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https://mathoverflow.net/questions/219529/on-the-zero-set-of-a-c2-function-on-0-12 | # On the zero set of a $C^2$ function on $[0,1]^2$
Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I_x\subset [0,1]$ such that $f(x,y)=0$ for all $y\in I_x$. Does it follow that there must exist an open ball in $[0, 1]^2$ where $f$ is identically $0$?
Yes. Each $I_x$ contains some segment $[p,q]$ with rational endpoints. Let $A(p,q)$ be a set of corresponding $x$. It is impossible by Baire category theorem that each $A(p,q)$ is nowhere dense. Thus there exist $p,q$ and a segment $\Delta$ of positive length such that $A(p,q)$ is dense in $\Delta$. Hence $f$ does vanish on $\Delta\times [p,q]$, continuity of $f$ is enough 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": 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": 0.9871925115585327, "perplexity": 27.34300032603796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2019-09/segments/1550247526282.78/warc/CC-MAIN-20190222200334-20190222222334-00222.warc.gz"} |
https://www.physicsforums.com/threads/forces-on-a-beam-replace-with-force-and-couple.796943/ | Forces on a beam, replace with force and couple
1. Feb 9, 2015
bnosam
1. The problem statement, all variables and given/known data
http://oi58.tinypic.com/2a5d4w3.jpg
2. Relevant equations
So I can just sum up all the forces then sum the moments?
3. The attempt at a solution
FBx = 0 lb
FCx = 800 cos30 = 692.82 lb
FEx = 0 lb
FBy = -2500 lb
FCy = 800sin30 = 400 lb
FEy = -1200 lb
Resultant force:
FAx = 0 + 692.82 + 0 = 692.82 lb
FAy = -2500 + 400 - 1200 = -3300 lb
MOB = (2500)(2) = 5000 lb*ft counterclockwise
MOC = 0 lb*ft
MOE = having trouble with this one
Does everything seem ok here so far?
2. Feb 9, 2015
SteamKing
Staff Emeritus
It's not clear why you are calculating moments about point C. The problem statement clearly states that the three forces are to be replaced by a force-couple combination acting at point A.
3. Feb 9, 2015
bnosam
I thought I could calculate moments anywhere?
4. Feb 9, 2015
SteamKing
Staff Emeritus
Yes, you can calculate moments anywhere, but when you are asked to find them at a particular point, it saves time to use that point as the reference for your moment calculations.
BTW, your moment calculations about point C were incorrect, anyway. Remember, the force must act perpendicular to the moment arm in order to calculate the magnitude of the moment.
5. Feb 9, 2015
bnosam
B: 2500*4 = 10000 ft*lb
That's correct, right?
The reason why I wanted to take them about C is because of the angling on C and to simplify it more.
6. Feb 9, 2015
SteamKing
Staff Emeritus
The magnitude of this moment is correct, but you need to establish a sign convention for your moment directions. Moments can be either clockwise or counterclockwise, and failing to distinguish between these two directions will lead to erroneous results.
The angle of the arm BCD w.r.t. the horizontal only affects the moment due to the force at C.
7. Feb 9, 2015
bnosam
Oh B should be counterclockwise, right?
so then C should be Moc = (800)(6)? If I get what you're saying?
8. Feb 9, 2015
SteamKing
Staff Emeritus
Right.
Point C is not 6' from point A. Remember the definition of what a moment is.
9. Feb 9, 2015
bnosam
So because it's on an angle we want only the horizontal component of where C is?
10. Feb 9, 2015
SteamKing
Staff Emeritus
Yes.
But it's not that simple. Because the force at C is also directed at an angle, it must be broken up into its components in order to calculate the moment acting about point A.
11. Feb 9, 2015
bnosam
so we want the cos of the angle? 2*cos(30) = 1.732 ft
12. Feb 9, 2015
SteamKing
Staff Emeritus
The distance from A to C would be 4' + 2' * cos (30°), but the force at C is 800 lbs. acting at 60° to the horizontal. You've got to compute the components of that force. Each force component will create a moment acting about point A.
13. Feb 9, 2015
bnosam
4+ 2 cos(30) = 5.73
Moc = (5.73) * 800cos60 = (5.73)*400 = 2292 clockwise, right?
If I'm understanding you, that is.
14. Feb 10, 2015
SteamKing
Staff Emeritus
Nope. You should draw the components of the force at C on your picture and see how they produce their moments.
15. Feb 10, 2015
bnosam
4+ 2 cos(30) = 5.73
Moc = (5.73) * 800sin60 = (5.73)*692.82 = 3969.86 clockwise, right?
Does that seem better?
16. Feb 10, 2015
SteamKing
Staff Emeritus
The magnitude looks OK. Are you sure the direction of this moment is clockwise w.r.t. point A?
What about the moment due to the horizontal component of the force at C?
17. Feb 10, 2015
bnosam
No it should be counterclockwise I believe. Is there a better way to tell despite intuition?
In the case of C:
10.46 * 1200 = 12552 lb*ft
18. Feb 10, 2015
SteamKing
Staff Emeritus
This is actually the moment due to the force applied at point E.
What about the moment due to the horizontal component of the force applied at point C?
Examination of the direction of the force in relation to the reference point is fine. There are different methods of calculating moments which take care of having to visualize which direction the moment will take, but you will learn those later.
19. Feb 10, 2015
bnosam
Sorry, I mistyped and meant E :p
Wouldn't the horizontal component of C cause a counterclockwise rotation also? Since it would push against the angle and push it upwards.
20. Feb 10, 2015
SteamKing
Staff Emeritus
Yes, it would.
Draft saved Draft deleted
Similar Discussions: Forces on a beam, replace with force and couple | {"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": 0.8172770142555237, "perplexity": 1601.4484572658296}, "config": {"markdown_headings": false, "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-2017-34/segments/1502886105334.20/warc/CC-MAIN-20170819085604-20170819105604-00586.warc.gz"} |
http://math-mprf.org/journal/articles/id1244/ | Classical Behavior of the Integrated Density of States for the Uniform Magnetic Field and a Randomly Perturbed Lattice
#### N. Ueki
2011, v.17, №3, 347-368
ABSTRACT
For the Schrodinger operators on $L^2(R^2)$ and $L^2(R^3)$ with the uniform magnetic field and the scalar potentials located at all sites of a randomly perturbed lattice, the asymptotic behavior of the integrated density of states at the infimum of the spectrum is investigated. The randomly perturbed lattice is the model considered by Fukushima and this describes an intermediate situation between the ordered lattice and the Poisson random field. In this paper the scalar potentials are assumed to decay slowly and its effect to the leading term of the asymptotics are determined explicitly. As the perturbed lattice tends to the Poisson model, the determined leading term tends to that for the Poisson model.
Keywords: Lifshitz tails,magnetic Schrodinger operator | {"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": 0.9246094822883606, "perplexity": 315.73461686519164}, "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-2018-26/segments/1529267863519.49/warc/CC-MAIN-20180620124346-20180620144346-00167.warc.gz"} |
https://www.arxiv-vanity.com/papers/gr-qc/9306008/ | \Pubnum
=KUNS-0.0pt 122-XX
1201 \date=May 28 , 1993 \titlepage\titleDeflationary Universe Scenario \authorBoris Spokoiny †† Address after June 23, 1993: Landau Institute for Theoretical Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow region, Russia.
\address
Department of Physics, Kyoto University, Kyoto 606-01, Japan
\abstract
We show that it is possible to realize an inflationary scenario even without conversion of the false vacuum energy to radiation. Such cosmological models have a deflationary stage in which is decreasing and radiation produced by particle creation in an expanding Universe becomes dominant. The preceding inflationary stage ends since the inflaton potential becomes steep. False vacuum energy is finally (partly) converted to the inflaton kinetic energy , the potential energy rapidly decreases and the Universe comes to the deflationary stage with a scale factor . Basic features and observational consequences of this scenario are indicated.
\REF\Star
A.A.Starobinsky,Phys.Lett.91B (1980)99; \nextlineA.D.Linde,Phys.Lett.108B(1982);129B (1983)177; \nextlineA.Albrecht and P.J.Steinhardt,Phys.Rev.Lett.48(1982)1220.
\REF\Guth
A.H.Guth,Phys.Rev.D 23 (1981) 347; \nextlineD.La & P.J. Steinhardt, Phys. Rev.Lett. 62 (1989) 376.
\REF\Rat
B.Ratra & P.J.E.Peebles, Phys.Rev.D37 (1988)3406.
\REF\Spo
B.L.Spokoiny, Vistas in Astronomy 37 (1993)481; Stochastic approach to a non de Sitter Inflation, Kyoto Univ. Preprint KUNS-1192, May (1993).
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Y.Fujii and T.Nishioka, Phys.Rev.D42 (1990)361;ibid. D45 (1992)2140; Phys.Lett. 254B (1991)347.
\REF\Dol
A.D.Dolgov,in The Very Early Universe, Proceedings of the Nuffield Workshop, Cambridge, UK, 1982, edited by G. W. Gibbons and S.T.Siklos.
\REF\Ste
R.Crittenden and P.J.Steinhardt, Phys.Lett.293B (1992)32.
\REF\Luc
F.Lucchin & S.Matarrese , Phys.Rev.D32(1985)1316; Phys.Lett. 164B (1985) 282.
\REF\Hal
J.Halliwell,Phys.Lett. B185 (1987) 341.
\REF\Bar
J.D.Barrow, Phys.Lett.B187 (1987) 12.
\chapter
Introduction It has become almost an inflationary paradigm that after the end of the superluminal (exponential) expansion stage driven by false - vacuum energy there is conversion of this energy to radiation. There are two standard ways to realize this conversion: by decay of the oscillations of the inflaton field near the minimum of its potential in the new or chaotic inflationary scenarios [\Star] or by collisions of the bubbles of new vacuum in the old or extended inflation [\Guth]. However for the solution of the famous problems such as horizon or flatness problems and others such a conversion is not a necessary condition. In fact what one needs is some production of radiation that will become dominant and finally will drive the evolution of the Universe. In this paper we investigate basic features of this possibility in detail.
We consider inflationary models in which an inflaton potential is almost flat in some region,say at so as the Universe expands (quasi)exponentially while the inflaton is in this region and in the region the inflaton potential becomes steep enough so as inflation ceases. In distinct to the standard models we consider the case when the potential just rapidly falls for large values of and does not have a minimum of the potential. To understand how the Universe could become a radiation dominated one it is enough to recall that in an expanding Universe there is particle production. At the inflationary stage the energy density of the produced particles is inflated out but in the post - inflationary epoch the situation could be changed. If the potential falls rapidly enough the energy density of the produced particles which are then thermalized can become dominated. Since for relativistic particles (and radiation) the stage in which is decreasing is necessary for the above phenomenon to occur. We call this stage a deflationary one . (On the contrary at the inflationary stage this quantity is rapidly increasing due to slow variation of .)
There is an essential difference between the deflationary realization of the inflationary scenario and the standard one. In the ordinary realization an inflationary stage only provides some (good) initial conditions for the following it standard radiation dominated stage . The inflaton becomes “frozen” in the minimum of the potential and does not participate in the dynamics at the radiation dominated stage. In principle this circumstance restricts greatly the possibility of comparison of different inflationary models from observations. This standard construction was greatly affected by the success of the nucleosynthesis theory and very severe restrictions which follow from this theory for the dynamics of the Universe during the nucleosynthesis epoch and for any ‘nonstandard’ contributions to the energy-momentum tensor at this time. So any alternative model in which an inflation ends in a different way should have some mechanism to pass through the nucleosynthesis test if one does not want a new fine tuning.
On the contrary in the deflationary realization an inflaton “survives” and its dynamics may be still important at later stages of the evolution of the Universe,even at the present time. For example inflaton can give us dark matter. In this case we have a natural bias since perturbations of the scalar field decay when the wavelength becomes less than the size of the horizon. This results in a more homogeneous distribution of mass in comparison with galaxy distribution (see [\Rat]). There is also a natural mechanism that provides very small inflaton energy density during the nucleosynthesis time which still allows to have (relatively) large inflaton energy density now. The idea is rather simple. If the inflaton potential is steep enough at the deflationary stage then the dynamics of the Universe is driven by the inflaton kinetic energy term and the potential energy term is rapidly decaying. (We call such a stage a critical deflationary one). After the moment of time when radiation becomes dominant the kinetic energy term starts to fall rapidly, the potential energy term is very small since it was falling at the preceding critical deflationary stage but it becomes almost frozen (since time derivative of the inflaton is rapidly decaying). So there is a big “quasipure” radiation dominated epoch in this model at which the contribution of the scalar field into the energy-momentum tensor is negligible. The nucleosynthesis period is supposed to be inside this epoch. When the (falling) radiation energy density becomes of the order of (frozen) inflaton potential energy density the Universe comes to the “combined” stage at which the contribution of the inflaton energy is com that of ordinary matter.
\chapter
Dynamics of the Universe with a deflationary stage
A simple inflationary model is given by the system of equations
H2=13M2(˙φ22+V(φ)),
¨φ+3H˙φ+V′=0,
where in some region ,say , the logarithm of the potential is flat:\nextline and in the other region it is steep. Here . So while the dynamics of the scalar field is slow and the Universe expands quasi-exponentially: (in the simplest model). In eq.(2) one takes into account only the second and the third terms disregarding the first one. When the scalar field reaches the region of the steep potential its dynamics becomes rapid with respect to the Universe expansion. In eq.(2) the first and the third terms become the most important ones,the second term being a small friction. In an ordinary inflationary scenario the potential has a minimum and the scalar field rapidly rolls down to the bottom of the potential and starts to oscillate near the minimum of the potential. These oscillations generate ordinary matter.
In this paper we consider another possibility of the transition to the radiation dominated stage and consider an inflaton potential that does not have a minimum and just falls at large . Matter is generated by particle production at the deflationary stage. In the standard scenario the potential energy of the scalar field is transformed into the energy of oscillations, some part of the energy being lost due to friction. The transition from the region of the flat logarithm of the potential to the region of the steep logarithm of the potential is sharp in the sense that the transition proceeds in a (small) region . In our case there could be two possibilities of this transition : a sharp transition and a smooth one with the transitional region . A simple example of the potential with a sharp transition is
V(φ)=exp(−λ1φ/M)1+exp(λ2φ/M),
where and is large enough. A general form of the potential with a smooth transition is
V(φ)=V0exp(−λ(φ)φM),
where is a slowly varying function of , satisfying the condition **Stochastic approach to the inflationary Universe driven by the scalar field with such type of potential was developed in [\Spo].
Mλ′≪1.
A simple example of this type of potential is just a gaussian potential \nextline with .
1)Sharp transition. First we consider the case that is closer to the standard one: sharp transition to the region of the steep potential. In this case potential energy is converted to the kinetic energy of the inflaton:
˙φ2∼V(φ∗),
and the Universe becomes driven by the kinetic energy term. This kinetic energy term is decaying due to friction. Assuming that the potential energy does not affect the inflaton dynamics we obtain from eq.(2)
˙φ=cφa3,
where is a constant and . Substituting (7) into (1) we get
a(t)∝t1/3,
φ=φ0+√6Mloga.
Using (9) we find the condition for the decay of the potential:
ϵ(φ)=~V(φ)~V(φ∗)≪1, where ~V(φ)=V(φ)exp(√6φM).
Thus if (10) is fulfilled the potential decays more rapidly than the inflaton kinetic energy term.
In an expanding Universe there is particle production. The energy density of particles produced by gravitational field for the characteristic time may be estimated as for the scalar particles non-conformally coupled to gravity. Here is the number of species and it is supposed that the mass of produced particles . The energy density of the particles produced at the inflationary stage is inflated out but the energy density of the above scalar particles produced at the deflationary stage (where decreases) and falls less rapidly than the main contribution (of the scalar field ) to the energy-momentum tensor. Although initially the energy density of the produced particles is negligible with respect to the main contribution due to the inflaton it becomes dominating after some time. To investigate dynamics of the Universe we are to add matter energy density to the inflaton energy density in eq.(1):
(˙aa)2=13M2(c2φa6+c0a4).
While the inflaton contribution dominates and the Universe expands according to (8). However at the contribution of radiation becomes of the same order and at the Universe is radiation dominated. The value of the Hubble parameter at the moment when the contributions of both types of matter become equal to each other (at ) gives us a characteristic time of the change of epochs. At the inflationary stage radiation energy density decays faster than that of inflaton but at the inflaton kinetic energy dominated stage (8) the situation is inverse . This is why we call this stage a deflationary one. Integration of eq.(11) gives us
tt0=aa0√1+(aa0)2−arcsinhaa0,
where At we have the deflationary stage
a(t)≃a0(3t/t0)1/3,
at - radiation dominated stage
a(t)≃a0(t/t0)1/2.
The length of the deflationary epoch may be estimated as follows. At the moment when the contributions of matter and the scalar field become equal to each other we have where and are the values of the Hubble parameter and the scale factor at the beginning of the deflationary stage and is the value of the scale factor at (at the beginning of the radiation dominated stage (14)). Thus we obtain
ara∗∼50n−1/2MH∗.
The created particles are thermalized and we may estimate the temperature just after the achievement of thermal equilibrium as follows. Assuming that the created scalar particles are Higgs bosons and interactions between them are mainly due to gauge bosons we may estimate the interaction rate as . Thermal equilibrium is achieved when . We also have relations and at the deflationary stage. As a result we get
Teq∼α(n100g∗)38H∗.
Taking as usual , , we obtain . As follows from the constraints on the amplitude of gravitational waves which follow from the small anisotropy of the microwave background . Constraints from adiabatic perturbations typically give a lower value of . Thus and is rather low. We find from (15) that at the beginning of the radiation dominated stage
Tr∼(n33g∗)1/410−2H2∗M2M∼10−2H2∗M2M.
Thus in the above estimates . On the other hand we need for the successful nucleosynthesis in any case, which means and .
At the radiation dominated stage (14) so the integral converges and the change of the value of the scalar field for the whole radiation dominated epoch is . This means that the potential at the end of the radiation dominated stage is of the same order of magnitude as that in the beginning of this stage if we suppose that the potential has only one characteristic scale of variation - the (reduced) Planck mass . Thus the potential does not change essentially at the radiation dominated stage contrary to the preceding deflationary stage where it was falling rapidly according to (8)-(10). Roughly speaking one may say that the potential is frozen at the radiation dominated stage. A similar analysis for a nonrelativistic matter dominated stage gives qualitatively the same results. This means that the above radiation dominated stage (or maybe already nonrelativistic matter stage) is not eternal and at last the Universe will get
ε∼V(φ)∼V(φr),
where is the value of the scalar field at the beginning of the radiation dominated stage, and as follows from (2)
˙φ2∼V(φ)∼ε∝1/t2
Since we want for the combined stage (19) to begin after the nucleosynthesis time we need for the potential to be steep enough:
V(φr)
where
λeff=√6log30V(φ∗)πg∗T4NUClog30V(φ∗)πg∗T4r.
Now we investigate the dynamics of the Universe with scalar field and matter. So we should add a matter energy term into eq.(1):
H2=13M2(ε+˙φ22+V(φ)),
Matter is supposed to satisfy the equation of state , where for radiation and for dust. Conservation of the energy-momentum tensor of the system “scalar field plus matter” and the equation for the scalar field (2) give us One may easily check that the transition solution from the matter dominated stage value to the combined solution (19) is
φ(t)≃φm+ct−1−k1+k−1+k2(3+k)V′(φr)t2,
where is a constant, for radiation and for dust (non- relativistic matter). The second term in (23) describes the initially dominated but now falling mode (7) and the third term describes the second mode of eq.(2) which is initially small but growing. Solution (23) is valid at . It is easy to check that
ε(tc)∼λ2V(φm)∼λ2˙φ2(tc),
where
λ≡λ(φ)=M(logV)′.
So is a characteristic time after which the solution (19) is achieved. From eq.(24) we can make some qualitative conclusions about the behavior of the Universe filled with matter and a scalar field. If then the scalar field dominates. This corresponds to the inflationary situation. On the contrary if then matter dominates. If then at least naively the contributions of matter and the scalar field are comparable. Since we are not able to solve the system of the scalar field - metric equations analytically for an arbitrary potential , to get some ideas about the behavior of the system in general case we consider some exactly solvable types of the potential that may be not the most interesti applications. Let be constant or a slowly varying function of satisfying the condition (5). This means that we consider potentials with a (pseudo)exponential asymptotic at behavior (4). Then we find from (19)
φ≃~φ0+Alogt, A=2M/λ.
For any equation of state the system of metric and scalar field equations has the solution (26) ****Solution (26)-(29) has been obtained in refs. [\Rat,\Fuj]. It was considered in the “model of a decaying cosmological constant” [\Fuj-\Dol] in [\Fuj] in the context different from ours. In ref.[\Rat] cosmological consequences of a rolling homogeneous scalar field were discussed without any relation to inflation. with
~φ0=A2log2(3p−1)M2λ2V0
and
a(t)∝tp,
where
p=2/3(1+k), ε=3p2M2t2(1−3(1+k)/λ2)
The solution (26)-(29) exists if , the ratio of the inflaton energy density to the ordinary matter energy density being
εφε=3(1+k)/λ2(1−3(1+k)/λ2).
So we see that if the potential is steep enough inclusion of the scalar field does not change the evolution behavior (28). On the contrary if the potential is flat enough (), the scalar field dominates , matter is inflated out and we have the solution (28) with
p=2/λ2, φ=φ′0+(2M/λ)log(Mt), ε∝t−2λc/λ.
2)Smooth transition.
