tunnels significantly increased immediately after surgery, especially in the first month (P0.01). There was no significant difference in tunnel enlargement rate between group A and B. Tunnel center location changed even in the first two weeks. Bone tunnel enlargement following double-bundle ACL reconstruction occurred at an earlier time point after surgery than anticipated. Postoperative immobilization could not prevent bone tunnel enlargement, but might prevent tunnel migration. Copyright © 2017 Elsevier B.V. All rights reserved.
20. Stochastic bifurcation in a model of love with colored noise
Science.gov (United States)
Yue, Xiaokui; Dai, Honghua; Yuan, Jianping
2015-07-01
In this paper, we wish to examine the stochastic bifurcation induced by multiplicative Gaussian colored noise in a dynamical model of love where the random factor is used to describe the complexity and unpredictability of psychological systems. First, the dynamics in deterministic love-triangle model are considered briefly including equilibrium points and their stability, chaotic behaviors and chaotic attractors. Then, the influences of Gaussian colored noise with different parameters are explored such as the phase plots, top Lyapunov exponents, stationary probability density function (PDF) and stochastic bifurcation. The stochastic P-bifurcation through a qualitative change of the stationary PDF will be observed and bifurcation diagram on parameter plane of correlation time and noise intensity is presented to find the bifurcation behaviors in detail. Finally, the top Lyapunov exponent is computed to determine the D-bifurcation when the noise intensity achieves to a critical value. By comparison, we find there is no connection between two kinds of stochastic bifurcation.
1. Bifurcation Behavior Analysis in a Predator-Prey Model
Directory of Open Access Journals (Sweden)
Nan Wang
2016-01-01
Full Text Available A predator-prey model is studied mathematically and numerically. The aim is to explore how some key factors influence dynamic evolutionary mechanism of steady conversion and bifurcation behavior in predator-prey model. The theoretical works have been pursuing the investigation of the existence and stability of the equilibria, as well as the occurrence of bifurcation behaviors (transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation, which can deduce a standard parameter controlled relationship and in turn provide a theoretical basis for the numerical simulation. Numerical analysis ensures reliability of the theoretical results and illustrates that three stable equilibria will arise simultaneously in the model. It testifies the existence of Bogdanov-Takens bifurcation, too. It should also be stressed that the dynamic evolutionary mechanism of steady conversion and bifurcation behavior mainly depend on a specific key parameter. In a word, all these results are expected to be of use in the study of the dynamic complexity of ecosystems.
2. Effects of Colored Noise on Periodic Orbits in a One-Dimensional Map
Science.gov (United States)
Li, Feng-Guo; Ai, Bao-Quan
2011-06-01
Noise can induce inverse period-doubling transition and chaos. The effects of the colored noise on periodic orbits, of the different periodic sequences in the logistic map, are investigated. It is found that the dynamical behaviors of the orbits, induced by an exponentially correlated colored noise, are different in the mergence of transition, and the effects of the noise intensity on their dynamical behaviors are different from the effects of the correlation time of noise. Remarkably, the noise can induce new periodic orbits, namely, two new orbits emerge in the period-four sequence at the bifurcation parameter value μ = 3.5, four new orbits in the period-eight sequence at μ = 3.55, and three new orbits in the period-six sequence at μ = 3.846, respectively. Moreover, the dynamical behaviors of the new orbits clearly show the resonancelike response to the colored noise.
3. Effects of Colored Noise on Periodic Orbits in a One-Dimensional Map
International Nuclear Information System (INIS)
Li Fengguo; Ai Baoquan
2011-01-01
Noise can induce inverse period-doubling transition and chaos. The effects of the colored noise on periodic orbits, of the different periodic sequences in the logistic map, are investigated. It is found that the dynamical behaviors of the orbits, induced by an exponentially correlated colored noise, are different in the mergence of transition, and the effects of the noise intensity on their dynamical behaviors are different from the effects of the correlation time of noise. Remarkably, the noise can induce new periodic orbits, namely, two new orbits emerge in the period-four sequence at the bifurcation parameter value μ = 3.5, four new orbits in the period-eight sequence at μ = 3.55, and three new orbits in the period-six sequence at μ = 3.846, respectively. Moreover, the dynamical behaviors of the new orbits clearly show the resonancelike response to the colored noise. (general)
4. Efficacy and safety of teneligliptin add-on to insulin monotherapy in Japanese patients with type 2 diabetes mellitus: a 16-week, randomized, double-blind, placebo-controlled trial with an open-label period.
Science.gov (United States)
Kadowaki, Takashi; Kondo, Kazuoki; Sasaki, Noriyuki; Miyayama, Kyoko; Yokota, Shoko; Terata, Ryuji; Gouda, Maki
2017-09-01
To assess the efficacy and safety of teneligliptin as add-on to insulin monotherapy in patients with type 2 diabetes mellitus (T2DM). In a 16-week, double-blind period, 148 Japanese T2DM patients with inadequate glycemic control with insulin and diet/exercise therapies were randomized to placebo or teneligliptin 20 mg. In a subsequent 36-week, open-label period, all patients received teneligliptin once daily. The primary outcome measure was change in HbA1c at the end of the double-blind period. The difference between placebo and teneligliptin in change in HbA1c in the double-blind period (least squares mean ± SE) was -0.80% ± 0.11%; teneligliptin was superior (ANCOVA, P 1). The HbA1c-lowering effect of teneligliptin was maintained throughout the open-label period. The incidence of adverse events was 53.5% with placebo and 44.2% with teneligliptin in the double-blind period, 66.7% in the placebo/teneligliptin group in the open-label period, and 77.9% in the teneligliptin/teneligliptin group over both double-blind/open-label periods. The incidence of hypoglycemic symptoms was 11.1% in the placebo/teneligliptin group in the open-label period and 27.3% in the teneligliptin/teneligliptin group over both double-blind/open-label periods. Teneligliptin was effective and well tolerated in Japanese T2DM patients with inadequate glycemic control. NCT02081599.
5. Equivariant bifurcation in a coupled complex-valued neural network rings
International Nuclear Information System (INIS)
Zhang, Chunrui; Sui, Zhenzhang; Li, Hongpeng
2017-01-01
Highlights: • Complex value Hopfield-type network with Z4 × Z2 symmetry is discussed. • The spatio-temporal patterns of bifurcating periodic oscillations are obtained. • The oscillations can be in phase or anti-phase depending on the parameters and delay. - Abstract: Network with interacting loops and time delays are common in physiological systems. In the past few years, the dynamic behaviors of coupled interacting loops neural networks have been widely studied due to their extensive applications in classification of pattern recognition, signal processing, image processing, engineering optimization and animal locomotion, and other areas, see the references therein. In a large amount of applications, complex signals often occur and the complex-valued recurrent neural networks are preferable. In this paper, we study a complex value Hopfield-type network that consists of a pair of one-way rings each with four neurons and two-way coupling between each ring. We discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. The existence of multiple branches of bifurcating periodic solution is obtained. We also found that the spatio-temporal patterns of bifurcating periodic oscillations alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural network oscillators. The oscillations of corresponding neurons in the two loops can be in phase or anti-phase depending on the parameters and delay. Some numerical simulations support our analysis results.
6. Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia. Part III: Where is the Joint Between Norms and Hazards from a Bifurcation Analysis Viewpoint?
Science.gov (United States)
Kolokolov, Yury; Monovskaya, Anna
2016-06-01
The paper continues the application of the bifurcation analysis in the research on local climate dynamics based on processing the historically observed data on the daily average land surface air temperature. Since the analyzed data are from instrumental measurements, we are doing the experimental bifurcation analysis. In particular, we focus on the discussion where is the joint between the normal dynamics of local climate systems (norms) and situations with the potential to create damages (hazards)? We illustrate that, perhaps, the criteria for hazards (or violent and unfavorable weather factors) relate mainly to empirical considerations from human opinion, but not to the natural qualitative changes of climate dynamics. To build the bifurcation diagrams, we base on the unconventional conceptual model (HDS-model) which originates from the hysteresis regulator with double synchronization. The HDS-model is characterized by a variable structure with the competition between the amplitude quantization and the time quantization. Then the intermittency between three periodical processes is considered as the typical behavior of local climate systems instead of both chaos and quasi-periodicity in order to excuse the variety of local climate dynamics. From the known specific regularities of the HDS-model dynamics, we try to find a way to decompose the local behaviors into homogeneous units within the time sections with homogeneous dynamics. Here, we present the first results of such decomposition, where the quasi-homogeneous sections (QHS) are determined on the basis of the modified bifurcation diagrams, and the units are reconstructed within the limits connected with the problem of shape defects. Nevertheless, the proposed analysis of the local climate dynamics (QHS-analysis) allows to exhibit how the comparatively modest temperature differences between the mentioned units in an annual scale can step-by-step expand into the great temperature differences of the daily
7. Experimental Study of Flow in a Bifurcation
Science.gov (United States)
2003-11-01
An instability known as the Dean vortex occurs in curved pipes with a longitudinal pressure gradient. A similar effect is manifest in the flow in a converging or diverging bifurcation, such as those found in the human respiratory airways. The goal of this study is to characterize secondary flows in a bifurcation. Particle image velocimetry (PIV) and laser-induced fluorescence (LIF) experiments were performed in a clear, plastic model. Results show the strength and migration of secondary vortices. Primary velocity features are also presented along with dispersion patterns from dye visualization. Unsteadiness, associated with a hairpin vortex, was also found at higher Re. This work can be used to assess the dispersion of particles in the lung. Medical delivery systems and pollution effect studies would profit from such an understanding.
8. Bifurcations and chaos of DNA solitonic dynamics
International Nuclear Information System (INIS)
Gonzalez, J.A.; Martin-Landrove, M.; Carbo, J.R.; Chacon, M.
1994-09-01
We investigated the nonlinear DNA torsional equations proposed by Yakushevich in the presence of damping and external torques. Analytical expressions for some solutions are obtained in the case of the isolated chain. Special attention is paid to the stability of the solutions and the range of soliton interaction in the general case. The bifurcation analysis is performed and prediction of chaos is obtained for some set of parameters. Some biological implications are suggested. (author). 11 refs, 13 figs
9. Torus bifurcations in multilevel converter systems
DEFF Research Database (Denmark)
Zhusubaliyev, Zhanybai T.; Mosekilde, Erik; Yanochkina, Olga O.
2011-01-01
embedded one into the other and with their basins of attraction delineated by intervening repelling tori. The paper illustrates the coexistence of three stable tori with different resonance behaviors and shows how reconstruction of these tori takes place across the borders of different dynamical regimes....... The paper also demonstrates how pairs of attracting and repelling tori emerge through border-collision torus-birth and border-collision torus-fold bifurcations. © 2011 World Scientific Publishing Company....
10. Sex differences in intracranial arterial bifurcations
DEFF Research Database (Denmark)
Lindekleiv, Haakon M; Valen-Sendstad, Kristian; Morgan, Michael K
2010-01-01
Subarachnoid hemorrhage (SAH) is a serious condition, occurring more frequently in females than in males. SAH is mainly caused by rupture of an intracranial aneurysm, which is formed by localized dilation of the intracranial arterial vessel wall, usually at the apex of the arterial bifurcation. T....... The female preponderance is usually explained by systemic factors (hormonal influences and intrinsic wall weakness); however, the uneven sex distribution of intracranial aneurysms suggests a possible physiologic factor-a local sex difference in the intracranial arteries....
11. Drift bifurcation detection for dissipative solitons
International Nuclear Information System (INIS)
Liehr, A W; Boedeker, H U; Roettger, M C; Frank, T D; Friedrich, R; Purwins, H-G
2003-01-01
We report on the experimental detection of a drift bifurcation for dissipative solitons, which we observe in the form of current filaments in a planar semiconductor-gas-discharge system. By introducing a new stochastic data analysis technique we find that due to a change of system parameters the dissipative solitons undergo a transition from purely noise-driven objects with Brownian motion to particles with a dynamically stabilized finite velocity
12. Bifurcation analysis of a delay reaction-diffusion malware propagation model with feedback control
Science.gov (United States)
Zhu, Linhe; Zhao, Hongyong; Wang, Xiaoming
2015-05-01
With the rapid development of network information technology, information networks security has become a very critical issue in our work and daily life. This paper attempts to develop a delay reaction-diffusion model with a state feedback controller to describe the process of malware propagation in mobile wireless sensor networks (MWSNs). By analyzing the stability and Hopf bifurcation, we show that the state feedback method can successfully be used to control unstable steady states or periodic oscillations. Moreover, formulas for determining the properties of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidences show that the linear term of the controller is enough to delay the onset of the Hopf bifurcation and the properties of the bifurcation can be regulated to achieve some desirable behaviors by choosing the appropriate higher terms of the controller. Furthermore, we obtain that the spatial-temporal dynamic characteristics of malware propagation are closely related to the rate constant for nodes leaving the infective class for recovered class and the mobile behavior of nodes.
13. Hopf bifurcation of an (n + 1) -neuron bidirectional associative memory neural network model with delays.
Science.gov (United States)
Xiao, Min; Zheng, Wei Xing; Cao, Jinde
2013-01-01
Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n + 1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.
14. Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap
International Nuclear Information System (INIS)
Qu Shixian; Lu Yongzhi; Zhang Lin; He Daren
2008-01-01
Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-11, period-6, chaotic band-12 and band-6 attractors. They are induced by different mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically. (general)
15. Energized Oxygen : Speiser Current Sheet Bifurcation
Science.gov (United States)
George, D. E.; Jahn, J. M.
2017-12-01
A single population of energized Oxygen (O+) is shown to produce a cross-tail bifurcated current sheet in 2.5D PIC simulations of the magnetotail without the influence of magnetic reconnection. Treatment of oxygen in simulations of space plasmas, specifically a magnetotail current sheet, has been limited to thermal energies despite observations of and mechanisms which explain energized ions. We performed simulations of a homogeneous oxygen background, that has been energized in a physically appropriate manner, to study the behavior of current sheets and magnetic reconnection, specifically their bifurcation. This work uses a 2.5D explicit Particle-In-a-Cell (PIC) code to investigate the dynamics of energized heavy ions as they stream Dawn-to-Dusk in the magnetotail current sheet. We present a simulation study dealing with the response of a current sheet system to energized oxygen ions. We establish a, well known and studied, 2-species GEM Challenge Harris current sheet as a starting point. This system is known to eventually evolve and produce magnetic reconnection upon thinning of the current sheet. We added a uniform distribution of thermal O+ to the background. This 3-species system is also known to eventually evolve and produce magnetic reconnection. We add one additional variable to the system by providing an initial duskward velocity to energize the O+. We also traced individual particle motion within the PIC simulation. Three main results are shown. First, energized dawn- dusk streaming ions are clearly seen to exhibit sustained Speiser motion. Second, a single population of heavy ions clearly produces a stable bifurcated current sheet. Third, magnetic reconnection is not required to produce the bifurcated current sheet. Finally a bifurcated current sheet is compatible with the Harris current sheet model. This work is the first step in a series of investigations aimed at studying the effects of energized heavy ions on magnetic reconnection. This work differs
16. Passive band-gap reconfiguration born from bifurcation asymmetry.
Science.gov (United States)
Bernard, Brian P; Mann, Brian P
2013-11-01
Current periodic structures are constrained to have fixed energy transmission behavior unless active control or component replacement is used to alter their wave propagation characteristics. The introduction of nonlinearity to generate multiple stable equilibria is an alternative strategy for realizing distinct energy propagation behaviors. We investigate the creation of a reconfigurable band-gap system by implementing passive switching between multiple stable states of equilibrium, to alter the level of energy attenuation in response to environmental stimuli. The ability to avoid potentially catastrophic loads is demonstrated by tailoring the bandpass and band-gap regions to coalesce for two stable equilibria and varying an external load parameter to trigger a bifurcation. The proposed phenomenon could be utilized in remote or autonomous applications where component modifications and active control are impractical.
17. Global Bifurcation of a Novel Computer Virus Propagation Model
Directory of Open Access Journals (Sweden)
Jianguo Ren
2014-01-01
Full Text Available In a recent paper by J. Ren et al. (2012, a novel computer virus propagation model under the effect of the antivirus ability in a real network is established. The analysis there only partially uncovers the dynamics behaviors of virus spread over the network in the case where around bifurcation is local. In the present paper, by mathematical analysis, it is further shown that, under appropriate parameter values, the model may undergo a global B-T bifurcation, and the curves of saddle-node bifurcation, Hopf bifurcation, and homoclinic bifurcation are obtained to illustrate the qualitative behaviors of virus propagation. On this basis, a collection of policies is recommended to prohibit the virus prevalence. To our knowledge, this is the first time the global bifurcation has been explored for the computer virus propagation. Theoretical results and corresponding suggestions may help us suppress or eliminate virus propagation in the network.
18. Nonlinear physical systems spectral analysis, stability and bifurcations
CERN Document Server
Kirillov, Oleg N
2013-01-01
Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems.Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynam
19. Bifurcation of rupture path by linear and cubic damping force
Science.gov (United States)
Dennis L. C., C.; Chew X., Y.; Lee Y., C.
2014-06-01
Bifurcation of rupture path is studied for the effect of linear and cubic damping. Momentum equation with Rayleigh factor was transformed into ordinary differential form. Bernoulli differential equation was obtained and solved by the separation of variables. Analytical or exact solutions yielded the bifurcation was visible at imaginary part when the wave was non dispersive. For the dispersive wave, bifurcation of rupture path was invisible.
20. A codimension two bifurcation in a railway bogie system
DEFF Research Database (Denmark)
Zhang, Tingting; True, Hans; Dai, Huanyun
2017-01-01
In this paper, a comprehensive analysis is presented to investigate a codimension two bifurcation that exists in a nonlinear railway bogie dynamic system combining theoretical analysis with numerical investigation. By using the running velocity V and the primary longitudinal stiffness (Formula...... coexist in a range of the bifurcation parameters which can lead to jumps in the lateral oscillation amplitude of the railway bogie system. Furthermore, reduce the values of the bifurcation parameters gradually. Firstly, the supercritical Hopf bifurcation turns into a subcritical one with multiple limit...
1. Predicting bifurcation angle effect on blood flow in the microvasculature.
Science.gov (United States)
Yang, Jiho; Pak, Y Eugene; Lee, Tae-Rin
2016-11-01
Since blood viscosity is a basic parameter for understanding hemodynamics in human physiology, great amount of research has been done in order to accurately predict this highly non-Newtonian flow property. However, previous works lacked in consideration of hemodynamic changes induced by heterogeneous vessel networks. In this paper, the effect of bifurcation on hemodynamics in a microvasculature is quantitatively predicted. The flow resistance in a single bifurcation microvessel was calculated by combining a new simple mathematical model with 3-dimensional flow simulation for varying bifurcation angles under physiological flow conditions. Interestingly, the results indicate that flow resistance induced by vessel bifurcation holds a constant value of approximately 0.44 over the whole single bifurcation model below diameter of 60μm regardless of geometric parameters including bifurcation angle. Flow solutions computed from this new model showed substantial decrement in flow velocity relative to other mathematical models, which do not include vessel bifurcation effects, while pressure remained the same. Furthermore, when applying the bifurcation angle effect to the entire microvascular network, the simulation results gave better agreements with recent in vivo experimental measurements. This finding suggests a new paradigm in microvascular blood flow properties, that vessel bifurcation itself, regardless of its angle, holds considerable influence on blood viscosity, and this phenomenon will help to develop new predictive tools in microvascular research. Copyright © 2016 Elsevier Inc. All rights reserved.
2. Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks.
Science.gov (United States)
Wang, Zhen; Campbell, Sue Ann
2017-11-01
We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z N symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.
3. Towards classification of the bifurcation structure of a spherical cavitation bubble.
Science.gov (United States)
Behnia, Sohrab; Sojahrood, Amin Jafari; Soltanpoor, Wiria; Sarkhosh, Leila
2009-12-01
We focus on a single cavitation bubble driven by ultrasound, a system which is a specimen of forced nonlinear oscillators and is characterized by its extreme sensitivity to the initial conditions. The driven radial oscillations of the bubble are considered to be implicated by the principles of chaos physics and owing to specific ranges of control parameters, can be periodic or chaotic. Despite the growing number of investigations on its dynamics, there is not yet an inclusive yardstick to sort the dynamical behavior of the bubble into classes; also, the response oscillations are so complex that long term prediction on the behavior becomes difficult to accomplish. In this study, the nonlinear dynamics of a bubble oscillator was treated numerically and the simulations were proceeded with bifurcation diagrams. The calculated bifurcation diagrams were compared in an attempt to classify the bubble dynamic characteristics when varying the control parameters. The comparison reveals distinctive bifurcation patterns as a consequence of driving the systems with unequal ratios of R(0)lambda (where R(0) is the bubble initial radius and lambda is the wavelength of the driving ultrasonic wave). Results indicated that systems having the equal ratio of R(0)lambda, share remarkable similarities in their bifurcating behavior and can be classified under a unit category.
4. Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks
Science.gov (United States)
Wang, Zhen; Campbell, Sue Ann
2017-11-01
We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with ZN symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.
5. The Growth Periods Responses of Double-season Paddy Rice to Climate Change in Hunan Province, China over the Past Two Decades
Science.gov (United States)
Wang, Y.; Li, Y.; Yi, M.; Ye, T.
2015-12-01
The shifts of timing and length of the growing season (TLGS) are important indicators of crop response to climate change. With the help of satellite image data, it becomes feasible to retrieve the TLGS in a spatially continuous manner, which also accommodates local variation of TGSs. In this article, the TGSs of paddy rice in Hunan Province, China since 1995 was retrieved using times-series curves of MODIS Normalized Difference Vegetation Index (NDVI), Enhanced Vegetation Index (EVI), and Land Surface Water Index (LSWI). The change in TLGS and its connection to regional climate change was discussed. The results showed the advance of TGSs of double-season paddy rice and the reduction of GSL in the past 20 years, which is believed to be linked to the rise in the temperature and precipitation in the growth periods. Understanding the local variation and trend of TLGS influenced by climate change is essential for making agricultural adaptive policies to reduce the risk of crop damaged, also can provide key information for studying how multi-hazards affect crop exposure.
6. Discovery of a Detached, Eclipsing 40 Minute Period Double White Dwarf Binary and a Friend: Implications for He+CO White Dwarf Mergers
International Nuclear Information System (INIS)
Brown, Warren R.; Kilic, Mukremin; Kosakowski, Alekzander; Gianninas, A.
2017-01-01
We report the discovery of two detached double white dwarf (WD) binaries, SDSS J082239.546+304857.19 and SDSS J104336.275+055149.90, with orbital periods of 40 and 46 minutes, respectively. The 40 minute system is eclipsing; it is composed of a 0.30 M ⊙ and a 0.52 M ⊙ WD. The 46 minute system is a likely LISA verification binary. The short 20 ± 2 Myr and ∼34 Myr gravitational-wave merger times of the two binaries imply that many more such systems have formed and merged over the age of the Milky Way. We update the estimated Milky Way He+CO WD binary merger rate and affirm our previously published result: He+CO WD binaries merge at a rate at least 40 times greater than the formation rate of stable mass-transfer AM CVn binaries, and so the majority must have unstable mass-transfer. The implication is that spin–orbit coupling in He+CO WD mergers is weak, or perhaps nova-like outbursts drive He+CO WDs into merger, as proposed by Shen.
7. Discovery of a Detached, Eclipsing 40 Minute Period Double White Dwarf Binary and a Friend: Implications for He+CO White Dwarf Mergers
Science.gov (United States)
Brown, Warren R.; Kilic, Mukremin; Kosakowski, Alekzander; Gianninas, A.
2017-09-01
We report the discovery of two detached double white dwarf (WD) binaries, SDSS J082239.546+304857.19 and SDSS J104336.275+055149.90, with orbital periods of 40 and 46 minutes, respectively. The 40 minute system is eclipsing; it is composed of a 0.30 M ⊙ and a 0.52 M ⊙ WD. The 46 minute system is a likely LISA verification binary. The short 20 ± 2 Myr and ˜34 Myr gravitational-wave merger times of the two binaries imply that many more such systems have formed and merged over the age of the Milky Way. We update the estimated Milky Way He+CO WD binary merger rate and affirm our previously published result: He+CO WD binaries merge at a rate at least 40 times greater than the formation rate of stable mass-transfer AM CVn binaries, and so the majority must have unstable mass-transfer. The implication is that spin-orbit coupling in He+CO WD mergers is weak, or perhaps nova-like outbursts drive He+CO WDs into merger, as proposed by Shen. Based on observations obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona, and on observations obtained with the Apache Point Observatory 3.5 m telescope, which is owned and operated by the Astrophysical Research Consortium.
8. Discovery of a Detached, Eclipsing 40 Minute Period Double White Dwarf Binary and a Friend: Implications for He+CO White Dwarf Mergers
Energy Technology Data Exchange (ETDEWEB)
Brown, Warren R. [Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138 (United States); Kilic, Mukremin; Kosakowski, Alekzander; Gianninas, A., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, 440 W. Brooks Street, Norman, OK 73019 (United States)
2017-09-20
We report the discovery of two detached double white dwarf (WD) binaries, SDSS J082239.546+304857.19 and SDSS J104336.275+055149.90, with orbital periods of 40 and 46 minutes, respectively. The 40 minute system is eclipsing; it is composed of a 0.30 M {sub ⊙} and a 0.52 M {sub ⊙} WD. The 46 minute system is a likely LISA verification binary. The short 20 ± 2 Myr and ∼34 Myr gravitational-wave merger times of the two binaries imply that many more such systems have formed and merged over the age of the Milky Way. We update the estimated Milky Way He+CO WD binary merger rate and affirm our previously published result: He+CO WD binaries merge at a rate at least 40 times greater than the formation rate of stable mass-transfer AM CVn binaries, and so the majority must have unstable mass-transfer. The implication is that spin–orbit coupling in He+CO WD mergers is weak, or perhaps nova-like outbursts drive He+CO WDs into merger, as proposed by Shen.
9. Bifurcation theory for toroidal MHD instabilities
International Nuclear Information System (INIS)
Maschke, E.K.; Morros Tosas, J.; Urquijo, G.
1992-01-01
Using a general representation of magneto-hydrodynamics in terms of stream functions and potentials, proposed earlier, a set of reduced MHD equations for the case of toroidal geometry had been derived by an appropriate ordering with respect to the inverse aspect ratio. When all dissipative terms are neglected in this reduced system, it has the same linear stability limits as the full ideal MHD equations, to the order considered. When including resistivity, thermal conductivity and viscosity, we can apply bifurcation theory to investigate nonlinear stationary solution branches related to various instabilities. In particular, we show that a stationary solution of the internal kink type can be found
10. Bifurcation, pattern formation and chaos in combustion
International Nuclear Information System (INIS)
Bayliss, A.; Matkowsky, B.J.
1991-01-01
In this paper problems in gaseous combustion and in gasless condensed phase combustion are studied both analytically and numerically. In gaseous combustion we consider the problem of a flame stabilized on a line source of fuel. The authors find both stationary and pulsating axisymmetric solutions as well as stationary and pulsating cellular solutions. The pulsating cellular solutions take the form of either traveling waves or standing waves. Transitions between these patterns occur as parameters related to the curvature of the flame front and the Lewis number are varied. In gasless condensed phase combustion both planar and nonplanar problems are studied. For planar condensed phase combustion we consider two models: accounts for melting and does not. Both models are shown to exhibit a transition from uniformly to pulsating propagating combustion when a parameter related to the activation energy is increased. Upon further increasing this parameter both models undergo a transition to chaos: by intermittency and by a period doubling sequence. In nonplanar condensed phase combustion the nonlinear development of a branch of standing wave solutions is studied and is shown to lead to relaxation oscillations and subsequently to a transition to quasi-periodicity
11. Bifurcation of Mobility, Bifurcation of Law : Externalization of migration policy before the EU Court of Justice
NARCIS (Netherlands)
Spijkerboer, T.P.
2017-01-01
The externalization of European migration policy has resulted in a bifurcation of global human mobility, which is divided along a North/South axis. In two judgments, the EU Court of Justice was confronted with cases challenging the exclusion of Syrian refugees from Europe. These cases concern core
12. Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans
Energy Technology Data Exchange (ETDEWEB)
Skardal, Per Sebastian, E-mail: [email protected] [Departament d' Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona (Spain); Restrepo, Juan G., E-mail: [email protected] [Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309 (United States)
2014-12-15
The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper, we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes—or phase reversals—low-periodic dynamics prevail, while away from the nodes, the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model.
13. Patient-reported outcomes in patients with overactive bladder treated with mirabegron and tolterodine in a prospective, double-blind, randomized, two-period crossover, multicenter study (PREFER).