Now we briefly consider dynamics in the case of a smooth transition to the deflationary phase. The inflaton potential is given by (4).
First let us consider for simplicity the case of a constant . In this case there is a power-law solution of the system (1)-(2)
a(t)∝tp, p=2/λ2, φ=φ0+(2M/λ)log(Mt),
If and (5) is fulfilled we have an approximate solution (32) where .
If then and (32) describes an inflationary stage. If then and we will call this stage “pseudo-inflation”. ****** Crittenden and Steinhardt [\Ste] recently considered inflation with a Brans-Dicke type dilaton and depending upon parameter . The tunneling of the inflaton to the true vacuum appears at the “pseudo-inflationary” stage thus avoiding large bubbles. In the Einstein frame the dilaton part of the CS model is equivalent to the model with the dilaton potential (4). In this paper we consider another possibility of the exit from the inflationary stage without any tunneling.
If then and we will call this stage deflation. Solution (32) was obtained in [\Luc] ( for a constant ) and considered mainly for an inflationary stage in fact[8-10,4]. On the contrary our focus is mainly on the deflationary case . A more careful analysis shows that solution (32) is valid only if which corresponds to . If then solution (32) corresponds to negative in (4) and is not applicable. In this case we should use an asymptotic at (critical deflationary) solution (8)-(9) and
V(φ)∝t−(2+q),
where In the above solution it is supposed that
One may easily check that the energy density of particles produced in the vicinity of the moment of time when is dominant. Indeed the energy density produced at the moment and it is easy to see that the maximal production is near the moment of time when is maximal. This moment corresponds to , where is the index of expansion (), and , thus just to the end of the superluminal expansion. The energy density of the produced particles is initially decreasing but at the deflationary stage where is decreasing () it becomes more and more dominant. The deflationary stage starts when . Radiation produced after thermalization of the particles created just after the end of the superluminal expansion must become dominant in the Universe before its temperature drops below . This is a necessary condition for the model.
Acknowledgments
I am very grateful to Misao Sasaki for very useful discussions concerning nucleosynthesis restrictions and growth of perturbations and to Ewan Stewart and Jun’ichi Yokoyama for important stimulating comments which allowed to clarify basic features of the model. This work was supported in part by a JSPS Fellowship and by Monbusho Grant-in-Aid for Encouragement of Young Scientists, No. 92010.
\refout
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For everything else, email us at [email protected]. | {"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": 0.9398189783096313, "perplexity": 564.0695699719545}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2022-33/segments/1659882571584.72/warc/CC-MAIN-20220812045352-20220812075352-00009.warc.gz"} |
https://www.scm.com/doc/ADF/Input/Charge_transfer_integrals.html?highlight=electrontransfer | Charge transfer integrals (transport properties)¶
ADF can provide input parameters, such as charge transfer integrals, that are needed in approximate methods that model charge transport properties. ADF has the unique feature that it can (also) calculate such transfer integrals based on the direct method by the use of its unique fragment approach.
In theoretical models of charge transport in organic materials, see Refs. [6] [7] [8], the whole system is divided into fragments, in which an electron or hole is localized on a fragment, and can hop from one fragment to another. In the tight-binding approximation that is used in these models the electron or hole is approximated with a single orbital, and it is assumed that only the nearest neighboring fragments can couple. The models require accurate values of electronic couplings for charge transfer (also referred to as charge transfer integrals or hopping matrix elements) and site energies (energy of a charge when it is localized at a particular molecule) as a function of the geometric conformation of adjacent molecules. Charge transfer integrals for hole transport can be calculated from the energetic splitting of the two highest-occupied molecular orbitals (HOMO and HOMO-1) in a system consisting of two adjacent molecules, also called “energy splitting in dimer” (ESID) method. For electron transport these can be calculated from the two lowest-unoccupied orbitals (LUMO and LUMO+1) in this ESID method. ADF can also calculate transfer integrals based on the direct method by the use of its unique fragment approach. see Refs. [7] [8]. ADF allows one to use molecular orbitals on individual molecules as a basis set in calculations on a system composed of two or more molecules. The charge transfer integrals obtained in this way differ significantly from values estimated from the energy splitting between the highest occupied molecular orbitals in a dimer. The difference is due to the nonzero spatial overlap between the molecular orbitals on adjacent molecules. Also, ADF’s methods are applicable in cases where an orbital on one molecule couples with two or more orbitals on another molecule.
Charge transfer integrals with the TRANSFERINTEGRALS key¶
In this method the matrix elements of the molecular Kohn-Sham Hamiltonian in the basis of fragment orbitals is used to calculate site energies and charge transfer integrals. Likewise the overlap integrals between fragment orbitals are calculated. No explicit electrons are removed or added in this method. For electron mobility calculations the fragment LUMO’s are considered. For hole mobility calculations the fragment HOMO’s are considered.
To calculate the charge transfer integrals, spatial overlap integrals and site energies, include the key TRANSFERINTEGRALS in the input for ADF. Symmetry NOSYM should be used. The molecular system typically should be build from 2 fragments. In the fragment calculation full symmetry can be used.
TRANSFERINTEGRALS
Symmetry NOSYM
Fragments
frag1 frag1.t21
frag2 frag2.t21
End
By default, integrals are calculated only for the HOMO (LUMO) of the fragments, and possibly HOMO-1, HOMO-2 (LUMO+1, LUMO+2) if the energy of those fragment orbitals are close to the HOMO (LUMO) of that fragment. To calculate the matrix elements and overlap integrals based on all fragment orbitals one can use the key:
PRINT FMATSFO
The method described here to calculate charge transfer integrals is more approximate than the next method that uses FDE. The major difference is how effects of a localized charge are included.
If 2 fragments are used the electronic coupling V (also known as effective (generalized) transfer integrals J_eff) for hole transfer or electron transfer is calculated as V = (J-S(e1+e2)/2)/(1-S^2). Here e1, e2, are the site energies of fragment 1 and 2, respectively. J is the charge transfer integral, and S the overlap integral. Note that the effect of a possible degeneracy of HOMO or LUMO is not taken into account. The electronic coupling between the HOMO of the donor fragment and the LUMO of the acceptor fragment and vice-versa is also calculated, which represent the probability of a charge recombination process.
Charge transfer integrals with FDE¶
Overview
The ELECTRONTRANSFER keyblock invokes the calculation of Hamiltonian (site energies and couplings) and overlap matrix elements with FDE-derived localized states. Two FDE calculations are (not strictly) needed before running the ELECTRONTRANSFER calculation. The calculated matrix elements are theoretically similar to the ones obtained with the TRANSFERINTEGRALS keyword. A recent review casts ELECTRONTRANSFER in the state-of-the-art of methods for computing charge transfer couplings, and provides a step-by-step guide for computing such couplings with ADF [1].
Features
• Effects of orbitals relaxation due to localized charges, Refs. [9] [10].
• Effects of polarization due to molecules in the environment, Ref. [2].
• ELECTRONTRANSFER is linear scaling in the number of fragments when the system is composed by more than one fragment.
• The code can tackle hole transfer, electron transfer, charge separation and charge recombination processes [3].
• Can compute transfer integrals from diabats obtained with the Constrained DFT method (i.e., invoked by using the experimental CDFT keyword of ADF). This includes calculation of couplings between diabats made of a single fragment.
• It is possible to include the effect of the environment on CDFT diabats by coupling it with an FDE calculation [4].
• The performance of ElectronTransfer in the evaluation of the hole transfer coupling, was benchmarked against wave functions methods, with an error below 7%. PBE, PW91, B3LYP and PBE0 functionals, with PW91k for the non-additive component of the kinetic energy and TZP basis function, are recommended to obtain the FDE-derived localized states [5].
• The code was parallelized in ADF2018.
Limitations
• Hybrid functionals are not yet supported
• TAPE21 files with charge or spin-localized states with specific names are needed
These two limitations do not apply to the method with the TRANSFERINTEGRALS key, but the TRANSFERINTEGRALS has many other limitations. Like the method with the TRANSFERINTEGRALS key fragments are needed.
ELECTRONTRANSFER input
The minimum input for the ELECTRONTRANSFER key is:
FRAGMENTS
frag1 FragFile1
...
fragN FragFileN
END
ELECTRONTRANSFER
NumFrag N
END
where frag1 … fragN are the labels of the fragments in the calculation, and FragFile1 … FragFileN are the TAPE21 files of spin RESTRICTED calculations of the isolated fragments, N is the total number of fragments employed in the calculation.
Files and file names
The fragment files to be used in the ELECTRONTRANSFER calculation are generally different from the TAPE21 files used in the FRAGMENTS key block. Two types of fragment TAPE21 files are needed by the calculation:
• The isolated closed-shell TAPE files for the FRAGMENTS keyblock
• The TAPE21 files of the charge or spin localized states (which can be obtained with an FDE calculation as done in the example below)
There are 2 charge localized states. They are labeled with A and B. The respective TAPE21 files must be names as follows:
fragA1.t21, fragA2.t21, ... , fragAN.t21 (for state A)
fragB1.t21, fragB2.t21, ... , fragBN.t21 (for state B)
The above files should be copied to the working folder of the ADF calculation prior to executing ADF.
Options
ELECTRONTRANSFER
NumFrag N
{Joint|Disjoint}
{Debug}
{Print EIGS|SAB}
{FDE}
{INVTHR threshold}
{CDFT}
END
Joint|Disjoint
The default is “Joint”. Joint is always recommended. The “Disjoint” formalism is described in Ref. [2] and is much faster than the “Joint” formalism when more than 2 fragments are considered. Joint and Disjoint are equivalent for systems composed of only 2 fragments. Disjoint should only be used if the fragment files are obtained in an FDE calculation (see FDE below).
Debug
The code performs additional checks (determinants, diagonalizations, inversions, traces, etc.). Substantial increase in the output should be expected.
Print
If EIGS, it will print the (unformatted) matrix of the MO coefficient in the AO representation. If SAB, it will print the (unformatted) matrix of the diagonal and transition overlap matrix in the MO representation.
FDE
An FDE calculation including more than 2 fragments must include the following key block:
ELECTRONTRANSFER
FDE
END
and the numerical integration precision in the last FDE calculation for every subsystem should be set to no less than:
BeckeGrid
Quality Good
End
if in the subsequent ELECTRONTRANSFER calculation the DISJOINT subkey is used.
Invthr threshold
Default 1.0e-3, is a threshold for the Penrose inversion of the transition overlap matrix. If warnings about density fitting are printed, invthr may be increased up to 1.0e-2. Larger invthr might affect the quality of the calculated couplings and excitation energies.
CDFT
If disjoint is selected, this keyword must be selected if the evaluation of the electronic coupling is sought between diabats located on the same CDFT fragment. In this case, the CDFT fragment has to be always first under the ATOMS keyword.
KNADD
If disjoint is selected, this allows the kinetic energy of each fragment to be obtained locally without using the full grid. This keyword is recommended when there are many tens of subsystems, such as systems with several solvent molecules.
Output
The output of the example in \$ADFHOME/examples/adf/ElectronTransfer_FDE_H2O is discussed here. This example involves the calculation of electronic coupling, site energies and charge-transfer excitation energy for the hole transfer in a water dimer.
============ Electron Transfer RESULTS ===================
Electronic Coupling = 0.000000 eV
Electronic Coupling = -0.003569 cm-1
H11-H22 = -1.396546 eV
Excitation Energy = 1.396546 eV
Overlap = 0.000000
H11 H22 H12 = -152.443000816341 -152.391678701092 -151.743979368040 Eh
S11 S22 S12 = 0.981795415192 0.981006454450 -0.000000023700
=========== END Electron Transfer RESULTS ================
Due to symmetry, the overlap is almost diagonal (Overlap = 0.00), thus the transition density is evaluated with one less electron as explained in Ref. [2].
The electronic coupling between the state with a positive charge localized on one water molecule and another with the charge localized on the other water molecule is given by “Electronic Coupling” and is reported in eV and cm^-1.
“H11-H22” is the difference of the site energies in eV. Values of the site energies are given by the first two values of “H11 H22 H12” in atomic units.
“Excitation Energy” reports the value of the transfer excitation energy as calculated by diagonalization of the 2X2 generalized eigenvalue problem in the basis of the charge-localized states, see Refs. [2] [9].
“S11 S22 S12” are the values of the non-normalized overlaps.
References
[1] P. Ramos, M. Mankarious, M. Pavanello, A critical look at methods for calculating charge transfer couplings fast and accurately, in Practical Aspects of Computational Chemistry IV, Jerzy Leszczynski and Manoj Shukla (eds.), 2016, Springer
[2] (1, 2, 3, 4) M. Pavanello, T. Van Voorhis, L. Visscher, and J. Neugebauer, An accurate and linear-scaling method for calculating charge-transfer excitation energies and diabatic couplings, Journal of Chemical Physics 138, 054101 (2013)
[3] A. Solovyeva, M. Pavanello, J. Neugebauer, Describing Long-Range Charge-Separation Processes with Subsystem Density-Functional Theory, J. Chem. Phys. 140, 164103 (2014)
[4] P. Ramos, M. Pavanello, Constrained Subsystem Density Functional Theory, Physical Chemistry Chemical Physics 18, 21172 (2016).
[5] P. Ramos, M. Papadakis, M. Pavanello, Performance of Frozen Density Embedding for Modeling Hole Transfer Reactions, Journal of Physical Chemistry B 119, 7541 (2015)
[6] M.D. Newton, Quantum chemical probes of electron-transfer kinetics: the nature of donor-acceptor interactions, Chemical Reviews 91, 767 (1991).
[7] (1, 2) K. Senthilkumar, F.C. Grozema, F.M. Bickelhaupt, and L.D.A. Siebbeles, Charge transport in columnar stacked triphenylenes: Effects of conformational fluctuations on charge transfer integrals and site energies, Journal of Chemical Physics 119, 9809 (2003).
[8] (1, 2) K. Senthilkumar, F.C. Grozema, C. Fonseca Guerra, F.M. Bickelhaupt, F.D. Lewis, Y.A. Berlin, M.A. Ratner, and L.D.A. Siebbeles, Absolute Rates of Hole Transfer in DNA, Journal of the American Chemical Society 127, 14894 (2005)
[9] (1, 2) M. Pavanello and J. Neugebauer, Modelling charge transfer reactions with the frozen density embedding formalism, Journal of Chemical Physics 135, 234103 (2011)
[10] M. Pavanello, T. Van Voorhis, L. Visscher, and J. Neugebauer, An accurate and linear-scaling method for calculating charge-transfer excitation energies and diabatic couplings, Journal of Chemical Physics 138, 054101 (2013) | {"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": 0.8250702619552612, "perplexity": 3084.266148788286}, "config": {"markdown_headings": false, "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/1581875144027.33/warc/CC-MAIN-20200219030731-20200219060731-00012.warc.gz"} |
http://mathhelpforum.com/discrete-math/141787-help-function-problem-print.html | # Help with Function problem
• Apr 27th 2010, 05:32 PM
holyravioli
Help with Function problem
Question: X is the set of senators serving in 1998. For x ∈ X, define g(x) to be the number of terms a senator has held. Is g(x) a function?
Thanks for any help!
• Apr 28th 2010, 10:00 AM
emakarov
If you say "terms held by a certain date", then for every senator such number of terms is unique. If you just ask how many term a senator x served, the answer may be one, and two, and three...
• Apr 28th 2010, 04:08 PM
rmorin
Quote:
Originally Posted by holyravioli
Question: X is the set of senators serving in 1998. For x ∈ X, define g(x) to be the number of terms a senator has held. Is g(x) a function?
Thanks for any help!
Yes.
if x $\in X \Rightarrow$ g(x)=1. Because x is a senator, and a senator is a one senator.
Maybe you have a mistake went write the problem.
(Sorry for my bad english, i learn english) | {"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": 1, "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": 0.9420849084854126, "perplexity": 2815.7208485902356}, "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-2017-17/segments/1492917123549.87/warc/CC-MAIN-20170423031203-00107-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://www.lofoya.com/Solved/1363/p-and-q-start-running-in-opposite-directions-towards-each-other | # Easy Time, Speed & Distance Solved QuestionAptitude Discussion
Q. $P$ and $Q$ start running in opposite directions (towards each other) on a circular track starting at diametrically opposite points. They first meet after $P$ has run for 75m and then they next meet after $Q$ has run 100 m after their first meeting. Assume that both of them are running a constant speed. The length of the track (in metre) is:
✖ A. 70 ✖ B. 175 ✔ C. 250 ✖ D. 350
Solution:
Option(C) is correct
Both $P$ and $Q$ are traveling at a constant speed, so the ratio of the distance covered by both $P$ and $Q$ will be same.
Now, assume that the length of track be $x$.
So, for the time of 1st meet,
$P$ travels $= 75 \text{ m}$ and $Q$ travels $=x/2-75 \text{ m}$
Thus, ratio $\dfrac{P}{Q}=\dfrac{75}{x/2-75}$ ---------(1)
Now, for the time of second meet.
$Q$ travels $= 100 \text{ m}$ and $P$ travels $=x-100 \text{ m}$
Thus, ratio $\dfrac{P}{Q}=\dfrac{x-100}{100}$ --------(2)
From (1) and (2),
$\dfrac{75}{x/2-75}=\dfrac{x-100}{100}$
Upon solving, we get,
$x=\textbf{250 m}$
Edit: Based on the comments from Ramesh and Paritosh Kumar, final answer has been changed from option(D) to option(C). The solution has been modified too.
## (7) Comment(s)
Ayushi Deep Srivastava
()
P and Q are traveling at a constant speed, so the ratio of the distance covered by both P and Q will be same.
Please can someone explain me this point , speed being constant , y the ratio of the distance covered by them always same?
Ketaki
()
Why have we done x/2 for Q when we are finding time for first meet ? Not able to get it.
Paritosh Kumar
()
250 has to be the answer, explanation as below :
Assume track length to be T.
For first meet A travels 75, B travels T/2 - 75.
For second meet B travels 100, A travels T-100.
Solving 75/(T/2-75)=(T-100)/100
substituting options or by solving, 250 is the solution.
Deepak
()
Thank you Paritosh for letting me know. Changed the correct option choice and modified the solution, taking inputs from your comment.
Ramesh
()
Ratio of the distance covered by both $P$ and $Q$ will be same
Suppose length of track is $2x$
$\dfrac{75}{(x-75)}= \dfrac{(2x-100)}{100}$
$x=125$
Length of track will be 250
Abc
()
yeah only speeds are constant, they have mentioned.
Pranav
()
Nothing being said abt their speeds are equal !!!!!!! | {"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": 0.8319365978240967, "perplexity": 1261.1935584645128}, "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-2018-05/segments/1516084889917.49/warc/CC-MAIN-20180121021136-20180121041136-00322.warc.gz"} |
http://math.stackexchange.com/questions/89052/average-value-theorem-and-calculus-integration | # average value theorem and calculus integration
i have sat for 2 hours trying to understand how the area of R1+R3=R2 cant really get this becuz R2 has a negative section and i think f(c) should be lower. Once again, how is R1 and R3= R2? R1 and R3 is all positive area whereas R2 has negative area so in my opinion the average area of R2 should be much lower because of the negative section of R2...Can someone please explain this to me? These are part of lecture notes.
-
As geometric regions, the areas of $R_1$, $R_2$, and $R_3$ must be positive. Geometric area must be positive. (Have you ever met a triangle whose area was $-10$ cm$^2$?). And so it makes sense to claim that $Area(R_1) + Area(R_3) = Area(R_2)$.
Where many students get confused is in the interpretation of $\int f(x) \;dx$ as a sum and difference of certain areas. If continuous $f < 0$ on $[a,b]$, then $\int_a^b f(x)\;dx$ has a negative sign, and it represents the negative of the area of the region below the $x$-axis but above the curve $y = f(x)$. Notice the terminology: "Negative of the area of that region". The implication is that the (geometric) area of the region is positive, and taking the negative of it gives you the value of $\int_a^b f(x)\;dx$.
Hope this helps!
-
i dont get how area can be negative, isnt all area suppose to be >=0? also, Area R1 and R3 arent half of the [a,b] interval so how can they say that R1+R3=R2? – Raynos Dec 6 '11 at 23:58
Yes, all area is supposed to be $\geq 0$, as I mentioned above. What do you mean by "Area R1 and R3 aren't half of the $[a, b]$ interval"? I'm only talking about area as in the amount of paint needed to paint those regions. Try this experiment: draw tiny squares in those regions $R_1, R_2$ and $R_3$, and count them. See how close the total number of squares for $R_1$ and $R_3$ match the total for just $R_2$ (you'll have to use tiny squares to get a reasonable approximation of the areas, though). – Shaun Ault Dec 8 '11 at 0:48
Also, the notation $\int_a^b f(x)\;dx$ must not be thought of as the same as a geometric area. $\int_a^b f(x) \;dx$ is often not an area calculation (for example, when finding work in physics: $W = \int_a^b F(x)\;dx$, where $F$ is the force acting at distance $x$). – Shaun Ault Dec 8 '11 at 0:51 | {"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": 0.9283372163772583, "perplexity": 299.7435096446075}, "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-2014-52/segments/1418815948154.77/warc/CC-MAIN-20141217113228-00111-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/105629/continuity-with-values-in-l2/105640 | # Continuity with values in L^2
Hi,
let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose $$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in L^2(0,T;W^{-1,2}(\Omega)),$$ where $W^{-1,2}(\Omega)=(W^{1,2}_0(\Omega))^*$. Is there a result stating that from those regularities one can deduce $u\in C([0,T];L^2(\Omega))$?
Thanks a lot, Richard
-
In the case $\Omega=\mathbb{R}^n$ we have $$L^2([0,T];W^{1,2}(\Omega))\cap W^{1,2}([0,T];W^{-1,2}(\Omega))\hookrightarrow BUC([0,T];X),$$ where $X$ is given via real interpolation: $$X=\big(W^{1,2}(\Omega),W^{-1,2}(\Omega)\big)_{1/2, 2}=B^0_{2,2}(\Omega).$$ This is basically contained in Linear and quasilinear parabolic problems I by H. Amann (Theorem III.4.10.2). Since $B^0_{2,2}(\Omega)=L^2(\Omega)$ the desired result follows. Now the case of smooth bounded $\Omega$ should follow via extension and restriction.
Part of your hypotheses are that $u \in L^2(I, W^{1,2}) \cap H^1(I, W^{-1,2})$, where $I = (0,T)$, $H^1 = W^{1,2}$ in $t$ (first Sobolev space), so interpolation gives that $u \in H^{1/2}(I, L^2)$ which just misses continuity. Adding in that $u \in L^\infty$ and using interpolation again probably does the trick, but you should check that.
Set $V:=L^2$ and $H:=W^{-1,2}$ and define the maximal regularity space $$MR_2(0,T;H,D):=W^{1,2}(0,T;H)\cap L^2(0,T;D),$$ where $D$ is the domain of the unbounded operator on $H$ associated with the quadratic form $Q(u):=\|u\|_V^2$. Then it is well-known (reference in some book by J. Lions, but now I cannot find an exact one) that $MR_2(0,T;H,D)$ is continuously embedded in $C([0,T];V)$. This is almost what you want. Not quite (I have not checked, but in this case it looks like $D=W^{1,2}_0$, unlike in your assumptions; and, worse, you are not assuming $u$ to be in $W^{1,2}(0,T;H)$), but also not far away, if you are able to use your assumption that $u\in L^\infty$ to define an equivalent norm on the set of those $$\{u\in L^2(0,T;W^{1,2}) \hbox{ s.t. } u_t\in L^2(0,T;W^{−1,2})\}.$$ | {"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9820713400840759, "perplexity": 185.82020292716163}, "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-2015-40/segments/1443738008122.86/warc/CC-MAIN-20151001222008-00025-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://k12.libretexts.org/Bookshelves/Science_and_Technology/Book%3A_Physics_-_From_Stargazers_to_Starships_(CK-12)/23%3A_The_Planets/23.04%3A_Early_History_False_Leads | Skip to main content
# 23.4: Early History, False Leads
As noted earlier (See the chapter “Does the Earth Revolve Around the Sun", Aristarchus of Samos proposed that the Earth revolved around the Sun, but the idea was rejected by later Greek astronomers, in particular Hipparchus. Ptolemy, living in Egypt in the 2nd century AD, expressed the consensus when he argued that all fixed stars were on some distant sphere which rotated around the Earth. Ptolemy tried to assemble and write down all that was known in his day about the heavens in “The Great Treatise," now known as the “Almagest," a corruption of its Arab name. (An annotated translation by G.J. Toomre was published in 1984 by Princeton University Press and is now available in paperback for \$39.50. See p. 120, Nature vol. 397, 14 January 1999.)
To explain the motion of planets, Ptolemy used a theory which started with Hipparchus. Following the work of Aristarchus (See the chapter “Estimating the Distance to the Moon") and Hipparchus (See the chapter “Distance to the Moon, Part 2"), it was already accepted that the Moon moved around Earth. Ptolemy assumed that the Sun, planets and the distant stars (whatever those were) also moved around the Earth. To the Greeks, the circle represented perfection, and Ptolemy assumed Moon, Sun, and stars moved in circles too. Since the motion was not exactly uniform (later explained by Kepler's laws --- http://www.phy6.org/stargaze/Skeplaws.htm), he assumed that these circles were centered some distance away from the Earth.
While the Sun moved around Earth, Venus and Mercury obviously moved around it, on circles of their own, centered near the Sun. But what about Mars, Jupiter and Saturn? Cleverly, Ptolemy proposed that like Venus and Mercury, each of them also rotated around a point in the sky that orbited around Earth like the Sun, except that those points were empty. The backtracking of the planets now looked similar to the backtracking of Venus and Mercury. The center carrying each of those planets accounted for the planet's regular motion, but to this the planet's own motion around that center had to be added, and sometimes the sum of the two made the planet appear (for a while) to advance backwards.
This “explanation" left open the question what the planets, Sun, and Moon were, and why they displayed such strange motions. But worse, it was also inaccurate. As the positions of the planets were measured more and more accurately, additional corrections had to be introduced.
Yet Ptolemy's view of the solar system dominated European astronomy for over 1000 years. One reason was that astronomy almost stopped in its tracks during the decline and fall of the Roman Empire and during the “dark ages" that followed. The study of the heavens continued in the Arab world, under Arab rulers, but of all the achievements of Arab astronomers, the one which exerted the greatest influence was the preservation and translation of Ptolemy's books and thus of his erroneous views.
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https://physics.stackexchange.com/questions/454109/if-we-know-the-distance-of-a-distant-light-emitting-object-how-could-we-know-th | # If we know the distance of a distant light emitting object, how could we know the age without knowing how long the light has been here?
I have been contemplating the age of our observable universe and I was unsure as to how we could tell the age by only divdeing the distance by the constent with out knowing how long the light has been here. Wouldn't that only tell us the minimum of elapsed time.
• without duration only the minimum of time can be calculated we do not know how long the light has been here> billy bob – Apprentice DR NormanERustJR. Jan 29 at 15:13
The age of the universe is not measured to be the radius of the observable universe divided by the speed of light. In fact, the observable universe is much, much larger than this. The age of the universe is measured by using certain cosmological models (usually the $$\Lambda$$CDM model) and reconstructing the time of the Big Bang from observed values of the known densities of different types of matter. | {"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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8123472929000854, "perplexity": 262.43380386358604}, "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-2019-26/segments/1560627999817.30/warc/CC-MAIN-20190625092324-20190625114324-00032.warc.gz"} |
http://math.stackexchange.com/questions/359286/distribution-of-stochastic-integral-w-r-t-to-centered-poisson-process | # Distribution of stochastic integral w.r.t. to centered Poisson process
Let $N(t)$ be a Poisson process with intensity $\lambda$ and define the centered process as $N_0(t):= N(t)-\lambda t$. A stochastic integral can be properly defined w.r.t. to $N_0(t)$ (but not w.r.t. $N(t)$).
Let $f$ be a nice function, for concreteness let's say $f(s)=s^2$. I want to calculate the distribution (also covariances in the time domain) of the process
$$X(t) := \int_0^t s^2 \;dN_0(s)$$
I have tried the following argument:
Consider the partial sums
$$S_n = \sum_{i=0}^{n-1} t_i^2 \;(N_0(t_{i+1}) - N_0(t_i)),$$
where $0=t_0 < t_1 < \dots < t_n=t$ is some partition of the interval $[0,t]$.
Then $\mathbb E[S_n] = 0$ and $Var[S_n] = \sum_{i=0}^{n-1} t_i^4 \lambda (t_{i+1} - t_i)$, using the independence of increments of the process $N_0(t)$.
Now, in analogy to the argument for an integral w.r.t. to Brownian motion being again normal, I tried to calculate the characteristic function of the distribution of $S_n$ hoping that by taking its limit I would get something I can interpret. Using the fact that the sum of two independent Poisson distributions (fulfilled by independence of increments) is again Poisson, I am left with the problem that I don't know how to deal with the prefactor $t_i^2$, since the distribution of $t_i^2 (N_0(t_{i+1}) - N_0(t_i))$ is nothing really nice, as far as I can tell.
Questions:
Can one properly finish my argument or is there a nicer way to calculate the above stochastic integral within $L^2$-theory without referring to Ito calculus?
What can one say about $CoV(X(t),X(s))$?
-
The stochastic integrals which interest you can be properly defined with respect to $N=(N(t))_{t\geqslant0}$ as well, for example, for every deterministic function $u$, $$X(t)=\int_0^tu(s)\mathrm dN(s)=\sum_{k=1}^{N(t)}u(T_k),$$ where $(T_k)_{k\geqslant1}$ enumerates the jump times of the process $N$. Conditioning on $N(t)=n$, $\{T_k\mid1\leqslant k\leqslant n\}$ is distributed like $\{U_k\mid1\leqslant k\leqslant n\}$ where $(U_k)_k$ is an i.i.d. sample uniformly distributed on $(0,t)$. Thus, $$E[X(t)\mid N(t)]=N(t)\int_0^tu(s)\frac{\mathrm ds}t,$$ and $$E[X(t)]=E[N(t)]\int_0^tu(s)\frac{\mathrm ds}t=\lambda\int_0^tu(s)\mathrm ds.$$ More generally, for every $a$, $$E[\mathrm e^{aX(t)}\mid N(t)]=\left(\int_0^t\mathrm e^{au(s)}\frac{\mathrm ds}t\right)^{N(t)},$$ hence $$E[\mathrm e^{aX(t)}]=\exp\left(\lambda\int_0^t(\mathrm e^{au(s)}-1)\mathrm ds\right).$$ This, together with the fact that the increments of $X=(X(t))_t$ are independent, fully determines the joint distribution of the process $X$. Now, $$X_0(t)=\int_0^tu(s)\mathrm dN_0(s)$$ is simply $$X_0(t)=X(t)-\lambda\int_0^tu(s)\mathrm ds,$$ hence the joint distribution of the process $X_0=(X_0(t))_t$ is fully determined by the fact that its increments are independent and by the identity $$E[\mathrm e^{aX_0(t)}]=\exp\left(\lambda\int_0^t(\mathrm e^{au(s)}-1-au(s))\mathrm ds\right).$$ For example, for every $0\leqslant s\leqslant t$, $$\mathrm{cov}(X_0(s),X_0(t))=\mathrm{var}(X_0(s))=E[X_0(s)^2]=\lambda\int_0^su(x)^2\mathrm dx.$$ | {"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": 0.9970740675926208, "perplexity": 98.4448107243367}, "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/1414119649133.8/warc/CC-MAIN-20141024030049-00007-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://socratic.org/questions/what-is-the-difference-between-a-lewis-dot-symbol-and-a-lewis-structure | Chemistry
Topics
# What is the difference between a Lewis dot symbol and a Lewis structure?
Dec 2, 2015
A "Lewis dot symbol" is used for atoms to show their valence electrons. A "Lewis structure" shows the connectivity of atoms within a molecule.
For instance, let's construct what methane looks like. ${\text{CH}}_{4}$.
Carbon can contribute four total valence electrons to make bonds, and it looks like this:
and hydrogen, contributing one valence electron, looks like this:
Combining one carbon with four hydrogens by moving them close together on paper gives you the Lewis dot structure.
But you're not done there.
Two electrons are in one bond line. In ${\text{CH}}_{4}$, there are four bond lines and thus four single, $\sigma$ bonds. So as ${\text{CH}}_{4}$, they combine to give:
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https://automaticaddison.com/difference-between-maximum-likelihood-and-maximum-a-posteriori-estimation/ | # Difference Between Maximum Likelihood and Maximum a Posteriori Estimation
In this blog post, we will take a look at the difference between maximum likelihood estimation and maximum a posteriori estimation. Both are methods that attempt to estimate unknown values for parameters.
# What is Maximum Likelihood Estimation?
I will explain the term maximum likelihood estimation by using a real-world example.
Suppose you went out to the store and bought a six-sided die. You have no prior information about the type of die that you have. It could be a standard die which has a uniform prior probability distribution (i.e. each face has 1/6 chance of being face-up on any given roll), or you could have a weighted die where some numbers are more likely to appear than others. How would you calculate the probability of getting each number for a given roll of the die?
One way to calculate the probabilities is to roll the die 10,000 times and keep track of how many times (i.e. frequency in the table below) each face appeared. Your table might look something like this:
What you see above is the basis of maximum likelihood estimation. In maximum likelihood estimation, you estimate the parameters by maximizing the “likelihood function.” Stated more simply, you choose the value of the parameters that were most likely to have generated the data that was observed in the table above.
So, what is the problem with maximum likelihood estimation? Well, as you saw above, we did not incorporate any prior knowledge (i.e. prior belief information) into our calculation. What if we knew that this was a weighted die where the probabilities of each face were as follows:
Maximum likelihood estimation does not take this into account. It assumes a uniform prior probability distribution. That is, it assumes that the probabilities of each face of the die are equal before we begin rolling the die 500 times. The problem with this assumption is that you would need to have a huge dataset (i.e. have to roll the die many times, perhaps until your arm gets too tired to continue rolling!) to get values for the appropriate probabilities that are close to what they should be.
More formally, maximum likelihood estimation looks like this:
Parameter = argmax P(Observed Data | Parameter)
where P = probability and | = given.
So what do we do? Maximum a posteriori (MAP) estimation to the rescue!
# What is Maximum a Posteriori (MAP) Estimation?
Maximum a Posteriori (MAP) Estimation is similar to Maximum Likelihood Estimation (MLE) with a couple major differences. MAP takes prior probability information into account.
For example, if we knew that the die in our example above was a weighted die with the probabilities noted in the table in the previous section, MAP estimation factors this information into the parameter estimation.
More formally, MAP estimation looks like this:
Parameter = argmax P(Observed Data | Parameter)P(Parameter)
where P = probability and | = given.
Contrast that equation above to the MLE equation from the previous section. Note that the likelihood is now weighted by the prior probability. This distinction is important.
Imagine in MLE if we did 500 rolls, and one of the faces appeared only one time (i.e. 1/500). MLE would generate incorrect parameter values. Having that extra nonzero prior probability factor makes sure that the model does not overfit to the observed data in the way that MLE does.
# Why Is Maximum a Posteriori (MAP) Estimation the Ideal Estimation Method for Smaller Datasets?
It is ideal because it takes into account prior knowledge of an event. MLE does not and is prone to overfitting. For this reason, MAP is considered a regularization of MLE. Adding the prior probability information reduces the overdependence on the observed data for parameter estimation. | {"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": 0.899495005607605, "perplexity": 345.1766152265022}, "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-2020-05/segments/1579250601628.36/warc/CC-MAIN-20200121074002-20200121103002-00200.warc.gz"} |
https://nips.cc/Conferences/2018/ScheduleMultitrack?event=11971 | Timezone: »
Poster
Predictive Approximate Bayesian Computation via Saddle Points
Yingxiang Yang · Bo Dai · Negar Kiyavash · Niao He
Wed Dec 05 07:45 AM -- 09:45 AM (PST) @ Room 210 #6
Approximate Bayesian computation (ABC) is an important methodology for Bayesian inference when the likelihood function is intractable. Sampling-based ABC algorithms such as rejection- and K2-ABC are inefficient when the parameters have high dimensions, while the regression-based algorithms such as K- and DR-ABC are hard to scale. In this paper, we introduce an optimization-based ABC framework that addresses these deficiencies. Leveraging a generative model for posterior and joint distribution matching, we show that ABC can be framed as saddle point problems, whose objectives can be accessed directly with samples. We present the predictive ABC algorithm (P-ABC), and provide a probabilistically approximately correct (PAC) bound that guarantees its learning consistency. Numerical experiment shows that P-ABC outperforms both K2- and DR-ABC significantly. | {"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": 0.9438759088516235, "perplexity": 3458.59743071151}, "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-2022-49/segments/1669446710933.89/warc/CC-MAIN-20221203143925-20221203173925-00477.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=4228873&postcount=1 | View Single Post
P: 12 Hello I may well be all wrong about this but I am trying to understand from a microscopic point of view why Entropy is a concave function of internal energy. I found this in the following .pdf: http://physics.technion.ac.il/ckfind...potentials.pdf I started from this wikipedia article and i understand why, if the particles composing the system have a limited number of available energy levels, then S(E) first increases and then decreases. But saying that S(E) is concave should mean: - when the temperature is T1, if i give a dE to the system its entropy increases of dS1 - when the tempereture is T2>T1, if I give the same dE to the system, its Entropy increases only of dS2 < dS1 I cannot see this with single particles. If I have N particles in their lowest energy state there is only one microstate: all the particles are still. If I give to this system the tiniest possible amount of energy, it will be taken by just one of the particle, so the possible microstates are N. If I add another dE, the possible microstates should be N + N(N-1) = N^2 ... that is or one particle gets both dE or two different particles get it. Every time I add a dE I should increase the power of N. Now, if the entropy is somehow proportional to the logarithm of the number of microstates, I should get S proportional to K ln(N^E), that is, something that is proportianl to E... taht is, no concavity I am sure I am getting all this wrong... could you please help me understand this? Thank You Wentu | {"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": 0.8256660103797913, "perplexity": 355.99128667144316}, "config": {"markdown_headings": false, "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-2014-41/segments/1410657113000.87/warc/CC-MAIN-20140914011153-00151-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://www.physicsforums.com/threads/em-waves-penetration.425727/ | # EM waves penetration
1. Sep 1, 2010
### Kyoma
When an EM wave has a higher frequency, it is able to penetrate materials more easily.
However, why does a microwave able to penetrate haze while visible light can't? Why visible light is able to enter our atmosphere while those of gamma can't?
2. Sep 1, 2010
### Staff: Mentor
This is not generally true. E.g.
http://en.wikipedia.org/wiki/File:Water_absorption_spectrum.png [Broken]
Last edited by a moderator: May 4, 2017
3. Sep 2, 2010
### krononauten
First assuming TEM mode (no field component in the propagation direction).
In addition to the above poster, the attenuation of a material could be viewed upon as the sum of three things
1) Reflection, at the first boundary of the material
2) Multi-reflection, inside the material between the boundaries of the material
3) Absorption in the material itself
1) and 3) tend to increase the attenuation and 2) increase it.
1) is often just a simple function of the different wave impedances of the two regions (that is from where the wave came from and into the material we are talking about).
The absorption (3) is highly dependent on the material parameters in our "shield material".
E.g. to attenuate low frequency magnetic fields, you need a material with a high permeability value. Also all of this is of course frequency dependent, HOWEVER you cannot assume it to be linear :-)
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https://www.albert.io/learn/trigonometry/question/inverse-function-area-volume-prism | Limited access
We are given the figure above with the given dimensions and we are given the formula for the volume of a triangular prism as:
$$\text{Volume= Length (or height)} \times \text{Area of Base}$$
Find the measure of angle BAC to the nearest tenth of a degree.
A
${ 45.0 }^{ o }$
B
${ 86.6 }^{ o }$
C
${ 3.4 }^{ o }$
D
${ 82.4 }^{ o }$
Select an assignment template | {"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": 0.9154543280601501, "perplexity": 284.94930932307926}, "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-2017-17/segments/1492917118713.1/warc/CC-MAIN-20170423031158-00227-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.science.gov/topicpages/m/modified+homotopy+perturbation.html | #### Sample records for modified homotopy perturbation
1. Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method
PubMed Central
Căruntu, Bogdan
2014-01-01
The paper presents the optimal homotopy perturbation method, which is a new method to find approximate analytical solutions for nonlinear partial differential equations. Based on the well-known homotopy perturbation method, the optimal homotopy perturbation method presents an accelerated convergence compared to the regular homotopy perturbation method. The applications presented emphasize the high accuracy of the method by means of a comparison with previous results. PMID:25003150
2. Application of the Homotopy Perturbation Method to the Nonlinear Pendulum
ERIC Educational Resources Information Center
Belendez, A.; Hernandez, A.; Belendez, T.; Marquez, A.
2007-01-01
The homotopy perturbation method is used to solve the nonlinear differential equation that governs the nonlinear oscillations of a simple pendulum, and an approximate expression for its period is obtained. Only one iteration leads to high accuracy of the solutions and the relative error for the approximate period is less than 2% for amplitudes as…
3. Series Expansion of Functions with He's Homotopy Perturbation Method
ERIC Educational Resources Information Center
Khattri, Sanjay Kumar
2012-01-01
Finding a series expansion, such as Taylor series, of functions is an important mathematical concept with many applications. Homotopy perturbation method (HPM) is a new, easy to use and effective tool for solving a variety of mathematical problems. In this study, we present how to apply HPM to obtain a series expansion of functions. Consequently,…
4. Analytical method for space-fractional telegraph equation by homotopy perturbation transform method
Prakash, Amit
2016-06-01
The object of the present article is to study spacefractional telegraph equation by fractional Homotopy perturbation transform method (FHPTM). The homotopy perturbation transform method is an innovative adjustment in Laplace transform algorithm. Three test examples are presented to show the efficiency of the proposed technique.
5. Laplace homotopy perturbation method for Burgers equation with space- and time-fractional order
Johnston, S. J.; Jafari, H.; Moshokoa, S. P.; Ariyan, V. M.; Baleanu, D.
2016-07-01
The fractional Burgers equation describes the physical processes of unidirectional propagation of weakly nonlinear acoustic waves through a gas-filled pipe. The Laplace homotopy perturbation method is discussed to obtain the approximate analytical solution of space-fractional and time-fractional Burgers equations. The method used combines the Laplace transform and the homotopy perturbation method. Numerical results show that the approach is easy to implement and accurate when applied to partial differential equations of fractional orders.
6. Numerical Solution of Problems in Calculus of Variations by Homotopy Perturbation Method
SciTech Connect
Jafari, M. A.; Aminataei, A.
2008-09-01
In this work we use Homotopy Perturbation Method (HPM) to solve differential equations that arise in variational problems. To illustrate the method some examples are provided. The results show the efficiency and accuracy of the HPM. HPM can be considered an alternative method to Adomian decomposition method. Both of these methods can obtain analytic form of the solution in some cases.