Science.gov (United States)
Herschorn, Sender; Staskin, David; Tu, Le Mai; Fialkov, Jonathan; Walsh, Terry; Gooch, Katherine; Schermer, Carol R
2018-04-19
The PREFER study was an assessment of medication tolerability, treatment preference and symptom improvement during treatment with mirabegron (M) and tolterodine (T) extended release (ER) in patients with overactive bladder (OAB). In this analysis of PREFER, patient-reported outcomes (PROs) were assessed during treatment. PREFER was a two-period, 8-week crossover, double-blind, phase IV study (NCT02138747) of treatment-naïve adults with OAB ≥3 months randomized to 1 of 4 treatment sequences (M/T; T/M; M/M; T/T), separated by a 2-week washout. Tolterodine ER was dosed at 4 mg for 8 weeks and mirabegron was dosed at 25 mg for 4 weeks then increased to 50 mg for the next 4 weeks. At each visit, PROs related to treatment satisfaction, quality of life and symptom bother were assessed using the OAB Satisfaction (OAB-S; 3 independent scales/5 single-item overall assessments), OAB-q (total health-related QoL [HRQoL] and subscales [Sleep, Social, Coping, Concern] and Symptom Bother scale) and Patient Perception of Bladder Condition (PPBC) questionnaires. Responder rates were reported for OAB-q subscales based on a minimal important difference (MID; ≥ 10-point improvement) and OAB-S Medication Tolerability score ≥ 90. In total, 358 randomized patients received ≥1 dose of double-blind study medication and completed ≥1 post-baseline value (OAB-S scale, OAB-q, PPBC): M/T (n = 154), T/M (n = 144), M/M (n = 30) or T/T (n = 30). At end of treatment (EoT), mirabegron and tolterodine ER were associated with similar mean improvements in 7 of the 8 OAB-S scores investigated, OAB-q scales and PPBC. A higher percentage of patients achieved clinically relevant improvements (MID) in OAB-q scales and OAB-S Medication Tolerability score during treatment with mirabegron than tolterodine ER. On average, patients with OAB experienced improvements in treatment satisfaction, HRQoL and symptom bother that were of a similar magnitude during treatment with
14. Study of intermittent bifurcations and chaos in boost PFC converters by nonlinear discrete models
International Nuclear Information System (INIS)
Zhang Hao; Ma Xikui; Xue Bianling; Liu Weizeng
2005-01-01
This paper mainly deals with nonlinear phenomena like intermittent bifurcations and chaos in boost PFC converters under peak-current control mode. Two nonlinear models in the form of discrete maps are derived to describe precisely the nonlinear dynamics of boost PFC converters from two points of view, i.e., low- and high-frequency regimes. Based on the presented discrete models, both the evolution of intermittent behavior and the periodicity of intermittency are investigated in detail from the fast and slow-scale aspects, respectively. Numerical results show that the occurrence of intermittent bifurcations and chaos with half one line period is one of the most distinguished dynamical characteristics. Finally, we make some instructive conclusions, which prove to be helpful in improving the performances of practical circuits
15. Bifurcation into functional niches in adaptation.
Science.gov (United States)
2004-01-01
One of the central questions in evolutionary biology concerns the dynamics of adaptation and diversification. This issue can be addressed experimentally if replicate populations adapting to identical environments can be investigated in detail. We have studied 501 such replicas using digital organisms adapting to at least two fundamentally different functional niches (survival strategies) present in the same environment: one in which fast replication is the way to live, and another where exploitation of the environment's complexity leads to complex organisms with longer life spans and smaller replication rates. While these two modes of survival are closely analogous to those expected to emerge in so-called r and K selection scenarios respectively, the bifurcation of evolutionary histories according to these functional niches occurs in identical environments, under identical selective pressures. We find that the branching occurs early, and leads to drastic phenotypic differences (in fitness, sequence length, and gestation time) that are permanent and irreversible. This study confirms an earlier experimental effort using microorganisms, in that diversification can be understood at least in part in terms of bifurcations on saddle points leading to peak shifts, as in the picture drawn by Sewall Wright.
16. Bifurcations and chaos of classical trajectories in a deformed nuclear potential
International Nuclear Information System (INIS)
Carbonell, J.; Arvieu, R.
1983-01-01
The organization of the phase space of a classical nucleon in an axially symmetric deformed potential with the restriction Lsub(z)=0 is studied by drawing the Poincare surfaces of section. In the limit of small deformations three simple limits help to understand this organization. Moreover important bifurcations of periodic trajectories occur. At higher deformations multifurcations and chaos are observed. Chaos is developed to a larger extent in the heavier nuclei. (author)
17. Analysis of Vehicle Steering and Driving Bifurcation Characteristics
Directory of Open Access Journals (Sweden)
Xianbin Wang
2015-01-01
Full Text Available The typical method of vehicle steering bifurcation analysis is based on the nonlinear autonomous vehicle model deriving from the classic two degrees of freedom (2DOF linear vehicle model. This method usually neglects the driving effect on steering bifurcation characteristics. However, in the steering and driving combined conditions, the tyre under different driving conditions can provide different lateral force. The steering bifurcation mechanism without the driving effect is not able to fully reveal the vehicle steering and driving bifurcation characteristics. Aiming at the aforementioned problem, this paper analyzed the vehicle steering and driving bifurcation characteristics with the consideration of driving effect. Based on the 5DOF vehicle system dynamics model with the consideration of driving effect, the 7DOF autonomous system model was established. The vehicle steering and driving bifurcation dynamic characteristics were analyzed with different driving mode and driving torque. Taking the front-wheel-drive system as an example, the dynamic evolution process of steering and driving bifurcation was analyzed by phase space, system state variables, power spectral density, and Lyapunov index. The numerical recognition results of chaos were also provided. The research results show that the driving mode and driving torque have the obvious effect on steering and driving bifurcation characteristics.
18. Sediment discharge division at two tidally influenced river bifurcations
NARCIS (Netherlands)
Sassi, M.G.; Hoitink, A.J.F.; Vermeulen, B.; Hidayat, H.
2013-01-01
[1] We characterize and quantify the sediment discharge division at two tidally influenced river bifurcations in response to mean flow and secondary circulation by employing a boat-mounted acoustic Doppler current profiler (ADCP), to survey transects at bifurcating branches during a semidiurnal
19. Views on the Hopf bifurcation with respect to voltage instabilities
Energy Technology Data Exchange (ETDEWEB)
Roa-Sepulveda, C A [Universidad de Concepcion, Concepcion (Chile). Dept. de Ingenieria Electrica; Knight, U G [Imperial Coll. of Science and Technology, London (United Kingdom). Dept. of Electrical and Electronic Engineering
1994-12-31
This paper presents a sensitivity study of the Hopf bifurcation phenomenon which can in theory appear in power systems, with reference to the dynamics of the process and the impact of demand characteristics. Conclusions are drawn regarding power levels at which these bifurcations could appear and concern the concept of the imaginary axis as a hard` limit eigenvalue analyses. (author) 20 refs., 31 figs.
20. Numerical bifurcation analysis of a class of nonlinear renewal equations
NARCIS (Netherlands)
Breda, Dimitri; Diekmann, Odo; Liessi, Davide; Scarabel, Francesca
2016-01-01
We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic-and Ricker-type population equations and exhibits
1. Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion
Science.gov (United States)
Li, Chengxian; Liu, Haihong; Zhang, Tonghua; Yan, Fang
2017-12-01
In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.
2. Transversus abdominis plane block reduces morphine consumption in the early postoperative period following microsurgical abdominal tissue breast reconstruction: a double-blind, placebo-controlled, randomized trial.
Science.gov (United States)
Zhong, Toni; Ojha, M; Bagher, Shaghayegh; Butler, Kate; Srinivas, Coimbatore; McCluskey, Stuart A; Clarke, Hance; O'Neill, Anne C; Novak, Christine B; Hofer, Stefan O P
2014-11-01
The analgesic efficacy of the transversus abdominis plane peripheral nerve block following abdominal tissue breast reconstruction has not been studied in a randomized controlled trial. The authors conducted a double-blind, placebo-controlled, 1:1 allocation, two-arm parallel group, superiority design, randomized controlled trial in patients undergoing microsurgical abdominally based breast reconstruction. Intraoperatively, epidural catheters were inserted under direct vision through the triangle of Petit on both sides of the abdomen into the transversus abdominis plane just before rectus fascial closure. Patients received either bupivacaine (study group) or saline (placebo group) through the catheters for 2 postoperative days. All patients received hydromorphone by means of a patient-controlled analgesic pump. The primary outcome was the difference in the parenteral opioid consumption on each postoperative day between the groups. The secondary outcome measures included the following: total in-hospital opioid; antinausea medication; pain, nausea, and sedation scores; Quality of Recovery Score; time to ambulation; and hospital stay duration. Between September of 2011 and June of 2013, 93 patients were enrolled: 49 received bupivacaine and 44 received saline. There were 11 postoperative complications (13 percent); none were related to the catheter. Primary outcomes were completed by 85 of 93 patients (91.3 percent); the mean parenteral morphine consumption was significantly reduced on postoperative day 1 in the bupivacaine group (20.7±20.1 mg) compared with 30.0±19.1 mg in the control group (p=0.02). There were no significant differences in secondary outcomes. Following abdominally based breast reconstruction, transversus abdominis plane peripheral nerve block is safe and significantly reduces morphine consumption in the early postoperative period. Therapeutic, II.
3. Universal Critical Power for Nonlinear Schroedinger Equations with a Symmetric Double Well Potential
International Nuclear Information System (INIS)
Sacchetti, Andrea
2009-01-01
Here we consider stationary states for nonlinear Schroedinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.
4. Bifurcation diagram of a cubic three-parameter autonomous system
Directory of Open Access Journals (Sweden)
Lenka Barakova
2005-07-01
Full Text Available In this paper, we study the cubic three-parameter autonomous planar system $$displaylines{ dot x_1 = k_1 + k_2x_1 - x_1^3 - x_2,cr dot x_2 = k_3 x_1 - x_2, }$$ where $k_2, k_3$ are greater than 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be applied to the macroeconomical model IS-LM with Kaldor's assumptions. In this model existence of a stable limit cycles has already been studied (Andronov-Hopf bifurcation. We present the whole bifurcation diagram and among others, we prove existence of more difficult bifurcations and existence of unstable cycles.
5. Critical bifurcation surfaces of 3D discrete dynamics
Directory of Open Access Journals (Sweden)
Michael Sonis
2000-01-01
Full Text Available This paper deals with the analytical representation of bifurcations of each 3D discrete dynamics depending on the set of bifurcation parameters. The procedure of bifurcation analysis proposed in this paper represents the 3D elaboration and specification of the general algorithm of the n-dimensional linear bifurcation analysis proposed by the author earlier. It is proven that 3D domain of asymptotic stability (attraction of the fixed point for a given 3D discrete dynamics is bounded by three critical bifurcation surfaces: the divergence, flip and flutter surfaces. The analytical construction of these surfaces is achieved with the help of classical Routh–Hurvitz conditions of asymptotic stability. As an application the adjustment process proposed by T. Puu for the Cournot oligopoly model is considered in detail.
6. Bifurcation of transition paths induced by coupled bistable systems.
Science.gov (United States)
Tian, Chengzhe; Mitarai, Namiko
2016-06-07
We discuss the transition paths in a coupled bistable system consisting of interacting multiple identical bistable motifs. We propose a simple model of coupled bistable gene circuits as an example and show that its transition paths are bifurcating. We then derive a criterion to predict the bifurcation of transition paths in a generalized coupled bistable system. We confirm the validity of the theory for the example system by numerical simulation. We also demonstrate in the example system that, if the steady states of individual gene circuits are not changed by the coupling, the bifurcation pattern is not dependent on the number of gene circuits. We further show that the transition rate exponentially decreases with the number of gene circuits when the transition path does not bifurcate, while a bifurcation facilitates the transition by lowering the quasi-potential energy barrier.
7. Hopf Bifurcation of Compound Stochastic van der Pol System
Directory of Open Access Journals (Sweden)
Shaojuan Ma
2016-01-01
Full Text Available Hopf bifurcation analysis for compound stochastic van der Pol system with a bound random parameter and Gaussian white noise is investigated in this paper. By the Karhunen-Loeve (K-L expansion and the orthogonal polynomial approximation, the equivalent deterministic van der Pol system can be deduced. Based on the bifurcation theory of nonlinear deterministic system, the critical value of bifurcation parameter is obtained and the influence of random strength δ and noise intensity σ on stochastic Hopf bifurcation in compound stochastic system is discussed. At last we found that increased δ can relocate the critical value of bifurcation parameter forward while increased σ makes it backward and the influence of δ is more sensitive than σ. The results are verified by numerical simulations.
8. Topological chaos, braiding and bifurcation of almost-cyclic sets.
Science.gov (United States)
Grover, Piyush; Ross, Shane D; Stremler, Mark A; Kumar, Pankaj
2012-12-01
In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., "Topological chaos and periodic braiding of almost-cyclic sets," Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or "ghost rods" around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.
9. Climate bifurcation during the last deglaciation?
Directory of Open Access Journals (Sweden)
T. M. Lenton
2012-07-01
Full Text Available There were two abrupt warming events during the last deglaciation, at the start of the Bølling-Allerød and at the end of the Younger Dryas, but their underlying dynamics are unclear. Some abrupt climate changes may involve gradual forcing past a bifurcation point, in which a prevailing climate state loses its stability and the climate tips into an alternative state, providing an early warning signal in the form of slowing responses to perturbations, which may be accompanied by increasing variability. Alternatively, short-term stochastic variability in the climate system can trigger abrupt climate changes, without early warning. Previous work has found signals consistent with slowing down during the last deglaciation as a whole, and during the Younger Dryas, but with conflicting results in the run-up to the Bølling-Allerød. Based on this, we hypothesise that a bifurcation point was approached at the end of the Younger Dryas, in which the cold climate state, with weak Atlantic overturning circulation, lost its stability, and the climate tipped irreversibly into a warm interglacial state. To test the bifurcation hypothesis, we analysed two different climate proxies in three Greenland ice cores, from the Last Glacial Maximum to the end of the Younger Dryas. Prior to the Bølling warming, there was a robust increase in climate variability but no consistent slowing down signal, suggesting this abrupt change was probably triggered by a stochastic fluctuation. The transition to the warm Bølling-Allerød state was accompanied by a slowing down in climate dynamics and an increase in climate variability. We suggest that the Bølling warming excited an internal mode of variability in Atlantic meridional overturning circulation strength, causing multi-centennial climate fluctuations. However, the return to the Younger Dryas cold state increased climate stability. We find no consistent evidence for slowing down during the Younger Dryas, or in a longer
10. The Aortic Bifurcation Angle as a Factor in Application of the Outback for Femoropopliteal Lesions in Ipsilateral Versus Contralateral Approaches.
Science.gov (United States)
Raskin, Daniel; Khaitovich, Boris; Balan, Shmuel; Silverberg, Daniel; Halak, Moshe; Rimon, Uri
2018-01-01
To assess the technical success of the Outback reentry device in contralateral versus ipsilateral approaches for femoropopliteal arterial occlusion. A retrospective review of patients treated for critical limb ischemia (CLI) using the Outback between January 2013 and July 2016 was performed. Age, gender, length and site of the occlusion, approach site, aortic bifurcation angle, and reentry site were recorded. Calcification score was assigned at both aortic bifurcation and reentry site. Technical success was assessed. During the study period, a total of 1300 endovascular procedures were performed on 489 patients for CLI. The Outback was applied on 50 femoropopliteal chronic total occlusions. Thirty-nine contralateral and 11 ipsilateral antegrade femoral were accessed. The device was used successfully in 41 patients (82%). There were nine failures, all in the contralateral approach group. Six due to inability to deliver the device due to acute aortic bifurcation angle and three due to failure to achieve luminal reentry. Procedural success was significantly affected by the aortic bifurcation angle (p = 0.013). The Outback has high technical success rates in treatment of femoropopliteal occlusion, when applied from either an ipsi- or contralateral approach. When applied in contralateral access, acute aortic bifurcation angle predicts procedural failure.
11. Bifurcations and Patterns in Nonlinear Dissipative Systems
Energy Technology Data Exchange (ETDEWEB)
Guenter Ahlers
2005-05-27
This project consists of experimental investigations of heat transport, pattern formation, and bifurcation phenomena in non-linear non-equilibrium fluid-mechanical systems. These issues are studies in Rayleigh-B\\'enard convection, using both pure and multicomponent fluids. They are of fundamental scientific interest, but also play an important role in engineering, materials science, ecology, meteorology, geophysics, and astrophysics. For instance, various forms of convection are important in such diverse phenomena as crystal growth from a melt with or without impurities, energy production in solar ponds, flow in the earth's mantle and outer core, geo-thermal stratifications, and various oceanographic and atmospheric phenomena. Our work utilizes computer-enhanced shadowgraph imaging of flow patterns, sophisticated digital image analysis, and high-resolution heat transport measurements.
12. The bifurcations of nearly flat origami
Science.gov (United States)
Santangelo, Christian
Self-folding origami structures provide one means of fabricating complex, three-dimensional structures from a flat, two-dimensional sheet. Self-folding origami structures have been fabricated on scales ranging from macroscopic to microscopic and can have quite complicated structures with hundreds of folds arranged in complex patterns. I will describe our efforts to understand the mechanics and energetics of self-folding origami structures. Though the dimension of the configuration space of an origami structure scales with the size of the boundary and not with the number of vertices in the interior of the structure, a typical origami structure is also floppy in the sense that there are many possible ways to assign fold angles consistently. I will discuss our theoretical progress in understanding the geometry of the configuration space of origami. For random origami, the number of possible bifurcations grows surprisingly quickly even when the dimension of the configuration space is small. EFRI ODISSEI-1240441, DMR-0846582.
13. Transport Bifurcation in a Rotating Tokamak Plasma
International Nuclear Information System (INIS)
Highcock, E. G.; Barnes, M.; Schekochihin, A. A.; Parra, F. I.; Roach, C. M.; Cowley, S. C.
2010-01-01
The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all nonzero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients.
14. Bifurcated SEN with Fluid Flow Conditioners
Directory of Open Access Journals (Sweden)
F. Rivera-Perez
2014-01-01
Full Text Available This work evaluates the performance of a novel design for a bifurcated submerged entry nozzle (SEN used for the continuous casting of steel slabs. The proposed design incorporates fluid flow conditioners attached on SEN external wall. The fluid flow conditioners impose a pseudosymmetric pattern in the upper zone of the mold by inhibiting the fluid exchange between the zones created by conditioners. The performance of the SEN with fluid flow conditioners is analyzed through numerical simulations using the CFD technique. Numerical results were validated by means of physical simulations conducted on a scaled cold water model. Numerical and physical simulations confirmed that the performance of the proposed SEN is superior to a traditional one. Fluid flow conditioners reduce the liquid free surface fluctuations and minimize the occurrence of vortexes at the free surface.
15. Oscillatory bifurcation for semilinear ordinary differential equations
Directory of Open Access Journals (Sweden)
Tetsutaro Shibata
2016-06-01
\\] where $f(u = u + (1/2\\sin^k u$ ($k \\ge 2$ and $\\lambda > 0$ is a bifurcation parameter. It is known that $\\lambda$ is parameterized by the maximum norm $\\alpha = \\Vert u_\\lambda\\Vert_\\infty$ of the solution $u_\\lambda$ associated with $\\lambda$ and is written as $\\lambda = \\lambda(k,\\alpha$. When we focus on the asymptotic behavior of $\\lambda(k,\\alpha$ as $\\alpha \\to \\infty$, it is natural to expect that $\\lambda(k, \\alpha \\to \\pi^2/4$, and its convergence rate is common to $k$. Contrary to this expectation, we show that $\\lambda(2n_1+1,\\alpha$ tends to $\\pi^2/4$ faster than $\\lambda(2n_2,\\alpha$ as $\\alpha \\to \\infty$, where $n_1\\ge 1,\\ n_2 \\ge 1$ are arbitrary given integers.
16. Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We consider a simplified bidirectional associated memory (BAM neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson's criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.
17. Clausius entropy for arbitrary bifurcate null surfaces
International Nuclear Information System (INIS)
Baccetti, Valentina; Visser, Matt
2014-01-01
Jacobson’s thermodynamic derivation of the Einstein equations was originally applied only to local Rindler horizons. But at least some parts of that construction can usefully be extended to give meaningful results for arbitrary bifurcate null surfaces. As presaged in Jacobson’s original article, this more general construction sharply brings into focus the questions: is entropy objectively ‘real’? Or is entropy in some sense subjective and observer-dependent? These innocent questions open a Pandora’s box of often inconclusive debate. A consensus opinion, though certainly not universally held, seems to be that Clausius entropy (thermodynamic entropy, defined via a Clausius relation dS=đQ/T) should be objectively real, but that the ontological status of statistical entropy (Shannon or von Neumann entropy) is much more ambiguous, and much more likely to be observer-dependent. This question is particularly pressing when it comes to understanding Bekenstein entropy (black hole entropy). To perhaps further add to the confusion, we shall argue that even the Clausius entropy can often be observer-dependent. In the current article we shall conclusively demonstrate that one can meaningfully assign a notion of Clausius entropy to arbitrary bifurcate null surfaces—effectively defining a ‘virtual Clausius entropy’ for arbitrary ‘virtual (local) causal horizons’. As an application, we see that we can implement a version of the generalized second law (GSL) for this virtual Clausius entropy. This version of GSL can be related to certain (nonstandard) integral variants of the null energy condition. Because the concepts involved are rather subtle, we take some effort in being careful and explicit in developing our framework. In future work we will apply this construction to generalize Jacobson’s derivation of the Einstein equations. (paper)
18. Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays
Science.gov (United States)
Lv, Qiuyu; Liao, Xiaofeng
2018-03-01
In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting τ1, τ2 respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.
19. Stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters
International Nuclear Information System (INIS)
Xu, X.; Hu, H.Y.; Wang, H.L.
2006-01-01
It is very common that neural network systems usually involve time delays since the transmission of information between neurons is not instantaneous. Because memory intensity of the biological neuron usually depends on time history, some of the parameters may be delay dependent. Yet, little attention has been paid to the dynamics of such systems. In this Letter, a detailed analysis on the stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters is given. Moreover, the direction and the stability of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. It shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that of systems with delay-independent parameters only
20. Pitchfork bifurcation and circuit implementation of a novel Chen hyper-chaotic system
International Nuclear Information System (INIS)
Dong En-Zeng; Chen Zeng-Qiang; Chen Zai-Ping; Ni Jian-Yun
2012-01-01
In this paper, a novel four dimensional hyper-chaotic system is coined based on the Chen system, which contains two quadratic terms and five system parameters. The proposed system can generate a hyper-chaotic attractor in wide parameters regions. By using the center manifold theorem and the local bifurcation theory, a pitchfork bifurcation is demonstrated to arise at the zero equilibrium point. Numerical analysis demonstrates that the hyper-chaotic system can generate complex dynamical behaviors, e.g., a direct transition from quasi-periodic behavior to hyper-chaotic behavior. Finally, an electronic circuit is designed to implement the hyper-chaotic system, the experimental results are consist with the numerical simulations, which verifies the existence of the hyper-chaotic attractor. Due to the complex dynamic behaviors, this new hyper-chaotic system is useful in the secure communication. (general)
1. Bifurcation and nonlinear dynamic analysis of a flexible rotor supported by relative short gas journal bearings
International Nuclear Information System (INIS)
Wang, C.-C.; Jang, M.-J.; Yeh, Y.-L.
2007-01-01
This paper studies the bifurcation and nonlinear behaviors of a flexible rotor supported by relative short gas film bearings. A time-dependent mathematical model for gas journal bearings is presented. The finite difference method with successive over relation method is employed to solve the Reynolds' equation. The system state trajectory, Poincare maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal center in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and subharmonic response of the rotor and journal center. This paper shows how the dynamic behavior of this type of system varies with changes in rotor mass and rotational velocity. The results of this study contribute to a further understanding of the nonlinear dynamics of gas film rotor-bearing systems
2. Uncertainty Quantification and Bifurcation Analysis of an Airfoil with Multiple Nonlinearities
Directory of Open Access Journals (Sweden)
Haitao Liao
2013-01-01
Full Text Available In order to calculate the limit cycle oscillations and bifurcations of nonlinear aeroelastic system, the problem of finding periodic solutions with maximum vibration amplitude is transformed into a nonlinear optimization problem. An algebraic system of equations obtained by the harmonic balance method and the stability condition derived from the Floquet theory are used to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the proposed approach is validated, and the effects of structural parameter uncertainty on the limit cycle oscillations and bifurcations of an airfoil with multiple nonlinearities are studied. Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation.
3. Reliability of cortical lesion detection on double inversion recovery MRI applying the MAGNIMS-Criteria in multiple sclerosis patients within a 16-months period.
Directory of Open Access Journals (Sweden)
Tobias Djamsched Faizy
Full Text Available In patients with multiple sclerosis (MS, Double Inversion Recovery (DIR magnetic resonance imaging (MRI can be used to identify cortical lesions (CL. We sought to evaluate the reliability of CL detection on DIR longitudinally at multiple subsequent time-points applying the MAGNIMs scoring criteria for CLs.26 MS patients received a 3T-MRI (Siemens, Skyra with DIR at 12 time-points (TP within a 16 months period. Scans were assessed in random order by two different raters. Both raters separately marked all CLs on each scan and total lesion numbers were obtained for each scan-TP and patient. After a retrospective re-evaluation, the number of consensus CLs (conL was defined as the total number of CLs, which both raters finally agreed on. CLs volumes, relative signal intensities and CLs localizations were determined. Both ratings (conL vs. non-consensus scoring were compared for further analysis.A total number of n = 334 CLs were identified by both raters in 26 MS patients with a first agreement of both raters on 160 out of 334 of the CLs found (κ = 0.48. After the retrospective re-evaluation, consensus agreement increased to 233 out of 334 CL (κ = 0.69. 93.8% of conL were visible in at least 2 consecutive TP. 74.7% of the conL were visible in all 12 consecutive TP. ConL had greater mean lesion volumes and higher mean signal intensities compared to lesions that were only detected by one of the raters (p<0.05. A higher number of CLs in the frontal, parietal, temporal and occipital lobe were identified by both raters than the number of those only identified by one of the raters (p<0.05.After a first assessment, slightly less than a half of the CL were considered as reliably detectable on longitudinal DIR images. A retrospective re-evaluation notably increased the consensus agreement. However, this finding is narrowed, considering the fact that retrospective evaluation steps might not be practicable in clinical routine. Lesions that were not reliably
4. A Prospective Randomized, Double-Blind, Two-Period Crossover Pharmacokinetic Trial Comparing Green Coffee Bean Extract-A Botanically Sourced Caffeine-With a Synthetic USP Control.
Science.gov (United States)
Morton, Kayce; Knight, Katelin; Kalman, Douglas; Hewlings, Susan
2018-04-16
Coffee is a primary dietary source of the chlorogenic acids (CGAs) of phenolic compounds. Coffee contains caffeine and other phytonutrients, including CGAs. Caffeine on its own has been well characterized and descried pharmacokinetically in the literature, less so for CGAs. The purpose of this double-blind crossover study was to determine the comparative pharmacokinetics of CGAs with caffeine (natural extract) with synthetic caffeine (US Pharmacopeia [USP] standard). Sixteen healthy male subjects were randomly assigned to take 1 dose of product 1, 60 mg of botanically sourced caffeine from 480 mg of green coffee bean extract, or product 2, 60 mg of synthetic USP caffeine, with 5 days between. Blood analysis was done to determine the levels of CGA compounds, more specifically 3-, 4-, and 5-caffeoylquinic acid (CQA), and serum caffeine. The natural caffeine extract exhibited mean peak concentrations (C max ) of 3-CQA (11.4 ng/mL), 4-CQA (6.84 ng/mL), and 5-CQA (7.20 ng/mL). The mean systemic 4-hour exposure (AUC 0-4 h ) was 3-CQA (27.3 ng·h/mL), 4-CQA (16.1 ng·h/mL), and 5-CQA (15.7 ng·h/mL). The median t max was 3-CQA (1.00 hour), 4-CQA (1.00 hour), and 5-CQA (1.50 hours). The t max of caffeine was 0.75 hours (natural extract) and 0.63 hours (synthetic caffeine). C max and AUC 0-4 h of serum caffeine were statistically equivalent between products. The geometric least-squares mean ratios (GMRs) of C max and AUC 0-4 h of caffeine were 97.77% (natural extract) and 98.33% (synthetic caffeine). It would appear that CGA compounds from the natural caffeine extract are bioavailable, and 3-CGA may be the compound most absorbed. In addition, caffeine sourced from natural extract versus synthetic were statistically similar for pharmacokinetic parameters. There were no adverse events or safety concerns. © 2018 The Authors. Clinical Pharmacology in Drug Development Published by Wiley Periodicals, Inc. on behalf of The American College of Clinical Pharmacology.
5. Theoretical modeling of the dynamics of a semiconductor laser subject to double-reflector optical feedback
Energy Technology Data Exchange (ETDEWEB)
Bakry, A. [King Abdulaziz University, 80203, Department of Physics, Faculty of Science (Saudi Arabia); Abdulrhmann, S. [Jazan University, 114, Department of Physics, Faculty of Sciences (Saudi Arabia); Ahmed, M., E-mail: [email protected] [King Abdulaziz University, 80203, Department of Physics, Faculty of Science (Saudi Arabia)
2016-06-15
We theoretically model the dynamics of semiconductor lasers subject to the double-reflector feedback. The proposed model is a new modification of the time-delay rate equations of semiconductor lasers under the optical feedback to account for this type of the double-reflector feedback. We examine the influence of adding the second reflector to dynamical states induced by the single-reflector feedback: periodic oscillations, period doubling, and chaos. Regimes of both short and long external cavities are considered. The present analyses are done using the bifurcation diagram, temporal trajectory, phase portrait, and fast Fourier transform of the laser intensity. We show that adding the second reflector attracts the periodic and perioddoubling oscillations, and chaos induced by the first reflector to a route-to-continuous-wave operation. During this operation, the periodic-oscillation frequency increases with strengthening the optical feedback. We show that the chaos induced by the double-reflector feedback is more irregular than that induced by the single-reflector feedback. The power spectrum of this chaos state does not reflect information on the geometry of the optical system, which then has potential for use in chaotic (secure) optical data encryption.
6. Chaotic behavior of current-carrying plasmas in external periodic oscillations
Energy Technology Data Exchange (ETDEWEB)
Ohno, Noriyasu; Tanaka, Masayoshi; Komori, Akio; Kawai, Yoshinobu
1989-01-01
A set of cascading bifurcations and a chaotic state in the presence of an external periodic oscillation are experimentally investigated in a current-carrying plasma. The measured bifurcation sequence leading to chaos, which is controlled by changing plasma densities and the frequencies of external oscillations, is in qualitative agreement with a theory which describes anharmonic systems in periodic fields. (author).
7. Stent implantation for the treatment of wide-necked aneurysms located at internal carotid artery bifurcation
International Nuclear Information System (INIS)
Xing Ming; Yang Pengfei; Huang Qinghai; Zhao Wenyuan; Hong Bo; Xu Yi; Liu Jianmin
2012-01-01
Objective: To preliminarily evaluate the feasibility, safety and efficacy of stent placement for the treatment of wide-necked aneurysms located at internal carotid artery bifurcation. Methods: Eleven patients with wide-necked aneurysms located at internal carotid artery bifurcation, who were encountered during the period from Jan. 2004 to Dec. 2010 in hospital, were collected. A total of 16 intracranial aneurysms were detected, of which 11 were wide-necked and were located at internal carotid artery bifurcation. The diameters of the aneurysms ranged from 2.5 mm to 18 mm. Individual stent type and stenting technique was employed for each patient. Follow-up at 1, 3, 6 and 12 months after the procedure was conducted. Results: A total of 11 different stents were successfully deployed in the eleven patients. The stents included balloon expandable stent (n=1) and self-expanding stent (n=10). According to Raymond grading for the immediate occlusion of the aneurysm, grade Ⅰ (complete obliteration) was obtained in 4, grade Ⅱ (residual neck) in 2 and grade Ⅲ (residual aneurysm) in 5 cases. No procedure-related complications occurred. At the time of discharge, the modified Rankin score was 0-1 in the eleven patients. During the follow-up period lasting for 1-108 months, all the patients were in stable condition and no newly-developed neurological dysfunction or bleeding observed. Follow-up examination with angiography (1-48 months) showed that the aneurysms were cured (no visualization) in 4 cases, improved in 2 cases and in stable condition in one case. Conclusion: For the treatment of wide-necked aneurysms located at internal carotid artery bifurcation, stent implantation is clinically feasible, safe and effective. Further studies are required to evaluate its long-term efficacy. (authors)
8. Fractional noise destroys or induces a stochastic bifurcation
Energy Technology Data Exchange (ETDEWEB)
Yang, Qigui, E-mail: [email protected] [School of Sciences, South China University of Technology, Guangzhou 510640 (China); Zeng, Caibin, E-mail: [email protected] [School of Sciences, South China University of Technology, Guangzhou 510640 (China); School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640 (China); Wang, Cong, E-mail: [email protected] [School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640 (China)
2013-12-15
Little seems to be known about the stochastic bifurcation phenomena of non-Markovian systems. Our intention in this paper is to understand such complex dynamics by a simple system, namely, the Black-Scholes model driven by a mixed fractional Brownian motion. The most interesting finding is that the multiplicative fractional noise not only destroys but also induces a stochastic bifurcation under some suitable conditions. So it opens a possible way to explore the theory of stochastic bifurcation in the non-Markovian framework.
9. Optimization Design and Application of Underground Reinforced Concrete Bifurcation Pipe
Directory of Open Access Journals (Sweden)
Chao Su
2015-01-01
Full Text Available Underground reinforced concrete bifurcation pipe is an important part of conveyance structure. During construction, the workload of excavation and concrete pouring can be significantly decreased according to optimized pipe structure, and the engineering quality can be improved. This paper presents an optimization mathematical model of underground reinforced concrete bifurcation pipe structure according to real working status of several common pipe structures from real cases. Then, an optimization design system was developed based on Particle Swarm Optimization algorithm. Furthermore, take the bifurcation pipe of one hydropower station as an example: optimization analysis was conducted, and accuracy and stability of the optimization design system were verified successfully.
10. Arctic melt ponds and bifurcations in the climate system
Science.gov (United States)
Sudakov, I.; Vakulenko, S. A.; Golden, K. M.
2015-05-01
Understanding how sea ice melts is critical to climate projections. In the Arctic, melt ponds that develop on the surface of sea ice floes during the late spring and summer largely determine their albedo - a key parameter in climate modeling. Here we explore the possibility of a conceptual sea ice climate model passing through a bifurcation point - an irreversible critical threshold as the system warms, by incorporating geometric information about melt pond evolution. This study is based on a bifurcation analysis of the energy balance climate model with ice-albedo feedback as the key mechanism driving the system to bifurcation points.
11. FFT Bifurcation Analysis of Routes to Chaos via Quasiperiodic Solutions
Directory of Open Access Journals (Sweden)
L. Borkowski
2015-01-01
Full Text Available The dynamics of a ring of seven unidirectionally coupled nonlinear Duffing oscillators is studied. We show that the FFT analysis presented in form of a bifurcation graph, that is, frequency distribution versus a control parameter, can provide a valuable and helpful complement to the corresponding typical bifurcation diagram and the course of Lyapunov exponents, especially in context of detailed identification of the observed attractors. As an example, bifurcation analysis of routes to chaos via 2-frequency and 3-frequency quasiperiodicity is demonstrated.
12. Bifurcation of learning and structure formation in neuronal maps
DEFF Research Database (Denmark)
Marschler, Christian; Faust-Ellsässer, Carmen; Starke, Jens
2014-01-01
to map formation in the laminar nucleus of the barn owl's auditory system. Using equation-free methods, we perform a bifurcation analysis of spatio-temporal structure formation in the associated synaptic-weight matrix. This enables us to analyze learning as a bifurcation process and follow the unstable...... states as well. A simple time translation of the learning window function shifts the bifurcation point of structure formation and goes along with traveling waves in the map, without changing the animal's sound localization performance....
13. Bifurcations in the response of a flexible rotor in squeeze-film dampers with retainer springs
International Nuclear Information System (INIS)
Inayat-Hussain, Jawaid I.
2009-01-01
Squeeze-film dampers are commonly used in conjunction with rolling-element or hydrodynamic bearings in rotating machinery. Although these dampers serve to provide additional damping to the rotor-bearing system, there have however been some cases of rotors mounted in these dampers exhibiting non-linear behaviour. In this paper a numerical study is undertaken to determine the effects of design parameters, i.e., gravity parameter, W, mass ratio, α, and stiffness ratio, K, on the bifurcations in the response of a flexible rotor mounted in squeeze-film dampers with retainer springs. The numerical simulations were undertaken for a range of speed parameter, Ω, between 0.1 and 5.0. Numerical results showed that increasing K causes the onset speed of bifurcation to increase, whilst an increase of α reduces the onset speed of bifurcation. For a specific combination of K and α values, the onset speed of bifurcation appeared to be independent of W. The instability of the rotor response at this onset speed was due to a saddle-node bifurcation for all the parameter values investigated in this work with the exception of the combination of α = 0.1 and K = 0.5, where a secondary Hopf bifurcation was observed. The speed range of non-synchronous response was seen to decrease with the increase of α; in fact non-synchronous rotor response was totally absent for α=0.4. With the exception of the case α = 0.1, the speed range of non-synchronous response was also seen to decrease with the increase of K. Multiple responses of the rotor were observed at certain values of Ω for various combinations of parameters W, α and K, where, depending on the values of the initial conditions the rotor response could be either synchronous or quasi-periodic. The numerical results presented in this work were obtained for an unbalance parameter, U, value of 0.1, which is considered as the upper end of the normal unbalance range of most practical rotor systems. These results provide some insights
14. Ion acoustic waves and double-layers in electronegative expanding plasmas
International Nuclear Information System (INIS)
Plihon, Nicolas; Chabert, Pascal
2011-01-01
Ion acoustic waves and double-layers are observed in expanding plasmas in electronegative gases, i.e., plasmas containing an appreciable fraction of negative ions. The reported experiments are performed in argon gas with a variable amount of SF 6 . When varying the amount of SF 6 , the negative ion fraction increases and three main regimes were identified previously: (i) the plasma smoothly expands at low negative ion fraction, (ii) a static double-layer (associated with an abrupt potential drop and ion acceleration) forms at intermediate negative ion fraction, (iii) double-layers periodically form and propagate (in the plasma expansion direction) at high negative ion fraction. In this paper, we show that transition phases exist in between these regimes, where fluctuations are observed. These fluctuations are unstable slow ion acoustic waves, propagating in the direction opposite to the plasma expansion. These fluctuations are excited by the most unstable eigenmodes and display turbulent features. It is suggested that the static double layer forms when the ion acoustic fluctuations become non-linearly unstable: the double layer regime being a bifurcated state of the smoothly expanding regime. For the highest negative ion fraction, a coexistence of (upstream propagating) slow ion acoustic fluctuations and (downstream) propagating double layers was observed.
15. Eckhaus and Benjamin-Feir instabilities near a weakly inverted bifurcation
International Nuclear Information System (INIS)
Brand, H.R.; Deissler, R.J.
1992-01-01
We investigate how the criteria for two prototype instabilities in one-dimensional pattern-forming systems, namely for the Eckhaus instability and for the Benjamin-Feir instability, change as one goes from a continuous bifurcation to a spatially periodic or spatially and/or time-periodic state to the corresponding weakly inverted, i.e., hysteretic, cases. We also give the generalization to two-dimensional patterns in systems with anisotropy as they arise, for example, for hydrodynamic instabilities in nematic liquid crystals
16. Bifurcation analysis of an aerodynamic journal bearing system considering the effect of stationary herringbone grooves
International Nuclear Information System (INIS)
Wang, C.-C.
2007-01-01
This paper investigates the bifurcation and nonlinear behavior of an aerodynamic journal bearing system taking into account the effect of stationary herringbone grooves. A finite difference method based on the successive over relation approach is employed to solve the Reynolds' equation. The analysis reveals a complex dynamical behavior comprising periodic and quasi-periodic responses of the rotor center. The dynamic behavior of the bearing system varies with changes in the bearing number and rotor mass. The results of this study provide a better understanding of the nonlinear dynamics of aerodynamic grooved journal bearing systems
17. Regularizations of two-fold bifurcations in planar piecewise smooth systems using blowup
DEFF Research Database (Denmark)
Kristiansen, Kristian Uldall; Hogan, S. J.
2015-01-01
type of limit cycle that does not appear to be present in the original PWS system. For both types of limit cycle, we show that the criticality of the Hopf bifurcation that gives rise to periodic orbits is strongly dependent on the precise form of the regularization. Finally, we analyse the limit cycles...... as locally unique families of periodic orbits of the regularization and connect them, when possible, to limit cycles of the PWS system. We illustrate our analysis with numerical simulations and show how the regularized system can undergo a canard explosion phenomenon...
18. CISM Session on Bifurcation and Stability of Dissipative Systems
CERN Document Server
1993-01-01
The first theme concerns the plastic buckling of structures in the spirit of Hill’s classical approach. Non-bifurcation and stability criteria are introduced and post-bifurcation analysis performed by asymptotic development method in relation with Hutchinson’s work. Some recent results on the generalized standard model are given and their connection to Hill’s general formulation is presented. Instability phenomena of inelastic flow processes such as strain localization and necking are discussed. The second theme concerns stability and bifurcation problems in internally damaged or cracked colids. In brittle fracture or brittle damage, the evolution law of crack lengths or damage parameters is time-independent like in plasticity and leads to a similar mathematical description of the quasi-static evolution. Stability and non-bifurcation criteria in the sense of Hill can be again obtained from the discussion of the rate response.
19. Bifurcation dynamics of the tempered fractional Langevin equation
Energy Technology Data Exchange (ETDEWEB)
Zeng, Caibin, E-mail: [email protected]; Yang, Qigui, E-mail: [email protected] [School of Mathematics, South China University of Technology, Guangzhou 510640 (China); Chen, YangQuan, E-mail: [email protected] [MESA LAB, School of Engineering, University of California, Merced, 5200 N. Lake Road, Merced, California 95343 (United States)
2016-08-15
Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.
20. Deformable 4DCT lung registration with vessel bifurcations
International Nuclear Information System (INIS)
Hilsmann, A.; Vik, T.; Kaus, M.; Franks, K.; Bissonette, J.P.; Purdie, T.; Beziak, A.; Aach, T.
2007-01-01
In radiotherapy planning of lung cancer, breathing motion causes uncertainty in the determination of the target volume. Image registration makes it possible to get information about the deformation of the lung and the tumor movement in the respiratory cycle from a few images. A dedicated, automatic, landmark-based technique was developed that finds corresponding vessel bifurcations. Hereby, we developed criteria to characterize pronounced bifurcations for which correspondence finding was more stable and accurate. The bifurcations were extracted from automatically segmented vessel trees in maximum inhale and maximum exhale CT thorax data sets. To find corresponding bifurcations in both data sets we used the shape context approach of Belongie et al. Finally, a volumetric lung deformation was obtained using thin-plate spline interpolation and affine registration. The method is evaluated on 10 4D-CT data sets of patients with lung cancer. (orig.)
1. Bifurcation theory for hexagonal agglomeration in economic geography
CERN Document Server
Ikeda, Kiyohiro
2014-01-01
This book contributes to an understanding of how bifurcation theory adapts to the analysis of economic geography. It is easily accessible not only to mathematicians and economists, but also to upper-level undergraduate and graduate students who are interested in nonlinear mathematics. The self-organization of hexagonal agglomeration patterns of industrial regions was first predicted by the central place theory in economic geography based on investigations of southern Germany. The emergence of hexagonal agglomeration in economic geography models was envisaged by Krugman. In this book, after a brief introduction of central place theory and new economic geography, the missing link between them is discovered by elucidating the mechanism of the evolution of bifurcating hexagonal patterns. Pattern formation by such bifurcation is a well-studied topic in nonlinear mathematics, and group-theoretic bifurcation analysis is a well-developed theoretical tool. A finite hexagonal lattice is used to express uniformly distri...
2. Defining Electron Bifurcation in the Electron-Transferring Flavoprotein Family.
Science.gov (United States)
Garcia Costas, Amaya M; Poudel, Saroj; Miller, Anne-Frances; Schut, Gerrit J; Ledbetter, Rhesa N; Fixen, Kathryn R; Seefeldt, Lance C; Adams, Michael W W; Harwood, Caroline S; Boyd, Eric S; Peters, John W
2017-11-01
Electron bifurcation is the coupling of exergonic and endergonic redox reactions to simultaneously generate (or utilize) low- and high-potential electrons. It is the third recognized form of energy conservation in biology and was recently described for select electron-transferring flavoproteins (Etfs). Etfs are flavin-containing heterodimers best known for donating electrons derived from fatty acid and amino acid oxidation to an electron transfer respiratory chain via Etf-quinone oxidoreductase. Canonical examples contain a flavin adenine dinucleotide (FAD) that is involved in electron transfer, as well as a non-redox-active AMP. However, Etfs demonstrated to bifurcate electrons contain a second FAD in place of the AMP. To expand our understanding of the functional variety and metabolic significance of Etfs and to identify amino acid sequence motifs that potentially enable electron bifurcation, we compiled 1,314 Etf protein sequences from genome sequence databases and subjected them to informatic and structural analyses. Etfs were identified in diverse archaea and bacteria, and they clustered into five distinct well-supported groups, based on their amino acid sequences. Gene neighborhood analyses indicated that these Etf group designations largely correspond to putative differences in functionality. Etfs with the demonstrated ability to bifurcate were found to form one group, suggesting that distinct conserved amino acid sequence motifs enable this capability. Indeed, structural modeling and sequence alignments revealed that identifying residues occur in the NADH- and FAD-binding regions of bifurcating Etfs. Collectively, a new classification scheme for Etf proteins that delineates putative bifurcating versus nonbifurcating members is presented and suggests that Etf-mediated bifurcation is associated with surprisingly diverse enzymes. IMPORTANCE Electron bifurcation has recently been recognized as an electron transfer mechanism used by microorganisms to maximize
3. Ergodicity-breaking bifurcations and tunneling in hyperbolic transport models
Science.gov (United States)
Giona, M.; Brasiello, A.; Crescitelli, S.
2015-11-01
One of the main differences between parabolic transport, associated with Langevin equations driven by Wiener processes, and hyperbolic models related to generalized Kac equations driven by Poisson processes, is the occurrence in the latter of multiple stable invariant densities (Frobenius multiplicity) in certain regions of the parameter space. This phenomenon is associated with the occurrence in linear hyperbolic balance equations of a typical bifurcation, referred to as the ergodicity-breaking bifurcation, the properties of which are thoroughly analyzed.
4. Hopf bifurcation of the stochastic model on business cycle
International Nuclear Information System (INIS)
Xu, J; Wang, H; Ge, G
2008-01-01
A stochastic model on business cycle was presented in thas paper. Simplifying the model through the quasi Hamiltonian theory, the Ito diffusion process was obtained. According to Oseledec multiplicative ergodic theory and singular boundary theory, the conditions of local and global stability were acquired. Solving the stationary FPK equation and analyzing the stationary probability density, the stochastic Hopf bifurcation was explained. The result indicated that the change of parameter awas the key factor to the appearance of the stochastic Hopf bifurcation
5. Iterative Controller Tuning for Process with Fold Bifurcations
DEFF Research Database (Denmark)
Huusom, Jakob Kjøbsted; Poulsen, Niels Kjølstad; Jørgensen, Sten Bay
2007-01-01
Processes involving fold bifurcation are notoriously difficult to control in the vicinity of the fold where most often optimal productivity is achieved . In cases with limited process insight a model based control synthesis is not possible. This paper uses a data driven approach with an improved...... version of iterative feedback tuning to optimizing a closed loop performance criterion, as a systematic tool for tuning process with fold bifurcations....
6. Bifurcated states of the error-field-induced magnetic islands
International Nuclear Information System (INIS)
Zheng, L.-J.; Li, B.; Hazeltine, R.D.
2008-01-01
We find that the formation of the magnetic islands due to error fields shows bifurcation when neoclassical effects are included. The bifurcation, which follows from including bootstrap current terms in a description of island growth in the presence of error fields, provides a path to avoid the island-width pole in the classical description. The theory offers possible theoretical explanations for the recent DIII-D and JT-60 experimental observations concerning confinement deterioration with increasing error field
7. Attractors, bifurcations, & chaos nonlinear phenomena in economics
CERN Document Server
Puu, Tönu
2003-01-01
The present book relies on various editions of my earlier book "Nonlinear Economic Dynamics", first published in 1989 in the Springer series "Lecture Notes in Economics and Mathematical Systems", and republished in three more, successively revised and expanded editions, as a Springer monograph, in 1991, 1993, and 1997, and in a Russian translation as "Nelineynaia Economicheskaia Dinamica". The first three editions were focused on applications. The last was differ ent, as it also included some chapters with mathematical background mate rial -ordinary differential equations and iterated maps -so as to make the book self-contained and suitable as a textbook for economics students of dynamical systems. To the same pedagogical purpose, the number of illus trations were expanded. The book published in 2000, with the title "A ttractors, Bifurcations, and Chaos -Nonlinear Phenomena in Economics", was so much changed, that the author felt it reasonable to give it a new title. There were two new math ematics ch...
8. Bifurcated equilibria in centrifugally confined plasma
International Nuclear Information System (INIS)
Shamim, I.; Teodorescu, C.; Guzdar, P. N.; Hassam, A. B.; Clary, R.; Ellis, R.; Lunsford, R.
2008-01-01
A bifurcation theory and associated computational model are developed to account for abrupt transitions observed recently on the Maryland Centrifugal eXperiment (MCX) [R. F. Ellis et al. Phys. Plasmas 8, 2057 (2001)], a supersonically rotating magnetized plasma that relies on centrifugal forces to prevent thermal expansion of plasma along the magnetic field. The observed transitions are from a well-confined, high-rotation state (HR-mode) to a lower-rotation, lesser-confined state (O-mode). A two-dimensional time-dependent magnetohydrodynamics code is used to simulate the dynamical equilibrium states of the MCX configuration. In addition to the expected viscous drag on the core plasma rotation, a momentum loss term is added that models the friction of plasma on the enhanced level of neutrals expected in the vicinity of the insulators at the throats of the magnetic mirror geometry. At small values of the external rotation drive, the plasma is not well-centrifugally confined and hence experiences the drag from near the insulators. Beyond a critical value of the external drive, the system makes an abrupt transition to a well-centrifugally confined state in which the plasma has pulled away from the end insulator plates; more effective centrifugal confinement lowers the plasma mass near the insulators allowing runaway increases in the rotation speed. The well-confined steady state is reached when the external drive is balanced by only the viscosity of the core plasma. A clear hysteresis phenomenon is shown.
9. Dansgaard–Oeschger events: bifurcation points in the climate system
Directory of Open Access Journals (Sweden)
A. A. Cimatoribus
2013-02-01
Full Text Available Dansgaard–Oeschger events are a prominent mode of variability in the records of the last glacial cycle. Various prototype models have been proposed to explain these rapid climate fluctuations, and no agreement has emerged on which may be the more correct for describing the palaeoclimatic signal. In this work, we assess the bimodality of the system, reconstructing the topology of the multi-dimensional attractor over which the climate system evolves. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We show that Dansgaard–Oeschger events have weak early warning signals if the ensemble of events is considered. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing (e.g. solar, ice sheets, either forcing directly the transition or pacing it through stochastic resonance. These findings are most consistent with a model that associates Dansgaard–Oeschger with changing boundary conditions, and with the presence of a bifurcation point.
10. Freeform inkjet printing of cellular structures with bifurcations.
Science.gov (United States)
Christensen, Kyle; Xu, Changxue; Chai, Wenxuan; Zhang, Zhengyi; Fu, Jianzhong; Huang, Yong
2015-05-01
Organ printing offers a great potential for the freeform layer-by-layer fabrication of three-dimensional (3D) living organs using cellular spheroids or bioinks as building blocks. Vascularization is often identified as a main technological barrier for building 3D organs. As such, the fabrication of 3D biological vascular trees is of great importance for the overall feasibility of the envisioned organ printing approach. In this study, vascular-like cellular structures are fabricated using a liquid support-based inkjet printing approach, which utilizes a calcium chloride solution as both a cross-linking agent and support material. This solution enables the freeform printing of spanning and overhang features by providing a buoyant force. A heuristic approach is implemented to compensate for the axially-varying deformation of horizontal tubular structures to achieve a uniform diameter along their axial directions. Vascular-like structures with both horizontal and vertical bifurcations have been successfully printed from sodium alginate only as well as mouse fibroblast-based alginate bioinks. The post-printing fibroblast cell viability of printed cellular tubes was found to be above 90% even after a 24 h incubation, considering the control effect. © 2014 Wiley Periodicals, Inc.
11. Bursting oscillations, bifurcation and synchronization in neuronal systems
Energy Technology Data Exchange (ETDEWEB)
Wang Haixia [School of Science, Nanjing University of Science and Technology, Nanjing 210094 (China); Wang Qingyun, E-mail: [email protected] [Department of Dynamics and Control, Beihang University, Beijing 100191 (China); Lu Qishao [Department of Dynamics and Control, Beihang University, Beijing 100191 (China)
2011-08-15
Highlights: > We investigate bursting oscillations and related bifurcation in the modified Morris-Lecar neuron. > Two types of fast-slow bursters are analyzed in detail. > We show the properties of some crucial bifurcation points. > Synchronization transition and the neural excitability are explored in the coupled bursters. - Abstract: This paper investigates bursting oscillations and related bifurcation in the modified Morris-Lecar neuron. It is shown that for some appropriate parameters, the modified Morris-Lecar neuron can exhibit two types of fast-slow bursters, that is 'circle/fold cycle' bursting and 'subHopf/homoclinic' bursting with class 1 and class 2 neural excitability, which have different neuro-computational properties. By means of the analysis of fast-slow dynamics and phase plane, we explore bifurcation mechanisms associated with the two types of bursters. Furthermore, the properties of some crucial bifurcation points, which can determine the type of the burster, are studied by the stability and bifurcation theory. In addition, we investigate the influence of the coupling strength on synchronization transition and the neural excitability in two electrically coupled bursters with the same bursting type. More interestingly, the multi-time-scale synchronization transition phenomenon is found as the coupling strength varies.
12. Bifurcation magnetic resonance in films magnetized along hard magnetization axis
Energy Technology Data Exchange (ETDEWEB)
Vasilevskaya, Tatiana M., E-mail: [email protected] [Ulyanovsk State University, Leo Tolstoy 42, 432017 Ulyanovsk (Russian Federation); Sementsov, Dmitriy I.; Shutyi, Anatoliy M. [Ulyanovsk State University, Leo Tolstoy 42, 432017 Ulyanovsk (Russian Federation)
2012-09-15
We study low-frequency ferromagnetic resonance in a thin film magnetized along the hard magnetization axis performing an analysis of magnetization precession dynamics equations and numerical simulation. Two types of films are considered: polycrystalline uniaxial films and single-crystal films with cubic magnetic anisotropy. An additional (bifurcation) resonance initiated by the bistability, i.e. appearance of two closely spaced equilibrium magnetization states is registered. The modification of dynamic modes provoked by variation of the frequency, amplitude, and magnetic bias value of the ac field is studied. Both steady and chaotic magnetization precession modes are registered in the bifurcation resonance range. - Highlights: Black-Right-Pointing-Pointer An additional bifurcation resonance arises in a case of a thin film magnetized along HMA. Black-Right-Pointing-Pointer Bifurcation resonance occurs due to the presence of two closely spaced equilibrium magnetization states. Black-Right-Pointing-Pointer Both regular and chaotic precession modes are realized within bifurcation resonance range. Black-Right-Pointing-Pointer Appearance of dynamic bistability is typical for bifurcation resonance.
13. Bifurcation magnetic resonance in films magnetized along hard magnetization axis
International Nuclear Information System (INIS)
Vasilevskaya, Tatiana M.; Sementsov, Dmitriy I.; Shutyi, Anatoliy M.
2012-01-01
We study low-frequency ferromagnetic resonance in a thin film magnetized along the hard magnetization axis performing an analysis of magnetization precession dynamics equations and numerical simulation. Two types of films are considered: polycrystalline uniaxial films and single-crystal films with cubic magnetic anisotropy. An additional (bifurcation) resonance initiated by the bistability, i.e. appearance of two closely spaced equilibrium magnetization states is registered. The modification of dynamic modes provoked by variation of the frequency, amplitude, and magnetic bias value of the ac field is studied. Both steady and chaotic magnetization precession modes are registered in the bifurcation resonance range. - Highlights: ► An additional bifurcation resonance arises in a case of a thin film magnetized along HMA. ► Bifurcation resonance occurs due to the presence of two closely spaced equilibrium magnetization states. ► Both regular and chaotic precession modes are realized within bifurcation resonance range. ► Appearance of dynamic bistability is typical for bifurcation resonance.
14. Periodic precursors of nonlinear dynamical transitions
International Nuclear Information System (INIS)
Jiang Yu; Dong Shihai; Lozada-Cassou, M.
2004-01-01
We study the resonant response of a nonlinear system to external periodic perturbations. We show by numerical simulation that the periodic resonance curve may anticipate the dynamical instability of the unperturbed nonlinear periodic system, at parameter values far away from the bifurcation points. In the presence of noise, the buried intrinsic periodic dynamics can be picked out by analyzing the system's response to periodic modulation of appropriate intensity
15. A study on stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point with the moment method
Energy Technology Data Exchange (ETDEWEB)
Zhang Guangjun [State Key Laboratory of Mechanical Structural Strength and Vibration, School of Architectural Engineering and Mechanics, Xi' an Jiaotong University, Xi' an, Shaanxi (China); Xu Jianxue [State Key Laboratory of Mechanical Structural Strength and Vibration, School of Architectural Engineering and Mechanics, Xi' an Jiaotong University, Xi' an, Shaanxi (China)] e-mail: [email protected]
2006-02-01
This paper analyzes the stochastic resonance induced by a novel transition of one-dimensional bistable system in the neighborhood of bifurcation point with the method of moment, which refer to the transition of system motion among a potential well of stable fixed point before bifurcation of original system and double-well potential of two coexisting stable fixed points after original system bifurcation at the presence of internal noise. The results show: the semi-analytical result of stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point may be obtained, and the semi-analytical result is in accord with the one of Monte Carlo simulation qualitatively, the occurrence of stochastic resonance is related to the bifurcation of noisy nonlinear dynamical system moment equations, which induce the transfer of energy of ensemble average (Ex) of system response in each frequency component and make the energy of ensemble average of system response concentrate on the frequency of input signal, stochastic resonance occurs.
16. A study on stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point with the moment method
International Nuclear Information System (INIS)
Zhang Guangjun; Xu Jianxue
2006-01-01
This paper analyzes the stochastic resonance induced by a novel transition of one-dimensional bistable system in the neighborhood of bifurcation point with the method of moment, which refer to the transition of system motion among a potential well of stable fixed point before bifurcation of original system and double-well potential of two coexisting stable fixed points after original system bifurcation at the presence of internal noise. The results show: the semi-analytical result of stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point may be obtained, and the semi-analytical result is in accord with the one of Monte Carlo simulation qualitatively, the occurrence of stochastic resonance is related to the bifurcation of noisy nonlinear dynamical system moment equations, which induce the transfer of energy of ensemble average (Ex) of system response in each frequency component and make the energy of ensemble average of system response concentrate on the frequency of input signal, stochastic resonance occurs
17. Torus-doubling process via strange nonchaotic attractors
International Nuclear Information System (INIS)
Mitsui, Takahito; Uenohara, Seiji; Morie, Takashi; Horio, Yoshihiko; Aihara, Kazuyuki
2012-01-01
Torus-doubling bifurcations typically occur only a finite number of times. It has been assumed that torus-doubling bifurcations in quasiperiodically forced systems are interrupted by the appearance of strange nonchaotic attractors (SNAs). In the present Letter, we study a quasiperiodically forced noninvertible map and report the occurrence of a torus-doubling process via SNAs. The mechanism of this process is numerically clarified. Furthermore, this process is experimentally demonstrated in a switched-capacitor integrated circuit. -- Highlights: ► We report the occurrence of a torus-doubling process via strange nonchaotic attractors (SNAs). ► The process consists of the gradual fractalization of a torus and the Heagy–Hammel transition. ► The torus-doubling process via SNAs is also experimentally demonstrated in an electronic circuit.
18. Bifurcation analysis and stability design for aircraft longitudinal motion with high angle of attack
Directory of Open Access Journals (Sweden)
Xin Qi
2015-02-01
Full Text Available Bifurcation analysis and stability design for aircraft longitudinal motion are investigated when the nonlinearity in flight dynamics takes place severely at high angle of attack regime. To predict the special nonlinear flight phenomena, bifurcation theory and continuation method are employed to systematically analyze the nonlinear motions. With the refinement of the flight dynamics for F-8 Crusader longitudinal motion, a framework is derived to identify the stationary bifurcation and dynamic bifurcation for high-dimensional system. Case study shows that the F-8 longitudinal motion undergoes saddle node bifurcation, Hopf bifurcation, Zero-Hopf bifurcation and branch point bifurcation under certain conditions. Moreover, the Hopf bifurcation renders series of multiple frequency pitch oscillation phenomena, which deteriorate the flight control stability severely. To relieve the adverse effects of these phenomena, a stabilization control based on gain scheduling and polynomial fitting for F-8 longitudinal motion is presented to enlarge the flight envelope. Simulation results validate the effectiveness of the proposed scheme.