7. Homotopy Perturbation Transform Method with He's Polynomial for Solution of Coupled Nonlinear Partial Differential Equations
Sharma, Dinkar; Singh, Prince; Chauhan, Shubha
2016-01-01
In this paper, a combined form of the Laplace transform method with the homotopy perturbation method (HPTM) is applied to solve nonlinear systems of partial differential equations viz. the system of third order KdV Equations and the systems of coupled Burgers' equations in one- and two- dimensions. The nonlinear terms can be easily handled by the use of He's polynomials. The results shows that the HPTM is very efficient, simple and avoids the round-off errors. Four test examples are considered to illustrate the present scheme. Further the results are compared with Homotopy perturbation method (HPM) which shows that this method is a suitable method for solving systems of partial differential equations.
8. Homotopy Perturbation Method-Based Analytical Solution for Tide-Induced Groundwater Fluctuations.
PubMed
Munusamy, Selva Balaji; Dhar, Anirban
2016-05-01
The groundwater variations in unconfined aquifers are governed by the nonlinear Boussinesq's equation. Analytical solution for groundwater fluctuations in coastal aquifers under tidal forcing can be solved using perturbation methods. However, the perturbation parameters should be properly selected and predefined for traditional perturbation methods. In this study, a new dimensional, higher-order analytical solution for groundwater fluctuations is proposed by using the homotopy perturbation method with a virtual perturbation parameter. Parameter-expansion method is used to remove the secular terms generated during the solution process. The solution does not require any predefined perturbation parameter and valid for higher values of amplitude parameter A/D, where A is the amplitude of the tide and D is the aquifer thickness. PMID:26340338
9. Modified hyperspheres algorithm to trace homotopy curves of nonlinear circuits composed by piecewise linear modelled devices.
PubMed
Vazquez-Leal, H; Jimenez-Fernandez, V M; Benhammouda, B; Filobello-Nino, U; Sarmiento-Reyes, A; Ramirez-Pinero, A; Marin-Hernandez, A; Huerta-Chua, J
2014-01-01
We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation. PMID:25184157
10. Modified Hyperspheres Algorithm to Trace Homotopy Curves of Nonlinear Circuits Composed by Piecewise Linear Modelled Devices
PubMed Central
Vazquez-Leal, H.; Jimenez-Fernandez, V. M.; Benhammouda, B.; Filobello-Nino, U.; Sarmiento-Reyes, A.; Ramirez-Pinero, A.; Marin-Hernandez, A.; Huerta-Chua, J.
2014-01-01
We present a homotopy continuation method (HCM) for finding multiple operating points of nonlinear circuits composed of devices modelled by using piecewise linear (PWL) representations. We propose an adaptation of the modified spheres path tracking algorithm to trace the homotopy trajectories of PWL circuits. In order to assess the benefits of this proposal, four nonlinear circuits composed of piecewise linear modelled devices are analysed to determine their multiple operating points. The results show that HCM can find multiple solutions within a single homotopy trajectory. Furthermore, we take advantage of the fact that homotopy trajectories are PWL curves meant to replace the multidimensional interpolation and fine tuning stages of the path tracking algorithm with a simple and highly accurate procedure based on the parametric straight line equation. PMID:25184157
11. Solving the Helmholtz equation in conformal mapped ARROW structures using homotopy perturbation method.
PubMed
Reck, Kasper; Thomsen, Erik V; Hansen, Ole
2011-01-31
The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution. PMID:21368995
12. Laplace transform homotopy perturbation method for the approximation of variational problems.
PubMed
Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R
2016-01-01
This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems. PMID:27006884
13. Calculation of the neutron diffusion equation by using Homotopy Perturbation Method
Koklu, H.; Ersoy, A.; Gulecyuz, M. C.; Ozer, O.
2016-03-01
The distribution of the neutrons in a nuclear fuel element in the nuclear reactor core can be calculated by the neutron diffusion theory. It is the basic and the simplest approximation for the neutron flux function in the reactor core. In this study, the neutron flux function is obtained by the Homotopy Perturbation Method (HPM) that is a new and convenient method in recent years. One-group time-independent neutron diffusion equation is examined for the most solved geometrical reactor core of spherical, cubic and cylindrical shapes, in the frame of the HPM. It is observed that the HPM produces excellent results consistent with the existing literature.
14. Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate
Esmaeilpour, M.; Ganji, D. D.
2007-12-01
In this Letter, the problem of forced convection over a horizontal flat plate is presented and the homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.
15. A Homotopy Perturbation-Based Method for Large Deflection of a Cantilever Beam Under a Terminal Follower Force
Wang, Yong-Gang; Lin, Wen-Hui; Liu, Ning
2012-05-01
The large deflection problem of a uniform cantilever beam subjected to a terminal concentrated follower force is investigated. The governing equations, which characterize a two-point boundary value problem, are transformed into an initial-value problem. A new algorithm based on the homotopy perturbation method is proposed and applied to the resulting problem and the characteristics of load versus displacement are obtained analytically. The convergence of this method is discussed and the details of load-deflection curves are present. Compared with other existing methods, the present scheme is shown to be highly accurate, while only lower order perturbation is required.
16. A novel solution procedure for a three-level atom interacting with one-mode cavity field via modified homotopy analysis method
Abdel Wahab, N. H.; Salah, Ahmed
2015-05-01
In this paper, the interaction of a three-level -configration atom and a one-mode quantized electromagnetic cavity field has been studied. The detuning parameters, the Kerr nonlinearity and the arbitrary form of both the field and intensity-dependent atom-field coupling have been taken into account. The wave function when the atom and the field are initially prepared in the excited state and coherent state, respectively, by using the Schrödinger equation has been given. The analytical approximation solution of this model has been obtained by using the modified homotopy analysis method (MHAM). The homotopy analysis method is mentioned summarily. MHAM can be obtained from the homotopy analysis method (HAM) applied to Laplace, inverse Laplace transform and Pade approximate. MHAM is used to increase the accuracy and accelerate the convergence rate of truncated series solution obtained by the HAM. The time-dependent parameters of the anti-bunching of photons, the amplitude-squared squeezing and the coherent properties have been calculated. The influence of the detuning parameters, Kerr nonlinearity and photon number operator on the temporal behavior of these phenomena have been analyzed. We noticed that the considered system is sensitive to variations in the presence of these parameters.
17. Density perturbations in general modified gravitational theories
SciTech Connect
De Felice, Antonio; Tsujikawa, Shinji; Mukohyama, Shinji
2010-07-15
We derive the equations of linear cosmological perturbations for the general Lagrangian density f(R,{phi},X)/2+L{sub c}, where R is a Ricci scalar, {phi} is a scalar field, and X=-{partial_derivative}{sup {mu}{phi}{partial_derivative}}{sub {mu}{phi}/}2 is a field kinetic energy. We take into account a nonlinear self-interaction term L{sub c}={xi}({phi}) {open_square}{phi}({partial_derivative}{sup {mu}{phi}{partial_derivative}}{sub {mu}{phi}}) recently studied in the context of ''Galileon'' cosmology, which keeps the field equations at second order. Taking into account a scalar-field mass explicitly, the equations of matter density perturbations and gravitational potentials are obtained under a quasistatic approximation on subhorizon scales. We also derive conditions for the avoidance of ghosts and Laplacian instabilities associated with propagation speeds. Our analysis includes most of modified gravity models of dark energy proposed in literature; and thus it is convenient to test the viability of such models from both theoretical and observational points of view.
18. Homotopy optimization methods for global optimization.
SciTech Connect
Dunlavy, Daniel M.; O'Leary, Dianne P.
2005-12-01
We define a new method for global optimization, the Homotopy Optimization Method (HOM). This method differs from previous homotopy and continuation methods in that its aim is to find a minimizer for each of a set of values of the homotopy parameter, rather than to follow a path of minimizers. We define a second method, called HOPE, by allowing HOM to follow an ensemble of points obtained by perturbation of previous ones. We relate this new method to standard methods such as simulated annealing and show under what circumstances it is superior. We present results of extensive numerical experiments demonstrating performance of HOM and HOPE.
19. Computing model independent perturbations in dark energy and modified gravity
SciTech Connect
Battye, Richard A.; Pearson, Jonathan A. E-mail: [email protected]
2014-03-01
We present a methodology for computing model independent perturbations in dark energy and modified gravity. This is done from the Lagrangian for perturbations, by showing how field content, symmetries, and physical principles are often sufficient ingredients for closing the set of perturbed fluid equations. The fluid equations close once ''equations of state for perturbations'' are identified: these are linear combinations of fluid and metric perturbations which construct gauge invariant entropy and anisotropic stress perturbations for broad classes of theories. Our main results are the proof of the equation of state for perturbations presented in a previous paper, and the development of the required calculational tools.
20. Modified contour-improved perturbation theory
SciTech Connect
Cvetic, Gorazd; Loewe, Marcelo; Martinez, Cristian; Valenzuela, Cristian
2010-11-01
The semihadronic tau decay width allows a clean extraction of the strong coupling constant at low energies. We present a modification of the standard ''contour-improved'' method based on a derivative expansion of the Adler function. The new approach has some advantages compared to contour-improved perturbation theory. The renormalization scale dependence is weaker by more than a factor of 2 and the last term of the expansion is reduced by about 10%, while the renormalization scheme dependence remains approximately equal. The extracted QCD coupling at the tau mass scale is by 2% lower than the contour-improved value. We find {alpha}{sub s}(M{sub Z}{sup 2})=0.1211{+-}0.0010.
1. Perturbations of single-field inflation in modified gravity theory
Qiu, Taotao; Xia, Jun-Qing
2015-05-01
In this paper, we study the case of single field inflation within the framework of modified gravity theory where the gravity part has an arbitrary form f (R). Via a conformal transformation, this case can be transformed into its Einstein frame where it looks like a two-field inflation model. However, due to the existence of the isocurvature modes in such a multi-degree-of-freedom (m.d.o.f.) system, the (curvature) perturbations are not equivalent in two frames, so despite of its convenience, it is illegal to treat the perturbations in its Einstein frame as the "real" ones as we always do for pure f (R) theory or single field with nonminimal coupling. Here by pulling the results of curvature perturbations back into its original Jordan frame, we show explicitly the power spectrum and spectral index of the perturbations in the Jordan frame, as well as how it differs from the Einstein frame. We also fit our results with the newest Planck data. Since there is large parameter space in these models, we show that it is easy to fit the data very well.
2. Modified artificial bee colony optimization with block perturbation strategy
Jia, Dongli; Duan, Xintao; Khurram Khan, Muhammad
2015-05-01
As a newly emerged swarm intelligence-based optimizer, the artificial bee colony (ABC) algorithm has attracted the interest of researchers in recent years owing to its ease of use and efficiency. In this article, a modified ABC algorithm with block perturbation strategy (BABC) is proposed. Unlike basic ABC, in the BABC algorithm, not one element but a block of elements from the parent solutions is changed while producing a new solution. The performance of the BABC algorithm is investigated and compared with that of the basic ABC, modified ABC, Brest's differential evolution, self-adaptive differential evolution and restart covariance matrix adaptation evolution strategy (IPOP-CMA-ES) over a set of widely used benchmark functions. The obtained results show that the performance of BABC is better than, or at least comparable to, that of the basic ABC, improved differential evolution variants and IPOP-CMA-ES in terms of convergence speed and final solution accuracy.
3. Vacuum structure for scalar cosmological perturbations in modified gravity models
SciTech Connect
Felice, Antonio De; Suyama, Teruaki E-mail: [email protected]
2009-06-01
We have found for the general class of Modified Gravity Models f(R, G) a new instability which can arise in vacuum for the scalar modes of the cosmological perturbations if the background is not de Sitter. In particular, the short-wavelength modes, if stable, in general have a group velocity which depends linearly in k, the wave number. Therefore these modes will be in general superluminal. We have also discussed the condition for which in general these scalar modes will be ghost-like. There is a subclass of these models, defined out of properties of the function f(R, G) and to which the f(R) and f(G) models belong, which however do not have this feature.
4. Noncommutative geometry modified non-Gaussianities of cosmological perturbation
SciTech Connect
Fang Kejie; Xue Wei; Chen Bin
2008-03-15
We investigate the noncommutative effect on the non-Gaussianities of primordial cosmological perturbation. In the lowest order of string length and slow-roll parameter, we find that in the models with small speed of sound the noncommutative modifications could be observable if assuming a relatively low string scale. In particular, the dominant modification of the non-Gaussianity estimator f{sub NL} could reach O(1) in Dirac-Born-Infeld (DBI) inflation and K-inflation. The corrections are sensitive to the speed of sound and the choice of string length scale. Moreover the shapes of the corrected non-Gaussianities are distinct from that of ordinary ones.
5. Practical approach to cosmological perturbations in modified gravity
Silvestri, Alessandra; Pogosian, Levon; Buniy, Roman V.
2013-05-01
The next generation of large scale surveys will not only measure cosmological parameters within the framework of general relativity, but will also allow for precision tests of the framework itself. At the order of linear perturbations, departures from the growth in the standard cosmological model can be quantified in terms of two functions of time and Fourier number k. We argue that in local theories of gravity, in the quasistatic approximation, these functions must be ratios of polynomials in k, with the numerator of one function being equal to the denominator of the other. Moreover, the polynomials are even and of second degree in practically all viable models considered today. This means that, without significant loss of generality, one can use data to constrain only five functions of a single variable, instead of two functions of two variables. Furthermore, since the five functions are expected to be slowly varying, one can fit them to data in a nonparametric way with the aid of an explicit smoothness prior. We discuss practical application of this parametrization to forecasts and fits.
6. Open-closed homotopy algebra in mathematical physics
SciTech Connect
Kajiura, Hiroshige; Stasheff, Jim
2006-02-15
In this paper we discuss various aspects of open-closed homotopy algebras (OCHAs) presented in our previous paper, inspired by Zwiebach's open-closed string field theory, but that first paper concentrated on the mathematical aspects. Here we show how an OCHA is obtained by extracting the tree part of Zwiebach's quantum open-closed string field theory. We clarify the explicit relation of an OCHA with Kontsevich's deformation quantization and with the B-models of homological mirror symmetry. An explicit form of the minimal model for an OCHA is given as well as its relation to the perturbative expansion of open-closed string field theory. We show that our open-closed homotopy algebra gives us a general scheme for deformation of open string structures (A{sub {infinity}} algebras) by closed strings (L{sub {infinity}} algebras)
7. Homotopy theory in toric topology
Grbić, J.; Theriault, S.
2016-04-01
In toric topology one associates with each simplicial complex K on m vertices two key spaces, the Davis-Januszkiewicz space DJK and the moment-angle complex \\mathscr{Z}K, which are related by a homotopy fibration \\mathscr{Z}K\\xrightarrow{\\tilde{w}}DJ_K\\to \\prodi=1m{C}P∞. A great deal of work has been done to study the properties of DJK and \\mathscr{Z}K, their generalizations to polyhedral products, and applications to algebra, combinatorics, and geometry. Chap. 1 surveys some of the main results in the homotopy theory of these spaces. Chap. 2 breaks new ground by initiating a study of the map \\tilde{w}. It is shown that, for a certain family of simplicial complexes K, the map \\tilde{w} is a sum of higher and iterated Whitehead products. Bibliography: 49 titles.
8. Perturbations of Schwarzschild black holes in Chern-Simons modified gravity
Yunes, Nicolás; Sopuerta, Carlos F.
2008-03-01
We study perturbations of a Schwarzschild black hole in Chern-Simons modified gravity. We begin by showing that Birkhoff’s theorem holds for a wide family of Chern-Simons coupling functions, a scalar field present in the theory that controls the strength of the Chern-Simons correction to the Einstein-Hilbert action. After decomposing the perturbations in spherical harmonics, we study the linearized modified field equations and find that axial and polar modes are coupled, in contrast to general relativity. The divergence of the modified equations leads to the Pontryagin constraint, which forces the vanishing of the Cunningham-Price-Moncrief master function associated with axial modes. We analyze the structure of these equations and find that the appearance of the Pontryagin constraint yields an overconstrained system that does not allow for generic black hole oscillations. We illustrate this situation by studying the case characterized by a canonical choice of the coupling function and pure-parity perturbative modes. We end with a discussion of how to extend Chern-Simons modified gravity to bypass the Pontryagin constraint and the suppression of perturbations.
9. A cohomological framework for homotopy moment maps
Frégier, Yaël; Laurent-Gengoux, Camille; Zambon, Marco
2015-11-01
Given a Lie group acting on a manifold M preserving a closed n + 1-form ω, the notion of homotopy moment map for this action was introduced in Fregier (0000), in terms of L∞-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of the results of Fregier (0000).
10. Homotopy Classification of Bosonic String Field Theory
Münster, Korbinian; Sachs, Ivo
2014-09-01
We prove the decomposition theorem for the loop homotopy Lie algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the open-closed homotopy algebra, we show that string field theory is background independent and locally unique in a very precise sense. Finally, we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories.
11. Homotopy invariance of η-invariants
PubMed Central
Weinberger, Shmuel
1988-01-01
Intersection homology and results related to the higher signature problem are applied to show that certain combinations of η-invariants of the signature operator are homotopy invariant in various circumstances. PMID:16593961
12. Matter density perturbations in modified gravity models with arbitrary coupling between matter and geometry
SciTech Connect
Nesseris, Savvas
2009-02-15
We consider theories with an arbitrary coupling between matter and gravity and obtain the perturbation equation of matter on subhorizon scales. Also, we derive the effective gravitational constant G{sub eff} and two parameters {sigma} and {eta}, which along with the perturbation equation of the matter density are useful to constrain the theory from growth factor and weak lensing observations. Finally, we use a completely solvable toy model which exhibits nontrivial phenomenology to investigate specific features of the theory. We obtain the analytic solution of the modified Friedmann equation for the scale factor a in terms of time t and use the age of the oldest star clusters and the primordial nucleosynthesis bounds in order to constrain the parameters of our toy model.
13. On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach
Van Gorder, Robert A.; Vajravelu, K.
2009-12-01
The Homotopy Analysis Method of Liao [Liao SJ. Beyond perturbation: introduction to the Homotopy Analysis Method. Boca Raton: Chapman & Hall/CRC Press; 2003] has proven useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the Homotopy Analysis Method, in a fairly general setting. Further, we discuss various convergence requirements on solutions.
14. An Efficient Algorithm for Perturbed Orbit Integration Combining Analytical Continuation and Modified Chebyshev Picard Iteration
Elgohary, T.; Kim, D.; Turner, J.; Junkins, J.
2014-09-01
Several methods exist for integrating the motion in high order gravity fields. Some recent methods use an approximate starting orbit, and an efficient method is needed for generating warm starts that account for specific low order gravity approximations. By introducing two scalar Lagrange-like invariants and employing Leibniz product rule, the perturbed motion is integrated by a novel recursive formulation. The Lagrange-like invariants allow exact arbitrary order time derivatives. Restricting attention to the perturbations due to the zonal harmonics J2 through J6, we illustrate an idea. The recursively generated vector-valued time derivatives for the trajectory are used to develop a continuation series-based solution for propagating position and velocity. Numerical comparisons indicate performance improvements of ~ 70X over existing explicit Runge-Kutta methods while maintaining mm accuracy for the orbit predictions. The Modified Chebyshev Picard Iteration (MCPI) is an iterative path approximation method to solve nonlinear ordinary differential equations. The MCPI utilizes Picard iteration with orthogonal Chebyshev polynomial basis functions to recursively update the states. The key advantages of the MCPI are as follows: 1) Large segments of a trajectory can be approximated by evaluating the forcing function at multiple nodes along the current approximation during each iteration. 2) It can readily handle general gravity perturbations as well as non-conservative forces. 3) Parallel applications are possible. The Picard sequence converges to the solution over large time intervals when the forces are continuous and differentiable. According to the accuracy of the starting solutions, however, the MCPI may require significant number of iterations and function evaluations compared to other integrators. In this work, we provide an efficient methodology to establish good starting solutions from the continuation series method; this warm start improves the performance of the
15. Nonlinear filters with log-homotopy
Daum, Fred; Huang, Jim
2007-09-01
We derive and test a new nonlinear filter that implements Bayes' rule using an ODE rather than with a pointwise multiplication of two functions. This avoids one of the fundamental and well known problems in particle filters, namely "particle collapse" as a result of Bayes' rule. We use a log-homotopy to construct this ODE. Our new algorithm is vastly superior to the classic particle filter, and we do not use any proposal density supplied by an EKF or UKF or other outside source. This paper was written for normal engineers, who do not have homotopy for breakfast.
16. Experiments with conjugate gradient algorithms for homotopy curve tracking
NASA Technical Reports Server (NTRS)
Irani, Kashmira M.; Ribbens, Calvin J.; Watson, Layne T.; Kamat, Manohar P.; Walker, Homer F.
1991-01-01
There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here, variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems.
17. Direct perturbation analysis on the localized waves of the modified nonlinear Schrödinger equation under nonvanishing boundary condition
Li, Min; Xu, Tao; Wang, Lei
2016-04-01
In this paper, the modified nonlinear Schrödinger equation is investigated via the direct perturbation method, which can describe the femtosecond optical pulse propagation in a monomodal optical fiber. Considering the quintic nonlinear perturbation, we obtain the approximate solution with the first-order correction, which can be expressed by the solution and symmetry of the derivative nonlinear Schrödinger equation. Under the nonvanishing boundary conditions, the approximate dark and anti-dark soliton solutions are derived and the existence conditions are also given. The effects of the perturbation on the propagations and interactions of the solitons on the nonzero background are discussed by comparing the physical quantities of solitons with the unperturbed case. It is found that the quintic nonlinear perturbation can lead to the change of the velocity as well as the pulse compression, but has no influence on the dynamics of the elastic interactions. Finally, numerical simulations are performed to support the theoretical results.