19. Complexity and Hopf Bifurcation Analysis on a Kind of Fractional-Order IS-LM Macroeconomic System
Science.gov (United States)
Ma, Junhai; Ren, Wenbo
On the basis of our previous research, we deepen and complete a kind of macroeconomics IS-LM model with fractional-order calculus theory, which is a good reflection on the memory characteristics of economic variables, we also focus on the influence of the variables on the real system, and improve the analysis capabilities of the traditional economic models to suit the actual macroeconomic environment. The conditions of Hopf bifurcation in fractional-order system models are briefly demonstrated, and the fractional order when Hopf bifurcation occurs is calculated, showing the inherent complex dynamic characteristics of the system. With numerical simulation, bifurcation, strange attractor, limit cycle, waveform and other complex dynamic characteristics are given; and the order condition is obtained with respect to time. We find that the system order has an important influence on the running state of the system. The system has a periodic motion when the order meets the conditions of Hopf bifurcation; the fractional-order system gradually stabilizes with the change of the order and parameters while the corresponding integer-order system diverges. This study has certain significance to policy-making about macroeconomic regulation and control.
20. Coupled large earthquakes in the Baikal rift system: Response to bifurcations in nonlinear resonance hysteresis
Directory of Open Access Journals (Sweden)
Anatoly V. Klyuchevskii
2013-11-01
Full Text Available The current lithospheric geodynamics and tectonophysics in the Baikal rift are discussed in terms of a nonlinear oscillator with dissipation. The nonlinear oscillator model is applicable to the area because stress change shows up as quasi-periodic inharmonic oscillations at rifting attractor structures (RAS. The model is consistent with the space-time patterns of regional seismicity in which coupled large earthquakes, proximal in time but distant in space, may be a response to bifurcations in nonlinear resonance hysteresis in a system of three oscillators corresponding to the rifting attractors. The space-time distribution of coupled MLH > 5.5 events has been stable for the period of instrumental seismicity, with the largest events occurring in pairs, one shortly after another, on two ends of the rift system and with couples of smaller events in the central part of the rift. The event couples appear as peaks of earthquake ‘migration’ rate with an approximately decadal periodicity. Thus the energy accumulated at RAS is released in coupled large events by the mechanism of nonlinear oscillators with dissipation. The new knowledge, with special focus on space-time rifting attractors and bifurcations in a system of nonlinear resonance hysteresis, may be of theoretical and practical value for earthquake prediction issues. Extrapolation of the results into the nearest future indicates the probability of such a bifurcation in the region, i.e., there is growing risk of a pending M ≈ 7 coupled event to happen within a few years.
1. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit
Energy Technology Data Exchange (ETDEWEB)
Kengne, J. [Laboratory of Automation and Applied Computer (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang (Cameroon); Njitacke Tabekoueng, Z.; Kamdoum Tamba, V.; Nguomkam Negou, A. [Laboratory of Automation and Applied Computer (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Dschang (Cameroon); Department of Physics, Laboratory of Electronics and Signal Processing (LETS), Faculty of Science, University of Dschang, Dschang (Cameroon)
2015-10-15
In this contribution, a novel memristor-based oscillator, obtained from Shinriki's circuit by substituting the nonlinear positive conductance with a first order memristive diode bridge, is introduced. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. The basic dynamical properties of the system are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponents' spectrum. It is found that in addition to the classical period-doubling and symmetry restoring crisis scenarios reported in the original circuit, the memristor-based oscillator experiences the unusual and striking feature of multiple attractors (i.e., coexistence of a pair of asymmetric periodic attractors with a pair of asymmetric chaotic ones) over a broad range of circuit parameters. Results of theoretical analyses are verified by laboratory experimental measurements.
2. Inverse bifurcation analysis: application to simple gene systems
Directory of Open Access Journals (Sweden)
Schuster Peter
2006-07-01
Full Text Available Abstract Background Bifurcation analysis has proven to be a powerful method for understanding the qualitative behavior of gene regulatory networks. In addition to the more traditional forward problem of determining the mapping from parameter space to the space of model behavior, the inverse problem of determining model parameters to result in certain desired properties of the bifurcation diagram provides an attractive methodology for addressing important biological problems. These include understanding how the robustness of qualitative behavior arises from system design as well as providing a way to engineer biological networks with qualitative properties. Results We demonstrate that certain inverse bifurcation problems of biological interest may be cast as optimization problems involving minimal distances of reference parameter sets to bifurcation manifolds. This formulation allows for an iterative solution procedure based on performing a sequence of eigen-system computations and one-parameter continuations of solutions, the latter being a standard capability in existing numerical bifurcation software. As applications of the proposed method, we show that the problem of maximizing regions of a given qualitative behavior as well as the reverse engineering of bistable gene switches can be modelled and efficiently solved.
3. Stochastic Bifurcation Analysis of an Elastically Mounted Flapping Airfoil
Directory of Open Access Journals (Sweden)
Bose Chandan
2018-01-01
Full Text Available The present paper investigates the effects of noisy flow fluctuations on the fluid-structure interaction (FSI behaviour of a span-wise flexible wing modelled as a two degree-of-freedom elastically mounted flapping airfoil. In the sterile flow conditions, the system undergoes a Hopf bifurcation as the free-stream velocity exceeds a critical limit resulting in a stable limit-cycle oscillation (LCO from a fixed point response. On the other hand, the qualitative dynamics changes from a stochastic fixed point to a random LCO through an intermittent state in the presence of irregular flow fluctuations. The probability density function depicts the most probable system state in the phase space. A phenomenological bifurcation (P-bifurcation analysis based on the transition in the topology associated with the structure of the joint probability density function (pdf of the response variables has been carried out. The joint pdf corresponding to the stochastic fixed point possesses a Dirac delta function like structure with a sharp single peak around zero. As the mean flow speed crosses the critical value, the joint pdf bifurcates to a crater-like structure indicating the occurrence of a P-bifurcation. The intermittent state is characterized by the co-existence of the unimodal as well as the crater like structure.
4. Transverse single-file diffusion and enhanced longitudinal diffusion near a subcritical bifurcation
Science.gov (United States)
Dessup, Tommy; Coste, Christophe; Saint Jean, Michel
2018-05-01
A quasi-one-dimensional system of repelling particles undergoes a configurational phase transition when the transverse confining potential decreases. Below a threshold, it becomes energetically favorable for the system to adopt one of two staggered raw patterns, symmetric with respect to the system axis. This transition is a subcritical pitchfork bifurcation for short range interactions. As a consequence, the homogeneous zigzag pattern is unstable in a finite zigzag amplitude range [hC 1,hC 2] . We exhibit strong qualitative effects of the subcriticality on the thermal motions of the particles. When the zigzag amplitude is close enough to the limits hC 1 and hC 2, a transverse vibrational soft mode occurs which induces a strongly subdiffusive behavior of the transverse fluctuations, similar to single-file diffusion. On the contrary, the longitudinal fluctuations are enhanced, with a diffusion coefficient which is more than doubled. Conversely, a simple measurement of the thermal fluctuations allows a precise determination of the bifurcation thresholds.
5. Photoinduced Intramolecular Bifurcate Hydrogen Bond: Unusual Mutual Influence of the Components.
Science.gov (United States)
Sigalov, Mark V; Shainyan, Bagrat A; Sterkhova, Irina V
2017-09-01
A series of 7-hydroxy-2-methylidene-2,3-dihydro-1H-inden-1-ones with 2-pyrrolyl (3), 4-dimethylaminophenyl (4), 4-nitrophenyl (5), and carboxyl group (6) as substituents at the exocyclic double bond was synthesized in the form of the E-isomers (4-6) or predominantly as the Z-isomer (3) which in solution is converted to the E-isomer. The synthesized compounds and their model analogues were studied by NMR spectroscopy, X-ray analysis, and MP2 theoretical calculations. The E-isomers having intramolecular O-H···O═C hydrogen bond are converted by UV irradiation to the Z-isomers having bifurcated O-H···O···H-X hydrogen bond. Unexpected shortening (and, thus, strengthening) of the O-H···O═C component of the bifurcated hydrogen bond upon the formation of the C═O···H-X hydrogen bond was found experimentally, proved theoretically (MP2), and explained by a roundabout interaction of the H-donor (HX) and H-acceptor (C═O) via the system of conjugated bonds.
6. Steady-state bifurcations of the three-dimensional Kolmogorov problem
Directory of Open Access Journals (Sweden)
Zhi-Min Chen
2000-08-01
Full Text Available This paper studies the spatially periodic incompressible fluid motion in $mathbb R^3$ excited by the external force $k^2(sin kz, 0,0$ with $kgeq 2$ an integer. This driving force gives rise to the existence of the unidirectional basic steady flow $u_0=(sin kz,0, 0$ for any Reynolds number. It is shown in Theorem 1.1 that there exist a number of critical Reynolds numbers such that $u_0$ bifurcates into either 4 or 8 or 16 different steady states, when the Reynolds number increases across each of such numbers.
7. A bifurcation giving birth to order in an impulsively driven complex system
Energy Technology Data Exchange (ETDEWEB)
2016-08-15
Nonlinear oscillations lie at the heart of numerous complex systems. Impulsive forcing arises naturally in many scenarios, and we endeavour to study nonlinear oscillators subject to such forcing. We model these kicked oscillatory systems as a piecewise smooth dynamical system, whereby their dynamics can be investigated. We investigate the problem of pattern formation in a turbulent combustion system and apply this formalism with the aim of explaining the observed dynamics. We identify that the transition of this system from low amplitude chaotic oscillations to large amplitude periodic oscillations is the result of a discontinuity induced bifurcation. Further, we provide an explanation for the occurrence of intermittent oscillations in the system.
8. Stability and Bifurcation Analysis in a Maglev System with Multiple Delays
Science.gov (United States)
Zhang, Lingling; Huang, Jianhua; Huang, Lihong; Zhang, Zhizhou
This paper considers the time-delayed feedback control for Maglev system with two discrete time delays. We determine constraints on the feedback time delays which ensure the stability of the Maglev system. An algorithm is developed for drawing a two-parametric bifurcation diagram with respect to two delays τ1 and τ2. Direction and stability of periodic solutions are also determined using the normal form method and center manifold theory by Hassard. The complex dynamical behavior of the Maglev system near the domain of stability is confirmed by exhaustive numerical simulation.
9. Sediment sorting at a side channel bifurcation
Science.gov (United States)
van Denderen, Pepijn; Schielen, Ralph; Hulscher, Suzanne
2017-04-01
Side channels have been constructed to reduce the flood risk and to increase the ecological value of the river. In various Dutch side channels large aggradation in these channels occurred after construction. Measurements show that the grain size of the deposited sediment in the side channel is smaller than the grain size found on the bed of the main channel. This suggest that sorting occurs at the bifurcation of the side channel. The objective is to reproduce with a 2D morphological model the fining of the bed in the side channel and to study the effect of the sediment sorting on morphodynamic development of the side channel. We use a 2D Delft3D model with two sediment fractions. The first fraction corresponds with the grain size that can be found on the bed of the main channel and the second fraction corresponds with the grain size found in the side channel. With the numerical model we compute several side channel configurations in which we vary the length and the width of the side channel, and the curvature of the upstream channel. From these computations we can derive the equilibrium state and the time scale of the morphodynamic development of the side channel. Preliminary results show that even when a simple sediment transport relation is used, like Engelund & Hansen, more fine sediment enters the side channel than coarse sediment. This is as expected, and is probably related to the bed slope effects which are a function of the Shields parameter. It is expected that by adding a sill at the entrance of the side channel the slope effect increases. This might reduce the amount of coarse sediment which enters the side channel even more. It is unclear whether the model used is able to reproduce the effect of such a sill correctly as modelling a sill and reproducing the correct hydrodynamic and morphodynamic behaviour is not straightforward in a 2D model. Acknowledgements: This research is funded by STW, part of the Dutch Organization for Scientific Research under
10. Type I intermittency related to the spatiotemporal dynamics of double layers and ion-acoustic instabilities in plasma
International Nuclear Information System (INIS)
Chiriac, S.; Dimitriu, D. G.; Sanduloviciu, M.
2007-01-01
Anodic double layer instabilities occur in low-temperature diffusion filament-type discharge plasma by applying a certain positive bias with respect to the plasma potential to an additional electrode. Periodic nonlinear regimes, characterized by proper dynamics of double layers, are sustained if excitation and ionization rates in front of the electrode reach the value for which current limitation effects appear in the static current-voltage characteristic. It was experimentally shown that under specific experimental conditions these ordered spatiotemporal phenomena can evolve into chaotic states by type I intermittency. This transition was verified by the evolution of time series, fast Fourier transform amplitude plots, three-dimensional reconstructed state spaces, power laws, and flickering phenomena spectrum, as well as by the return map and tangent bifurcation
11. EXPERIMENTAL STUDY ON SEDIMENT DISTRIBUTION AT CHANNEL BIFURCATION
Institute of Scientific and Technical Information of China (English)
G.M. Tarekul ISLAM; M.R. KABIR; Ainun NISHAT
2002-01-01
This paper presents the experimental results on the distribution of sediments at channel bifurcation.The experiments have been conducted in a physical model of channel bifurcation. It consists of a straight main channel which bifurcates into two branch channels of different widths. The test rig is a mobile bed with fixed bank. Four different noses have been used to study the phenomenon. For each nose, three upstream discharges viz. 20 l/s, 30 l/s and 40 l/s have been employed. From the measured data, discharges and sediment transport ratios per unit width are calculated in the downstream branches.These data have been set to the general nodal point relation and a set of equations has been developed to describe the distribution of sediments to the downstream branches for different nose angles.
12. Adaptive Control of Electromagnetic Suspension System by HOPF Bifurcation
Directory of Open Access Journals (Sweden)
Aming Hao
2013-01-01
Full Text Available EMS-type maglev system is essentially nonlinear and unstable. It is complicated to design a stable controller for maglev system which is under large-scale disturbance and parameter variance. Theory analysis expresses that this phenomenon corresponds to a HOPF bifurcation in mathematical model. An adaptive control law which adjusts the PID control parameters is given in this paper according to HOPF bifurcation theory. Through identification of the levitated mass, the controller adjusts the feedback coefficient to make the system far from the HOPF bifurcation point and maintain the stability of the maglev system. Simulation result indicates that adjusting proportion gain parameter using this method can extend the state stability range of maglev system and avoid the self-excited vibration efficiently.
13. Bifurcated equilibria in two-dimensional MHD with diamagnetic effects
International Nuclear Information System (INIS)
Ottaviani, M.; Tebaldi, C.
1998-12-01
In this work we analyzed the sequence of bifurcated equilibria in two-dimensional reduced magnetohydrodynamics. Diamagnetic effects are studied under the assumption of a constant equilibrium pressure gradient, not altered by the formation of the magnetic island. The formation of an island when the symmetric equilibrium becomes unstable is studied as a function of the tearing mode stability parameter Δ' and of the diamagnetic frequency, by employing fixed-points numerical techniques and an initial value code. At larger values of Δ' a tangent bifurcation takes place, above which no small island solutions exist. This bifurcation persists up to fairly large values of the diamagnetic frequency (of the order of one tenth of the Alfven frequency). The implications of this phenomenology for the intermittent MHD dynamics observed in tokamaks is discussed. (authors)
14. Stochastic stability and bifurcation in a macroeconomic model
International Nuclear Information System (INIS)
Li Wei; Xu Wei; Zhao Junfeng; Jin Yanfei
2007-01-01
On the basis of the work of Goodwin and Puu, a new business cycle model subject to a stochastically parametric excitation is derived in this paper. At first, we reduce the model to a one-dimensional diffusion process by applying the stochastic averaging method of quasi-nonintegrable Hamiltonian system. Secondly, we utilize the methods of Lyapunov exponent and boundary classification associated with diffusion process respectively to analyze the stochastic stability of the trivial solution of system. The numerical results obtained illustrate that the trivial solution of system must be globally stable if it is locally stable in the state space. Thirdly, we explore the stochastic Hopf bifurcation of the business cycle model according to the qualitative changes in stationary probability density of system response. It is concluded that the stochastic Hopf bifurcation occurs at two critical parametric values. Finally, some explanations are given in a simply way on the potential applications of stochastic stability and bifurcation analysis
15. Dynamical systems V bifurcation theory and catastrophe theory
CERN Document Server
1994-01-01
Bifurcation theory and catastrophe theory are two of the best known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Moreover, understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, the first printing of w...
16. Bifurcations in the optimal elastic foundation for a buckling column
International Nuclear Information System (INIS)
Rayneau-Kirkhope, Daniel; Farr, Robert; Ding, K.; Mao, Yong
2010-01-01
We investigate the buckling under compression of a slender beam with a distributed lateral elastic support, for which there is an associated cost. For a given cost, we study the optimal choice of support to protect against Euler buckling. We show that with only weak lateral support, the optimum distribution is a delta-function at the centre of the beam. When more support is allowed, we find numerically that the optimal distribution undergoes a series of bifurcations. We obtain analytical expressions for the buckling load around the first bifurcation point and corresponding expansions for the optimal position of support. Our theoretical predictions, including the critical exponent of the bifurcation, are confirmed by computer simulations.
17. Bifurcations in the optimal elastic foundation for a buckling column
Energy Technology Data Exchange (ETDEWEB)
Rayneau-Kirkhope, Daniel, E-mail: [email protected] [School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD (United Kingdom); Farr, Robert [Unilever R and D, Olivier van Noortlaan 120, AT3133, Vlaardingen (Netherlands); London Institute for Mathematical Sciences, 22 South Audley Street, Mayfair, London (United Kingdom); Ding, K. [Department of Physics, Fudan University, Shanghai, 200433 (China); Mao, Yong [School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD (United Kingdom)
2010-12-01
We investigate the buckling under compression of a slender beam with a distributed lateral elastic support, for which there is an associated cost. For a given cost, we study the optimal choice of support to protect against Euler buckling. We show that with only weak lateral support, the optimum distribution is a delta-function at the centre of the beam. When more support is allowed, we find numerically that the optimal distribution undergoes a series of bifurcations. We obtain analytical expressions for the buckling load around the first bifurcation point and corresponding expansions for the optimal position of support. Our theoretical predictions, including the critical exponent of the bifurcation, are confirmed by computer simulations.
18. Bifurcation-free design method of pulse energy converter controllers
International Nuclear Information System (INIS)
Kolokolov, Yury; Ustinov, Pavel; Essounbouli, Najib; Hamzaoui, Abdelaziz
2009-01-01
In this paper, a design method of pulse energy converter (PEC) controllers is proposed. This method develops a classical frequency domain design, based on the small signal modeling, by means of an addition of a nonlinear dynamics analysis stage. The main idea of the proposed method consists in fact that the PEC controller, designed with an application of the small signal modeling, is tuned after with taking into the consideration an essentially nonlinear nature of the PEC that makes it possible to avoid bifurcation phenomena in the PEC dynamics at the design stage (bifurcation-free design). Also application of the proposed method allows an improvement of the designed controller performance. The application of this bifurcation-free design method is demonstrated on an example of the controller design of direct current-direct current (DC-DC) buck converter with an input electromagnetic interference filter.
19. Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system
International Nuclear Information System (INIS)
Zhang Jiangang; Li Xianfeng; Chu Yandong; Yu Jianning; Chang Yingxiang
2009-01-01
In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincare maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincare sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.
20. Longitudinal traveling waves bifurcating from Vlasov plasma equilibria
International Nuclear Information System (INIS)
Holloway, J.P.
1989-01-01
The kinetic equations governing longitudinal motion along a straight magnetic field in a multi-species collisionless plasma are investigated. A necessary condition for the existence of small amplitude spatially periodic equilibria and traveling waves near a given spatially uniform background equilibrium is derived, and the wavelengths which such solutions must approach as their amplitude decreases to zero are discussed. A sufficient condition for the existence of these small amplitude waves is also established. This is accomplished by studying the nonlinear ODE for the potential which arises when the distribution functions are represented in a BGK form; the arbitrary functions of energy that describe the BGK representation are tested as an infinite dimensional set of parameters in a bifurcation theory for the ODE. The positivity and zero current condition in the wave frame of the BGK distribution functions are maintained. The undamped small amplitude nonlinear waves so constructed can be made to satisfy the Vlasov dispersion relation exactly, but in general they need only satisfy it approximately. Numerical calculations reveal that even a thermal equilibrium electron-proton plasma with equal ion and electron temperatures will support undamped traveling waves with phase speeds greater than 1.3 times the electron velocity; the dispersion relation for this case exhibits both Langmuir and ion-acoustic branches as long wavelength limits, and shows how these branches are in fact connected by short wavelength waves of intermediate frequency. In apparent contradiction to the linear theory of Landau, these exact solutions of the kinetic equations do not damp; this contradiction is explained by observing that the linear theory is, in general, fundamentally incapable of describing undamped traveling waves
1. Periodic motions and chaos for a damped mobile piston system in a high pressure gas cylinder with P control
International Nuclear Information System (INIS)
Wang, Donghua; Huang, Jianzhe
2017-01-01
In this paper, the complex motions for a moving piston in a closed gas cylinder will be analyzed using the discrete implicit maps method. The strong nonlinearity of such system will be observed due to the large quadratic and cubic stiffness. Period-1 motions which contain high order of harmonic components will be presented. The periodic motions will be discretized into multiple continuous mappings, and such mapping can be analyzed via Newton–Raphson iteration. The stability analysis will be given and the analytic conditions for the saddle-node and period-doubling bifurcation will be determined. From the semi-analytic solution route, the possible motions without considering the impact of the piston with the end wall of the cylinder will be obtained analytically. The scheme to reduce the vibration of the piston can be obtained through the parameter studies.
2. An Approach to Robust Control of the Hopf Bifurcation
Directory of Open Access Journals (Sweden)
Giacomo Innocenti
2011-01-01
Full Text Available The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third-order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.
3. Three dimensional nilpotent singularity and Sil'nikov bifurcation
International Nuclear Information System (INIS)
Li Xindan; Liu Haifei
2007-01-01
In this paper, by using the normal form, blow-up theory and the technique of global bifurcations, we study the singularity at the origin with threefold zero eigenvalue for nonsymmetric vector fields with nilpotent linear part and 4-jet C ∼ -equivalent toy-bar -bar x+z-bar -bar y+ax 3 y-bar -bar z,with a 0, and analytically prove the existence of Sil'nikov bifurcation, and then of the strange attractor for certain subfamilies of the nonsymmetric versal unfoldings of this singularity under some conditions
4. Transportation and concentration inequalities for bifurcating Markov chains
DEFF Research Database (Denmark)
Penda, S. Valère Bitseki; Escobar-Bach, Mikael; Guillin, Arnaud
2017-01-01
We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful...... concentration inequalities.We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive...
5. Bifurcated transition of radial transport in the HIEI tandem mirror
International Nuclear Information System (INIS)
Sakai, O.; Yasaka, Y.
1995-01-01
Transition to a high radial confinement mode in a mirror plasma is triggered by limiter biasing. Sheared plasma rotation is induced in the high confinement phase which is characterized by reduction of edge turbulence and a confinement enhancement factor of 2-4. Edge plasma parameters related to radial confinement show a hysteresis phenomenon as a function of bias voltage or bias current, leading to the fact that transition from low to high confinement mode occurs between the bifurcated states. A transition model based on azimuthal momentum balance is employed to clarify physics of the observed bifurcation. copyright 1995 American Institute of Physics
6. Flow Topology Transition via Global Bifurcation in Thermally Driven Turbulence
Science.gov (United States)
Xie, Yi-Chao; Ding, Guang-Yu; Xia, Ke-Qing
2018-05-01
We report an experimental observation of a flow topology transition via global bifurcation in a turbulent Rayleigh-Bénard convection. This transition corresponds to a spontaneous symmetry breaking with the flow becomes more turbulent. Simultaneous measurements of the large-scale flow (LSF) structure and the heat transport show that the LSF bifurcates from a high heat transport efficiency quadrupole state to a less symmetric dipole state with a lower heat transport efficiency. In the transition zone, the system switches spontaneously and stochastically between the two long-lived metastable states.
7. Discretizing the transcritical and pitchfork bifurcations – conjugacy results
KAUST Repository
Lóczi, Lajos
2015-01-07
© 2015 Taylor & Francis. We present two case studies in one-dimensional dynamics concerning the discretization of transcritical (TC) and pitchfork (PF) bifurcations. In the vicinity of a TC or PF bifurcation point and under some natural assumptions on the one-step discretization method of order (Formula presented.) , we show that the time- (Formula presented.) exact and the step-size- (Formula presented.) discretized dynamics are topologically equivalent by constructing a two-parameter family of conjugacies in each case. As a main result, we prove that the constructed conjugacy maps are (Formula presented.) -close to the identity and these estimates are optimal.
8. Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures
International Nuclear Information System (INIS)
Fefferman, C L; Lee-Thorp, J P; Weinstein, M I
2016-01-01
Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or ‘edge’ and are localized transverse to it. This paper summarizes and extends the authors’ work on the bifurcation of topologically protected edge states in continuous two-dimensional (2D) honeycomb structures. We consider a family of Schrödinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous 2D honeycomb structures (http://arxiv.org/abs/1506.06111). The topologically protected edge state bifurcation is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schrödinger operator. Numerical simulations for honeycomb structures of varying contrasts and ‘rational edges’ (zigzag, armchair and others), support the following scenario: (a) for low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived edge quasi-modes which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge. (paper)
9. Bifurcations of edge states—topologically protected and non-protected—in continuous 2D honeycomb structures
Science.gov (United States)
Fefferman, C. L.; Lee-Thorp, J. P.; Weinstein, M. I.
2016-03-01
Edge states are time-harmonic solutions to energy-conserving wave equations, which are propagating parallel to a line-defect or ‘edge’ and are localized transverse to it. This paper summarizes and extends the authors’ work on the bifurcation of topologically protected edge states in continuous two-dimensional (2D) honeycomb structures. We consider a family of Schrödinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous 2D honeycomb structures (http://arxiv.org/abs/1506.06111). The topologically protected edge state bifurcation is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schrödinger operator. Numerical simulations for honeycomb structures of varying contrasts and ‘rational edges’ (zigzag, armchair and others), support the following scenario: (a) for low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived edge quasi-modes which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge.
10. A note on tilted Bianchi type VIh models: the type III bifurcation
Science.gov (United States)
Coley, A. A.; Hervik, S.
2008-10-01
In this note we complete the analysis of Hervik, van den Hoogen, Lim and Coley (2007 Class. Quantum Grav. 24 3859) of the late-time behaviour of tilted perfect fluid Bianchi type III models. We consider models with dust, and perfect fluids stiffer than dust, and eludicate the late-time behaviour by studying the centre manifold which dominates the behaviour of the model at late times. In the dust case, this centre manifold is three-dimensional and can be considered a double bifurcation as the two parameters (h and γ) of the type VIh model are varied. We therefore complete the analysis of the late-time behaviour of tilted ever-expanding Bianchi models of types I VIII.
11. Iterative methods for the detection of Hopf bifurcations in finite element discretisation of incompressible flow problems
International Nuclear Information System (INIS)
Cliffe, K.A.; Garratt, T.J.; Spence, A.
1992-03-01
This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalised eigenvalue problems arising from mixed finite element discretisations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and can be used in a scheme to determine the stability of steady state solutions and to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalised eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem to compute approximations to the eigenvalue of smallest real part. Numerical experiments are performed using a model of double diffusive convection. (author)
12. Bifurcation structure of positive stationary solutions for a Lotka-Volterra competition model with diffusion I
Science.gov (United States)
Kan-On, Yukio
2007-04-01
This paper is concerned with the bifurcation structure of positive stationary solutions for a generalized Lotka-Volterra competition model with diffusion. To establish the structure, the bifurcation theory and the interval arithmetic are employed.
13. Necessary and sufficient conditions for Hopf bifurcation in tri-neuron equation with a delay
International Nuclear Information System (INIS)
Liu Xiaoming; Liao Xiaofeng
2009-01-01
In this paper, we consider the delayed differential equations modeling three-neuron equations with only a time delay. Using the time delay as a bifurcation parameter, necessary and sufficient conditions for Hopf bifurcation to occur are derived. Numerical results indicate that for this model, Hopf bifurcation is likely to occur at suitable delay parameter values.
14. Local bifurcation analysis in nuclear reactor dynamics by Sotomayor’s theorem
International Nuclear Information System (INIS)
Pirayesh, Behnam; Pazirandeh, Ali; Akbari, Monireh
2016-01-01
Highlights: • When the feedback reactivity is considered as a nonlinear function some complex behaviors may emerge in the system such as local bifurcation phenomenon. • The qualitative behaviors of a typical nuclear reactor near its equilibrium points have been studied analytically. • Comprehensive analytical bifurcation analyses presented in this paper are transcritical bifurcation, saddle- node bifurcation and pitchfork bifurcation. - Abstract: In this paper, a qualitative approach has been used to explore nuclear reactor behaviors with nonlinear feedback. First, a system of four dimensional ordinary differential equations governing the dynamics of a typical nuclear reactor is introduced. These four state variables are the relative power of the reactor, the relative concentration of delayed neutron precursors, the fuel temperature and the coolant temperature. Then, the qualitative behaviors of the dynamical system near its equilibria have been studied analytically by using local bifurcation theory and Sotomayor’s theorem. The results indicated that despite the uncertainty of the reactivity, we can analyze the qualitative behavior changes of the reactor from the bifurcation point of view. Notably, local bifurcations that were considered in this paper include transcritical bifurcation, saddle-node bifurcation and pitchfork bifurcation. The theoretical analysis showed that these three types of local bifurcations may occur in the four dimensional dynamical system. In addition, to confirm the analytical results the numerical simulations are given.