18. Power System Transient Stability Analysis through a Homotopy Analysis Method
SciTech Connect
Wang, Shaobu; Du, Pengwei; Zhou, Ning
2014-04-01
As an important function of energy management systems (EMSs), online contingency analysis plays an important role in providing power system security warnings of instability. At present, N-1 contingency analysis still relies on time-consuming numerical integration. To save computational cost, the paper proposes a quasi-analytical method to evaluate transient stability through time domain periodic solutions’ frequency sensitivities against initial values. First, dynamic systems described in classical models are modified into damping free systems whose solutions are either periodic or expanded (non-convergent). Second, because the sensitivities experience sharp changes when periodic solutions vanish and turn into expanded solutions, transient stability is assessed using the sensitivity. Third, homotopy analysis is introduced to extract frequency information and evaluate the sensitivities only from initial values so that time consuming numerical integration is avoided. Finally, a simple case is presented to demonstrate application of the proposed method, and simulation results show that the proposed method is promising.
19. Modified fractional variational iteration method for solving the generalized time-space fractional Schrödinger equation.
PubMed
Hong, Baojian; Lu, Dianchen
2014-01-01
Based on He's variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations. PMID:25276865
20. Parameter estimation for metabolic networks with two stage Bregman regularization homotopy inversion algorithm.
PubMed
Wang, Hong; Wang, Xi-cheng
2014-02-21
Metabolism is a very important cellular process and its malfunction contributes to human disease. Therefore, building dynamic models for metabolic networks with experimental data in order to analyze biological process rationally has attracted a lot of attention. Owing to the technical limitations, some unknown parameters contained in models need to be estimated effectively by means of the computational method. Generally, problems of parameter estimation of nonlinear biological network are known to be ill condition and multimodal. In particular, with the increasing amount and enlarging the scope of parameters, many optimization algorithms often fail to find a global solution. In this paper, two-stage variable factor Bregman regularization homotopy method is proposed. Discrete homotopy is used to identify the possible extreme region and continuous homotopy is executed for the purpose of stability of path tracing in the special region. Meanwhile, Latin hypercube sampling is introduced to get the good initial guess value and a perturbation strategy is developed to jump out of the local optimum. Three metabolic network inverse problems are investigated to demonstrate the effectiveness of the proposed method. PMID:24060619
1. Co-contraction modifies the stretch reflex elicited in muscles shortened by a joint perturbation
PubMed Central
Lewis, Gwyn N.; MacKinnon, Colum D.; Trumbower, Randy; Perreault, Eric J.
2011-01-01
Simultaneous contraction of agonist and antagonist muscles acting about a joint influences joint stiffness and stability. Although several studies have shown that reflexes in the muscle lengthened by a joint perturbation are modulated during co-contraction, little attention has been given to reflex regulation in the antagonist (shortened) muscle. The goal of the present study was to determine whether co-contraction gives rise to altered reflex regulation across the joint by examining reflexes in the muscle shortened by a joint perturbation. Reflexes were recorded from electromyographic activity in elbow flexors and extensors while positional perturbations to the elbow joint were applied. Perturbations were delivered during isolated activation of the flexor or extensor muscles as well as during flexor and extensor co-contraction. Across the group, the shortening reflex in the elbow extensor switched from suppression during isolated extensor muscle activation to facilitation during co-contraction. The shortening reflex in the elbow flexor remained suppressive during co-contraction but was significantly smaller compared to the response obtained during isolated elbow flexor activation. This response in the shortened muscle was graded by the level of activation in the lengthened muscle. The lengthening reflex did not change during co-contraction. These results support the idea that reflexes are regulated across multiple muscles around a joint. We speculate that the facilitatory response in the shortened muscle arises through a fast-conducting oligosynaptic pathway involving Ib interneurons. PMID:20878148
2. From Atiyah Classes to Homotopy Leibniz Algebras
Chen, Zhuo; Stiénon, Mathieu; Xu, Ping
2016-01-01
A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold X makes T X [-1] into a Lie algebra object in D + ( X), the bounded below derived category of coherent sheaves on X. Furthermore, Kapranov proved that, for a Kähler manifold X, the Dolbeault resolution {Ω^{bullet-1}(T_X^{1, 0})} of T X [-1] is an L ∞ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair ( L, A), i.e. a Lie algebroid L together with a Lie subalgebroid A, we define the Atiyah class α E of an A-module E as the obstruction to the existence of an A- compatible L-connection on E. We prove that the Atiyah classes α L/ A and α E respectively make L/ A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the bounded below derived category {D^+(A)} , where {A} is the abelian category of left {U(A)} -modules and {U(A)} is the universal enveloping algebra of A. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/ A and E, and inducing the aforesaid Lie structures in {D^+(A)}.
3. New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity via modified trigonometric function series method
Zhang, Zai-yun; Li, Yun-xiang; Liu, Zhen-hai; Miao, Xiu-jin
2011-08-01
In this paper, the modified trigonometric function series method is employed to solve the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. Exact traveling wave solutions are obtained.
4. Experimental Perturbations Modify the Performance of Early Warning Indicators of Regime Shift.
PubMed
Benedetti-Cecchi, Lisandro; Tamburello, Laura; Maggi, Elena; Bulleri, Fabio
2015-07-20
Ecosystems may shift abruptly between alternative states in response to environmental perturbations. Early warning indicators have been proposed to anticipate such regime shifts, but experimental field tests of their validity are rare. We exposed rocky intertidal algal canopies to a gradient of press perturbations and recorded the response of associated assemblages over 7 years. Reduced cover and biomass of algal canopies promoted the invasion of algal turfs, driving understory assemblages toward collapse upon total canopy removal. A dynamic model indicated the existence of a critical threshold separating the canopy- and turf-dominated states. We evaluated common indicators of regime shift as the system approached the threshold, including autocorrelation, SD, and skewness. These indicators captured changes in understory cover due to colonization of algal turfs. All indicators increased significantly as the system approached the critical threshold, in agreement with theoretical predictions. The performance of indicators changed when we superimposed a pulse disturbance on the press perturbation that amplified environmental noise. This treatment caused several experimental units to switch repeatedly between the canopy- and the turf-dominated state, resulting in a significant increase in overall variance of understory cover, a negligible effect on skewness and no effect on autocorrelation. Power analysis indicated that autocorrelation and SD were better suited at anticipating a regime shift under mild and strong fluctuations of the state variable, respectively. Our results suggest that regime shifts may be anticipated under a broad range of fluctuating conditions using the appropriate indicator. PMID:26166776
5. Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory
Hoefel, Eduardo; Livernet, Muriel
2012-08-01
Open-closed homotopy algebras (OCHA) and strong homotopy Leibniz pairs (SHLP) were introduced by Kajiura and Stasheff in 2004. In an appendix to their paper, Markl observed that an SHLP is equivalent to an algebra over the minimal model of a certain operad, without showing that the operad is Koszul. In the present paper, we show that both OCHA and SHLP are algebras over the minimal model of the zeroth homology of two versions of the Swiss-cheese operad and prove that these two operads are Koszul. As an application, we show that the OCHA operad is non-formal as a 2-colored operad but is formal as an algebra in the category of 2-collections.
6. Efficient Homotopy Continuation Algorithms with Application to Computational Fluid Dynamics
Brown, David A.
New homotopy continuation algorithms are developed and applied to a parallel implicit finite-difference Newton-Krylov-Schur external aerodynamic flow solver for the compressible Euler, Navier-Stokes, and Reynolds-averaged Navier-Stokes equations with the Spalart-Allmaras one-equation turbulence model. Many new analysis tools, calculations, and numerical algorithms are presented for the study and design of efficient and robust homotopy continuation algorithms applicable to solving very large and sparse nonlinear systems of equations. Several specific homotopies are presented and studied and a methodology is presented for assessing the suitability of specific homotopies for homotopy continuation. . A new class of homotopy continuation algorithms, referred to as monolithic homotopy continuation algorithms, is developed. These algorithms differ from classical predictor-corrector algorithms by combining the predictor and corrector stages into a single update, significantly reducing the amount of computation and avoiding wasted computational effort resulting from over-solving in the corrector phase. The new algorithms are also simpler from a user perspective, with fewer input parameters, which also improves the user's ability to choose effective parameters on the first flow solve attempt. Conditional convergence is proved analytically and studied numerically for the new algorithms. The performance of a fully-implicit monolithic homotopy continuation algorithm is evaluated for several inviscid, laminar, and turbulent flows over NACA 0012 airfoils and ONERA M6 wings. The monolithic algorithm is demonstrated to be more efficient than the predictor-corrector algorithm for all applications investigated. It is also demonstrated to be more efficient than the widely-used pseudo-transient continuation algorithm for all inviscid and laminar cases investigated, and good performance scaling with grid refinement is demonstrated for the inviscid cases. Performance is also demonstrated
7. Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint
SciTech Connect
Hermant, Audrey
2010-02-15
This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points.
8. Benchmark of a modified iterated perturbation theory approach on the fcc lattice at strong coupling
Arsenault, Louis-François; Sémon, Patrick; Tremblay, A.-M. S.
2012-08-01
The dynamical mean-field theory approach to the Hubbard model requires a method to solve the problem of a quantum impurity in a bath of noninteracting electrons. Iterated perturbation theory (IPT) has proven its effectiveness as a solver in many cases of interest. Based on general principles and on comparisons with an essentially exact continuous-time quantum Monte Carlo (CTQMC) solver, here we show that the standard implementation of IPT fails away from half-filling when the interaction strength is much larger than the bandwidth. We propose a slight modification to the IPT algorithm that replaces one of the equations by the requirement that double occupancy calculated with IPT gives the correct value. We call this method IPT-D. We recover the Fermi liquid ground state away from half-filling. The Fermi liquid parameters, density of states, chemical potential, energy, and specific heat on the fcc lattice are calculated with both IPT-D and CTQMC as benchmark examples. We also calculated the resistivity and the optical conductivity within IPT-D. Particle-hole asymmetry persists even at coupling twice the bandwidth. A generalization to the multiorbital case is suggested. Several algorithms that speed up the calculations are described in appendixes.
9. A Choice Reaction Time Index of Callosal Anatomical Homotopy
ERIC Educational Resources Information Center
Desjardins, Sameul; Braun, Claude M. J.; Achim, Andre; Roberge, Carl
2009-01-01
Tachistoscopically presented bilateral stimulus pairs not parallel to the meridian produced significantly longer RTs on a task requiring discrimination of shapes (Go/no-Go) than pairs emplaced symmetrically on each side of the meridian in Desjardins and Braun [Desjardins, S., & Braun, C. M. J. (2006). Homotopy and heterotopy and the bilateral…
10. Splitting a simple homotopy equivalence along a submanifold with filtration
SciTech Connect
Bak, A; Muranov, Yu V
2008-06-30
A simple homotopy equivalence f:M{sup n}{yields}X{sup n} of manifolds splits along a submanifold Y subset of X if it is homotopic to a map that is a simple homotopy equivalence on the transversal preimage of the submanifold and on the complement of this preimage. The problem of splitting along a submanifold with filtration is a natural generalization of this problem. In this paper we define groups LSF{sub *} of obstructions to splitting along a submanifold with filtration and describe their properties. We apply the results obtained to the problem of the realization of surgery and splitting obstructions by maps of closed manifolds and consider several examples. Bibliography: 36 titles.
11. Homotopy theory of strong and weak topological insulators
Kennedy, Ricardo; Guggenheim, Charles
2015-06-01
We use homotopy theory to extend the notion of strong and weak topological insulators to the nonstable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly d -dimensional," i.e., not realizable by stacking lower-dimensional insulators, a more restrictive definition of "strong" is required outside the stable regime. However, this does not exclude weak topological insulators from being "truly d -dimensional," which we demonstrate by an example. Additionally, we prove some useful technical results, including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime, as well as the rigorous justification of the parameter space replacements Td→Sd and Tdk×Sdx→Sdk+dx used widely in the current literature.
12. Dynamic homotopy and landscape dynamical set topology in quantum control
SciTech Connect
Dominy, Jason; Rabitz, Herschel
2012-08-15
We examine the topology of the subset of controls taking a given initial state to a given final state in quantum control, where 'state' may mean a pure state Double-Vertical-Line {psi}>, an ensemble density matrix {rho}, or a unitary propagator U(0, T). The analysis consists in showing that the endpoint map acting on control space is a Hurewicz fibration for a large class of affine control systems with vector controls. Exploiting the resulting fibration sequence and the long exact sequence of basepoint-preserving homotopy classes of maps, we show that the indicated subset of controls is homotopy equivalent to the loopspace of the state manifold. This not only allows us to understand the connectedness of 'dynamical sets' realized as preimages of subsets of the state space through this endpoint map, but also provides a wealth of additional topological information about such subsets of control space.
13. Particle flow for nonlinear filters with log-homotopy
Daum, Fred; Huang, Jim
2008-04-01
We describe a new nonlinear filter that is vastly superior to the classic particle filter. In particular, the computational complexity of the new filter is many orders of magnitude less than the classic particle filter with optimal estimation accuracy for problems with dimension greater than 2 or 3. We consider nonlinear estimation problems with dimensions varying from 1 to 20 that are smooth and fully coupled (i.e. dense not sparse). The new filter implements Bayes' rule using particle flow rather than with a pointwise multiplication of two functions; this avoids one of the fundamental and well known problems in particle filters, namely "particle collapse" as a result of Bayes' rule. We use a log-homotopy to derive the ODE that describes particle flow. This paper was written for normal engineers, who do not have homotopy for breakfast.
14. On the singular perturbations for fractional differential equation.
PubMed
Atangana, Abdon
2014-01-01
The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method. PMID:24683357
15. Homotopy of rational maps and the quantization of Skyrmions
Krusch, Steffen
2003-04-01
The Skyrme model is a classical field theory which models the strong interaction between atomic nuclei. It has to be quantized in order to compare it to nuclear physics. When the Skyrme model is semi-classically quantized it is important to take the Finkelstein-Rubinstein constraints into account. The aim of this paper is to show how to calculate these FR constraints directly from the rational map ansatz using basic homotopy theory. We then apply this construction in order to quantize the Skyrme model in the simplest approximation, the zero mode quantization. This is carried out for up to 22 nucleons and the results are compared to experiment.
16. Topological phases: Isomorphism, homotopy and K-theory
Thiang, Guo Chuan
2015-06-01
Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modeled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and K-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of K-theory.
17. An Investigation of How a Meteor Light Curve is Modified by Meteor Shape and Atmospheric Density Perturbations
NASA Technical Reports Server (NTRS)
Stokan, E.; Campbell-Brown, M. D.
2011-01-01
This is a preliminary investigation of how perturbations to meteoroid shape or atmospheric density affect a meteor light curve. A simple equation of motion and ablation are simultaneously solved numerically to give emitted light intensity as a function of height. It is found that changing the meteoroid shape, by changing the relationship between the cross-section area and the mass, changes the curvature and symmetry of the light curve, while making a periodic oscillation in atmospheric density gives a small periodic oscillation in the light curve.
18. Modified Iterated perturbation theory in the strong coupling regime and its application to the 3d FCC lattice
Arsenault, Louis-François; Sémon, Patrick; Shastry, B. Sriram; Tremblay, A.-M. S.
2012-02-01
The Dynamical Mean-Field theory(DMFT) approach to the Hubbard model requires a method to solve the problem of a quantum impurity in a bath of non-interacting electrons. Iterated Perturbation Theory(IPT)[1] has proven its effectiveness as a solver in many cases of interest. Based on general principles and on comparisons with an essentially exact Continuous-Time Quantum Monte Carlo (CTQMC)[2], here we show that the standard implementation of IPT fails when the interaction is much larger than the bandwidth. We propose a slight modification to the IPT algorithm by requiring that double occupancy calculated with IPT gives the correct value. We call this method IPT-D. We show how this approximate impurity solver compares with respect to CTQMC. We consider a face centered cubic lattice(FCC) in 3d for different physical properties. We also use IPT-D to study the thermopower using two recently proposed approximations[3]S^* and SKelvin that do not require analytical continuation and show how thermopower is essentially the entropy per particle in the incoherent regime but not in the coherent one.[1]H.Kajueter et al. Phys. Rev. Lett. 77, 131(1996)[2]P. Werner, et al. Phys. Rev. Lett. 97, 076405(2006)[3]B.S. Sriram Shastry Rep. Prog. Phys. 72 016501(2009)
19. Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation.
PubMed
Wen, Xiao-Yong; Yang, Yunqing; Yan, Zhenya
2015-07-01
In this paper, a simple and constructive method is presented to find the generalized perturbation (n,M)-fold Darboux transformations (DTs) of the modified nonlinear Schrödinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1,N-1)-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc., and may be useful to study the physical mechanism of multirogue waves in optics. The dynamical behaviors of these multi-rogue-wave solutions are illustrated using numerical simulations. The same Darboux matrix can also be used to investigate the Gerjikov-Ivanov equation such that its multi-rogue-wave solutions and their wave structures are also found. The method can also be extended to find multi-rogue-wave solutions of other nonlinear integrable equations. PMID:26274257
20. The order of a homotopy invariant in the stable case
SciTech Connect
Podkorytov, Semen S
2011-08-31
Let X, Y be cell complexes, let U be an Abelian group, and let f:[X,Y]{yields}U be a homotopy invariant. By definition, the invariant f has order at most r if the characteristic function of the rth Cartesian power of the graph of a continuous map a:X{yields}Y determines the value f([a]) Z-linearly. It is proved that, in the stable case (that is, when dimX<2n-1, and Y is (n-1)-connected for some natural number n), for a finite cell complex X the order of the invariant f is equal to its degree with respect to the Curtis filtration of the group [X,Y]. Bibliography: 9 titles.
1. A monolithic homotopy continuation algorithm with application to computational fluid dynamics
Brown, David A.; Zingg, David W.
2016-09-01
A new class of homotopy continuation methods is developed suitable for globalizing quasi-Newton methods for large sparse nonlinear systems of equations. The new continuation methods, described as monolithic homotopy continuation, differ from the classical predictor-corrector algorithm in that the predictor and corrector phases are replaced with a single phase which includes both a predictor and corrector component. Conditional convergence and stability are proved analytically. Using a Laplacian-like operator to construct the homotopy, the new algorithm is shown to be more efficient than the predictor-corrector homotopy continuation algorithm as well as an implementation of the widely-used pseudo-transient continuation algorithm for some inviscid and turbulent, subsonic and transonic external aerodynamic flows over the ONERA M6 wing and the NACA 0012 airfoil using a parallel implicit Newton-Krylov finite-difference flow solver.
2. Note on unit tangent vector computation for homotopy curve tracking on a hypercube
NASA Technical Reports Server (NTRS)
Chakraborty, A.; Allison, D. C. S.; Ribbens, C. J.; Watson, L. T.
1991-01-01
Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map and subsequent tracking of some smooth curve in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vector at different points along the zero curve, which amounts to calculating the kernel of the n x (n + 1) Jacobian matrix. While computing the tangent vector is just one part of the curve tracking algorithm, it can require a significant percentage of the total tracking time. This note presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms for the tangent vector computation on a hypercube.
3. Single-shooting homotopy method for parameter identification in dynamical systems.
PubMed
Vyasarayani, C P; Uchida, Thomas; McPhee, John
2012-03-01
An algorithm for identifying parameters in dynamical systems is developed in this work using homotopy transformations and the single-shooting method. The equations governing the dynamics of the mathematical model are augmented with observer-like homotopy terms that smooth the objective function. As a result, premature convergence to a local minimum is avoided and the obtained parameter estimates are globally optimal. Numerical examples are presented to demonstrate the application of the proposed approach to chaotic systems. PMID:22587155
4. Stable homotopy classification of A n 4 -polyhedra with 2- torsion free homology
Pan, JianZhong; Zhu, ZhongJian
2016-06-01
In this paper, we study the stable homotopy types of $\\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.
5. On the complexity of a combined homotopy interior method for convex programming
Yu, Bo; Xu, Qing; Feng, Guochen
2007-03-01
In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush-Kuhn-Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761-768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan-China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9-16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193-211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K-K-T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solveE However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.
6. Electronic and optical properties of pure and modified diamondoids studied by many-body perturbation theory and time-dependent density functional theory.