15. Bifurcations of nonlinear ion acoustic travelling waves in the frame of a Zakharov-Kuznetsov equation in magnetized plasma with a kappa distributed electron
International Nuclear Information System (INIS)
Kumar Samanta, Utpal; Saha, Asit; Chatterjee, Prasanta
2013-01-01
Bifurcations of nonlinear propagation of ion acoustic waves (IAWs) in a magnetized plasma whose constituents are cold ions and kappa distributed electron are investigated using a two component plasma model. The standard reductive perturbation technique is used to derive the Zakharov-Kuznetsov (ZK) equation for IAWs. By using the bifurcation theory of planar dynamical systems to this ZK equation, the existence of solitary wave solutions and periodic travelling wave solutions is established. All exact explicit solutions of these travelling waves are determined. The results may have relevance in dense space plasmas
16. Coronary bifurcation lesions treated with simple or complex stenting
DEFF Research Database (Denmark)
Behan, Miles W; Holm, Niels R; de Belder, Adam J
2016-01-01
AIMS: Randomized trials of coronary bifurcation stenting have shown better outcomes from a simple (provisional) strategy rather than a complex (planned two-stent) strategy in terms of short-term efficacy and safety. Here, we report the 5-year all-cause mortality based on pooled patient-level data...
17. The Boundary-Hopf-Fold Bifurcation in Filippov Systems
NARCIS (Netherlands)
Efstathiou, Konstantinos; Liu, Xia; Broer, Henk W.
2015-01-01
This paper studies the codimension-3 boundary-Hopf-fold (BHF) bifurcation of planar Filippov systems. Filippov systems consist of at least one discontinuity boundary locally separating the phase space to disjoint components with different dynamics. Such systems find applications in several fields,
18. Stability and Hopf bifurcation analysis of a new system
International Nuclear Information System (INIS)
Huang Kuifei; Yang Qigui
2009-01-01
In this paper, a new chaotic system is introduced. The system contains special cases as the modified Lorenz system and conjugate Chen system. Some subtle characteristics of stability and Hopf bifurcation of the new chaotic system are thoroughly investigated by rigorous mathematical analysis and symbolic computations. Meanwhile, some numerical simulations for justifying the theoretical analysis are also presented.
19. Nonintegrability of the unfolding of the fold-Hopf bifurcation
Science.gov (United States)
Yagasaki, Kazuyuki
2018-02-01
We consider the unfolding of the codimension-two fold-Hopf bifurcation and prove its meromorphic nonintegrability in the meaning of Bogoyavlenskij for almost all parameter values. Our proof is based on a generalized version of the Morales-Ramis-Simó theory for non-Hamiltonian systems and related variational equations up to second order are used.
20. Bifurcations and complete chaos for the diamagnetic Kepler problem
Science.gov (United States)
Hansen, Kai T.
1995-03-01
We describe the structure of bifurcations in the unbounded classical diamagnetic Kepler problem. We conjecture that this system does not have any stable orbits and that the nonwandering set is described by a complete trinary symbolic dynamics for scaled energies larger than ɛc=0.328 782. . ..
1. Bifurcations and Complete Chaos for the Diamagnetic Kepler Problem
OpenAIRE
Hansen, Kai T.
1995-01-01
We describe the structure of bifurcations in the unbounded classical Diamagnetic Kepler problem. We conjecture that this system does not have any stable orbits and that the non-wandering set is described by a complete trinary symbolic dynamics for scaled energies larger then $\\epsilon_c=0.328782\\ldots$.
2. Experimental Investigation of Bifurcations in a Thermoacoustic Engine
Directory of Open Access Journals (Sweden)
Vishnu R. Unni
2015-06-01
Full Text Available In this study, variation in the characteristics of the pressure oscillations in a thermoacoustic engine is explored as the input heat flux is varied. A bifurcation diagram is plotted to study the variation in the qualitative behavior of the acoustic oscillations as the input heat flux changes. At a critical input heat flux (60 Watt, the engine begins to produce acoustic oscillations in its fundamental longitudinal mode. As the input heat flux is increased, incommensurate frequencies appear in the power spectrum. The simultaneous presence of incommensurate frequencies results in quasiperiodic oscillations. On further increase of heat flux, the fundamental mode disappears and second mode oscillations are observed. These bifurcations in the characteristics of the pressure oscillations are the result of nonlinear interaction between multiple modes present in the thermoacoustic engine. Hysteresis in the bifurcation diagram suggests that the bifurcation is subcritical. Further, the qualitative analysis of different dynamic regimes is performed using nonlinear time series analysis. The physical reason for the observed nonlinear behavior is discussed. Suggestions to avert the variations in qualitative behavior of the pressure oscillations in thermoacoustic engines are also provided.
3. Stability of Bifurcating Stationary Solutions of the Artificial Compressible System
Science.gov (United States)
Teramoto, Yuka
2018-02-01
The artificial compressible system gives a compressible approximation of the incompressible Navier-Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number ɛ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small ɛ . In general, the range of ɛ shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of ɛ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.
4. Long term results of kissing stents in the aortic bifurcation
NARCIS (Netherlands)
Hinnen, J.W.; Konickx, M.A.; Meerwaldt, Robbert; Kolkert, J.L.P.; van der Palen, Jacobus Adrianus Maria; Huisman, A.B.
2015-01-01
BACKGROUND: To evaluate the long-term outcome after aortoiliac kissing stent placement and to analyze variables, which potentially influence the outcome of endovascular reconstruction of the aortic bifurcation. METHODS: All patients treated with aortoiliac kissing stents at our institution between
5. Femoral bifurcation with ipsilateral tibia hemimelia: Early outcome of ...
African Journals Online (AJOL)
Hereby, we present a case report of a 2-year-old boy who first presented in our orthopedic clinic as a 12-day-old neonate, with a grossly deformed right lower limb from a combination of complete tibia hemimelia and ipsilateral femoral bifurcation. Excision of femoral exostosis, knee disarticulation and prosthetic fitting gives ...
6. Hopf bifurcation formula for first order differential-delay equations
Science.gov (United States)
Rand, Richard; Verdugo, Anael
2007-09-01
This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using Lindstedt's perturbation method.
7. Direction and stability of bifurcating solutions for a Signorini problem
Czech Academy of Sciences Publication Activity Database
Eisner, J.; Kučera, Milan; Recke, L.
2015-01-01
Roč. 113, January (2015), s. 357-371 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : Signorini problem * variational inequality * bifurcation direction Subject RIV: BA - General Mathematics Impact factor: 1.125, year: 2015 http://www.sciencedirect.com/science/article/pii/S0362546X14003228
8. Smooth bifurcation for a Signorini problem on a rectangle
Czech Academy of Sciences Publication Activity Database
Eisner, J.; Kučera, Milan; Recke, L.
2012-01-01
Roč. 137, č. 2 (2012), s. 131-138 ISSN 0862-7959 R&D Projects: GA AV ČR IAA100190805 Institutional research plan: CEZ:AV0Z10190503 Keywords : Signorini problem * smooth bifurcation * variational inequality Subject RIV: BA - General Mathematics http://dml.cz/dmlcz/142859
9. Bifurcation analysis and the travelling wave solutions of the Klein
In this paper, we investigate the bifurcations and dynamic behaviour of travelling wave solutions of the Klein–Gordon–Zakharov equations given in Shang et al, Comput. Math. Appl. 56, 1441 (2008). Under different parameter conditions, we obtain some exact explicit parametric representations of travelling wave solutions by ...
10. Epidemic model with vaccinated age that exhibits backward bifurcation
International Nuclear Information System (INIS)
Yang Junyuan; Zhang Fengqin; Li Xuezhi
2009-01-01
Vaccination of susceptibilities is included in a transmission model for a disease that confers immunity. In this paper, interplay of vaccination strategy together with vaccine efficacy and the vaccinated age is studied. In particular, vaccine efficacy can lead to a backward bifurcation. At the same time, we also discuss an abstract formulation of the problem, and establish the well-posedness of the model.
11. Bifurcation methods of dynamical systems for handling nonlinear ...
physics pp. 863–868. Bifurcation methods of dynamical systems for handling nonlinear wave equations. DAHE FENG and JIBIN LI. Center for Nonlinear Science Studies, School of Science, Kunming University of Science and Technology .... (b) It can be shown from (15) and (18) that the balance between the weak nonlinear.
12. Bifurcation analysis of wind-driven flows with MOM4
NARCIS (Netherlands)
Bernsen, E.; Dijkstra, H.A.; Wubs, F.W.
2009-01-01
In this paper, the methodology of bifurcation analysis is applied to the explicit time-stepping ocean model MOM4 using a Jacobian–Free Newton–Krylov (JFNK) approach. We in detail present the implementation of the JFNK method in MOM4 but restrict the preconditioning technique to the case for which
13. Chemical reaction systems with a homoclinic bifurcation: an inverse problem
Czech Academy of Sciences Publication Activity Database
Plesa, T.; Vejchodský, Tomáš; Erban, R.
2016-01-01
Roč. 54, č. 10 (2016), s. 1884-1915 ISSN 0259-9791 EU Projects: European Commission(XE) 328008 - STOCHDETBIOMODEL Institutional support: RVO:67985840 Keywords : nonnegative dynamical systems * bifurcations * oscillations Subject RIV: BA - General Mathematics Impact factor: 1.308, year: 2016 http://link.springer.com/article/10.1007%2Fs10910-016-0656-1
14. Bifurcation Analysis of Spiral Growth Processes in Plants
DEFF Research Database (Denmark)
Andersen, C.A.; Ernstsen, C.N.; Mosekilde, Erik
1999-01-01
In order to examine the significance of different assumptions about the range of the inhibitory forces, we have performed a series of bifurcation analyses of a simple model that can explain the formation of helical structures in phyllotaxis. Computer simulations are used to illustrate the role...
15. Smooth bifurcation for variational inequalities based on Lagrange multipliers
Czech Academy of Sciences Publication Activity Database
Eisner, Jan; Kučera, Milan; Recke, L.
2006-01-01
Roč. 19, č. 9 (2006), s. 981-1000 ISSN 0893-4983 R&D Projects: GA AV ČR(CZ) IAA100190506 Institutional research plan: CEZ:AV0Z10190503 Keywords : abstract variational inequality * bifurcation * Lagrange multipliers Subject RIV: BA - General Mathematics
16. Experimental bifurcation analysis of an impact oscillator – Determining stability
DEFF Research Database (Denmark)
Bureau, Emil; Schilder, Frank; Elmegård, Michael
2014-01-01
We propose and investigate three different methods for assessing stability of dynamical equilibrium states during experimental bifurcation analysis, using a control-based continuation method. The idea is to modify or turn off the control at an equilibrium state and study the resulting behavior...
17. Regularization of the Boundary-Saddle-Node Bifurcation
Directory of Open Access Journals (Sweden)
Xia Liu
2018-01-01
Full Text Available In this paper we treat a particular class of planar Filippov systems which consist of two smooth systems that are separated by a discontinuity boundary. In such systems one vector field undergoes a saddle-node bifurcation while the other vector field is transversal to the boundary. The boundary-saddle-node (BSN bifurcation occurs at a critical value when the saddle-node point is located on the discontinuity boundary. We derive a local topological normal form for the BSN bifurcation and study its local dynamics by applying the classical Filippov’s convex method and a novel regularization approach. In fact, by the regularization approach a given Filippov system is approximated by a piecewise-smooth continuous system. Moreover, the regularization process produces a singular perturbation problem where the original discontinuous set becomes a center manifold. Thus, the regularization enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation.
18. Topography of Aortic Bifurcation in a Black Kenyan Population ...
African Journals Online (AJOL)
After removal of abdominal viscera, peritoneum, fibrofatty connective tissue, inferior vena cava was removed to expose the termination of abdominal aorta. Vertebral level, angle and asymmetry of bifurcation were recorded. Data were analysed by SPSS version 17.0 for windows and are presented in tables and bar charts.
19. Can a brief period of double J stenting improve the outcome of extracorporeal shock wave lithotripsy for renal calculi sized 1 to 2 cm?
Science.gov (United States)
Sharma, Rakesh; Das, Ranjit Kumar; Basu, Supriya; Dey, Ranjan Kumar; Gupta, Rupesh; Deb, Partha Pratim
2017-01-01
Purpose Extracorporeal shock wave lithotripsy (ESWL) is an established modality for renal calculi. Its role for large stones is being questioned. A novel model of temporary double J (DJ) stenting followed by ESWL was devised and outcomes were assessed. Materials and Methods The study included 95 patients with renal calculi sized 1 to 2 cm. Patients were randomized into 3 groups. Group 1 received ESWL only, whereas group 2 underwent stenting followed by ESWL. In group 3, a distinct model was applied in which the stent was kept for 1 week and then removed, followed by ESWL. Procedural details, analgesic requirements, and outcome were analyzed. Results Eighty-eight patients (male, 47; female, 41) were available for analysis. The patients' mean age was 37.9±10.9 years. Stone profile was similar among groups. Group 3 received fewer shocks (mean, 3,155) than did group 1 (mean, 3,859; p=0.05) or group 2 (mean, 3,872; p=0.04). The fragmentation rate was similar in group 3 (96.7%) and groups 1 (81.5%, p=0.12) and 2 (87.1%, p=0.16). Overall clearance in group 3 was significantly improved (83.3%) compared with that in groups 1 (63.0%, p=0.02) and 2 (64.5%, p=0.02) and was maintained even in lower pole stones. The percentage successful outcome in groups 1, 2, and 3 was 66.7%, 64.5%, and 83.3%, respectively (p=0.21). The analgesic requirement in group 2 was higher than in the other groups (p=0.00). Group 2 patients also had more grade IIIa (2/3) and IIIB (1/2) complications. Conclusions Stenting adversely affects stone clearance and also makes the later course uncomfortable. Our model of brief stenting followed by ESWL provided better clearance, comfort, and a modest improvement in outcome with fewer sittings and steinstrasse in selected patients with large renal calculi. PMID:28261679
20. Can a brief period of double J stenting improve the outcome of extracorporeal shock wave lithotripsy for renal calculi sized 1 to 2 cm?
Directory of Open Access Journals (Sweden)
Rakesh Sharma
2017-03-01
Full Text Available Purpose: Extracorporeal shock wave lithotripsy (ESWL is an established modality for renal calculi. Its role for large stones is being questioned. A novel model of temporary double J (DJ stenting followed by ESWL was devised and outcomes were assessed. Materials and Methods: The study included 95 patients with renal calculi sized 1 to 2 cm. Patients were randomized into 3 groups. Group 1 received ESWL only, whereas group 2 underwent stenting followed by ESWL. In group 3, a distinct model was applied in which the stent was kept for 1 week and then removed, followed by ESWL. Procedural details, analgesic requirements, and outcome were analyzed. Results: Eighty-eight patients (male, 47; female, 41 were available for analysis. The patients’ mean age was 37.9±10.9 years. Stone profile was similar among groups. Group 3 received fewer shocks (mean, 3,155 than did group 1 (mean, 3,859; p=0.05 or group 2 (mean, 3,872; p=0.04. The fragmentation rate was similar in group 3 (96.7% and groups 1 (81.5%, p=0.12 and 2 (87.1%, p=0.16. Overall clearance in group 3 was significantly improved (83.3% compared with that in groups 1 (63.0%, p=0.02 and 2 (64.5%, p=0.02 and was maintained even in lower pole stones. The percentage successful outcome in groups 1, 2, and 3 was 66.7%, 64.5%, and 83.3%, respectively (p=0.21. The analgesic requirement in group 2 was higher than in the other groups (p=0.00. Group 2 patients also had more grade IIIa (2/3 and IIIB (1/2 complications. Conclusions: Stenting adversely affects stone clearance and also makes the later course uncomfortable. Our model of brief stenting followed by ESWL provided better clearance, comfort, and a modest improvement in outcome with fewer sittings and steinstrasse in selected patients with large renal calculi.
1. Numerical bifurcation analysis of conformal formulations of the Einstein constraints
International Nuclear Information System (INIS)
Holst, M.; Kungurtsev, V.
2011-01-01
The Einstein constraint equations have been the subject of study for more than 50 years. The introduction of the conformal method in the 1970s as a parametrization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental nonuniqueness problems with the conformal method as a parametrization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods. We discuss these results and their physical significance, which lead to some interesting remaining questions to
2. Clinical Outcome After DK Crush Versus Culotte Stenting of Distal Left Main Bifurcation Lesions: The 3-Year Follow-Up Results of the DKCRUSH-III Study.
Science.gov (United States)
Chen, Shao-Liang; Xu, Bo; Han, Ya-Ling; Sheiban, Imad; Zhang, Jun-Jie; Ye, Fei; Kwan, Tak W; Paiboon, Chitprapai; Zhou, Yu-Jie; Lv, Shu-Zheng; Dangas, George D; Xu, Ya-Wei; Wen, Shang-Yu; Hong, Lang; Zhang, Rui-Yan; Wang, Hai-Chang; Jiang, Tie-Ming; Wang, Yan; Sansoto, Teguh; Chen, Fang; Yuan, Zu-Yi; Li, Wei-Min; Leon, Martin B
2015-08-24
The present study aimed to investigate the difference in major adverse cardiac events (MACE) at 3 years after double-kissing (DK) crush versus culotte stenting for unprotected left main distal bifurcation lesions (LMDBLs). The multicenter and randomized DKCRUSH-III (Comparison of double kissing crush versus culotte stenting for unprotected distal left main bifurcation lesions: results from a multicenter, randomized, prospective study) showed that DK crush stenting was associated with fewer MACE at 1-year follow-up in patients with LMDBLs compared with culotte stenting. Here, we report the 3-year clinical outcome of the DKCRUSH-III study. A total of 419 patients with LMDBLs who were randomly assigned to either the DK crush or culotte group in the DKCRUSH-III study were followed for 3 year. The primary endpoint was the occurrence of a MACE at 3 years. Stent thrombosis (ST) was the safety endpoint. Patients were classified by simple and complex LMDBLs according to the DEFINITION (Definition and Impact of Complex Bifurcation Lesions on Clinical Outcomes After Percutaneous Coronary Intervention Using Drug-Eluting Stents) study criteria. At 3 years, MACE occurred in 49 patients the culotte group and in 17 patients in the DK crush group (cumulative event rates of 23.7% and 8.2%, respectively; p DK crush group (p = 0.007). Complex LMDBLs were associated with a higher rate of MACE (35.3%) at 3 years compared with a rate of 8.1% in patients with simple LMDBLs (p DK] Crush Versus Culotte Stenting for the Treatment of Unprotected Distal Left Main Bifurcation Lesions: DKCRUSH-III, a Multicenter Randomized Study Comparing Double-Stent Techniques; ChiCTR-TRC-11001877). Copyright © 2015 American College of Cardiology Foundation. Published by Elsevier Inc. All rights reserved.
3. Reduced order models, inertial manifolds, and global bifurcations: searching instability boundaries in nuclear power systems
International Nuclear Information System (INIS)
Suarez Antola, R.
2011-01-01
One of the goals of nuclear power systems design and operation is to restrict the possible states of certain critical subsystems, during steady operation and during transients, to remain inside a certain bounded set of admissible states and state variations. Also, during transients, certain restrictions must be imposed on the time scale of evolution of the critical subsystem's state. A classification of the different solution types concerning their relation with the operational safety of the power plant is done by distributing the different solution types in relation with the exclusion region of the power-flow map. In the framework of an analytic or numerical modeling process of a boiling water reactor (BWR) power plant, this could imply first to find an suitable approximation to the solution manifold of the differential equations describing the stability behavior of this nonlinear system, and then a classification of the different solution types concerning their relation with the operational safety of the power plant, by distributing the different solution types in relation with the exclusion region of the power-flow map. Inertial manifold theory gives a foundation for the construction and use of reduced order models (ROM's) of reactor dynamics to discover and characterize meaningful bifurcations that may pass unnoticed during digital simulations done with full scale computer codes of the nuclear power plant. The March-Leuba's BWR ROM is used to exemplify the analytical approach developed here. The equation for excess void reactivity of this ROM is generalized. A nonlinear integral-differential equation in the logarithmic power is derived, including the generalized thermal-hydraulics feedback on the reactivity. Introducing a Krilov- Bogoliubov-Mitropolsky (KBM) ansatz with both amplitude and phase being slowly varying functions of time relative to the center period of oscillation, a coupled set of nonlinear ordinary differential equations for amplitude and phase
4. Analysis and control of complex dynamical systems robust bifurcation, dynamic attractors, and network complexity
CERN Document Server
Imura, Jun-ichi; Ueta, Tetsushi
2015-01-01
This book is the first to report on theoretical breakthroughs on control of complex dynamical systems developed by collaborative researchers in the two fields of dynamical systems theory and control theory. As well, its basic point of view is of three kinds of complexity: bifurcation phenomena subject to model uncertainty, complex behavior including periodic/quasi-periodic orbits as well as chaotic orbits, and network complexity emerging from dynamical interactions between subsystems. Analysis and Control of Complex Dynamical Systems offers a valuable resource for mathematicians, physicists, and biophysicists, as well as for researchers in nonlinear science and control engineering, allowing them to develop a better fundamental understanding of the analysis and control synthesis of such complex systems.
5. Mortality and complications after aortic bifurcated bypass procedures for chronic aortoiliac occlusive disease
DEFF Research Database (Denmark)
Bredahl, Kim; Jensen, Leif Panduro; Schroeder, Torben V
2015-01-01
skills, particularly because open surgery is increasingly used in those patients who are unsuitable for endovascular repair and hence technically more demanding. We assessed the early outcome after aortic bifurcated bypass procedures during two decades of growing endovascular activity and identified...... preoperative risk factors. METHODS: Data on patients with chronic limb ischemia were prospectively collected during a 20-year period (1993 to 2012). The data were obtained from the Danish Vascular Registry, assessed, and merged with data from The Danish Civil Registration System. RESULTS: We identified 3623...... aortobifemoral and 144 aortobiiliac bypass procedures. The annual caseload fell from 323 to 106 during the study period, but the 30-day mortality at 3.6% (95% confidence interval [CI], 3.0-4.1) and the 30-day major complication rate remained constant at 20% (95% CI, 18-21). Gangrene (odds ratio [OR], 3.3; 95% CI...
6. Numerical bifurcation analysis of delay differential equations arising from physiological modeling.
Science.gov (United States)
Engelborghs, K; Lemaire, V; Bélair, J; Roose, D
2001-04-01
This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of steady state solutions and periodic solutions of systems of delay differential equations with an arbitrary number of fixed, discrete delays. Second, we demonstrate how these methods can be used to obtain insight into complex biological regulatory systems in which interactions occur with time delays: for this, we consider a system of two equations for the plasma glucose and insulin concentrations in a diabetic patient subject to a system of external assistance. The model has two delays: the technological delay of the external system, and the physiological delay of the patient's liver. We compute stability of the steady state solution as a function of two parameters, compare with analytical results and compute several branches of periodic solutions and their stability. These numerical results allow to infer two categories of diabetic patients for which the external system has different efficiency.
7. Bifurcation analysis of oscillating network model of pattern recognition in the rabbit olfactory bulb
Science.gov (United States)
Baird, Bill
1986-08-01
A neural network model describing pattern recognition in the rabbit olfactory bulb is analysed to explain the changes in neural activity observed experimentally during classical Pavlovian conditioning. EEG activity recorded from an 8×8 arry of 64 electrodes directly on the surface on the bulb shows distinct spatial patterns of oscillation that correspond to the animal's recognition of different conditioned odors and change with conditioning to new odors. The model may be considered a variant of Hopfield's model of continuous analog neural dynamics. Excitatory and inhibitory cell types in the bulb and the anatomical architecture of their connection requires a nonsymmetric coupling matrix. As the mean input level rises during each breath of the animal, the system bifurcates from homogenous equilibrium to a spatially patterned oscillation. The theory of multiple Hopf bifurcations is employed to find coupled equations for the amplitudes of these unstable oscillatory modes independent of frequency. This allows a view of stored periodic attractors as fixed points of a gradient vector field and thereby recovers the more familiar dynamical systems picture of associative memory.
8. Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation.
Science.gov (United States)
Krauskopf, Bernd; Sieber, Jan
2014-09-08
Models of global climate phenomena of low to intermediate complexity are very useful for providing an understanding at a conceptual level. An important aspect of such models is the presence of a number of feedback loops that feature considerable delay times, usually due to the time it takes to transport energy (for example, in the form of hot/cold air or water) around the globe. In this paper, we demonstrate how one can perform a bifurcation analysis of the behaviour of a periodically forced system with delay in dependence on key parameters. As an example, we consider the El-Niño Southern Oscillation (ENSO), which is a sea-surface temperature (SST) oscillation on a multi-year scale in the basin of the Pacific Ocean. One can think of ENSO as being generated by an interplay between two feedback effects, one positive and one negative, which act only after some delay that is determined by the speed of transport of SST anomalies across the Pacific. We perform here a case study of a simple delayed-feedback oscillator model for ENSO, which is parametrically forced by annual variation. More specifically, we use numerical bifurcation analysis tools to explore directly regions of delay-induced resonances and other stability boundaries in this delay-differential equation model for ENSO.
9. Simulation of Few Bifurcation Phase Diagrams of Belousov-Zhabotinsky Reaction with Eleven Variable Chaotic Model in CSTR
Directory of Open Access Journals (Sweden)
B. Swathi
2009-01-01
Full Text Available Simulation of the Gyorgyi, Rempe and Field eleven variable chaotic model in CSTR [Continuously Stirred Tank Reactor] is performed with respect to the concentrations of malonic acid and [Ce(III]. These simulation studies show steady state, periodic and non-periodic regions. These studies have been presented as two variable bifurcation phase diagrams. We also have observed the bursting phenomenon under different set of constraints. We have given much importance on computer simulation work but not included the experimental methods in this paper.
10. Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors
Science.gov (United States)
Ibrahim, K. M.; Jamal, R. K.; Ali, F. H.
2018-05-01
The behaviour of certain dynamical nonlinear systems are described in term as chaos, i.e., systems’ variables change with the time, displaying very sensitivity to initial conditions of chaotic dynamics. In this paper, we study archetype systems of ordinary differential equations in two-dimensional phase spaces of the Rössler model. A system displays continuous time chaos and is explained by three coupled nonlinear differential equations. We study its characteristics and determine the control parameters that lead to different behavior of the system output, periodic, quasi-periodic and chaos. The time series, attractor, Fast Fourier Transformation and bifurcation diagram for different values have been described.
11. Efficient algorithm for bifurcation problems of variational inequalities
International Nuclear Information System (INIS)
Mittelmann, H.D.
1983-01-01
For a class of variational inequalities on a Hilbert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that is compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient interaction. So some results are included for the path following in turning-point problems
12. Bifurcation software in Matlab with applications in neuronal modeling.
Science.gov (United States)
Govaerts, Willy; Sautois, Bart
2005-02-01
Many biological phenomena, notably in neuroscience, can be modeled by dynamical systems. We describe a recent improvement of a Matlab software package for dynamical systems with applications to modeling single neurons and all-to-all connected networks of neurons. The new software features consist of an object-oriented approach to bifurcation computations and the partial inclusion of C-code to speed up the computation. As an application, we study the origin of the spiking behaviour of neurons when the equilibrium state is destabilized by an incoming current. We show that Class II behaviour, i.e. firing with a finite frequency, is possible even if the destabilization occurs through a saddle-node bifurcation. Furthermore, we show that synchronization of an all-to-all connected network of such neurons with only excitatory connections is also possible in this case.
13. Hybrid intravenous digital subtraction angiography of the carotid bifurcation
International Nuclear Information System (INIS)
Burbank, F.H.; Enzmann, D.; Keyes, G.S.; Brody, W.R.
1984-01-01
A hybrid digital subtraction angiography technique and noise-reduction algorithm were used to evaluate the carotid bifurcation. Temporal, hybrid, and reduced-noise hybrid images were obtained in right and left anterior oblique projections, and both single- and multiple-frame images were created with each method. The resulting images were graded on a scale of 1 to 5 by three experienced neuroradiologists. Temporal images were preferred over hybrid images. The percentage of nondiagnostic examinations, as agreed upon by two readers, was higher for temporal alone than temporal + hybrid. In addition, also by agreement between two readers, temporal + hybrid images significantly increased the number of bifurcations seen in two views (87%) compared to temporal subtraction alone
14. Local bifurcations in differential equations with state-dependent delay.
Science.gov (United States)
Sieber, Jan
2017-11-01
A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.
15. Local bifurcations in differential equations with state-dependent delay
Science.gov (United States)
Sieber, Jan
2017-11-01
A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.
16. Synchronization of diffusively coupled oscillators near the homoclinic bifurcation
International Nuclear Information System (INIS)
Postnov, D.; Han, Seung Kee; Kook, Hyungtae
1998-09-01
It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the inphase synchronization and also that it is the only stable state in the weak coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the synchronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis. (author)
17. Model Reduction of Nonlinear Aeroelastic Systems Experiencing Hopf Bifurcation
KAUST Repository
Abdelkefi, Abdessattar
2013-06-18
In this paper, we employ the normal form to derive a reduced - order model that reproduces nonlinear dynamical behavior of aeroelastic systems that undergo Hopf bifurcation. As an example, we consider a rigid two - dimensional airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We apply the center manifold theorem on the governing equations to derive its normal form that constitutes a simplified representation of the aeroelastic sys tem near flutter onset (manifestation of Hopf bifurcation). Then, we use the normal form to identify a self - excited oscillator governed by a time - delay ordinary differential equation that approximates the dynamical behavior while reducing the dimension of the original system. Results obtained from this oscillator show a great capability to predict properly limit cycle oscillations that take place beyond and above flutter as compared with the original aeroelastic system.
18. Fold points and singularity induced bifurcation in inviscid transonic flow
International Nuclear Information System (INIS)
Marszalek, Wieslaw
2012-01-01
Transonic inviscid flow equation of elliptic–hyperbolic type when written in terms of the velocity components and similarity variable results in a second order nonlinear ODE having several features typical of differential–algebraic equations rather than ODEs. These features include the fold singularities (e.g. folded nodes and saddles, forward and backward impasse points), singularity induced bifurcation behavior and singularity crossing phenomenon. We investigate the above properties and conclude that the quasilinear DAEs of transonic flow have interesting properties that do not occur in other known quasilinear DAEs, for example, in MHD. Several numerical examples are included. -- Highlights: ► A novel analysis of inviscid transonic flow and its similarity solutions. ► Singularity induced bifurcation, singular points of transonic flow. ► Projection method, index of transonic flow DAEs, linearization via matrix pencil.
19. Bifurcation theory applied to buckling states of a cylindrical shell
Science.gov (United States)
1995-01-01
Veins, bronchii, and many other vessels in the human body are flexible enough to be capable of collapse if submitted to suitable applied external and internal loads. One way to describe this phenomenon is to consider an inextensible elastic and infinite tube, with a circular cross section in the reference configuration, subjected to a uniform external pressure. In this paper, we establish that the nonlinear equilibrium equation for this model has nontrivial solutions which appear for critical values of the pressure. To this end, the tools we use are the Liapunov-Schmidt decomposition and the bifurcation theorem for simple multiplicity. We conclude with the bifurcation diagram, showing the dependence between the cross-sectional area and the pressure.
20. Structural bifurcation of microwave helium jet discharge at atmospheric pressure
International Nuclear Information System (INIS)
Takamura, Shuichi; Kitoh, Masakazu; Soga, Tadasuke
2008-01-01
Structural bifurcation of microwave-sustained jet discharge at atmospheric gas pressure was found to produce a stable helium plasma jet, which may open the possibility of a new type of high-flux test plasma beam for plasma-wall interactions in fusion devices. The fundamental discharge properties are presented including hysteresis characteristics, imaging of discharge emissive structure, and stable ignition parameter area. (author) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8011890649795532, "perplexity": 2075.648384797102}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145746.24/warc/CC-MAIN-20200223032129-20200223062129-00194.warc.gz"} |
http://mathoverflow.net/questions/38155?sort=oldest | ## Rational solutions of homogeneous equations
Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\times n$-matrix $A$ (possibly $m>n$) with integer entries (say all entries $1,0,-1$ for simplicity). Given a solution $x\in {\Bbb R}^n$ and $\epsilon>0$, do there exist solutions in ${\Bbb Q}^n$ within distance $< \epsilon$ from $x$?
I am sure this kind of question has been considered somewhere. However, as a topologist, I have no idea where to look this up. Apart from answers also hints to literature about this genre of questions would be appreciated.
-
What is your definition of distance, and in what space does A live? – Per Alexandersson Sep 9 2010 at 8:22 A slightly more general and interesting question is given a variety, whether the set of rational solutions is dense in the reals and the p-adics. This is related to weak and strong approximation. – Daniel Loughran Sep 9 2010 at 10:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702829122543335, "perplexity": 214.67211850003443}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698104521/warc/CC-MAIN-20130516095504-00012-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/133308/functions-of-random-variables | # Functions of Random Variables
Let $X$ be a discrete random variable with probability function $f_X$ . Find formulas for the probability function and the distribution function of $Y =(X − a)^2$, where $a$ is an arbitrary constant.
-
Is this homework? What have you tried? Where are you getting stuck? – Matthew Conroy Apr 18 '12 at 4:07
Are you in touch with the other poster who is posting questions just like this one? Maybe the two of you should get together and have a chat. – Gerry Myerson Apr 18 '12 at 4:08
– Henry Apr 18 '12 at 11:32
We address both the case where the random variable has a discrete distribution, and the case where the random variable $X$ has continuous distribution, with density function $f(x)$. The continuous case seems to come up more often, so we deal with it first.
Let $Y=(X-a)^2$. We find an expression for $P(Y \le y)$. This is only interesting if $y \ge 0$. We have $Y>y$ iff $(X-a)^2>y$ iff $X>a+\sqrt{y}$ or $X<a-\sqrt{y}$. Thus the cumulative distribution function $F_Y(y)$ of $Y$ is given, for positive $y$, by $$F_Y(y)=1-\int_{-\infty}^{a-\sqrt{y}}f(x)\,dx -\int_{a+\sqrt{y}}^{\infty} f(x)\,dx.$$ Differentiate with respect to $y$ to find the density function $f_Y(y)$ of $Y$. For $y>0$ this is given by $$f_Y(y)=\frac{1}{2\sqrt{y}}\left(f(a-\sqrt{y})+f(a+\sqrt{y}\right).$$
In the general discrete case, for $y \ge 0$, we have $$P(Y=y)=P((X-a)^2=y)=P(X=a-\sqrt{y})+P(X=a+\sqrt{y}).\tag{\ast}$$ The required values in $(\ast)$ are then obtained from the probability distribution function of $X$. Let $f_X(x)=P(X=x)$. Then $P(Y=y)=f_X(a-\sqrt{y})+f_X(a+\sqrt{y})$. From this an expression for the cumulative distribution function $F_Y(y)$ can be written down. For $y \ge 0$ it is $$F_Y(y)=\sum_{x\le a+\sqrt{y}} f_X(x)-\sum_{x<a-\sqrt{y}} f_X(x).$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9702088832855225, "perplexity": 103.08495653723354}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931010166.36/warc/CC-MAIN-20141125155650-00216-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://www.crookedbrains.net/2007/09/t-shirt-with-attitude-so-whats-all-this.html | ## Sep 20, 2007
### T-Shirt With An Attitude.
So what's all this about? It's a pickup line using calculus. Try to solve the equation & you will come up with a interesting pickup line.
Though to the 99% of the people who will see this, it wont make any sense as what is the result of evaluation of this math expression, & apart from this a further 0.4% (plus or minus a bit) are more likely to jump on the conclusion that the integral evaluates to 42 without trying to plug in the values.
Still confused? So here is how it works:
2xdx integrates as x squared; to evaluate the integral of 2xdx over the interval from 10 to 13, you subtract the value at the bottom of the interval from the value at the top of the interval.
At the bottom of the interval, setting x equal to 10 yields 100, because 10 x 10 = 100. At the top of the interval, set x equal to 13, calculate x squared (13 x 13), and then subtract 100 from it. At this point, it is essential to note the presence of the question mark.
That's what makes this an interesting pickup line.
Source
Other Posts:
Water Bridge: Now This Is Engineering
Cool Magazine Illusions
What would you do with your old PC!
Sport Car Drawn In MS Paint
Related Posts:
Math Problems | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8745097517967224, "perplexity": 708.2654176857375}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131303461.20/warc/CC-MAIN-20150323172143-00012-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.codecademy.com/learn/mlef-math-and-statistics-for-ml-ai-engineers/modules/mlef-differential-calculus/cheatsheet | # Differential Calculus
### Limits
Limits quantify what happens to the values of a function as we approach a given point. This can be defined notationally as:
$\lim_{x \rightarrow 6} f(x) = L$
We can read this in simple terms as “the limit as x goes to 6 of f(x) approaches some value L”.
### Limit Definition of the Derivative
The limit definition of the derivative proves how to measure the instantaneous rate of change of a function at a specific point by looking at an infinitesimally small range of x values.
$instantaneous\ rate\ of\ change\ = \lim_{h \rightarrow 0} \frac{f(x+h)}{h}$
The animation provided shows that as we look at a smaller range of x values, we approach the instantaneous range of a point.
### Derivative Properties
The derivative is the slope of a tangent line at a specific point, and the derivative of a function f(x) is denoted as f’(x). We can use the derivative of a function to determine where the function is increasing, decreasing, at a minimum or maximum value, or at an inflection point.
If f’(x) = 0, then the function is not changing. This can mean one of a few things.
• It may mean that the function has reached a local maximum (or minimum). A local maximum is a value of x where f’(x) changes from positive to negative and thus hits 0 along the way. In f(x), the local maximum is lower than all the points around it.
• It may also mean that the function has reached what is called a local maximum. Our local maximum is higher than the points around it. When f’(x) goes from negative values to 0 to positive values, a local maximum forms.
• It may be an inflection point. This is a point where a function has a change in the direction of curvature. For example, the curve of the function goes from “facing down” to “facing up.” Finding inflection points involves a second derivative test, which we will not get to in this lesson.
If f’(x) > 0, the function is increasing, and if f’(x) < 0, the function is decreasing.
### Derivatives in Python
We can use the np.gradient() function from the NumPy library to calculate derivatives of functions represented by arrays. The code block shown shows how to calculate the derivative of the function f(x) = x3 + 2 using the gradient() function.
from math import pow
# dx is the "step" between each x valuedx = 0.05def f(x): # to calculate the y values of the function return pow(x, 3) + 2# x valuesf_array_x = [x for x in np.arange(0,4,dx)]# y valuesf_array_y = [f(x) for x in np.arange(0,4,dx)]
# derivative calculationf_array_deriv = np.gradient(f_array_y, dx)
### Calculating Derivatives
To take the derivative of polynomial functions, we use the power rule. This states the following:
$\frac{d}{dx}x^{n} = nx^{n-1}$
There are rules even beyond the power rule. Many common functions have defined derivatives. Here are some common ones:
\begin{aligned} \frac{d}{dx}ln(x) = \frac{1}{x} \\ \frac{d}{dx}e^x = e^x \\ \frac{d}{dx}sin(x) = cos(x) \\ \frac{d}{dx}cos(x) = -sin(x) \end{aligned}
### Derivative Rules
There are general rules we can use to calculate derivatives.
The derivative of a constant is equal to zero:
$\frac{d}{dx}c = 0$
Derivatives are linear operators, meaning that we can pull constants out of derivative calculations:
$\frac{d}{dx} c f(x) = c f'(x)$
The derivative of a sum is the sum of the derivatives, meaning we can say the following:
$\frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$
We define the derivative of two products as the following:
\begin{aligned} \frac{d}{dx}(f(x) + g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \\ f(x) = u(x)v(x) \rightarrow f'(x) =u(x)v'(x) + v(x)u'(x) \end{aligned} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9914430379867554, "perplexity": 495.6936070689491}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500357.3/warc/CC-MAIN-20230206181343-20230206211343-00490.warc.gz"} |
https://www.physicsforums.com/threads/sum-of-getometric-sequence-with-alternating-signs.541683/ | # Sum of Getometric Sequence with alternating signs
1. Oct 18, 2011
### notSomebody
1. The problem statement, all variables and given/known data
5^2 - 5^3 + 5^4 - ... + (-1)^k*5^k whre k is an integer with k >= 2
2. Relevant equations
3. The attempt at a solution
I know (5^(k-1) - 5^2)/2 gives you the sum if they were all positive. I tried multiplying it by (-1)^k or something but that just changes the sign. I wish I could give you more but I can't.
2. Oct 18, 2011
### HallsofIvy
Staff Emeritus
A geometric sequence is $\sum_{n=0}^N ar^n$. And the sum is:
$$\frac{1- r^{N+1}}{1- r}$$
r does not have to be positive. Your sequence has a= 1, r= -5.
3. Oct 18, 2011
### notSomebody
$$\frac{1- (-5)^{2+1}}{1- (-5)}$$ = 21 though and not 25
Last edited: Oct 18, 2011
4. Oct 18, 2011
### SammyS
Staff Emeritus
The sum that HallsofIvy gave includes (-5)0 and (-5)1
5. Oct 18, 2011
### HallsofIvy
Staff Emeritus
There are two ways to handle the fact that your sum starts with $r^2$ rather than $r^0= 1$.
1) Factor out an $r^2$ $(-5)^2+ (-5)^3\cdot\cdot\cdot+ (-5)^k= (-5)^2(1+ (-5)+ \cdot\cdot\cdot+ (-5)^{k-2})$
Use the formula I gave with n= k- 2 and then multiply by $(-5)^2= 25$.
2) Use the formula with n= k and then subtract of $(-5)^0+ (-5)^1= 1- 6= -4$.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
Similar Discussions: Sum of Getometric Sequence with alternating signs | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9412338137626648, "perplexity": 2461.509832070263}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688158.27/warc/CC-MAIN-20170922022225-20170922042225-00055.warc.gz"} |
http://mathoverflow.net/questions/23202/explicit-big-linearly-independent-sets/32710 | # explicit big linearly independent sets
In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or AC) are needed. Feel free to use similar (perhaps more precise) notions of "explicit", but reasonable ones! To be honest, I'm not so interested in a discussion about mathematical logic. If no example is there, well, then there is no example. ;-)
Can you give explicit large linearly independent subsets of $\mathbb{R}$ over $\mathbb{Q}$? For example, $\{ln(p) : p \text{ prime}\}$ is such a set, but it's only countable and surely is no basis. You can find more numbers which are linearly independent, but I cannot find uncountably many. AC implies $dim_\mathbb{Q} \mathbb{R} = |\mathbb{R}|$. Perhaps $ZF$ has a model in which every linearly independant subset of $\mathbb{R}$ is countable?
The same question for algebraically independent subsets of $\mathbb{R}$ over $\mathbb{Q}$? Perhaps the set above is such a subset? But anyway, it is too small.
Closely related problems: Can you give an explicit proper subspace of $\mathbb{R}$ over $\mathbb{Q}$, which is isomorphic to $\mathbb{R}$? If so, is the isomorphism explicit? Same question for subfields.
That would be great if there were explicit examples. :-)
-
Here is a linearly independent subset of $\mathbb{R}$ with size $2^{\aleph_0}$.
Let $q_0, q_1, \ldots$ be an enumeration of $\mathbb{Q}$. For every real number $r$, let $$T_r = \sum_{q_n < r} \frac{1}{n!}$$ The proof that these numbers are linearly independent is similar to the usual proof that $e$ is irrational. (It's a cute problem; there's spoiler below.)
I think a similar trick might work for algebraic independence, but I don't recall having seen such a construction. Actually, John von Neumann showed that the numbers $$A_r = \sum_{n=0}^\infty \frac{2^{2^{[nr]}}}{2^{2^{n^2}}}$$ are algebraically independent for $r > 0$. [Ein System algebraisch unabhängiger zahlen, Math. Ann. 99 (1928), no. 1, 134–141.] A more general result due to Jan Mycielski seems to go through in ZF + DC perhaps just ZF in some cases. [Independent sets in topological algebras, Fund. Math. 55 (1964), 139–147.]
As for subspaces and subfields isomorphic to $\mathbb{R}$, the answer is no. (Since I'm not allowed to post any logic here, I'll refer you to this answer and let you figure it out.)
Well, I'll bend the rules a little... Consider a $\mathbb{Q}$-linear isomorphism $h:\mathbb{R}\to H$, where $H$ is a $\mathbb{Q}$-linear subspace of $\mathbb{R}$ (i.e. $h$ is an additive group isomorphism onto the divisible subgroup $H$ of $\mathbb{R}$). If $h$ Baire measurable then it must be continuous by an ancient theorem of Banach and Pettis. It follows that $h(x) = xh(1)$ for all $x \in \mathbb{R}$ and therefore $H = \mathbb{R}$. Shelah has produced a model of ZF + DC where all sets of reals have the Baire property, so any such $h$ in this model must be Baire measurable. A similar argument works if Baire measurable is replaced by Lebesgue measurable, but Solovay's model of ZF + DC where all sets of reals are Lebesgue measurable uses the existence of an inaccessible cardinal, and this hypothesis was shown necessary by Shelah.
Spoiler
Suppose for the sake of contradiction that $r_1 > r_2 > \cdots > r_k$ and $a_1,a_2,\ldots,a_k \in \mathbb{Z}$ are such that $a_1T_{r_1} + a_2T_{r_2} + \cdots + a_kT_{r_k} = 0$. Choose a very large $n$ such that $r_1 > q_n > r_2$. If $n$ is large enough that $$(|a_1| + |a_2| + \cdots + |a_k|) \sum_{m=n+1}^\infty \frac{n!}{m!} < 1$$ then the tail terms of $n!(a_1T_{r_1}+\cdots+a_kT_{r_k}) = 0$ must cancel out, and we're left with $$a_1 = -\sum_{m=0}^{n-1} \sum_{q_m < r_i} a_i \frac{n!}{m!} \equiv 0 \pmod{n}$$ If moreover $n > |a_1|$, this means that $a_1 = 0$. Repeat to conclude that $a_1 = a_2 = \cdots a_k = 0$.
-
Very nice!!!!!! – Steven Gubkin May 1 '10 at 21:49
It depends where you put the word "explicit" ... $\mathbb{R}$ is explicitly ismorphic to a subspace of $\mathbb{R}^2$, right? Well, according to AC, $\mathbb{R}^2$ is also (non-explicitely) isomorphic to $\mathbb{R}$. – Gerald Edgar May 1 '10 at 23:40
perfect ! – Martin Brandenburg May 2 '10 at 9:57
@FrançoisG.Dorais: Dear François, does this trick (for constructing a large $\mathbb{Q}$-linearly independent subset) above belong to you? If not, do you know who should be credited if someone is to use this cute trick? – Burak Apr 1 '15 at 22:49
@Burak: I couldn't honestly claim ownership of this trick. I might have extracted it from earlier (and broader reaching) work by Jan Mycielski. I do recall that the first time I presented the trick was at a graduate student seminar at Dartmouth (over a decade ago). Unfortunately, I don't recall why or how I came up with the trick. At best, I rediscovered a trick which was known decades before I was even born... – François G. Dorais Apr 1 '15 at 23:40
Here's an answer that's similar in spirit to Pietro Majer's, but not quite the same.
As a first step, choose an uncountable family of infinite subsets of positive integers such that any two distinct sets in the family have finite intersection. This can be done explicitly in many ways. One I like is as follows. Since one can take an explicit bijection between $\mathbb{N}$ and $\mathbb{Z}^2$, it's good enough to create a family of subsets of $\mathbb{Z}^2$ instead. And to do that, for each real number $\alpha\in[0,\pi)$ take the set of all points in $\mathbb{Z}^2$ that are within a distance 2 (say) of the line that makes an angle $\alpha$ with the x-axis.
Once we have such a family F, we define a real number $r_X$ for each X in F as follows. It is a number between 0 and 1 that has only 0s and 1s in its decimal expansion. And it has a 1 at the nth place if and only if $n=m^2$ for some $m\in X$. (The reason for restricting to the squares is simply that we want the gaps between successive places where there might be a 1 to get larger and larger, so that we can ignore carrying problems.)
The numbers $r_X$ are linearly dependent over $\mathbb{Q}$ only if we can find a non-zero integer combination of finitely many of them that gives zero. But we can't: if we've got a non-zero coefficient t, then after a while the gaps will be longer than the number of digits of t (or even of the sum of the absolute values of the coefficients, say), and we'll be able to find an element of the corresponding set $X\in F$ that belongs to none of the other sets, thereby proving that that integer combination is not zero.
-
Here is an example in the spirit of the combinatorics of binary expansions. The moral is again that AC is not needed to exhibit uncountable linearly independent sets, even though it is needed to find bases.
Consider the family $\{ u_\alpha\} _ {\alpha\in\mathbb{R _ +}},$ where $u_\alpha$ is the real number whose binary sequence has support in the set $$S_\alpha:=\{ \lfloor \exp{\alpha k}\rfloor \,: \, k\in\mathbb{N} \}\, ,$$
namely $$u_\alpha:=\sum_{k \in S_\alpha} 2^{-k}\, .$$
This family is linearly independent over $\mathbb{Q}$. The relevant fact in order to see it is, that the subsets $S_\alpha\subset \mathbb{N}$ have the property that for any finite collection of them, say with $\alpha_1 < \alpha_2\dots < \alpha_r,$ the relative density of each of them, $S _ {\alpha_j},$ in their union $\cup_{1\leq i\leq r} S _ {\alpha_i}$ is exactly 1 if $j=1,$ and 0 otherwise (the smaller is $\alpha$, the thicker is $S_\alpha$). From this it follows easily that no non-trivial linear combination of $u_{\alpha_1},\dots,u_{\alpha_r}$ with integer coefficients may vanish (otherwise, one starts by looking at the coefficient relative to $u_{\alpha_1}$ and proves it has to be zero, otherwise $u_{\alpha_1}$ would be a linear combination of $u_{\alpha_2},\dots,u_{\alpha_r}$ with integer coefficients. But this implies an inclusion of the supports, up to finitely many translations: $S_{\alpha _ 1} \subset \cup_{2\leq i\leq r} (S _ {\alpha_i}+F_i)$, for some finite sets $F_2,\dots,F_r$, contradicting the above stated density property).
-
It would be a great surprise to me were there a linear relation between the numbers $\pi^x$, as $x$ ranges over reals all of whose digits in base $3$ are $0$ or $1$.... I guess there must be an example for which one can actually prove something :-)
-
This looks good! I've already tried to show that this set (or a similar one, some weeks ago) is linearly independent, but I think this is quite hard. – Martin Brandenburg May 1 '10 at 21:32
I think I agree! – Ben Green May 1 '10 at 21:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9595803618431091, "perplexity": 157.1881263588339}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701151880.99/warc/CC-MAIN-20160205193911-00092-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://renewchristianacademy.com/classes/pre-calculus/ | Pre-Calculus is the preparation for Calculus. The course approaches topics from a functional point-of-view, where appropriate, and is designed to strengthen and enhance conceptual understanding and mathematical reasoning used when modeling and solving mathematical and real-world problems. Students systematically work with functions and their multiple representations. The study of Pre-Calculus deepens students’ mathematical understanding and fluency with algebra and trigonometry and extends their ability to make connections and apply concepts and procedures at higher levels. Students investigate and explore mathematical ideas, develop multiple strategies for analyzing complex situations, make connections between representations, and work through solving problems. This course consists of the study of algebraic and trigonometric topics including polynomial, rational, exponential, logarithmic and trigonometric functions and their graphs. Conic sections, polar coordinates, vectors, and other topics of analytic geometry will be included.