PubMed
Demján, Tamás; Vörös, Márton; Palummo, Maurizia; Gali, Adam
2014-08-14
Diamondoids are small diamond nanoparticles (NPs) that are built up from diamond cages. Unlike usual semiconductor NPs, their atomic structure is exactly known, thus they are ideal test-beds for benchmarking quantum chemical calculations. Their usage in spintronics and bioimaging applications requires a detailed knowledge of their electronic structure and optical properties. In this paper, we apply density functional theory (DFT) based methods to understand the electronic and optical properties of a few selected pure and modified diamondoids for which accurate experimental data exist. In particular, we use many-body perturbation theory methods, in the G0W0 and G0W0+BSE approximations, and time-dependent DFT in the adiabatic local density approximation. We find large quasiparticle gap corrections that can exceed thrice the DFT gap. The electron-hole binding energy can be as large as 4 eV but it is considerably smaller than the GW corrections and thus G0W0+BSE optical gaps are about 50% larger than the Kohn-Sham (KS) DFT gaps. We find significant differences between KS time-dependent DFT and GW+BSE optical spectra on the selected diamondoids. The calculated G0W0 quasiparticle levels agree well with the corresponding experimental vertical ionization energies. We show that nuclei dynamics in the ionization process can be significant and its contribution may reach about 0.5 eV in the adiabatic ionization energies. PMID:25134572
7. Numerical Polynomial Homotopy Continuation Method and String Vacua
DOE PAGESBeta
Mehta, Dhagash
2011-01-01
Finding vmore » acua for the four-dimensional effective theories for supergravity which descend from flux compactifications and analyzing them according to their stability is one of the central problems in string phenomenology. Except for some simple toy models, it is, however, difficult to find all the vacua analytically. Recently developed algorithmic methods based on symbolic computer algebra can be of great help in the more realistic models. However, they suffer from serious algorithmic complexities and are limited to small system sizes. In this paper, we review a numerical method called the numerical polynomial homotopy continuation (NPHC) method, first used in the areas of lattice field theories, which by construction finds all of the vacua of a given potential that is known to have only isolated solutions. The NPHC method is known to suffer from no major algorithmic complexities and is embarrassingly parallelizable , and hence its applicability goes way beyond the existing symbolic methods. We first solve a simple toy model as a warm-up example to demonstrate the NPHC method at work. We then show that all the vacua of a more complicated model of a compactified M theory model, which has an S U ( 3 ) structure, can be obtained by using a desktop machine in just about an hour, a feat which was reported to be prohibitively difficult by the existing symbolic methods. Finally, we compare the various technicalities between the two methods.« less
8. A homotopy algorithm for digital optimal projection control GASD-HADOC
NASA Technical Reports Server (NTRS)
Collins, Emmanuel G., Jr.; Richter, Stephen; Davis, Lawrence D.
1993-01-01
The linear-quadratic-gaussian (LQG) compensator was developed to facilitate the design of control laws for multi-input, multi-output (MIMO) systems. The compensator is computed by solving two algebraic equations for which standard closed-loop solutions exist. Unfortunately, the minimal dimension of an LQG compensator is almost always equal to the dimension of the plant and can thus often violate practical implementation constraints on controller order. This deficiency is especially highlighted when considering control-design for high-order systems such as flexible space structures. This deficiency motivated the development of techniques that enable the design of optimal controllers whose dimension is less than that of the design plant. A homotopy approach based on the optimal projection equations that characterize the necessary conditions for optimal reduced-order control. Homotopy algorithms have global convergence properties and hence do not require that the initializing reduced-order controller be close to the optimal reduced-order controller to guarantee convergence. However, the homotopy algorithm previously developed for solving the optimal projection equations has sublinear convergence properties and the convergence slows at higher authority levels and may fail. A new homotopy algorithm for synthesizing optimal reduced-order controllers for discrete-time systems is described. Unlike the previous homotopy approach, the new algorithm is a gradient-based, parameter optimization formulation and was implemented in MATLAB. The results reported may offer the foundation for a reliable approach to optimal, reduced-order controller design.
9. Molecular Dynamics Investigations of the Local Structural Characteristics of DNA Oligonucleotides: Studies of Helical Axis Deformations, Conformational Sequence Dependence and Modified Nucleoside Perturbations.
Louise-May, Shirley
The present DNA studies investigate the local structure of DNA oligonucleotides in order to characterize helical axis deformations, sequence dependent fine structure and modified nucleoside perturbations of selected oligonucleotide sequences. The molecular dynamics method is used to generate an ensemble of energetically feasible DNA conformations which can then be analyzed for dynamical conformational properties, some of which can be compared to experimentally derived values. A theory and graphical presentation for the analysis of helical deformations of DNA based on the configurational statistics of polymers, called "Persistence Analysis", was designed. The results of the analysis on prototype forms, static crystal structures and two solvated MD simulations of the sequence d(CGCGAATTCGCG) indicate that all of the expected features of bending can be sensitively and systematically identified by this approach. Comparison of the relative performance of three molecular dynamics potential functions commonly used for dynamical modeling of biological macromolecules; CHARMm, AMBER and GROMOS was investigated via in vacuo MD simulations on the dodecamer sequence d(CGCGAATTCGCG)_2 with respect to the conformational properties of each dynamical model and their ability to support A and B families of DNA. Vacuum molecular dynamics simulations using the CHARMm force field carried out on simple homo- and heteropolymers of DNA led to the conclusion that sequence dependent fine structure appears to be well defined for adenine-thymine rich sequences both at the base pair and base step level whereas much of the the fine structure found in cytosine -guanine rich sequences appears to be context dependent. The local conformational properties of the homopolymer poly (dA) -poly (dT) revealed one dynamical model which was found in general agreement with fiber models currently available. Investigation of the relative structural static and dynamical effect of the misincorporation of
10. Electronic and optical properties of pure and modified diamondoids studied by many-body perturbation theory and time-dependent density functional theory
SciTech Connect
Demján, Tamás; Vörös, Márton; Palummo, Maurizia; Gali, Adam
2014-08-14
Diamondoids are small diamond nanoparticles (NPs) that are built up from diamond cages. Unlike usual semiconductor NPs, their atomic structure is exactly known, thus they are ideal test-beds for benchmarking quantum chemical calculations. Their usage in spintronics and bioimaging applications requires a detailed knowledge of their electronic structure and optical properties. In this paper, we apply density functional theory (DFT) based methods to understand the electronic and optical properties of a few selected pure and modified diamondoids for which accurate experimental data exist. In particular, we use many-body perturbation theory methods, in the G{sub 0}W{sub 0} and G{sub 0}W{sub 0}+BSE approximations, and time-dependent DFT in the adiabatic local density approximation. We find large quasiparticle gap corrections that can exceed thrice the DFT gap. The electron-hole binding energy can be as large as 4 eV but it is considerably smaller than the GW corrections and thus G{sub 0}W{sub 0}+BSE optical gaps are about 50% larger than the Kohn-Sham (KS) DFT gaps. We find significant differences between KS time-dependent DFT and GW+BSE optical spectra on the selected diamondoids. The calculated G{sub 0}W{sub 0} quasiparticle levels agree well with the corresponding experimental vertical ionization energies. We show that nuclei dynamics in the ionization process can be significant and its contribution may reach about 0.5 eV in the adiabatic ionization energies.
11. Global Study of the Simple Pendulum by the Homotopy Analysis Method
ERIC Educational Resources Information Center
Bel, A.; Reartes, W.; Torresi, A.
2012-01-01
Techniques are developed to find all periodic solutions in the simple pendulum by means of the homotopy analysis method (HAM). This involves the solution of the equations of motion in two different coordinate representations. Expressions are obtained for the cycles and periods of oscillations with a high degree of accuracy in the whole range of…
12. Control optimization of a lifting body entry problem by an improved and a modified method of perturbation function. Ph.D. Thesis - Houston Univ.
NASA Technical Reports Server (NTRS)
Garcia, F., Jr.
1974-01-01
A study of the solution problem of a complex entry optimization was studied. The problem was transformed into a two-point boundary value problem by using classical calculus of variation methods. Two perturbation methods were devised. These methods attempted to desensitize the contingency of the solution of this type of problem on the required initial co-state estimates. Also numerical results are presented for the optimal solution resulting from a number of different initial co-states estimates. The perturbation methods were compared. It is found that they are an improvement over existing methods.
13. Tracing structural optima as a function of available resources by a homotopy method
NASA Technical Reports Server (NTRS)
Haftka, Raphael T.; Watson, Layne T.; Plaut, Raymond H.; Shin, Yung S.
1988-01-01
Optimization problems are typically solved by starting with an initial estimate and proceeding iteratively to improve it until the optimum is found. The design points along the path from the initial estimate to the optimum are usually of no value. The present work proposes a strategy for tracing a path of optimum solutions parameterized by the amount of available resources. The paper specifically treats the optimum design of a structure to maximize its buckling load. Equations for the optimum path are obtained using Lagrange multipliers, and solved by a homotopy method. The solution path has several transitions from unimodal to bimodal solutions. The Lagrange multipliers and second-order optimality conditions are used to detect branching points and to switch to the optimum solution path. The procedure is applied to the design of a foundation which supports a column for maximum buckling load. Using the total available foundation stiffness as a homotopy parameter, a set of optimum foundation designs is obtained.
14. REVIEWS OF TOPICAL PROBLEMS: Defects in liquid crystals: homotopy theory and experimental studies
Kurik, Mikhail V.; Lavrentovich, O. D.
1988-03-01
The fundamental concepts of the homotopy theory of defects in liquid crystals and the results of experimental studies in this field are presented. The concepts of degeneracy space, homotopy groups, and topological charge, which are used for classifying the topologically stable inhomogeneous distributions in different liquid-crystalline phases are examined (uni and biaxial nematics, cholesterics, smectics, and columnar phases). Experimental data are given for the different mesophases on the structure and properties of dislocations, disclinations, point defects in the volume (hedgehogs) and on the surface of the medium (boojums), monopoles, domain formations, and solitons. Special attention is paid to the results of studies of defects in closed volumes (spherical drops, cylindrical capillaries), and to the connection between the topological charges of these defects and the character of the orientation of the molecules of the liquid crystal at the surface. A set of fundamentally new effects that can occur in studying the topological properties of defects in liquid crystals is discussed.
15. Solution of the Falkner-Skan wedge flow by a revised optimal homotopy asymptotic method.
PubMed
Madaki, A G; Abdulhameed, M; Ali, M; Roslan, R
2016-01-01
In this paper, a revised optimal homotopy asymptotic method (OHAM) is applied to derive an explicit analytical solution of the Falkner-Skan wedge flow problem. The comparisons between the present study with the numerical solutions using (fourth order Runge-Kutta) scheme and with analytical solution using HPM-Padé of order [4/4] and order [13/13] show that the revised form of OHAM is an extremely effective analytical technique. PMID:27186477
16. Multivalued behavior for a two-level system using Homotopy Analysis Method
Aquino, A. I.; Bo-ot, L. Ma. T.
2016-02-01
We use the Homotopy Analysis Method (HAM) to solve the system of equations modeling the two-level system and extract results which will pinpoint to turbulent behavior. We look at multivalued solutions as indicative of turbulence or turbulent-like behavior. We take different specific cases which result in multivalued velocities. The solutions are in series form and application of HAM ensures convergence in some region.
17. PERTURBING LIGNIFICATION
Technology Transfer Automated Retrieval System (TEKTRAN)
Perturbing lignification is possible in multiple and diverse ways. Without obvious growth/development phenotypes, transgenic angiosperms can have lignin levels reduced to half the normal level, can have compositions ranging from very high-guaiacyl/low-syringyl to almost totally syringyl, and can eve...
18. Cosmological Perturbations
Lesgourges, J.
2013-08-01
We present a self-contained summary of the theory of linear cosmological perturbations. We emphasize the effect of the six parameters of the minimal cosmological model, first, on the spectrum of Cosmic Microwave Background temperature anisotropies, and second, on the linear matter power spectrum. We briefly review at the end the possible impact of a few non-minimal dark matter and dark energy models.
19. Development of homotopy algorithms for fixed-order mixed H2/H(infinity) controller synthesis
NASA Technical Reports Server (NTRS)
Whorton, M.; Buschek, H.; Calise, A. J.
1994-01-01
A major difficulty associated with H-infinity and mu-synthesis methods is the order of the resulting compensator. Whereas model and/or controller reduction techniques are sometimes applied, performance and robustness properties are not preserved. By directly constraining compensator order during the optimization process, these properties are better preserved, albeit at the expense of computational complexity. This paper presents a novel homotopy algorithm to synthesize fixed-order mixed H2/H-infinity compensators. Numerical results are presented for a four-disk flexible structure to evaluate the efficiency of the algorithm.
20. Communication: Newton homotopies for sampling stationary points of potential energy landscapes
SciTech Connect
Mehta, Dhagash; Chen, Tianran; Hauenstein, Jonathan D.; Wales, David J.
2014-09-28
One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they all exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor ϕ{sup 4} model and atomic clusters.
1. Homotopy Algorithm for Fixed Order Mixed H2/H(infinity) Design
NASA Technical Reports Server (NTRS)
Whorton, Mark; Buschek, Harald; Calise, Anthony J.
1996-01-01
Recent developments in the field of robust multivariable control have merged the theories of H-infinity and H-2 control. This mixed H-2/H-infinity compensator formulation allows design for nominal performance by H-2 norm minimization while guaranteeing robust stability to unstructured uncertainties by constraining the H-infinity norm. A key difficulty associated with mixed H-2/H-infinity compensation is compensator synthesis. A homotopy algorithm is presented for synthesis of fixed order mixed H-2/H-infinity compensators. Numerical results are presented for a four disk flexible structure to evaluate the efficiency of the algorithm.
2. A homotopy analysis method for the option pricing PDE in illiquid markets
E-Khatib, Youssef
2012-09-01
One of the shortcomings of the Black and Scholes model on option pricing is the assumption that trading the underlying asset does not affect the underlying asset price. This can happen in perfectly liquid markets and it is evidently not viable in markets with imperfect liquidity (illiquid markets). It is well-known that markets with imperfect liquidity are more realistic. Thus, the presence of price impact while studying options is very important. This paper investigates a solution for the option pricing PDE in illiquid markets using the homotopy analysis method.
3. A homotopy algorithm for synthesizing robust controllers for flexible structures via the maximum entropy design equations
NASA Technical Reports Server (NTRS)
Collins, Emmanuel G., Jr.; Richter, Stephen
1990-01-01
One well known deficiency of LQG compensators is that they do not guarantee any measure of robustness. This deficiency is especially highlighted when considering control design for complex systems such as flexible structures. There has thus been a need to generalize LQG theory to incorporate robustness constraints. Here we describe the maximum entropy approach to robust control design for flexible structures, a generalization of LQG theory, pioneered by Hyland, which has proved useful in practice. The design equations consist of a set of coupled Riccati and Lyapunov equations. A homotopy algorithm that is used to solve these design equations is presented.
4. Discrete reductive perturbation technique
SciTech Connect
Levi, Decio; Petrera, Matteo
2006-04-15
We expand a partial difference equation (P{delta}E) on multiple lattices and obtain the P{delta}E which governs its far field behavior. The perturbative-reductive approach is here performed on well-known nonlinear P{delta}Es, both integrable and nonintegrable. We study the cases of the lattice modified Korteweg-de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra-Kac-Van Moerbeke equation and a nonintegrable lattice KdV equation. Such reductions allow us to obtain many new P{delta}Es of the nonlinear Schroedinger type.
5. Perturbative fragmentation
SciTech Connect
Kopeliovich, B. Z.; Pirner, H.-J.; Potashnikova, I. K.; Schmidt, Ivan; Tarasov, A. V.
2008-03-01
The Berger model of perturbative fragmentation of quarks to pions is improved by providing an absolute normalization and keeping all terms in a (1-z) expansion, which makes the calculation valid at all values of fractional pion momentum z. We also replace the nonrelativistic wave function of a loosely bound pion by the more realistic procedure of projecting to the light-cone pion wave function, which in turn is taken from well known models. The full calculation does not confirm the (1-z){sup 2} behavior of the fragmentation function (FF) predicted in [E. L. Berger, Z. Phys. C 4, 289 (1980); Phys. Lett. 89B, 241 (1980] for z>0.5, and only works at very large z>0.95, where it is in reasonable agreement with phenomenological FFs. Otherwise, we observe quite a different z-dependence which grossly underestimates data at smaller z. The disagreement is reduced after the addition of pions from decays of light vector mesons, but still remains considerable. The process dependent higher twist terms are also calculated exactly and found to be important at large z and/or p{sub T}.
6. Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein-Gordon-Zakharov equations in plasma physics
Saha Ray, S.; Sahoo, S.
2016-07-01
In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein-Gordon-Zakharov equations. We consider fractional coupled Klein-Gordon-Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein-Gordon-Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order α ∈ (1, 2], β ∈ (1, 2]. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.
7. Homotopy-Theoretic Study &Atomic-Scale Observation of Vortex Domains in Hexagonal Manganites.
PubMed
Li, Jun; Chiang, Fu-Kuo; Chen, Zhen; Ma, Chao; Chu, Ming-Wen; Chen, Cheng-Hsuan; Tian, Huanfang; Yang, Huaixin; Li, Jianqi
2016-01-01
Essential structural properties of the non-trivial "string-wall-bounded" topological defects in hexagonal manganites are studied through homotopy group theory and spherical aberration-corrected scanning transmission electron microscopy. The appearance of a "string-wall-bounded" configuration in RMnO3 is shown to be strongly linked with the transformation of the degeneracy space. The defect core regions (~50 Å) mainly adopt the continuous U(1) symmetry of the high-temperature phase, which is essential for the formation and proliferation of vortices. Direct visualization of vortex strings at atomic scale provides insight into the mechanisms and macro-behavior of topological defects in crystalline materials. PMID:27324701
8. Homotopy-Theoretic Study & Atomic-Scale Observation of Vortex Domains in Hexagonal Manganites
PubMed Central
Li, Jun; Chiang, Fu-Kuo; Chen, Zhen; Ma, Chao; Chu, Ming-Wen; Chen, Cheng-Hsuan; Tian, Huanfang; Yang, Huaixin; Li, Jianqi
2016-01-01
Essential structural properties of the non-trivial “string-wall-bounded” topological defects in hexagonal manganites are studied through homotopy group theory and spherical aberration-corrected scanning transmission electron microscopy. The appearance of a “string-wall-bounded” configuration in RMnO3 is shown to be strongly linked with the transformation of the degeneracy space. The defect core regions (~50 Å) mainly adopt the continuous U(1) symmetry of the high-temperature phase, which is essential for the formation and proliferation of vortices. Direct visualization of vortex strings at atomic scale provides insight into the mechanisms and macro-behavior of topological defects in crystalline materials. PMID:27324701
9. On the homotopy type of spaces of Morse functions on surfaces
SciTech Connect
Kudryavtseva, Elena A
2013-01-31
Let M be a smooth closed orientable surface. Let F be the space of Morse functions on M with a fixed number of critical points of each index such that at least {chi}(M)+1 critical points are labelled by different labels (numbered). The notion of a skew cylindric-polyhedral complex is introduced, which generalizes the notion of a polyhedral complex. The skew cylindric-polyhedral complex K-tilde ('the complex of framed Morse functions') associated with the space F is defined. In the case M=S{sup 2} the polytope K-tilde is finite; its Euler characteristic {chi}(K-tilde) is calculated and the Morse inequalities for its Betti numbers {beta}{sub j}(K-tilde) are obtained. The relation between the homotopy types of the polytope K-tilde and the space F of Morse functions equipped with the C{sup {infinity}}-topology is indicated. Bibliography: 51 titles.
10. Application of Homotopy analysis method for mechanical model of deepwater SCR installation
You, Xiangcheng; Xu, Hang
2012-09-01
In this paper, considering the process of deepwater SCR installation with the limitations of small deformation theory of beam and catenary theory, a mechanical model of deepwater SCR installation is given based on large deformation beam model. In the following model, getting the relation of the length of the riser, bending stiffness and the unit weight by dimensional analysis, the simple approximate analytical expressions are obtained by using Homotopy Analysis Method. In the same condition, the calculated results are compared with the proposed approximate analytical expressions, the catenary theory or the commercial software of nonlinear finite element program ORCAFLEX. Hopefully, a convenient and effective method for mechanical model of deepwater SCR installation is provided.
11. Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory
Ospina, Juan
2015-05-01
The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called "open-closed stringy topological quantum computer" which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided.
12. The influence of mild carbon dioxide on brain functional homotopy using resting-state fMRI.
PubMed
Marshall, Olga; Uh, Jinsoo; Lurie, Daniel; Lu, Hanzhang; Milham, Michael P; Ge, Yulin
2015-10-01
Homotopy reflects the intrinsic functional architecture of the brain through synchronized spontaneous activity between corresponding bilateral regions, measured as voxel mirrored homotopic connectivity (VMHC). Hypercapnia is known to have clear impact on brain hemodynamics through vasodilation, but have unclear effect on neuronal activity. This study investigates the effect of hypercapnia on brain homotopy, achieved by breathing 5% carbon dioxide (CO2 ) gas mixture. A total of 14 healthy volunteers completed three resting state functional MRI (RS-fMRI) scans, the first and third under normocapnia and the second under hypercapnia. VMHC measures were calculated as the correlation between the BOLD signal of each voxel and its counterpart in the opposite hemisphere. Group analysis was performed between the hypercapnic and normocapnic VMHC maps. VMHC showed a diffused decrease in response to hypercapnia. Significant regional decreases in VMHC were observed in all anatomical lobes, except for the occipital lobe, in the following functional hierarchical subdivisions: the primary sensory-motor, unimodal, heteromodal, paralimbic, as well as in the following functional networks: ventral attention, somatomotor, default frontoparietal, and dorsal attention. Our observation that brain homotopy in RS-fMRI is affected by arterial CO2 levels suggests that caution should be used when comparing RS-fMRI data between healthy controls and patients with pulmonary diseases and unusual respiratory patterns such as sleep apnea or chronic obstructive pulmonary disease. PMID:26138728
13. Piecewise-homotopy analysis method (P-HAM) for first order nonlinear ODE
Chin, F. Y.; Lem, K. H.; Chong, F. S.
2013-09-01
In homotopy analysis method (HAM), the determination for the value of the auxiliary parameter h is based on the valid region of the h-curve in which the horizontal segment of the h-curve will decide the valid h-region. All h-value taken from the valid region, provided that the order of deformation is large enough, will in principle yield an approximation series that converges to the exact solution. However it is found out that the h-value chosen within this valid region does not always promise a good approximation under finite order. This paper suggests an improved method called Piecewise-HAM (P-HAM). In stead of a single h-value, this method suggests using many h-values. Each of the h-values comes from an individual h-curve while each h-curve is plotted by fixing the time t at a different value. Each h-value is claimed to produce a good approximation only about a neighborhood centered at the corresponding t which the h-curve is based on. Each segment of these good approximations is then joined to form the approximation curve. By this, the convergence region is enhanced further. The P-HAM is illustrated and supported by examples.
14. A Fast and Accurate Sparse Continuous Signal Reconstruction by Homotopy DCD with Non-Convex Regularization
PubMed Central
Wang, Tianyun; Lu, Xinfei; Yu, Xiaofei; Xi, Zhendong; Chen, Weidong
2014-01-01
In recent years, various applications regarding sparse continuous signal recovery such as source localization, radar imaging, communication channel estimation, etc., have been addressed from the perspective of compressive sensing (CS) theory. However, there are two major defects that need to be tackled when considering any practical utilization. The first issue is off-grid problem caused by the basis mismatch between arbitrary located unknowns and the pre-specified dictionary, which would make conventional CS reconstruction methods degrade considerably. The second important issue is the urgent demand for low-complexity algorithms, especially when faced with the requirement of real-time implementation. In this paper, to deal with these two problems, we have presented three fast and accurate sparse reconstruction algorithms, termed as HR-DCD, Hlog-DCD and Hlp-DCD, which are based on homotopy, dichotomous coordinate descent (DCD) iterations and non-convex regularizations, by combining with the grid refinement technique. Experimental results are provided to demonstrate the effectiveness of the proposed algorithms and related analysis. PMID:24675758
15. A fast and accurate sparse continuous signal reconstruction by homotopy DCD with non-convex regularization.
PubMed
Wang, Tianyun; Lu, Xinfei; Yu, Xiaofei; Xi, Zhendong; Chen, Weidong
2014-01-01
In recent years, various applications regarding sparse continuous signal recovery such as source localization, radar imaging, communication channel estimation, etc., have been addressed from the perspective of compressive sensing (CS) theory. However, there are two major defects that need to be tackled when considering any practical utilization. The first issue is off-grid problem caused by the basis mismatch between arbitrary located unknowns and the pre-specified dictionary, which would make conventional CS reconstruction methods degrade considerably. The second important issue is the urgent demand for low-complexity algorithms, especially when faced with the requirement of real-time implementation. In this paper, to deal with these two problems, we have presented three fast and accurate sparse reconstruction algorithms, termed as HR-DCD, Hlog-DCD and Hlp-DCD, which are based on homotopy, dichotomous coordinate descent (DCD) iterations and non-convex regularizations, by combining with the grid refinement technique. Experimental results are provided to demonstrate the effectiveness of the proposed algorithms and related analysis. PMID:24675758
16. Cosmological perturbations in unimodular gravity
SciTech Connect
Gao, Caixia; Brandenberger, Robert H.; Cai, Yifu; Chen, Pisin E-mail: [email protected] E-mail: [email protected]
2014-09-01
We study cosmological perturbation theory within the framework of unimodular gravity. We show that the Lagrangian constraint on the determinant of the metric required by unimodular gravity leads to an extra constraint on the gauge freedom of the metric perturbations. Although the main equation of motion for the gravitational potential remains the same, the shift variable, which is gauge artifact in General Relativity, cannot be set to zero in unimodular gravity. This non-vanishing shift variable affects the propagation of photons throughout the cosmological evolution and therefore modifies the Sachs-Wolfe relation between the relativistic gravitational potential and the microwave temperature anisotropies. However, for adiabatic fluctuations the difference between the result in General Relativity and unimodular gravity is suppressed on large angular scales. Thus, no strong constraints on the theory can be derived.
17. Density perturbation theory
SciTech Connect
Palenik, Mark C.; Dunlap, Brett I.
2015-07-28
Despite the fundamental importance of electron density in density functional theory, perturbations are still usually dealt with using Hartree-Fock-like orbital equations known as coupled-perturbed Kohn-Sham (CPKS). As an alternative, we develop a perturbation theory that solves for the perturbed density directly, removing the need for CPKS. This replaces CPKS with a true Hohenberg-Kohn density perturbation theory. In CPKS, the perturbed density is found in the basis of products of occupied and virtual orbitals, which becomes ever more over-complete as the size of the orbital basis set increases. In our method, the perturbation to the density is expanded in terms of a series of density basis functions and found directly. It is possible to solve for the density in such a way that it makes the total energy stationary even if the density basis is incomplete.
18. A homotopy-based sparse representation for fast and accurate shape prior modeling in liver surgical planning.
PubMed
Wang, Guotai; Zhang, Shaoting; Xie, Hongzhi; Metaxas, Dimitris N; Gu, Lixu
2015-01-01
Shape prior plays an important role in accurate and robust liver segmentation. However, liver shapes have complex variations and accurate modeling of liver shapes is challenging. Using large-scale training data can improve the accuracy but it limits the computational efficiency. In order to obtain accurate liver shape priors without sacrificing the efficiency when dealing with large-scale training data, we investigate effective and scalable shape prior modeling method that is more applicable in clinical liver surgical planning system. We employed the Sparse Shape Composition (SSC) to represent liver shapes by an optimized sparse combination of shapes in the repository, without any assumptions on parametric distributions of liver shapes. To leverage large-scale training data and improve the computational efficiency of SSC, we also introduced a homotopy-based method to quickly solve the L1-norm optimization problem in SSC. This method takes advantage of the sparsity of shape modeling, and solves the original optimization problem in SSC by continuously transforming it into a series of simplified problems whose solution is fast to compute. When new training shapes arrive gradually, the homotopy strategy updates the optimal solution on the fly and avoids re-computing it from scratch. Experiments showed that SSC had a high accuracy and efficiency in dealing with complex liver shape variations, excluding gross errors and preserving local details on the input liver shape. The homotopy-based SSC had a high computational efficiency, and its runtime increased very slowly when repository's capacity and vertex number rose to a large degree. When repository's capacity was 10,000, with 2000 vertices on each shape, homotopy method cost merely about 11.29 s to solve the optimization problem in SSC, nearly 2000 times faster than interior point method. The dice similarity coefficient (DSC), average symmetric surface distance (ASD), and maximum symmetric surface distance measurement
19. Automated Lattice Perturbation Theory
SciTech Connect
Monahan, Christopher
2014-11-01
I review recent developments in automated lattice perturbation theory. Starting with an overview of lattice perturbation theory, I focus on the three automation packages currently "on the market": HiPPy/HPsrc, Pastor and PhySyCAl. I highlight some recent applications of these methods, particularly in B physics. In the final section I briefly discuss the related, but distinct, approach of numerical stochastic perturbation theory.
20. Frame independent cosmological perturbations
SciTech Connect
Prokopec, Tomislav; Weenink, Jan E-mail: [email protected]
2013-09-01
We compute the third order gauge invariant action for scalar-graviton interactions in the Jordan frame. We demonstrate that the gauge invariant action for scalar and tensor perturbations on one physical hypersurface only differs from that on another physical hypersurface via terms proportional to the equation of motion and boundary terms, such that the evolution of non-Gaussianity may be called unique. Moreover, we demonstrate that the gauge invariant curvature perturbation and graviton on uniform field hypersurfaces in the Jordan frame are equal to their counterparts in the Einstein frame. These frame independent perturbations are therefore particularly useful in relating results in different frames at the perturbative level. On the other hand, the field perturbation and graviton on uniform curvature hypersurfaces in the Jordan and Einstein frame are non-linearly related, as are their corresponding actions and n-point functions.
1. Minimum-fuel station-change for geostationary satellites using low-thrust considering perturbations
Zhao, ShuGe; Zhang, JingRui
2016-10-01
The objective of this paper is to find the minimum-fuel station change for geostationary satellites with low-thrust while considering significant perturbation forces for geostationary Earth orbit (GEO). The effect of Earth's triaxiality, lunisolar perturbations, and solar radiation pressure on the terminal conditions of a long duration GEO transfer is derived and used for establishing the station change model with consideration of significant perturbation forces. A method is presented for analytically evaluating the effect of Earth's triaxiality on the semimajor axis and longitude during a station change. The minimum-fuel problem is solved by the indirect optimization method. The easier and related minimum-energy problem is first addressed and then the energy-to-fuel homotopy is employed to finally obtain the solution of the minimum-fuel problem. Several effective techniques are employed in solving the two-point boundary-value problem with a shooting method to overcome the problem of the small convergence radius and the sensitivity of the initial costate variables. These methods include normalization of the initial costate vector, computation of the analytic Jacobians matrix, and switching detection. The simulation results show that the solution of the minimum-fuel station change with low-thrust considering significant perturbation forces can be obtained by applying these preceding techniques.
2. The Perturbed Puma Model
Rong, Shu-Jun; Liu, Qiu-Yu
2012-04-01
The puma model on the basis of the Lorentz and CPT violation may bring an economical interpretation to the conventional neutrinos oscillation and part of the anomalous oscillations. We study the effect of the perturbation to the puma model. In the case of the first-order perturbation which keeps the (23) interchange symmetry, the mixing matrix element Ue3 is always zero. The nonzero mixing matrix element Ue3 is obtained in the second-order perturbation that breaks the (23) interchange symmetry.
3. Chiral Perturbation Theory
Tiburzi, Brian C.
The era of high-precision lattice QCD has led to synergy between lattice computations and phenomenological input from chiral perturbation theory. We provide an introduction to chiral perturbation theory with a bent towards understanding properties of the nucleon and other low-lying baryons. Four main topics are the basis for this chapter. We begin with a discussion of broken symmetries and the procedure to construct the chiral Lagrangian. The second topic concerns specialized applications of chiral perturbation theory tailored to lattice QCD, such as partial quenching, lattice discretization, and finite-volume effects. We describe inclusion of the nucleon in chiral perturbation theory using a heavy-fermion Euclidean action. Issues of convergence are taken up as our final topic. We consider expansions in powers of the strange-quark mass, and the appearance of unphysical singularities in the heavy-particle formulation. Our aim is to guide lattice practitioners in understanding the predictions chiral perturbation theory makes for baryons, and show how the lattice will play a role in testing the rigor of the chiral expansion at physical values of the quark masses.
4. Vortex perturbation dynamics
NASA Technical Reports Server (NTRS)
Criminale, W. O.; Lasseigne, D. G.; Jackson, T. L.
1995-01-01
An initial value approach is used to examine the dynamics of perturbations introduced into a vortex under strain. Both the basic vortex considered and the perturbations are taken as fully three-dimensional. An explicit solution for the time evolution of the vorticity perturbations is given for arbitrary initial vorticity. Analytical solutions for the resulting velocity components are found when the initial vorticity is assumed to be localized. For more general initial vorticity distributions, the velocity components are determined numerically. It is found that the variation in the radial direction of the initial vorticity disturbance is the most important factor influencing the qualitative behavior of the solutions. Transient growth in the magnitude of the velocity components is found to be directly attributable to the compactness of the initial vorticity.
5. Cosmological perturbations in antigravity
Oltean, Marius; Brandenberger, Robert
2014-10-01
We compute the evolution of cosmological perturbations in a recently proposed Weyl-symmetric theory of two scalar fields with oppositely signed conformal couplings to Einstein gravity. It is motivated from the minimal conformal extension of the standard model, such that one of these scalar fields is the Higgs while the other is a new particle, the dilaton, introduced to make the Higgs mass conformally symmetric. At the background level, the theory admits novel geodesically complete cyclic cosmological solutions characterized by a brief period of repulsive gravity, or "antigravity," during each successive transition from a big crunch to a big bang. For simplicity, we consider scalar perturbations in the absence of anisotropies, with potential set to zero and without any radiation. We show that despite the necessarily wrong-signed kinetic term of the dilaton in the full action, these perturbations are neither ghostlike nor tachyonic in the limit of strongly repulsive gravity. On this basis, we argue—pending a future analysis of vector and tensor perturbations—that, with respect to perturbative stability, the cosmological solutions of this theory are viable.
6. Semi-computational simulation of magneto-hemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods.
PubMed
Basiri Parsa, A; Rashidi, M M; Anwar Bég, O; Sadri, S M
2013-09-01
In this paper, the semi-numerical techniques known as the optimal homotopy analysis method (HAM) and Differential Transform Method (DTM) are applied to study the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field. The two-dimensional momentum conservation partial differential equations are reduced to ordinary form incorporating Lorentizian magnetohydrodynamic body force terms. These ordinary differential equations are solved by the homotopy analysis method, the differential transform method and also a numerical method (fourth-order Runge-Kutta quadrature with a shooting method), under physically realistic boundary conditions. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series. The differential transform method (DTM) does not require an auxiliary parameter and is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. The influence of Hartmann number (Ha) and transpiration Reynolds number (mass transfer parameter, Re) on the velocity profiles in the channel are studied in detail. Interesting fluid dynamic characteristics are revealed and addressed. The HAM and DTM solutions are shown to both correlate well with numerical quadrature solutions, testifying to the accuracy of both HAM and DTM in nonlinear magneto-hemodynamics problems. Both these semi-numerical techniques hold excellent potential in modeling nonlinear viscous flows in biological systems. PMID:23930807
7. Liouvillian perturbations of black holes
Couch, W. E.; Holder, C. L.
2007-10-01
We apply the well-known Kovacic algorithm to find closed form, i.e., Liouvillian solutions, to the differential equations governing perturbations of black holes. Our analysis includes the full gravitational perturbations of Schwarzschild and Kerr, the full gravitational and electromagnetic perturbations of Reissner-Nordstrom, and specialized perturbations of the Kerr-Newman geometry. We also include the extreme geometries. We find all frequencies ω, in terms of black hole parameters and an integer n, which allow Liouvillian perturbations. We display many classes of black hole parameter values and their corresponding Liouvillian perturbations, including new closed-form perturbations of Kerr and Reissner-Nordstrom. We also prove that the only type 1 Liouvillian perturbations of Schwarzschild are the known algebraically special ones and that type 2 Liouvillian solutions do not exist for extreme geometries. In cases where we do not prove the existence or nonexistence of Liouvillian perturbations we obtain sequences of Diophantine equations on which decidability rests.
8. Aspects of perturbative unitarity
Anselmi, Damiano
2016-07-01
We reconsider perturbative unitarity in quantum field theory and upgrade several arguments and results. The minimum assumptions that lead to the largest time equation, the cutting equations and the unitarity equation are identified. Using this knowledge and a special gauge, we give a new, simpler proof of perturbative unitarity in gauge theories and generalize it to quantum gravity, in four and higher dimensions. The special gauge interpolates between the Feynman gauge and the Coulomb gauge without double poles. When the Coulomb limit is approached, the unphysical particles drop out of the cuts and the cutting equations are consistently projected onto the physical subspace. The proof does not extend to nonlocal quantum field theories of gauge fields and gravity, whose unitarity remains uncertain.
9. Topology and perturbation theory
Manjavidze, J.
2000-08-01
This paper contains description of the fields nonlinear modes successive quantization scheme. It is shown that the path integrals for absorption part of amplitudes are defined on the Dirac (δ-like) functional measure. This permits arbitrary transformation of the functional integral variables. New form of the perturbation theory achieved by mapping the quantum dynamics in the space WG of the (action, angle)-type collective variables. It is shown that the transformed perturbation theory contributions are accumulated exactly on the boundary ∂WG. Abilities of the developed formalism are illustrated by the Coulomb problem. This model is solved in the WC=(angle, angular momentum, Runge-Lentz vector) space and the reason of its exact integrability is emptiness of ∂WC.
10. Renormalized Lie perturbation theory
SciTech Connect
Rosengaus, E.; Dewar, R.L.
1981-07-01
A Lie operator method for constructing action-angle transformations continuously connected to the identity is developed for area preserving mappings. By a simple change of variable from action to angular frequency a perturbation expansion is obtained in which the small denominators have been renormalized. The method is shown to lead to the same series as the Lagrangian perturbation method of Greene and Percival, which converges on KAM surfaces. The method is not superconvergent, but yields simple recursion relations which allow automatic algebraic manipulation techniques to be used to develop the series to high order. It is argued that the operator method can be justified by analytically continuing from the complex angular frequency plane onto the real line. The resulting picture is one where preserved primary KAM surfaces are continuously connected to one another.
11. Intermolecular perturbation theory
Hayes, I. C.; Hurst, G. J. B.; Stone, A. J.
The new intermolecular perturbation theory described in the preceding papers is applied to some van der Waals molecules. HeBe is used as a test case, and the perturbation method converges well at interatomic distances down to about 4 a0, giving results in excellent agreement with supermolecule calculations. ArHF and ArHCl have been studied using large basis sets, and the results agree well with experimental data. The ArHX configuration is favoured over the ArXH configuration mainly because of larger polarization and charge-transfer contributions. In NeH2 the equilibrium geometry is determined by a delicate balance between opposing effects; with a double-zeta-polarization basis the correct configuration is predicted.
12. Homotopy semi-numerical simulation of peristaltic flow of generalised Oldroyd-B fluids with slip effects.
PubMed
Tripathi, Dharmendra; Bég, O Anwar; Curiel-Sosa, J L
2014-01-01
This investigation deals with the peristaltic flow of generalised Oldroyd-B fluids (with the fractional model) through a cylindrical tube under the influence of wall slip conditions. The analysis is carried out under the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions are obtained by using the highly versatile and rigorous semi-numerical procedure known as the homotopy analysis method. It is assumed that the cross section of the tube varies sinusoidally along the length of the tube. The effects of the dominant hydromechanical parameters, i.e. fractional parameters, material constants, slip parameter, time and amplitude on the pressure difference across one wavelength, are studied. Graphical plots reveal that the influence of both fractional parameters on pressure is opposite to each other. Interesting responses to a variation in the constants are obtained. Pressure is shown to be reduced by increasing the slip parameter. Furthermore, the pressure in the case of fractional models (fractional Oldroyd-B model and fractional Maxwell model) of viscoelastic fluids is considerably more substantial than that in the corresponding classical viscoelastic models (Oldroyd-B and Maxwell models). Applications of the study arise in biophysical food processing, embryology and gastro-fluid dynamics. PMID:22616875
13. Study of a homotopy continuation method for early orbit determination with the Tracking and Data Relay Satellite System (TDRSS)
NASA Technical Reports Server (NTRS)
Smith, R. L.; Huang, C.
1986-01-01
A recent mathematical technique for solving systems of equations is applied in a very general way to the orbit determination problem. The study of this technique, the homotopy continuation method, was motivated by the possible need to perform early orbit determination with the Tracking and Data Relay Satellite System (TDRSS), using range and Doppler tracking alone. Basically, a set of six tracking observations is continuously transformed from a set with known solution to the given set of observations with unknown solutions, and the corresponding orbit state vector is followed from the a priori estimate to the solutions. A numerical algorithm for following the state vector is developed and described in detail. Numerical examples using both real and simulated TDRSS tracking are given. A prototype early orbit determination algorithm for possible use in TDRSS orbit operations was extensively tested, and the results are described. Preliminary studies of two extensions of the method are discussed: generalization to a least-squares formulation and generalization to an exhaustive global method.
14. Dynamic Compressed HRRP Generation for Random Stepped-Frequency Radar Based on Complex-Valued Fast Sequential Homotopy
PubMed Central
You, Peng; Liu, Zhen; Wang, Hongqiang; Wei, Xizhang; Li, Xiang
2014-01-01
Compressed sensing has been applied to achieve high resolution range profiles (HRRPs) using a stepped-frequency radar. In this new scheme, much fewer pulses are required to recover the target's strong scattering centers, which can greatly reduce the coherent processing interval (CPI) and improve the anti-jamming capability. For practical applications, however, the required number of pulses is difficult to determine in advance and any reduction of the transmitted pulses is attractive. In this paper, a dynamic compressed sensing strategy for HRRP generation is proposed, in which the estimated HRRP is updated with sequentially transmitted and received pulses until the proper stopping rules are satisfied. To efficiently implement the sequential update, a complex-valued fast sequential homotopy (CV-FSH) algorithm is developed based on group sparse recovery. This algorithm performs as an efficient recursive procedure of sparse recovery, thus avoiding solving a new optimization problem from scratch. Furthermore, the proper stopping rules are presented according to the special characteristics of HRRP. Therefore, the optimal number of pulses required in each CPI can be sought adapting to the echo signal. The results using simulated and real data show the effectiveness of the proposed approach and demonstrate that the established dynamic strategy is more suitable for uncooperative targets. PMID:24815679
15. Entropy perturbations in N-flation
SciTech Connect
Cai Ronggen; Hu Bin; Piao Yunsong
2009-12-15
In this paper we study the entropy perturbations in N-flation by using the {delta}N formalism. We calculate the entropy corrections to the power spectrum of the overall curvature perturbation P{sub {zeta}}. We obtain an analytic form of the transfer coefficient T{sub RS}{sup 2}, which describes the correlation between the curvature and entropy perturbations, and investigate its behavior numerically. It turns out that the entropy perturbations cannot be neglected in N-flation because the amplitude of entropy components is approximately in the same order as the adiabatic one at the end of inflation T{sub RS}{sup 2}{approx}O(1). The spectral index n{sub S} is calculated and it becomes smaller after the entropy modes are taken into account, i.e., the spectrum becomes redder, compared to the pure adiabatic case. Finally we study the modified consistency relation of N-flation, and find that the tensor-to-scalar ratio (r{approx_equal}0.006) is greatly suppressed by the entropy modes, compared to the pure adiabatic one (r{approx_equal}0.017) at the end of inflation.