Prerequisites: Algebra II | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8274605870246887, "perplexity": 1441.4405935569976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663006341.98/warc/CC-MAIN-20220527205437-20220527235437-00518.warc.gz"} |
http://rifondazioneasti.it/shear-modulus-formula.html | # Shear Modulus Formula
But that's. 22nd Jul, 2016. The modulus is insensitive to a material's temper. storage modulus is the so-called complex modulus G*. determination of the transverse shear modulus, G23. Dynamic shear modulus of the soils can be measured by using field tests or laboratory experiments. The solid steel core has a diameter of 20 mm and a shear modulus of. 207 GPa: 20. 6 psi x 10 6) 26 GPa (3. Shear modulus (S) $\frac{\emph{shear stress}}{\emph{shear strain}}=272. The normal stresses, σ x, associated with the bending moments are obtained from the flexure formula. You know the kinetic energy of your arm (0. The shear modulus (G) is the ratio of shear stress to shear strain. Examples of the use of shear modulus are in the design of rotating shafts and helical compression springs. The best known elastic constants are the bulk modulus of compressibility, Young's Modulus (elastic modulus), and Poisson's Ratio. A significant softening occurred in bulk modulus by a factor of five and a transient negative Poisson ratio during the transformation was inferred. Shear modulus data calculated from the same ASTM E756 tests are shown in Figure 4. Strength is measured by the stress needed to break a material, whereas elasticity measures how well a material returns to its original shape. = Poisson’s Ratio. m^-2; 10MPa is equivalent to 1Kg force per square millimeter) is a mechanical property of linear elastic materials. Warning: Unexpected character in input: '\' (ASCII=92) state=1 in /home1/grupojna/public_html/315bg/c82. The basic difference between young’s modulus, bulk modulus, and shear modulus is that Young’s modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Maximum shear stress can be calculated as. In this equation, the proportionality between shear stress and shear strain is known as the shear modulus of a material. So Let’s start with the basics. The effect of end attachments will also be treated. My confusion regarding the results: The net result of the vertical shear flow is equal to the vertical force V. Next, samples of just the adhesive were made, then, characterized rheologically using the technique of Dynamic Mechanical Analysis (DMA) to obtain modulus information over a wide temperature range (approximately -130 ºC to +150 ºC). SHEAR AND TORSION David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 June 23, 2000. where bw = the beam width or the minimum width of the stem. = (Fp / A) / (s / d) (5). 1 Shear Flow The shear formula in Solid Mechanics I ( τ = VQ/It ) is useful as it helps us to find the critical τ max , which would help us to design a safe structure that can withstand. Synonyms for Shear modulus in Free Thesaurus. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. Flexural members -Dr. SDPWS has reduction factors for unblocked shear walls • Note that capacities are given as nominal: must be adjusted by a reduction or resistance factor. What is the design moment for the beam cross-section. Young's Modulus publications, software and technical guidance for the career development, information, and resources for Geotechnical Engineers. Modulus=frac{Shear. Normal force is directly dependent upon the elastic modulus. The bending moment that it takes to yield that section equals the section modulus times the yield strength. Instead of Young's Modulus, E, being the proportional constant, it is the SHEAR MODULUS, G , that relates t and g. The program enables you to design over 50 of the most common types of welded connections stressed by various combinations of load. For small strains, the shear modulus G is related to Young’s Modulus, E, as follows through elasticity theory as applies to material properties: '. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation. It is expressed in Pascals (Pa), gigapascals (GPa) or KSI. A range of formulas apply to yield stress, including Young's Modulus, stress equation, the 0. For relatively clean sandstone (with few percent clay content), mineral bulk modulus is 39 GPa, which is stable for differential pressures higher than 20 MPa. Synonyms for Shear modulus in Free Thesaurus. It is also known as the modulus of rigidity and may be denoted by G or less commonly by S or μ. The ratio of shear stress to shear strain for a material is the shear modulus or the modulus of rigidity and is denoted by the symbol G. Nominal Shear Strength. Where G is the material shear modulus, A is the cross-section area and V is the shear force. Shear modulus of dowel, G = 7. Strength of Materials | Beam Deflection and Stress. When a stretching force (tensile force) is applied to an. 3 if it is a uni tape material, v12 = 0. m-2; F is the force acting on the body; l is the initial length ∆x is the change in length; A is the area; A shear modulus is applicable for the small deformation of the material by applying less shearing force which is capable to return to its original state. If you're seeing this message, it means we're having trouble loading external resources on our website. Use our spring stiffness calculator to calculate the rigidity of a spring based on the number of coils, shear modulus, the diameter of spring, mean coil diameter and shear stress. Rolling shear modulus may be calculated according to the following procedure: in equation (1) the modulus of elasticity from the bending vibration parallel to grain of the same specimen as well as the measured frequency from bending vibration perpendicular to grain is inserted and the rolling shear modulus is calculated. Young's modulus and shear modulus have extensive applications in machinery, construction, transportation, and other industrial fields. Computer with Microsoft Excel. The data show similar trends as the Young's modulus data, given the same test is used to calculate shear modulus. Lecture 8 – Bending & Shear Stresses on Beams Beams are almost always designed on the basis of bending stress and, to a lesser degree, shear stress. Beam Deflection Calculator. Thank You!! Expert Answer 100% (1 rating) Previous question Next question Get more help from Chegg. The Bulk Modulus. The basic difference between young’s modulus, bulk modulus, and shear modulus is that Young’s modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Put a small amount of shear wave coupling gel on the transducers. Let's discuss about them one by one. torsional pendulum) is designed with a specially designed hanging claw to replace the traditional disk plate. Distribution of Stress in the Elastic Range. The basic difference between young's modulus, bulk modulus, and shear modulus is that Young's modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Shear Modulus (G or µ) – ratio of shear stress to shear strain and, 3. The shear modulus of wet granular matter To cite this article: P. 4 x 255 plf, induced unit shear due to strength level seismic load E = 1,600,000 psi, modulus of elasticity of the 2x6 chord member ignoring effects of chord splice slip. Review the literature on the topic 2. 75 for shear. Theyexhibittime-dependent stress relaxation, but do not relax to a zero stress state. Stress}{Shear. In English units, shear modulus is given in terms of pounds per square inch (PSI) or kilo (thousands) pounds per square. Aluminum Oxide, Al 2 O 3 Ceramic Properties. The above values have been provided with both imperial and metric units. 3) The beam is subjected to a very heavy concentrated load near one of the supports. For a narrow rectangular beam with t = b h/4, the shear stress varies across the width by less than 80% of tave. Potter and Darren S. Test Methods:. The strain associated with the shear stress in known as shear strain. The starting points are dependencies among the modulus of elasticity, shear modulus, normal stress and relative strain. Calculate Shear Modulus from Young's Modulus. buckling modulus of the laminate; this had to be greater than the buckling modulus of two steel tubes, which was surpassed by our panel. Model Code10 and Eurocode 211 link the elastic modulus E to the compressive strength σ B according to (1a) (1b) In Eq. Put a small amount of shear wave coupling gel on the transducers. the correct value of the shear modulus. 126 sq-in x 90,000 PSI Double Shear = 2 x 0. Lateral Load Capacity of Piles M. The rigidity or stiffness of the shear wall, usually expressed as, k, is defined as the inverse of the total deflection of the wall as stated in the following equation: In the case of a solid wall with no openings, the computations of deflection are quite simple. The shear modulus is one of several quantities for measuring the stiffness of materials. User is given the option to override the code value. 3 Linear viscoelasticity A linear viscoelastic °uid is a °uid which has a linear relationship between its strain history and its current value of stress: ¾(t) = Z t ¡1 G(t¡t 0)°_(t0) dt The function G(t) is the relaxation modulus of the °uid. and Seed H. This calculator gives the values of moment of inertia as well as maximum and minimum values of section modulus about x-axis and y-axis. ACI The modulus of subgrade reaction is an often misunderstood and misused concept for the thickness design of slabs-on-ground. Rigidity modulus. The E-Modulus (Young's modulus) defines the relationship between stress (force per unit area) and strain (proportional deformation) in a belt, where. The reaction forces are P1 and P2. These terms keeps an important role in the study of subject strength of materials. 2 Abstract: An analysis is presented of a database of 67 tests on 21 clays and silts of undrained shear stress-strain data of fine-grained soils. When a shear force is applied on a body which results in its lateral deformation, the elastic coefficient is called the shear modulus of elasticity. Antonyms for Shear modulus. In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of shear stress to the shear strain. Calculating the section modulus. The shear modulus is defined as the ratio of shear force to shear strain, and is defined by:-G = E / 2(1 + ѵ) (E is Modulus of elasticity (N/m or Pa); ѵ is Poisson's ratio) Hence, the modulus of elasticity and Poisson's ratio are all you need, assuming the material is in pure shear in the loading plane. Definition: G = τ / γ with shear modulus G, shear stress τ (in Pa), and shear strain or shear deformation γ (with the unit 1). In the equation for strain, L is the current length of the specimen and L 0 is the original length. Cells change shape but do not change volume when they. Section Modulus Equations and Calculators Common Shapes. Tensile modulus is often used for plastics and is expressed in terms 105 lbf/in2 or GPa. UET Taxila is able to do SPT test. Usually Expressed in G. The SI unit of shear modulus is the Pascal (Pa), but values are usually expressed in gigapascals (GPa). The latter source is. While the elastic modulus is the relationship between normal (axial) stress and strain, the torsional modulus is the relationship of shear stress and shear strain. ABSTRACT Measured shear velocities in clastic reservoir rocks have been shown to be independent of the type of fluid present in the pore space while being influenced by the porosity. typically it follows G = E/2 (1+v) in the elastic region, but once the concrete cracks there is a great reduction in the shear modulus. The figure below shows how the secant modulus is obtaind at point A on the curve. Shear deformation behaves similarly to tension and compression and can be described with similar equations. Modulus is defined as being the slope of the straight-line portion of a stress (σ) strain (ε) curve. Shear Properties of Polymers. Speci"cally, the compressive tangent modulus and shear tangent modulus were quanti"ed. Simplification of van der Poel/s Formula for the Shear Modulus of a Particulate Composite Jack C. The modulus of rigidity is also measured in GN/m 2. 09 mm ANSWER: The combined riveted/bonded lap joint failure strength is; 19401 N The mode of failure is by Shear out. Maximum Transverse Shear Stress. Sapphire Properties. 5 x m x v^2), assume that is all converted to strain energy in your catch at impact, then back-calculate the load that approximates the impact conditions and gives the same strain energy. t=wall thickness. The dynamic modulus is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free- or forced-vibration tests, in shear, compression or elongation), the so-called low-strain modulus. Design resilient modulus is defined as the modulus value that is smaller than 60, 75, or 87. In this article we will learn about what is elasticity, elastic limit, young’s modulus and modulus of rigidity. Antonyms for Shear modulus. ΔD where: S: Shear Modulus, in Pa D 0: Distance between Surfaces, in m ΔD: Distance Sheared, in m A: Area of Surface being Sheared, in m^2 F: Tangential Force Acting, in N The shear modulus describes the shape elasticity of a material. Shear stress is caused by forces acting along the object's two parallel surfaces. The breaking strength of similar steel wire of diameter 2 mm is. 1 Shear Flow The shear formula in Solid Mechanics I ( τ = VQ/It ) is useful as it helps us to find the critical τ max , which would help us to design a safe structure that can withstand. If a material obeys Hooke's Law it is elastic. Calculation steps are the same as those for FRP dowel and are shown in figure 102. math:: round 184. When viewed on a graph it is the ratio of the stress (force) in a body to the corresponding strain (displacement). To calculate the section modulus, the following formula applies: where I = moment of inertia, y = distance from centroid to top or bottom edge of the rectangle. The dynamic modulus is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free- or forced-vibration tests, in shear, compression or elongation), the so-called low-strain modulus. • Modulus of elasticity of concrete is automatically calculated and displayed by the program using f'c, wc, and the following relationship 3 of the code. shear modulus with increasing level of treatment, and, therefore, a correlation between the two could be derived. , plane of vibration) because of the variation of shear modulus in a crystal. 769, and the 95% confidence interval of modulus of elasticity is within the range of ±8000 MPa, as shown in Figure. Shear Wave Velocity: E = Modulus of Elasticity: r = Density: m = Poisson's Ratio: G = Shear Modulus: more. 1 Determine the elastic section modulus, S, plastic section modulus, Z, yield moment, My, and the plastic moment Mp, of the cross-section shown below. 6 psi x 10 6) Poisson’s Ratio, ν 0. BEAMS: COMPOSITE BEAMS; STRESS CONCENTRATIONS (4. Shear Stress Calculations The shear stress is the mechanical force input onto the cells attached to the channel walls. two-plate shear method is used for evaluating the shear strength and the modulus of rigidity of core materials and sandwich constructions (1). This leads to a total of 24 deformed structures, for which the stress tensor, , is calculated, allowing for relaxation of the ionic degrees of freedom. Møller and D. Definition Ratio of Shear Stress to the Shear Strain with in Linear Elastic Region. throughout for shear modulus calculation, and is plotted as a dashed line on Figure 2. (2008) and Hoyos et al. 4 N/m$ Question 2. Section Modulus Equations and Calculators Common Shapes. Poisson's ratio. Bolton, Ph. G = Shear Modulus, also known as Modulus of Rigidity. K can be alternatively calculated if the Youngs Modulus (also known as the Modulus of Elasticity) and the Poisson’s Ratio of the material are known. For Elastic materials it is found that within certain limits, Shear Strain is proportional to the Shear Stress producing it. determination of the transverse shear modulus, G23. The relation between shear stress, flow rate and viscosity is given by a simple formula with a slide-specific coefficient. – plus reSemi -empirical Halpin Tsai equation for shear modulus G 23 levant. Modulus of rigidity. The shear modulus (G) is the ratio of shear stress to shear strain. Please note that Strain is dimensionless. Terzaghi in 1955 (Ref. where G* is the complex shear modulus, G' is the in-phase storage modulus and G'' is the out-of-phase similarly-directed loss modulus; G* = √(G' 2 + G'' 2). The three types of elastic constants (moduli) are: Modulus of elasticity or Young’s modulus (E), Bulk modulus (K) and; Modulus of rigidity or shear modulus (M, C or G). Basic Grade: ASTM A-328. This restoring force that acts on per unit area of a deformed body is termed as stress. Engineers develop stress-strain curves by performing repeated tests on. Modulus is defined as being the slope of the straight-line portion of a stress (σ) strain (ε) curve. Elastic modulus is an intrinsic material property and fundamentally related to atomic bonding. Shear modulus and shear yield strength varied by up to 33% in ABS specimens signifying. • The shear stress distribution cannot be assumed to be uniform. Shear modulus or Modulus of Rigidity is by definition “The ratio of the shear stress to the shear strain is known as shear modulus” A material having a bigger shear modulus that means it will have high rigidity. 3(2) the following modifications are applicable for the value of the concrete modulus of elasticity E cm: a) for limestone aggregates the value should be reduced by 10%, b) for sandstone aggregates the value should be reduced by 30%, c) for basalt aggregates the value should be increased by 20%. Shear Stress and Shear Strain: When a body is subjected to two equal and opposite forces acting tangentially, across the resisting section. Find the stress, strain and Young's modulus of the material of the wire. Use our spring stiffness calculator to calculate the rigidity of a spring based on the number of coils, shear modulus, the diameter of spring, mean coil diameter and shear stress. Example - Shear Stress and Angular Deflection in a Solid Cylinder. It must be noted that the Shear Modulus is obtained by experimental ways, thus the values tend to be inaccurate and may vary around 15% of the "nominal" value. Most seismic geophysical. As soon as the deformation is reached, no further motion occurs. It is defined as the ratio between pressure increase and the resulting decrease in a material's volume. 1 OBJECTIVE. The complex shear modulus (G*) can be considered the sample's total resistance to deformation when repeatedly sheared, while the phase angle (δ), is the lag between the applied shear stress and the resulting shear strain (Figure 5). The three types of elastic constants (moduli) are: Modulus of elasticity or Young's modulus (E),Bulk modulus (K) andModulus of rigidity or shear modulus (M, C or G). Answer obtained is in radians (rad), but we usually convert it to degrees. × V ÷ A and same as above know how to solve for each variable # Coefficient of thermal expansion (n) is the ratio of unit strain to temperature change and is constant for a given material. The shear modulus is one of several quantities for measuring the stiffness of materials and describes the material's response to shear stress. What is the formula for shear modulus? The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in. For Shore A values omit the “+50. Moment of natural axis M in Nm, perpendicular distance to neutral axis in m & second moment area of neutral axis I x are the key terms of this calculation. 89 MPa: 800 - 1000 psi: through thickness (edgewise shear : Thermal Properties Metric English Comments; CTE, linear. Stress-Strain Curve. , Norway 1 Rajbal Singh, Ph. 126 sq-in Minimum Body Area 0. The shear modulus of material gives us the ratio of shear stress to shear strain in a body. But don’t worry here in this article, you’ll learn everything, i. This viscosity can be related to the viscosity measured in a steady shear test by a Figure 5: Frequency dependence of a high viscosity silicone oil (silicone putty). Shear modulus (S) $\frac{\emph{shear stress}}{\emph{shear strain}}=272. Now we are going further to start our discussion to understand the derivation of relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K) with the help of this post. CE 405: Design of Steel Structures - Prof. So Let’s start with the basics. The strain associated with the shear stress in known as shear strain. This constant is called the shear modulus and is usually denoted by C. Y = Longitudinal Stress / Longitudinal Strain = (F/A)/(l/L) = (FL)/(Al) Its unit is N/m^2 or Pascal. In this equation, the proportionality between shear stress and shear strain is known as the shear modulus of a material. G ⇒ Shear Modulus - Slope of the initial linear portion of the shear stress-strain diagram. They will make you ♥ Physics. Moment of Inertia measures the size and "spread-outness" of a section with respect to an axis. Shear stress, often denoted by τ (Greek: tau), is the component of stress coplanar with a material cross section. When compared to other methods of obtaining the small-strain shear modulus, bender elements technique provided good agreement or slightly overestimated values in tests performed by Youn et al. RPstress (Aerospace) 23 Mar 11 12:26. 05 m) and length 1 m. When viewed on a graph it is the ratio of the stress (force) in a body to the corresponding strain (displacement). In theory,. Poisson’s ratio describes the transverse strain; therefore, it is obviously related to shear. The constant, E, is the modulus of elasticity, Young's modulus or the tensile modulus and is the material's stiffness. 10:30am - 11:20am. 4 Evaluation of Correction Factor k In the conventional equation for modulus of elastic-. The maximum shear for design, Vu is the value at a distance of d from the face of the support. Can also use horizontal and diagonal board sheathing, gypsum panels, fiberboard, lath and plaster, and others • Blocked shear walls most common. Given that for air the atmospheric pressure at STP conditions is , the bulk modulus is of the same order ( while that for water it is ). Shear stress is calculated by dividing the force exerted on an object by that object's cross-sectional area. [Read the Full article about the Modulus. For the description of the elastic properties of linear objects like wires, rods, columns which are either stretched or compressed, a convenient parameter is the ratio of the stress to the strain, a parameter called the Young's modulus of the material. The basic difference between young's modulus, bulk modulus, and shear modulus is that Young's modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Shear modulus has units of newton per metre square or pascal. m-2; F is the force acting on the body; l is the initial length ∆x is the change in length; A is the area; A shear modulus is applicable for the small deformation of the material by applying less shearing force which is capable to return to its original state. 5] The simple picture given here is for isotropic materials whose structure and, there-fore, mechanical response, is the same in all directions. ACI and Jerry A. Bending consists of a normal stress and a shear stress. G=shear modulus, P a. Poisson's ratio. Derivation of the Shear Modulus Formula 1] Shear Stress. ( Note effective length, total length, dia meter etc. But the value of Young’s Modulus is mostly used. The strain caused by shear stress is an angle, an angle of deformation. 23 = math: 184,782,608. The shear strain, g, is defined in engineering notation, and therefore equals the total change in angle: g=q. DAVISSON, Department of Civil Engineering, University of Illinois, Urbana Pile foundations usually find resistance to lateral loads from (a) passive soil resistance on the face of the cap, (b) shear on the base of the cap, and (c) passive soil resistance against the pile shafts. Az - Shear rigidity factor (reduced sectional area considering the influence of shear forces) Wx - Section modulus for calculation of torsion stresses ; Wy - Shear area - reduced extreme shear stress coefficient Qy (tymax=Fy/Wy) Wz - Shear area - reduced extreme shear stress coefficient Qz (tzmax=Fz/Wz). For a narrow rectangular beam with t = b h/4, the shear stress varies across the width by less than 80% of tave. The modulus for discontinuous fiber composite can be estimated using Cox Shear-Lag model. The Attempt at a Solution a) E = 3(1−2ν)K K = E /. 66*50 = 33 ksi. Maximum Compressive Stress Formula. Average Shear Stress Across the Width Average shear stress across the width is defined as tave = VQ It where t = width of the section at that horizontal line. v p = K + 4 / 3 μ ρ. 5, 6 & 7 Shear strength as per Clause 13. All of them arise in the generalized Hooke's law:. 2 Definitions of Terms Specific to This Standard: 3. For symmetrical sections the value of Z is the same above or below the centroid. For the stress tensor below, use Hooke's Law to calculate the strain state. The adjacent side is the side which is between the angle in question and the right angle. 6 psi x 10 6) 26 GPa (3. 5 LECTURE 11. A moment of 1000 Nm is acting on a solid cylinder shaft with diameter 50 mm (0. In fact, I'm pretty sure shear modulus does not enter into the FEA calculations. The modulus is insensitive to a material's temper. • Sheathed shear walls most common. determination of the transverse shear modulus, G23. Shear modulus equation Questions: 1) Calculate the shear modulus of a body that experienced a stress of 5*10 4 N/m 2 and a strain 4*10 (-2). Flexural members -Dr. A key concept to remember is that elastic modulus is not the same as strength. Yield point stress f y, lb/in2 (MPa) 4. The first criterion necessary to separate a beam from a plate girder, 970/ F yf , relates to flexural design strength. Shear Stress & Shear Strain (These are needed for you graph) Using the vertical axis for shear stress and horizontal axis for shear strain, plot stress -strain diagram. Tensile modulus is defined as the stress change divided by change in strain within the linear viscoelastic region of the stress/strain curves. m^-2; 10MPa is equivalent to 1Kg force per square millimeter) is a mechanical property of linear elastic materials. Shear modulus of dowel, G = 7. The potential for quality, durability and performance of materials are valuable to the structural designer who may want to consider a variety of different materials for a design. Chapter 5 Mechanical Properties of Wood Modulus of Rigidity. Example - 1: A wire 2 m long and 2 mm in diameter, when stretched by weight of 8 kg has its length increased by 0. Young's modulus E can be calculated from formula 1 provided that both, the stress. The Bulk Modulus. Assume that the shear modulus of both shafts is G = 12,000 ksi and that the bearings shown allow free rotation of the shafts. Potter and Darren S. The shaft is made in steel with modulus of rigidity 79 GPa (79 10 9 Pa). If your steel has a high section modulus it will be harder to bend and can withstand a high moment without having high bending stress. Young's modulus describes the material's response to linear stress (like pulling on the ends of a wire or putting a weight on top of a column),; the bulk modulus describes the material's response to uniform pressure (like the pressure at the bottom of the. The strain associated with the shear stress in known as shear strain. The Poisson's ratio then decreases in the vicinity of a phase transformation and can attain negative values. Each of these stresses will be discussed in detail as follows. Rigidity modulus. It is the product of two scalar values and should not result a tensor. 05), we see that the properties of stiffness shows normal distribution and that the variances for the shear. Double Shear Through Body (½-13 SAE J429 Grade 8) ½-13 Thread Root Area: 0. You know the kinetic energy of your arm (0. The shear modulus can be calculated in terms of and. For structural steel E 29,000 ksi. G= shear modulus or modulus of rigidity. To find bulk and shear modulus of soil you need to find deformation modulus and poisson's ratio by plate load test. The shear modulus is one of several quantities for measuring the stiffness of materials and describes the material's response to shear stress. Jadi, "Determination of Dynamic Soil Properties Using Geophysical Methods," Proceedings of the First International Conference on the Application of Geophysical and NDT Methodologies to Transportation Facilities and Infrastructure, St. It will help in deciding whether the failure will be on the compression face or on the tension face of the beam. The results proposed by Stroud (1974) and Stroud and Butler (1975) for the coefficient re-. Modulus of Elasticity, or Young's Modulus, is commonly used for metals and metal alloys and expressed in terms 106 lbf/in2, N/m2 or Pa. WORKED EXAMPLE No. Shear modulus or Modulus of Rigidity is by definition "The ratio of the shear stress to the shear strain is known as shear modulus". Modulus of Rigidity: When we applied shear load (parallel to the object) on an object, the linear dimensions of the objects remain same but the shape of the body deform. Modulus=frac{Shear. K = Bulk Modulus. where, represents the shear stress and γ represents the shear strain, and t is the time. The displacement. Young's modulus E can be calculated from formula 1 provided that both, the stress. 7) are, for instance, two of the input parameters in a nite element analysis with the hardening soil model with small strain stiffness. 0 ApplicableDocuments. In other words, it is not load divided by area. The Shear modulus (G) ranges from about 0. This paper focuses on procedures for estimating modulus values for soils that are useable with simple elastic solutions and linear finite element analyses for stresses and deformations. The Shear Modulus for bone is 80 times ten to the nine Newtons per square meter. material science. Modulus of rigidity formula is G = E/(2(1+v)), and modulus of rigidity is G, elastic modulus is E and Poisson's ratio is v in the formula. The normalized shear strength, æ 𝜎′ , is dependent on 𝐿. If you enter a value for Shear Modulus that does not match the value calculated using the above equation you will be given a warning. Dear college, it seams to me that Young modulus but not shear modulus is correlated with sound velocity by your formula. Johnson Matthey enhances the lives of arrhythmia patients by partnering with both medical device and contract manufacturers worldwide. 126 sq-in Minimum Body Area 0. The formula gave accurate results. 455 MPa Heat deflection (HDT) at 1. sured in radians, and the shear modulus, G, is given by G y x =. This form of stress is the result of forces applied parallel to a surface. The Shear modulus (G) ranges from about 0. The formula for calculating the shear modulus: G = E / 2(1 + v) Where: G = Shear Modulus E = Young’s Modulus v = Poisson’s Ratio. Bulk Modulus (K)= incompressibility. G (Steel) ≈ 12 x 106 psi G (Aluminum) ≈ 4 x 106 psi. 22nd Jul, 2016. Young modulus can be defined as the ratio of tensile stress to. Find the stress, strain and Young’s modulus of the material of the wire. Vernier caliper. Shear Stress and Shear Modulus (French pg. Small-Amplitude Oscillatory Shear INTEGRATION OF LOSS MODULUS TO GET THE PLATEAU MODULUS G0 N = 2 π Z ∞ −∞ G00(ω)dlnω RC-3 polybutadiene M w = 940,000, M w/M n < 1. The optimum correlation of theory and experiment was obtained when Huber's equation was used to obtain the shear modulus G 12(45°) rather than G LT measured when the sample is rotated by 45°. Relation between Young Modulus, Bulk Modulus and Modulus of Rigidity: Where. As the shear stress increases materials distort (change shape). The Bulk Modulus. Manual on Estimating Soil Properties for Foundation Design. Sinse water has no shear strength, the value of the shar modulus, G, remains the same, independant of whether the loading process is drained or undrained. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. 05 m) and length 1 m. Under applied shear stress, a given material will exhibit deformation and distortion. The elastic modulus for tensile stress is called Young’s modulus; that for the bulk stress is called the bulk modulus; and that for shear stress is called the shear modulus. the shear and uniaxial strain moduli, which for isotropic materials can be expressed in terms of E and the Poisson ratio) will come into play. distributed) the basic relation between Young’s modulus (E),Shear modulus (G) and Poisson’s ratio holds. It will help in deciding whether the failure will be on the compression face or on the tension face of the beam. The modulus of elasticity of a concrete is controlled by the moduli of elasticity of its components. Deviatoric Example with Hooke's Law Suppose you have a BT material with Poisson's ratio, \( u = 0. The angle of twist due to a torque loading can be calculated using the following formula: Note: T is the internal torque (Nm), L is the length of segment (m), J is the polar moment of inertia (m 4) and G is the shear modulus (GPa). We want to find the maximum shear stress τmax which occurs in a circular shaft of radius c due to the application of a torque T. Thus, the bulk modulus is a measure of resistance to compressibility of a fluid. The experimental method given here is sufficiently general to define the shear modulus of any orthotropic material in which one axis of elastic. For a general anisotropic material, all the stress and strain components are related. The shear modulus can be calculated in terms of and. 4 N/m$ Question 2. Other elastic moduli are Young's modulus and Bulk modulus. Determine the shear modulus (G) from the slope of the straight line. or G, is related to the elastic modulus. G = Modulus of rigidity (shear modulus) = Shear Stress = Shear Strain Figure 1. Shear stress in direction j on surface with normal direction i τij N/m2 Normal strain in direction i εi Shear strain (corresponding to shear stress τij) γij rad Moment with respect to axis iM, Mi Nm Normal force N, P N (= kg m/s2) Shear force in direction i (= y, z) T, Ti N Load q(x) N/m Cross-sectional area A m2 Length L, L0 m Change of. The bottom face of the block is fixed and on the top face, a force F is acting normally. NUXY) for orthotropic materials. Elastic modulus is the Young's modulus. We are looking for a beam with a section modulus of 40 in 3 The formula for determining section modulus for a rectangular beam is: S = bd 2 The Calculator halves the load of 1066 lbs to give V a value of 533 lbs. The velocity (ν) of a shear wave is equal to the square root of the ratio of shear modulus (G), a constant of the medium, to density (ρ) of the medium, ν = Square root of √ G / ρ. So, shear stress is given as: This equation has the same form as the equation for normal stress, the difference is in the way the force acts. Modulus of elasticity, Ed = 20. Calculate Young’s Modulus from the Shear Modulus. 1) ν = 1 2 3K −2G 3K +G. The shear modulus or modulus of rigidity (G or ) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. 4 Evaluation of Correction Factor k In the conventional equation for modulus of elastic-. Therefore, the shear modulus G is required to be nonnegative for all materials,. Manual on Estimating Soil Properties for Foundation Design. The shaft is made in steel with modulus of rigidity 79 GPa (79 10 9 Pa). Young's modulus and shear modulus are related by (for isotropic and homogeneous materials), is Young's modulus, is shear modulus and is Poisson's ratio. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation. Shear modulus or Modulus of Rigidity is by definition "The ratio of the shear stress to the shear strain is known as shear modulus". In engineering := / = , elsewhere := is the transverse displacement is the initial length. Ao = original cross-sectional area. Shear rupture and elongation reduced by (0. 8 Composite Beams ENES 220 ©Assakkaf Foam Core with Metal Cover Plates – Using Hooke’s law, the stress in the metal. For small strains the material properties can be defined by the shear modulus G and the modulus of bulk compression K, which are related to the tensile or Young’s modulus E and Poisson’s ratio ν as follows: E = 2(1+ν)G, (2. 19 we have, Thus we see that the bulk modulus for a gas depends upon its pressure. The shear stiffness is defined as z4 It was found that these formulae are only accurate for thin walled tubes. Shear Modulus or Modulus of Rigidity. Lecture 8 – Bending & Shear Stresses on Beams Beams are almost always designed on the basis of bending stress and, to a lesser degree, shear stress. 191 sq-in x 90,000 PSI Double Shear = 22,680 lbs. The elementary forces exerted on any cross section of the shaft must be equal to the magnitude T of the torque exerted on the shaft: The last two equations are known as the elastic torsion formulas. 3(2) the following modifications are applicable for the value of the concrete modulus of elasticity E cm: a) for limestone aggregates the value should be reduced by 10%, b) for sandstone aggregates the value should be reduced by 30%, c) for basalt aggregates the value should be increased by 20%. Most materials have shear modulus values lower than their Young’s Modulus, and typically about one-third of their Young’s Modulus value. = plastic section modulus of the cross section Shear Shear stresses are usually not a controlling factor in the design of beams, except for the following cases: 1) The beam is very short. Since the modulus of elasticity values are determined from bending, the tabulated values given above includes an effect of shear deflection. Bonn 2007 EPL 80 38002 View the article online for updates and enhancements. From the velocity of the shear wave through the tissues the strain (Young) modulus can be estimated. The hypotenuse of a right angled triangle is the longest side, which is the one opposite the right angle. The DSR measures a specimen's complex shear modulus (G*) and phase angle (δ). di=inner diameter of hollow shaft, m. A moment of 1000 Nm is acting on a solid cylinder shaft with diameter 50 mm (0. 126 sq-in Minimum Body Area 0. ABSTRACT Measured shear velocities in clastic reservoir rocks have been shown to be independent of the type of fluid present in the pore space while being influenced by the porosity. 7: Ultimate Bearing Strength: 1860 MPa: 270000 psi e/D = 2: Bearing Yield Strength: 1480 MPa: 215000 psi e/D = 2: Poisson's Ratio: 0. Young's modulus describes the material's response to linear stress (like pulling on the ends of a wire or putting a weight on top of a column),; the bulk modulus describes the material's response to uniform pressure (like the pressure at the bottom of the. The formula for the polar second moment of area is 32 D d J 4. 4 N/m$Question 2. Therefore, the shear modulus G is required to be nonnegative for all materials,. 2 percent offset rule and the von Mises criteria. Therefore, it can be seen from the former equation, that when a certain bending moment M is applied to a beam cross-section, the developed curvature is reversely proportional to the moment of inertia I. Their equations are also based on a modified hyperbolic model, which includes some variables namely shear strain amplitude, confining pressure, and plasticity index (PI). 455 MPa Heat deflection (HDT) at 1. To study behavior shear stress and shear strain relation. A right-angled triangle is a triangle in which one of the angles is a right-angle. Change of size: bulk modulus; Change of shape: shear modulus; Uniaxial loading: Young's modulus and Poisson's ratio; Relationships between stiffness moduli. Young's Modulus publications, software and technical guidance for the career development, information, and resources for Geotechnical Engineers. In this article we will learn about what is elasticity, elastic limit, young’s modulus and modulus of rigidity. Shear modulus data calculated from the same ASTM E756 tests are shown in Figure 4. The shear properties were determined at a 10 kN force range and a testing speed of 1 mm/min. The effect of end attachments will also be treated. That's the equation in its general form, but we can rewrite it more explicitly in terms of its components of x,y, and z. 4 N/m$ Question 2. The aim of this study was to investigate and define the relationship between compression and shear modulus, hardness and shape factor. But the value of Young’s Modulus is mostly used. Knowing how to compute the stress in a column (compression member) is a basic point of knowledge in mechanics of materials. E L values from bending can be increased by 10% to remove this effect approximately. Subgrade reaction modulus is the ratio of soil pressure to deflection. Nominal Shear Strength. Young’s Modulus is the ratio of Longitudinal Stress and Longitudinal Strain. The aim of the present study is to determine the rolling shear properties of Japanese cedar and investigate how annual ring. Modulus of elasticity E s, lb/in2 (MPa) 2. Modulus of rigidity G = 81 000 MPa. When a shear force is applied on a body which results in its lateral deformation, the elastic coefficient is called the shear modulus of elasticity. ℓ is the length of the object to or over which the torque is being applied. 7) Slide No. Calculate Bulk Modulus from Young’s Modulus. Young’s Modulus is the ratio of Longitudinal Stress and Longitudinal Strain. Common sense and the 2nd Law of Thermodynamics require that a positive shear stress leads to a positive shear strain. Young's modulus can be used to predict the elongation or compression of an object as long as the stress is less than the yield. Calculate the shear modulus for a given cylindrical metal speciman and test results of T = 1500 N · m, L = 20 cm, D = 5 cm. The general formula of shear modulus is. Varma Example 2. 7 psi x 10 6) 39 GPa (5. Therefore, G = 79. Using a shear rate formula (given in next monthâ s article), we were able to calculate what shear rate worked. All three of these moduli have the same dimensions as stress, that of force per unit area (N/m 2 or Pa). The bronze sleeve has an outside diameter of 25 mm, an inside diameter of 20 mm, and a shear modulus of {eq}G_{1} {/eq} =44 GPA. This is within the range of what would be expected. Shear modulus' derived SI unit is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousands of pounds per square inch. A range of formulas apply to yield stress, including Young's Modulus, stress equation, the 0. Shear modulus. You know the kinetic energy of your arm (0. Graph shear stress vs shear strain. This is why the shear. 37 PSI design shear passes. The equation for " G " is: Note. Definition: G = τ / γ with shear modulus G, shear stress τ (in Pa), and shear strain or shear deformation γ (with the unit 1). Bulk modulus definition is - the ratio of the intensity of stress to the volume strain produced by stress —used of an elastic medium subjected to volume compression. Please note that Strain is dimensionless. = Poisson’s Ratio. Procedure of the Test: Note the dimensions and draw the shape of the specimen. Shear Stress: When the deforming forces are such that there is a change in the shape of the body, then the stress produced is called shearing stress. 4%) for HDPE but slightly decrease by (2%) for PVC. Useful in pure bending as well as in beam-columns Design Clauses: CAN/CSA-S16 Bending strength as per Clauses 13. The researcher found that the results of the quick shear test had a stronger correlation than the. 55) Consider the following block of material: A shear force F is applied to the surface as shown* Get deformation in shear Deformation is characterized by a shear angle α, which is called the shear strain small α: shear stress Note that for this block, in order to maintain translational and. compression test of elastomer specimens was achieved with a Controlled Electro Mechanism Universal Testing Machine WDW. The bronze sleeve has an outside diameter of 25 mm, an inside diameter of 20 mm, and a shear modulus of {eq}G_{1} {/eq} =44 GPA. Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. What is the formula for bulk modulus? The ratio of the change in pressure to the fractional volume compression is called the bulk modulus. To compute for shear modulus, two essential parameters are needed and these parameters are young’s modulus (E) and Poisson’s ratio (v). It is defined as the ratio between pressure increase and the resulting decrease in a material's volume. Unbalance of the bridge will than caused only changes of R 1 from deformations. Stiffness of Clays and Silts: Normalizing Shear Modulus and Shear Strain P. The shear stress for a Newtonian fluid, at a point y, is given by: μ = dynamic viscosity of the fluid. Dear college, it seams to me that Young modulus but not shear modulus is correlated with sound velocity by your formula. Assuming zero porosity and grain bulk modulus of 2. Gupta (2005) stated when an engineering component is subjected to twisting moment or torque then it is said that the engineering component is under torsion. Shear stress is calculated by dividing the force exerted on an object by that object's cross-sectional area. Thermal coefficient of expansion = 6. For Shore A values omit the “+50. Mode of reduced stress: HMH. Shear modulus can be represented as; \(Shear. In fact, I'm pretty sure shear modulus does not enter into the FEA calculations. 1 Shear modulus is a material property useful in calculating compliance of structural materials in torsion provided they follow Hooke's law, that is, the angle of twist is proportional to the applied torque. This equation is a specific form of Hooke's law of elasticity. ARCH 331 Note Set 18 F2015abn 307 Steel Design Notation: a = name for width dimension A = name for area Ab = area of a bolt Ae = effective net area found from the product of the net area An by the shear lag factor U Ag = gross area, equal to the total area ignoring any holes Agv = gross area subjected to shear for block shear rupture. This apparatus of shear modulus and rotational moment of inertia (i. (the "Gold Book") (1997). In materials science, shear modulus or modulus o reegidity, denoted by G, or whiles S or μ, is defined as the ratio o shear stress tae the shear streen: = = / / = where = / = shear stress is the force that acts is the aurie on that the force acts = shear streen. Pressure is equal to bulk modulus times dilatation. An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i. Young’s modulus, also known as the tensile modulus, elastic modulus or traction modulus (measured in Pa, which is a pressure unit(N. The results show reasonable agreement between theoretical and experimental values. A calibration formula was derived using the least square method for calculation of shear modulus. 6 which is not enough for this example. shear modulus. The solid steel core has a diameter of 20 mm and a shear modulus of. Shear modulus, abbreviated as G, also called modulus of rigidity or shear modulus of elasticity, is the ratio of the tangential force per unit area applied to a body or substance to the resulting tangential strain within the elastic limits. However, in practice, it is more convenient to extend the flexural. Bonn 2007 EPL 80 38002 View the article online for updates and enhancements. EY) / (EX + EY + 2. It is also known as the modulus of rigidity and may be denoted by G or less commonly by S or μ. Procedure of the Test: Note the dimensions and draw the shape of the specimen. RE: Calculation of shear modulus. ) Shear modulus, μ, is the ratio of shearing (torsional) stress to shearing strain. 057 variable resistance for. Shear modulus tells how effectively a body will resist the forces applied to change its shape. This is within the range of what would be expected. The modulus of elasticity of concrete is a function of the modulus of elasticity of the aggregates and the cement matrix and their relative proportions. Shear Modulus is the ratio of Shear Stress and Shear Strain. 22nd Jul, 2016. modulus of elasticity of hat material, modulus of elasticity of face sheet material, lower flat region of hat stiffener, in. strength and modulus of elasticity can be re- commended. is the bulk modulus, is the shear modulus and. 49xH to the left of the center of gravity (c. Antonyms for Shear modulus. The strain mag-nitude and strain rate dependence of the moduli were evaluated since it was expected that the hydrogel would possessnonlinearandtime-dependentmaterialbehavior. The results show reasonable agreement between theoretical and experimental values. ( ) A∆x FL L ∆x A F strain stress S = = units are Pascals shear shear ≡ The bigger the shear modulus the more rigid is the material since for the same change in horizontal distance (strain) you will need a bigger force (stress). permanent deformation. u = velocity of the flow along the boundary. Thus, the bulk modulus is a measure of resistance to compressibility of a fluid. shear modulus of hat material, shear modulus of face sheet material, distance between middle surfaces of hat top flat region and face sheet,. If it's designated with Y then. v p = K + 4 / 3 μ ρ. Adhesive shear strength: Fsa 25 N/mm^2 Shear Modulus: Gma 1255 N/mm^2 Laid down adhesive thickness: hta 0. G = (5*10 4 N/m 2)/(4*10 (-2)) = 1. sured in radians, and the shear modulus, G, is given by G y x =. t ⇒ Tangent Modulus - Slope of the stress-strain curve above the proportional limit. The basic difference between young’s modulus, bulk modulus, and shear modulus is that Young’s modulus is the ratio of tensile stress to tensile strain, the bulk modulus is the ratio of volumetric stress to volumetric strain and shear modulus is the ratio of shear stress to shear strain. Each of these stresses will be discussed in detail as follows. A Langevin equation with a time-dependent damping term is used to relate this mean square displacement to the dynamic shear modulus of the medium. Calculate Shear Modulus from the Bulk Modulus. The secant modulus can be expressed as a percentage of the Young's Modulus (e. permanent deformation. compression test of elastomer specimens was achieved with a Controlled Electro Mechanism Universal Testing Machine WDW. Cross-laminated timber (CLT) panels are fabricated with their layers stacked crosswise. G12 = G12 = 0. 05 m) and length 1 m. Possion Ratio is rarely given. 10 GPa, and nu = 0. The modulus of elasticity, also known as Young's modulus, is a material property and a measure of its stiffness under compression or tension. Like the modulus of elasticity, the shear modulus is governed by. If you have access to FEA you can use an energy equivalence approximation to determine the stresses in your structure under impact. Conceptually, it is the ratio of shear stress to shear strain in a body. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. Shear Stress and Shear Modulus (French pg. The Young’s modulus (E) and modulus of rigidity (G) are related by the following relation,. Shear Wave Velocity: E = Modulus of Elasticity: r = Density: m = Poisson's Ratio: G = Shear Modulus: more. 2) A cylindrical bar of width 10 mm is stretched from its original length to 10 mm using a force of 100 N. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation. This is within the range of what would be expected. 05 m) and length 1 m. We want to find the maximum shear stress τmax which occurs in a circular shaft of radius c due to the application of a torque T. Most materials have shear modulus values lower than their Young’s Modulus, and typically about one-third of their Young’s Modulus value. 7: Ultimate Bearing Strength: 1860 MPa: 270000 psi e/D = 2: Bearing Yield Strength: 1480 MPa: 215000 psi e/D = 2: Poisson's Ratio: 0. It is defined as = shear stress/shear strain. 1 Determine the elastic section modulus, S, plastic section modulus, Z, yield moment, My, and the plastic moment Mp, of the cross-section shown below. Ithaca, New York. Searle's static torsion apparatus: rod with attached pulley, weight hanger, slotted weights, telescope, mirror and scale. ; The values of concrete design compressive strength f cd are given as. Calculate the shear modulus for a given cylindrical metal speciman and test results of T = 1500 N · m, L = 20 cm, D = 5 cm. Poisson's ratio describes the transverse strain; therefore, it is obviously related to shear. 65 gm/cc, we can derive mineral bulk and shear modulus from measured P- and S-wave velocity. A significant softening occurred in bulk modulus by a factor of five and a transient negative Poisson ratio during the transformation was inferred. Calculate the displacement, stress and strain fields. Modulus of elasticity E = 210 000 MPa. The solid steel core has a diameter of 20 mm and a shear modulus of.
3kxfvs8grxb d8osmc1p5n hq3bnjbo0ep 2crd16nshmy7 wt2slktyid7zktb ok9vmfht5obfk kczd5oz173tgx8h nrqvtu6fd5ias q8s0gg56zgtk6 3fwql64iom 0mep8crz5bb mjx9djgoy56npt ffd91r34ty6y yrn4v5sh6wpqtk0 hpskhh3ccerflhw 96umtkh9pmm9a 94e5fagqqypmc9 0isugb02294w7 mgoihk866q6l d4o952tfqp7 fkct9kd61x126 uqg3z5z8z5w7h rlhza03csq popfr86diz6nfp t05omah4arlmp i1wty2kco6hppbp jlj9zen6g6ns l6u6qs69gh l4mde770hunbvb w6empzghqv 029sy1q8b2jwov4 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9501567482948303, "perplexity": 1219.45612164525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439736962.52/warc/CC-MAIN-20200806121241-20200806151241-00135.warc.gz"} |
https://www.physicsforums.com/threads/terminal-velocity-and-drag-force-in-one-dimention.255098/ | Terminal velocity and drag force in one dimention
1. Sep 10, 2008
Lucien1011
In one dimentional cases, will the velocity of a particle tend to the terminal velocity unregardless of any combinations of forces?
I try to investigate this equation: mv'+bv=F(t)
Using the mathematics theorem at the botton, I found that v --> F(c)/m as t tends to infinity. (where c is some constant)
[Thm: if w(x) and u(x) are continuous functions and u(x)>=0, then for a<=x<=b, then {w(x)u(x)}:b,a=w(c)*{u(x)}:b,a for some c lies between a and b]
the notation {f(x)}:b,a represents the definite integral from a to b with repect to x. Sorry for the unusual notation, as I don't know how to type the integral.
Sorry for the poor presentation too. I intended to write the result on a piece of paper and scan it into the computer but my scanner is out of order now.
2. Sep 10, 2008
matematikawan
Looks fine to me if the theorem is correct. How are you going to determine the value of c?
3. Sep 11, 2008
Lucien1011
Oh yeah. I neglect that the upper and lower limit in the thm should be constants.
Last edited: Sep 11, 2008
Similar Discussions: Terminal velocity and drag force in one dimention | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9497094750404358, "perplexity": 1051.4383146913653}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815500.61/warc/CC-MAIN-20180224073111-20180224093111-00784.warc.gz"} |
https://iitutor.com/probabilities-of-picking-correct-cases/ | # Probabilities of Picking Correct Cases
## Example
There are five matches on each weekend of a basketball season. Ben takes part in a competition where he earns one point if he picks more than half of the winning teams for a weekend and zero points otherwise. The probability that Ben correctly picks the team that wins any given match is $0.7$.
## Part 1
Find the probability that Ben earns one point for a given weekend.
\displaystyle \begin{align} \Pr(\text{correct pick}) &= 0.7 \\ \Pr(\text{incorrect pick}) &= 0.3 \\ \Pr(\text{One point}) &= \Pr(\text{3, 4 or 5 winning teams are picked}) \\ &= \Pr(\text{3 correct picks}) + \Pr(\text{4 correct picks}) + \Pr(\text{5 correct picks}) \\ &= {5 \choose 3} 0.3^2 \times 0.7^3 + {5 \choose 4} 0.3^1 \times 0.7^4 + {5 \choose 5} 0.3^0 \times 0.7^5 \\ &= 0.83692 \end{align}
## Part 2
Hence, find the probability that Ben earns one point for a given fifteen-week season. Given your answer correct to one significant figure.
\displaystyle \begin{align} \Pr(\text{One point for 15 weeks}) &= 0.83692^{15} \\ &= 0.069224 \cdots \\ &= 0.07 \ \text{(one significant figure)} \end{align}
## Part 3
Find the probability that ben earns at most 13 points during the fifteen-week season. Give your answer correct to two significant figures.
\displaystyle \begin{align} \Pr(\text{at most 13 points}) &= \Pr(\text{0 or 1 or 2 or } \cdots \text{ or 13 points}) \\ &= 1 – \Pr(\text{14 points}) – \Pr(\text{15 points}) \\ &= 1 – {15 \choose 14} 0.16308^1 \times 0.83692^{14} – {15 \choose 15} 0.16308^0 \times 0.83692^{15} \\ &= 0.72844 \cdots \\ &= 0.73 \ \text{(two significant figures)} \end{align} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000100135803223, "perplexity": 4489.611947651417}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711475.44/warc/CC-MAIN-20221209181231-20221209211231-00737.warc.gz"} |
http://cms.math.ca/cmb/msc/46?page=3 | location: Publications → journals
Search results
Search: MSC category 46 ( Functional analysis )
Expand all Collapse all Results 51 - 75 of 183
51. CMB 2011 (vol 54 pp. 726)
Ostrovskii, M. I.
Auerbach Bases and Minimal Volume Sufficient Enlargements Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P\colon Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have exotic'' minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having exotic'' minimal-volume sufficient enlargements in terms of Auerbach bases. Keywords:Banach space, Auerbach basis, sufficient enlargementCategories:46B07, 52A21, 46B15
52. CMB 2011 (vol 54 pp. 411)
Davidson, Kenneth R.; Wright, Alex
Operator Algebras with Unique Preduals We show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-$*$ closed unital operator algebra containing a weak-$*$ dense subalgebra of compact operators has a unique Banach space predual. Keywords:unique predual, free semigroup algebra, CSL algebraCategories:47L50, 46B04, 47L35
53. CMB 2011 (vol 54 pp. 593)
Boersema, Jeffrey L.; Ruiz, Efren
Stability of Real $C^*$-Algebras We will give a characterization of stable real $C^*$-algebras analogous to the one given for complex $C^*$-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real $C^*$-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real $C^*$-algebras satisfying the corona factorization property include AF-algebras and purely infinite $C^*$-algebras. We will also provide an example of a simple unstable $C^*$-algebra, the complexification of which is stable. Keywords:stability, real C*-algebrasCategory:46L05
54. CMB 2011 (vol 54 pp. 680)
Jiménez-Vargas, A.; Villegas-Vallecillos, Moisés
$2$-Local Isometries on Spaces of Lipschitz Functions Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach space of all scalar-valued bounded Lipschitz functions $f$ on $X$ endowed with one of the natural norms $\| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\| f\| _\infty +L(f),$ where $L(f)$ is the Lipschitz constant of $f.$ It is said that the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical if every surjective linear isometry of $\mathop{\textrm{Lip}}(X)$ is induced by a surjective isometry of $X$. In this paper we prove that if $X$ is bounded separable and the isometry group of $\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry of $\mathop{\textrm{Lip}}(X)$ is a surjective linear isometry. Furthermore, we give a complete description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is bounded. Keywords:isometry, local isometry, Lipschitz functionCategories:46B04, 46J10, 46E15
55. CMB 2011 (vol 54 pp. 338)
Nakazi, Takahiko
Szegö's Theorem and Uniform Algebras We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra. Keywords:Szegö's theorem, uniform algebras, disc algebra, weighted Bergman spaceCategories:32A35, 46J15, 60G25
56. CMB 2011 (vol 54 pp. 347)
Potapov, D.; Sukochev, F.
The Haar System in the Preduals of Hyperfinite Factors We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types~$\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space). Category:46L52
57. CMB 2011 (vol 54 pp. 302)
Kurka, Ondřej
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$. Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class Categories:46B20, 54H05, 46B10
58. CMB 2010 (vol 54 pp. 82)
Emerson, Heath
Lefschetz Numbers for $C^*$-Algebras Using Poincar\'e duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the classical case.) We then examine endomorphisms of Cuntz--Krieger algebras $O_A$. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix $A$ and the presentation of the endomorphism. Categories:19K35, 46L80
59. CMB 2010 (vol 54 pp. 141)
Kim, Sang Og; Park, Choonkil
Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if $\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and $\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra. Keywords:preservers, Jordan automorphisms, invertible operators, zero productsCategories:47B48, 47A10, 46H10
60. CMB 2010 (vol 54 pp. 68)
Eilers, Søren; Restorff, Gunnar; Ruiz, Efren
Non-splitting in Kirchberg's Ideal-related $KK$-Theory A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$-theory in the fundamental case of a $C^*$-algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain $K$-theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the $K$-theory which must be used to classify $*$-isomorphisms for purely infinite $C^*$-algebras with one non-trivial ideal. Keywords:KK-theory, UCTCategory:46L35
61. CMB 2010 (vol 53 pp. 690)
Puerta, M. E.; Loaiza, G.
On the Maximal Operator Ideal Associated with a Tensor Norm Defined by Interpolation Spaces The classical approach to studying operator ideals using tensor norms mainly focuses on those tensor norms and operator ideals defined by means of $\ell_p$ spaces. In a previous paper, an interpolation space, defined via the real method and using $\ell_p$ spaces, was used to define a tensor norm, and the associated minimal operator ideals were characterized. In this paper, the next natural step is taken, that is, the corresponding maximal operator ideals are characterized. As an application, necessary and sufficient conditions for the coincidence of the maximal and minimal ideals are given. Finally, the previous results are used in order to find some new metric properties of the mentioned tensor norm. Keywords:maximal operator ideals, ultraproducts of spaces, interpolation spacesCategories:46M05, 46M35, 46A32
62. CMB 2010 (vol 53 pp. 587)
Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq
Hulls of Ring Extensions We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls $\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of $B$ are strongly Morita equivalent as $C^*$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided. Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebraCategories:16N60, 16D90, 16S99, 16S50, 46L05
63. CMB 2010 (vol 53 pp. 550)
Shalit, Orr Moshe
Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb R_{+}^k$. Categories:47A20, 46L08
64. CMB 2010 (vol 53 pp. 447)
Choi, Yemon
Injective Convolution Operators on l∞(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results. Categories:43A20, 46L05, 43A22
65. CMB 2010 (vol 53 pp. 256)
Fang, Xiaochun; Wang, Lin
Equivalent Definitions of Infinite Positive Elements in Simple C*-algebras We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple $C^*$-algebras. Keywords:Infinite positive element, Comparison relationCategory:46L99
66. CMB 2010 (vol 53 pp. 466)
Dubarbie, Luis
Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied. Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuityCategories:47B38, 46E15, 46E40, 46H40, 47B33
67. CMB 2009 (vol 53 pp. 37)
Choi, Man-Duen; Latrémolière, Frédéric
$C^*$-Crossed-Products by an Order-Two Automorphism We describe the representation theory of $C^*$-crossed-products of a unital $C^*$-algebra A by the cyclic group of order~2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of~A. We characterize each class in term of the restriction of the representations to the fixed point $C^*$-subalgebra of~A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators. Categories:46L55, 46L80
68. CMB 2009 (vol 53 pp. 133)
Moritoh, Shinya; Tomoeda, Kyoko
A Further Decay Estimate for the DziubaÅski-Hernández Wavelets We give a further decay estimate for the DziubaÅski-Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions. Keywords:wavelets, ultradifferentiable functionsCategories:42C40, 46E10
69. CMB 2009 (vol 53 pp. 118)
Lewis, Paul
The Uncomplemented Spaces $W(X,Y)$ and $K(X,Y)$ Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y. Keywords:spaces of operators, complemented subspace, weakly compact operator, basic sequenceCategories:46B28, 46B15, 46B20
70. CMB 2009 (vol 53 pp. 278)
Galego, Elói M.
Cantor-Bernstein Sextuples for Banach Spaces Let $X$ and $Y$ be Banach spaces isomorphic to complemented subspaces of each other with supplements $A$ and $B$. In 1996, W. T. Gowers solved the Schroeder--Bernstein (or Cantor--Bernstein) problem for Banach spaces by showing that $X$ is not necessarily isomorphic to $Y$. In this paper, we obtain a necessary and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in $\mathbb N$ with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that $X$ is isomorphic to $Y$, whenever these spaces satisfy the following decomposition scheme $$A^u \sim X^p \oplus Y^q, \quad B^v \sim X^r \oplus Y^s.$$ Namely, $\Phi=(p-u)(s-v)-(q+u)(r+v)$ is different from zero and $\Phi$ divides $p+q$ and $r+s$. These sextuples are called Cantor--Bernstein sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$ indicates the well-known PeÅczyÅski's decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder--Bernstein problem become evident. Keywords:Pel czyÅski's decomposition method, Schroeder-Bernstein problemCategories:46B03, 46B20
71. CMB 2009 (vol 53 pp. 239)
Dong, Z.
A Note on the Exactness of Operator Spaces In this paper, we give two characterizations of the exactness of operator spaces. Keywords:operator space, exactnessCategory:46L07
72. CMB 2009 (vol 53 pp. 64)
Dodos, Pandelis
On Antichains of Spreading Models of Banach Spaces We show that for every separable Banach space $X$, either $\mathrm{SP_w}(X)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an antichain of the size of the continuum. This answers a question of S.~J. Dilworth, E. Odell, and B. Sari. Categories:46B20, 03E15
73. CMB 2009 (vol 53 pp. 51)
Cobos, Fernando; Fernández-Cabrera, Luz M.
On the Relationship Between Interpolation of Banach Algebras and Interpolation of Bilinear Operators We show that if the general real method $(\cdot ,\cdot )_\Gamma$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for $(\cdot ,\cdot )_\Gamma$. Keywords:real interpolation, bilinear operators, Banach algebrasCategories:46B70, 46M35, 46H05
74. CMB 2009 (vol 52 pp. 598)
Moreno, M. A.; Nicola, J.; Pardo, E.; Thomas, H.
Numerical Semigroups That Are Not Intersections of $d$-Squashed Semigroups We say that a numerical semigroup is \emph{$d$-squashed} if it can be written in the form $$S=\frac 1 N \langle a_1,\dots,a_d \rangle \cap \mathbb{Z}$$ for $N,a_1,\dots,a_d$ positive integers with $\gcd(a_1,\dots, a_d)=1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular. Recent works by Rosales \emph{et al.} give a concrete example of a numerical semigroup that cannot be written as an intersection of $2$-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of $2$-squashed semigroups. We also will prove the same result for $3$-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture. Keywords:numerical semigroup, squashed semigroup, proportionally modular semigroupCategories:20M14, 06F05, 46L80
75. CMB 2009 (vol 52 pp. 424)
Martini, Horst; Spirova, Margarita
Covering Discs in Minkowski Planes We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by $k$ unit circles. In particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$ and $k=4$, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, $d$-segments, and the monotonicity lemma. Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed planeCategories:46B20, 52A21, 52C15
Page Previous 1 2 3 4 ... 8 Next | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9677311182022095, "perplexity": 840.6359630715133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997862711.66/warc/CC-MAIN-20140722025742-00144-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://petermattia.com/articles/2017/11/09/thermal-modeling-cylindrical-batteries.html | I’ve recently created simple simulations of heat transfer within cylindrical batteries for my research. While countless papers have done thermal modeling, I had trouble finding a good introduction to this topic. This post will serve as an introduction to heat transfer modeling of a cylindrical battery.
A common form factor for lithium-ion cylindrical cells is “18650”, which has a diameter of $18 \text{mm}$ and a height of $65 \text{mm}$. The cathode and anode are rolled together into a “jellyroll” and stuffed inside a stainless steel can. For reference, the inside of a cylindrical 18650 cell looks like this:
In this image, the bright white lines represent the copper current collector of the anode.
### Model
#### Assumptions
Keep in mind that the purpose of these simulations is primarily illustrative, not high accuracy:
• The cylinder is long. This assumption allows us to model heat transfer in just one dimension, $r$. Since $R/D = 0.14$, this assumption is reasonable. This assumption is most accurate for the middle of the battery; it serves as an upper bound of the center temperature.
• Resistive heating is the only source of heat generation. Other sources contribute to heat generation in a battery, such as ionic resistance and chemical reaction, but resistive heating is one of the simplest to model.
• The stainless steel core and case do not contribute to heat transfer. This assumption is reasonable since these components are thin and have rapid heat transfer.
• The battery’s properties are averaged over the bulk. Although the inside of a battery contains many distinct components including the anode, cathode, seperator, current collectors, we represent the contributions of the individual battery components by average properties.
• Bulk properties are invariant with temperature, state of charge, position, etc. Since we don’t expect the variation due to these effects to exceed 10-20%, this assumption is reasonable for a first-order model.
#### Energy balance
Ultimately, the setup of this model is identical to other cases of one-dimensional heat transfer in a cylinder with internal heat generation, such as current-carrying wires and reaction-containing pipes. The derivation for the 1D cylindrical case is a classic chemical & mechanical engineering problem, so I won’t repeat it here. This textbook derives it nicely in equations (2-18) to (2-26).
Basically, we approach this problem with an energy balance:
$\text{(rate of energy in) − (rate of energy out) + (rate of heat generation) = (rate of change in energy)}$
The end result is below:
$\frac{1}{\alpha} \frac{\partial T}{\partial t} = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+ \frac{\dot{e}_{gen}}{k}$
This equation is a partial differential equation (PDE), which generally requires a numerical solution (as opposed to an analytical solution). This PDE includes a few parameters:
• $k$ is the thermal conductivity
• $\alpha = {k}/{\rho c_p}$ is the thermal diffusivity
• $\dot{e}_{gen} = \left(I^2 R_{int}\right)/\left(\pi R^2 L\right)$ is the volumetric heat generation rate. Here, we assume a constant heat generation rate due to resistive heating, given by $I^2 R_{int}$. The volume is simply the volume of an “18650” cylinder with $R = 9 \text{ mm}$ and $L = 65 \text{ mm}$.
#### Initial and boundary conditions
Since this equation is second-order in space ($r$) and first-order in time ($t$), we need two boundary conditions and one initial condition. They are given by:
• IC: $T(x,t=0)=T_{init}$. This basically means that the whole cell starts at some uniform temperature $T_{init}$. I’ve set $T_{init} = 30°C$ here.
• BC1: $\frac{\partial T}{\partial r} \bigr|_{r=0} = 0$. This is the thermal symmetry boundary condition; since the cell is symmetric across $r$, the maximum temperature is at $r = 0$ and thus the first derivative is $0$. See Eqn 2-50 and Fig 2-30 in this textbook for a more detailed description.
• BC2: $-k \frac{\partial T(R,t)}{\partial r} = h\big(T(R,t) - T_{\infty} \big)$. This is the boundary condition for convective heat transfer, which represents how a cell exchanges heat with its environment.
We also need to set limits of integration for both space and time. For space, we integrate between $r = 0$ and $r = R = 0.009 \text{ m}$, since an 18650 cell has a diameter of $9 \text{ mm}$. For time, we integrate between $t = 0$ and the total (dis)charging time, which varies depending on the C rate.
#### Parameter estimation
We now have a few parameters that require estimation. Fortunately, Drake et al (DOI) did a careful analysis of these parameters for an LFP/graphite 18650 cell. I mostly use his values in my analysis:
Parameter Value Units Source
$k$ $0.2$ $\text{ W/mK}$ Drake et al 2014
$\rho$ $2362$ $\text{ kg/m}^3$ Drake et al 2014
$c_p$ $1000$ $\text{ J/kgK}$ Maleki et al 1998
$h$ $10$ $\text{ W/m}^2\text{K}$ Engineering Toolbox (air convection)
$R_{int}$ $0.017$ $\text{ }\Omega$ My measurements
For $c_p$, Drake et al had a value of $1720 \text{ J/kgK}$. This value is much higher than other values of $c_p$ I found in literature. $1000 \text{ J/kgK}$ seems more reasonable.
One interesting point raised by Drake et al is that the radial heat transfer coefficient, $k_r$, is much lower than the axial heat transfer coefficient, $k_z$, since heat transfer through the polymeric seperator is limiting in the radial direction.
#### Solving the PDE
MATLAB has a built-in function called pdepe designed to solve one-dimensional PDEs like this one. I use this function with little additional modification. MATLAB’s own documentation for this function is quite good. If you’re interested in seeing my implementation, check out my GitHub repository for this code. Unfortunately Python doesn’t appear to have a nice built-in PDE solver yet, although one could solve it manually using an iterative finite-element model.
#### Biot number analysis
One of the major motivations for this type of work is estimating the internal temperature during cell operation, particularly during fast charging and discharging. We can quickly estimate the expected difference between surface and center temperatures by the dimensionless Biot number:
$Bi = \frac{L_c h}{k}$
For a cylinder, $L_c = R/2$. Thus:
$Bi = \frac{R h}{2k}$
Our best values for the required parameters are $R = 9 \text{mm} = 0.009 \text{m}$, $k = 0.2 \text{W/mK}$, and $h = 10 \text{W/m}^2\text{K}$ (air convection). With these values, we obtain:
$Bi = 0.225$
In thermal modeling, we can often assume that the difference between the surface and bulk temperature is small if $Bi < 0.1$. Our value is only a factor of two larger than this criterion. Thus, we should still account for spatial variation, but we should expect a small difference between the surface and core temperatures.
As an aside, this analysis changes significantly if we consider water or oil cooling ($h = 500 \text{W/m}^2\text{K}$):
$Bi = 11.25$
Now, we should expect a much larger difference between the center and surface temperatures.
### Results
Check out the results of charging at 1C, 5C, and 10C below:
#### 10C (charging time = 6 minutes)
While the temperature rise during 1C charging is less than one degree, the temperature rise during 10C charging is nearly 20°C! However, even in this case, the temperature difference between the center and surface of the battery is only a few degrees, as predicted by the Biot number analysis.
### Future work
This simulation is essentially the simplest thermal model of a battery you can create. Additional refinements include:
• Solve the PDE including the $z$ dependence of heat generation to include cooling from the caps
• Develop more sophisticated models for heat generation, $c_p$, and $k$ that includes additional heat generation terms (ionic resistance, chemical reaction, etc) and dependencies on state-of-charge, direction of charge, and temperature
• Account for the stainless steel core and can (neglected in this model)
I’ve enjoyed creating this model, and I think it nicely illustrates the power of simple simulations to guide understanding of a problem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8043238520622253, "perplexity": 740.1175075735199}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103941562.52/warc/CC-MAIN-20220701125452-20220701155452-00423.warc.gz"} |
https://readtiger.com/wkp/en/Angular_frequency | # Angular frequency
Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π. This figure uses the symbol ν, rather than f to denote frequency.
A sphere rotating around an axis. Points farther from the axis move faster, satisfying ω=v/r.
In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.
Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector ${\displaystyle {\vec {\omega }}}$ is sometimes used as a synonym for the vector quantity angular velocity.[1]
One revolution is equal to 2π radians, hence[1][2]
${\displaystyle \omega ={{2\pi } \over T}={2\pi f},}$
where:
ω is the angular frequency or angular speed (measured in radians per second),
T is the period (measured in seconds),
f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν).
## Units
In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. From the perspective of dimensional analysis, the unit hertz (Hz) is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention helps avoid confusion.[3]
In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency.
## Circular motion
In a rotating or orbiting object, there is a relation between distance from the axis, tangential speed, and the angular frequency of the rotation:
${\displaystyle \omega =v/r.}$
### Oscillations of a spring
An object attached to a spring can oscillate. If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by[4]
${\displaystyle \omega ={\sqrt {\frac {k}{m}}},}$
where
k is the spring constant,
m is the mass of the object.
ω is referred to as the natural frequency (which can sometimes be denoted as ω0).
As the object oscillates, its acceleration can be calculated by
${\displaystyle a=-\omega ^{2}x,}$
where x is displacement from an equilibrium position.
Using "ordinary" revolutions-per-second frequency, this equation would be
${\displaystyle a=-4\pi ^{2}f^{2}x.}$
### LC circuits
The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance (C measured in farads) and the inductance of the circuit (L, with SI unit henry):[5]
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}.}$
Adding series resistance (for example, due to the resistance of the wire in a coil) does not change the resonate frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonate frequency does depend on the losses of parallel elements. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9878824949264526, "perplexity": 690.3153022349326}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583879117.74/warc/CC-MAIN-20190123003356-20190123025356-00079.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.