16. Primordial power spectrum of tensor perturbations in Finsler spacetime
Li, Xin; Wang, Sai
2016-02-01
We first investigate the gravitational wave in the flat Finsler spacetime. In the Finslerian universe, we derive the perturbed gravitational field equation with tensor perturbations. The Finslerian background spacetime breaks rotational symmetry and induces parity violation. Then we obtain the modified primordial power spectrum of the tensor perturbations. The parity violation feature requires that the anisotropic effect contributes to the TT, TE, EE, BB angular correlation coefficients with l'=l+1 and TB, EB with l'=l. The numerical results show that the anisotropic contributions to the angular correlation coefficients depend on m, and TE and ET angular correlation coefficients are different.
17. Gauge-invariant perturbation theory for trans-Planckian inflation
SciTech Connect
Shankaranarayanan, S.; Lubo, Musongela
2005-12-15
The possibility that the scale-invariant inflationary spectrum may be modified due to the hidden assumptions about the Planck scale physics--dubbed as trans-Planckian inflation--has received considerable attention. To mimic the possible trans-Planckian effects, among various models, modified dispersion relations have been popular in the literature. In almost all the earlier analyses, unlike the canonical scalar field driven inflation, the trans-Planckian effects are introduced to the scalar/tensor perturbation equations in an ad hoc manner--without calculating the stress tensor of the cosmological perturbations from the covariant Lagrangian. In this work, we perform the gauge-invariant cosmological perturbations for the single scalar-field inflation with the Jacobson-Corley dispersion relation by computing the fluctuations of all the fields including the unit-timelike vector field which defines a preferred rest frame. We show that: (i) The nonlinear effects introduce corrections only to the perturbed energy density. The corrections to the energy density vanish in the super-Hubble scales. (ii) The scalar perturbations, in general, are not purely adiabatic. (iii) The equation of motion of the Mukhanov-Sasaki variable corresponding to the inflaton field is different from those presumed in the earlier analyses. (iv) The tensor perturbation equation remains unchanged. We perform the classical analysis for the resultant system of equations and also compute the power spectrum of the scalar perturbations in a particular limit. We discuss the implications of our results and compare with the earlier results.
18. Adjustments after an ankle dorsiflexion perturbation during human running.
PubMed
Scohier, M; De Jaeger, D; Schepens, B
2012-01-01
In this study we investigated the effect of a mechanical perturbation of unexpected timing during human running. With the use of a powered exoskeleton, we evoked a dorsiflexion of the right ankle during its swing phase while subjects ran on a treadmill. The perturbation resulted in an increase of the right ankle dorsiflexion of at least 5°. The first two as well as the next five steps after the perturbation were analyzed to observe the possible immediate and late biomechanical adjustments. In all cases subjects continued to run after the perturbation. The immediate adjustments were the greatest and the most frequent when the delay between the right ankle perturbation and the subsequent right foot touch-down was the shortest. For example, the vertical impact peak force was strongly modified on the first step after the perturbations and this adjustment was correlated to a right ankle angle still clearly modified at touch-down. Some late adjustments were observed in the subsequent steps predominantly occurring during left steps. Subjects maintained the step length and the step period as constant as possible by adjusting other step parameters in order to avoid stumbling and continue running at the speed imposed by the treadmill. To our knowledge, our experiments are the first to investigate perturbations of unexpected timing during human running. The results show that humans have a time-dependent, adapted strategy to maintain their running pattern. PMID:21872474
19. Baryon chiral perturbation theory
Scherer, S.
2012-03-01
We provide a short introduction to the one-nucleon sector of chiral perturbation theory and address the issue of power counting and renormalization. We discuss the infrared regularization and the extended on-mass-shell scheme. Both allow for the inclusion of further degrees of freedom beyond pions and nucleons and the application to higher-loop calculations. As applications we consider the chiral expansion of the nucleon mass to order Script O(q6) and the inclusion of vector and axial-vector mesons in the calculation of nucleon form factors. Finally, we address the complex-mass scheme for describing unstable particles in effective field theory.
20. Perturbative theory for Brownian vortexes
Moyses, Henrique W.; Bauer, Ross O.; Grosberg, Alexander Y.; Grier, David G.
2015-06-01
Brownian vortexes are stochastic machines that use static nonconservative force fields to bias random thermal fluctuations into steadily circulating currents. The archetype for this class of systems is a colloidal sphere in an optical tweezer. Trapped near the focus of a strongly converging beam of light, the particle is displaced by random thermal kicks into the nonconservative part of the optical force field arising from radiation pressure, which then biases its diffusion. Assuming the particle remains localized within the trap, its time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian vortexes, such as the biased Brownian pendulum, which circulate preferentially in the direction of the bias, the general Brownian vortex can change direction and even topology in response to temperature changes. Here we introduce a theory based on a perturbative expansion of the Fokker-Planck equation for weak nonconservative driving. The first-order solution takes the form of a modified Boltzmann relation and accounts for the rich phenomenology observed in experiments on micrometer-scale colloidal spheres in optical tweezers.
1. Chiral perturbation theory with nucleons
SciTech Connect
Meissner, U.G.
1991-09-01
I review the constraints posed on the interactions of pions, nucleons and photons by the spontaneously broken chiral symmetry of QCD. The framework to perform these calculations, chiral perturbation theory, is briefly discussed in the meson sector. The method is a simultaneous expansion of the Greens functions in powers of external moments and quark masses around the massless case, the chiral limit. To perform this expansion, use is made of a phenomenological Lagrangian which encodes the Ward-identities and pertinent symmetries of QCD. The concept of chiral power counting is introduced. The main part of the lectures of consists in describing how to include baryons (nucleons) and how the chiral structure is modified by the fact that the nucleon mass in the chiral limit does not vanish. Particular emphasis is put on working out applications to show the strengths and limitations of the methods. Some processes which are discussed are threshold photopion production, low-energy compton scattering off nucleons, {pi}N scattering and the {sigma}-term. The implications of the broken chiral symmetry on the nuclear forces are briefly described. An alternative approach, in which the baryons are treated as very heavy fields, is touched upon.
2. Perturbative theory for Brownian vortexes.
PubMed
Moyses, Henrique W; Bauer, Ross O; Grosberg, Alexander Y; Grier, David G
2015-06-01
Brownian vortexes are stochastic machines that use static nonconservative force fields to bias random thermal fluctuations into steadily circulating currents. The archetype for this class of systems is a colloidal sphere in an optical tweezer. Trapped near the focus of a strongly converging beam of light, the particle is displaced by random thermal kicks into the nonconservative part of the optical force field arising from radiation pressure, which then biases its diffusion. Assuming the particle remains localized within the trap, its time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian vortexes, such as the biased Brownian pendulum, which circulate preferentially in the direction of the bias, the general Brownian vortex can change direction and even topology in response to temperature changes. Here we introduce a theory based on a perturbative expansion of the Fokker-Planck equation for weak nonconservative driving. The first-order solution takes the form of a modified Boltzmann relation and accounts for the rich phenomenology observed in experiments on micrometer-scale colloidal spheres in optical tweezers. PMID:26172698
3. Canonical floquet perturbation theory
Pohlen, David J.
1992-12-01
Classical Floquet theory is examined in order to generate a canonical transformation to modal variables for periodic system. This transformation is considered canonical if the periodic matrix of eigenvectors is symplectic at the initial time. Approaches for symplectic normalization of the eigenvectors had to be examined for each of the different Poincare eigenvalue cases. Particular attention was required in the degenerate case, which depended on the solution of a generalized eigenvector. Transformation techniques to ensure real modal variables and real periodic eigenvectors were also needed. Periodic trajectories in the restricted three-body case were then evaluated using the canonical Floquet solution. The system used for analyses is the Sun-Jupiter system. This system was especially useful since it contained two of the more difficult Poincare eigenvalue cases, the degenerate case and the imaginary eigenvalue case. The perturbation solution to the canonical modal variables was examined using both an expansion of the Hamiltonian and using a representation that was considered exact. Both methods compared quite well for small perturbations to the initial condition. As expected, the expansion solution failed first due to truncation after the third order term of the expansion.
4. Numerical and perturbative computations of solitary waves of the Benjamin-Ono equation with higher order nonlinearity using Christov rational basis functions
Boyd, John P.; Xu, Zhengjie
2012-02-01
Computation of solitons of the cubically-nonlinear Benjamin-Ono equation is challenging. First, the equation contains the Hilbert transform, a nonlocal integral operator. Second, its solitary waves decay only as O(1/∣ x∣ 2). To solve the integro-differential equation for waves traveling at a phase speed c, we introduced the artificial homotopy H( uXX) - c u + (1 - δ) u2 + δu3 = 0, δ ∈ [0, 1] and solved it in two ways. The first was continuation in the homotopy parameter δ, marching from the known Benjamin-Ono soliton for δ = 0 to the cubically-nonlinear soliton at δ = 1. The second strategy was to bypass continuation by numerically computing perturbation series in δ and forming Padé approximants to obtain a very accurate approximation at δ = 1. To further minimize computations, we derived an elementary theorem to reduce the two-parameter soliton family to a parameter-free function, the soliton symmetric about the origin with unit phase speed. Solitons for higher order Benjamin-Ono equations are also computed and compared to their Korteweg-deVries counterparts. All computations applied the pseudospectral method with a basis of rational orthogonal functions invented by Christov, which are eigenfunctions of the Hilbert transform.
5. Flatbands under Correlated Perturbations
Bodyfelt, Joshua D.; Leykam, Daniel; Danieli, Carlo; Yu, Xiaoquan; Flach, Sergej
2014-12-01
Flatband networks are characterized by the coexistence of dispersive and flatbands. Flatbands (FBs) are generated by compact localized eigenstates (CLSs) with local network symmetries, based on destructive interference. Correlated disorder and quasiperiodic potentials hybridize CLSs without additional renormalization, yet with surprising consequences: (i) states are expelled from the FB energy EFB, (ii) the localization length of eigenstates vanishes as ξ ˜1 /ln (E -EFB) , (iii) the density of states diverges logarithmically (particle-hole symmetry) and algebraically (no particle-hole symmetry), and (iv) mobility edge curves show algebraic singularities at EFB . Our analytical results are based on perturbative expansions of the CLSs and supported by numerical data in one and two lattice dimensions.
6. Flatbands under correlated perturbations.
PubMed
Bodyfelt, Joshua D; Leykam, Daniel; Danieli, Carlo; Yu, Xiaoquan; Flach, Sergej
2014-12-01
Flatband networks are characterized by the coexistence of dispersive and flatbands. Flatbands (FBs) are generated by compact localized eigenstates (CLSs) with local network symmetries, based on destructive interference. Correlated disorder and quasiperiodic potentials hybridize CLSs without additional renormalization, yet with surprising consequences: (i) states are expelled from the FB energy E_{FB}, (ii) the localization length of eigenstates vanishes as ξ∼1/ln(E-E_{FB}), (iii) the density of states diverges logarithmically (particle-hole symmetry) and algebraically (no particle-hole symmetry), and (iv) mobility edge curves show algebraic singularities at E_{FB}. Our analytical results are based on perturbative expansions of the CLSs and supported by numerical data in one and two lattice dimensions. PMID:25526142
7. Discrete Newtonian cosmology: perturbations
Ellis, George F. R.; Gibbons, Gary W.
2015-03-01
In a previous paper (Gibbons and Ellis 2014 Discrete Newtonian cosmology Class. Quantum Grav. 31 025003), we showed how a finite system of discrete particles interacting with each other via Newtonian gravitational attraction would lead to precisely the same dynamical equations for homothetic motion as in the case of the pressure-free Friedmann-Lemaître-Robertson-Walker cosmological models of general relativity theory, provided the distribution of particles obeys the central configuration equation. In this paper we show that one can obtain perturbed such Newtonian solutions that give the same linearized structure growth equations as in the general relativity case. We also obtain the Dmitriev-Zel’dovich equations for subsystems in this discrete gravitational model, and show how it leads to the conclusion that voids have an apparent negative mass.
8. Perturbed effects at radiation physics
Külahcı, Fatih; Şen, Zekâi
2013-09-01
Perturbation methodology is applied in order to assess the linear attenuation coefficient, mass attenuation coefficient and cross-section behavior with random components in the basic variables such as the radiation amounts frequently used in the radiation physics and chemistry. Additionally, layer attenuation coefficient (LAC) and perturbed LAC (PLAC) are proposed for different contact materials. Perturbation methodology provides opportunity to obtain results with random deviations from the average behavior of each variable that enters the whole mathematical expression. The basic photon intensity variation expression as the inverse exponential power law (as Beer-Lambert's law) is adopted for perturbation method exposition. Perturbed results are presented not only in terms of the mean but additionally the standard deviation and the correlation coefficients. Such perturbation expressions provide one to assess small random variability in basic variables.
9. Homotopy Analysis Method for the heat transfer of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall
Esmaeilpour, M.; Domairry, G.; Sadoughi, N.; Davodi, A. G.
2010-09-01
In this article, a powerful analytical method, called the Homotopy Analysis Method (HAM) is introduced to obtain the exact solutions of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications. The HAM is employed to obtain the expressions for velocity and temperature fields. Tables are presented for various parameters on the velocity and temperature fields. These results are compared with the solutions which are obtained by Numerical Methods (NM). Also the convergence of the obtained HAM solution is discussed explicitly. These comparisons show that this analytical method is strongly powerful to solve nonlinear problems arising in heat transfer.
10. Cosmological perturbations in massive bigravity
SciTech Connect
Lagos, Macarena; Ferreira, Pedro G. E-mail: [email protected]
2014-12-01
We present a comprehensive analysis of classical scalar, vector and tensor cosmological perturbations in ghost-free massive bigravity. In particular, we find the full evolution equations and analytical solutions in a wide range of regimes. We show that there are viable cosmological backgrounds but, as has been found in the literature, these models generally have exponential instabilities in linear perturbation theory. However, it is possible to find stable scalar cosmological perturbations for a very particular choice of parameters. For this stable subclass of models we find that vector and tensor perturbations have growing solutions. We argue that special initial conditions are needed for tensor modes in order to have a viable model.
11. Canonical density matrix perturbation theory.
PubMed
Niklasson, Anders M N; Cawkwell, M J; Rubensson, Emanuel H; Rudberg, Elias
2015-12-01
Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free-energy ensembles in tight-binding, Hartree-Fock, or Kohn-Sham density-functional theory. The canonical density matrix perturbation theory can be used to calculate temperature-dependent response properties from the coupled perturbed self-consistent field equations as in density-functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large nonmetallic materials and metals at high temperatures. PMID:26764847
12. Perturbed atoms in molecules and solids: The PATMOS model
Røeggen, Inge; Gao, Bin
2013-09-01
A new computational method for electronic-structure studies of molecules and solids is presented. The key element in the new model - denoted the perturbed atoms in molecules and solids model - is the concept of a perturbed atom in a complex. The basic approximation of the new model is unrestricted Hartree Fock (UHF). The UHF orbitals are localized by the Edmiston-Ruedenberg procedure. The perturbed atoms are defined by distributing the orbitals among the nuclei in such a way that the sum of the intra-atomic UHF energies has a minimum. Energy corrections with respect to the UHF energy, are calculated within the energy incremental scheme. The most important three- and four-electron corrections are selected by introducing a modified geminal approach. Test calculations are performed on N2, Li2, and parallel arrays of hydrogen atoms. The character of the perturbed atoms is illustrated by calculations on H2, CH4, and C6H6.
13. Perturbed atoms in molecules and solids: The PATMOS model.
PubMed
Røeggen, Inge; Gao, Bin
2013-09-01
A new computational method for electronic-structure studies of molecules and solids is presented. The key element in the new model - denoted the perturbed atoms in molecules and solids model - is the concept of a perturbed atom in a complex. The basic approximation of the new model is unrestricted Hartree Fock (UHF). The UHF orbitals are localized by the Edmiston-Ruedenberg procedure. The perturbed atoms are defined by distributing the orbitals among the nuclei in such a way that the sum of the intra-atomic UHF energies has a minimum. Energy corrections with respect to the UHF energy, are calculated within the energy incremental scheme. The most important three- and four-electron corrections are selected by introducing a modified geminal approach. Test calculations are performed on N2, Li2, and parallel arrays of hydrogen atoms. The character of the perturbed atoms is illustrated by calculations on H2, CH4, and C6H6. PMID:24028099
14. Life expectancy change in perturbed communities: derivation and qualitative analysis.
PubMed
Dambacher, Jeffrey M; Levins, Richard; Rossignol, Philippe A
2005-09-01
Pollution, loss of habitat, and climate change are introducing dramatic perturbations to natural communities and affecting public health. Populations in perturbed communities can change dynamically, in both abundance and age structure. While analysis of the community matrix can predict changes in population abundance arising from a sustained or press perturbation, perturbations also have the potential to modify life expectancy, which adds yet another means to falsify experimental hypotheses and to monitor management interventions in natural systems. In some instances, an input to a community will produce no change in the abundance of a population but create a major shift in its mean age. We present an analysis of change in both abundance and life expectancy, leading to a formal quantitative assessment as well as qualitative predictions, and illustrate the usefulness of the technique through general examples relating to vector-borne disease and fisheries. PMID:16043195
15. Resumming the string perturbation series
Grassi, Alba; Mariño, Marcos; Zakany, Szabolcs
2015-05-01
We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local , which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.
16. On dark energy isocurvature perturbation
SciTech Connect
Liu, Jie; Zhang, Xinmin; Li, Mingzhe E-mail: [email protected]
2011-06-01
Determining the equation of state of dark energy with astronomical observations is crucially important to understand the nature of dark energy. In performing a likelihood analysis of the data, especially of the cosmic microwave background and large scale structure data the dark energy perturbations have to be taken into account both for theoretical consistency and for numerical accuracy. Usually, one assumes in the global fitting analysis that the dark energy perturbations are adiabatic. In this paper, we study the dark energy isocurvature perturbation analytically and discuss its implications for the cosmic microwave background radiation and large scale structure. Furthermore, with the current astronomical observational data and by employing Markov Chain Monte Carlo method, we perform a global analysis of cosmological parameters assuming general initial conditions for the dark energy perturbations. The results show that the dark energy isocurvature perturbations are very weakly constrained and that purely adiabatic initial conditions are consistent with the data.
17. Computing singularities of perturbation series
SciTech Connect
Kvaal, Simen; Jarlebring, Elias; Michiels, Wim
2011-03-15
Many properties of current ab initio approaches to the quantum many-body problem, both perturbational and otherwise, are related to the singularity structure of the Rayleigh-Schroedinger perturbation series. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the Hamiltonian matrix on a vector and does not rely on the terms in the perturbation series. The method can be useful for studying perturbation series of typical systems of moderate size, for fundamental development of resummation schemes, and for understanding the structure of singularities for typical systems. Some illustrative model problems are studied, including a helium-like model with {delta}-function interactions for which Moeller-Plesset perturbation theory is considered and the radius of convergence found.
18. Statistical anisotropy of the curvature perturbation from vector field perturbations
SciTech Connect
Dimopoulos, Konstantinos; Karciauskas, Mindaugas; Lyth, David H.; Rodriguez, Yeinzon E-mail: [email protected] E-mail: [email protected]
2009-05-15
The {delta}N formula for the primordial curvature perturbation {zeta} is extended to include vector as well as scalar fields. Formulas for the tree-level contributions to the spectrum and bispectrum of {zeta} are given, exhibiting statistical anisotropy. The one-loop contribution to the spectrum of {zeta} is also worked out. We then consider the generation of vector field perturbations from the vacuum, including the longitudinal component that will be present if there is no gauge invariance. Finally, the {delta}N formula is applied to the vector curvaton and vector inflation models with the tensor perturbation also evaluated in the latter case.
19. Cosmological perturbations during radion stabilization
Ashcroft, P. R.; van de Bruck, C.; Davis, A.-C.
2005-01-01
We consider the evolution of cosmological perturbations during radion stabilization, which we assume to happen after a period of inflation in the early universe. Concentrating on the Randall-Sundrum brane world scenario, we find that, if matter is present both on the positive and negative tension branes, the coupling of the radion to matter fields could have significant impact on the evolution of the curvature perturbation and on the production of entropy perturbations. We investigate both the case of a long-lived and a short-lived radion and outline similarities and differences to the curvaton scenario.
20. Causal compensated perturbations in cosmology
NASA Technical Reports Server (NTRS)
Veeraraghavan, Shoba; Stebbins, Albert
1990-01-01
A theoretical framework is developed to calculate linear perturbations in the gravitational and matter fields which arise causally in response to the presence of stiff matter sources in a FRW cosmology. It is shown that, in order to satisfy energy and momentum conservation, the gravitational fields of the source must be compensated by perturbations in the matter and gravitational fields, and the role of such compensation in containing the initial inhomogeneities in their subsequent evolution is discussed. A complete formal solution is derived in terms of Green functions for the perturbations produced by an arbitrary source in a flat universe containing cold dark matter. Approximate Green function solutions are derived for the late-time density perturbations and late-time gravitational waves in a universe containing a radiation fluid. A cosmological energy-momentum pseudotensor is defined to clarify the nature of energy and momentum conservation in the expanding universe.
1. Isocurvature perturbations in extra radiation
SciTech Connect
Kawasaki, Masahiro; Miyamoto, Koichi; Nakayama, Kazunori; Sekiguchi, Toyokazu E-mail: [email protected] E-mail: [email protected]
2012-02-01
Recent cosmological observations, including measurements of the CMB anisotropy and the primordial helium abundance, indicate the existence of an extra radiation component in the Universe beyond the standard three neutrino species. In this paper we explore the possibility that the extra radiation has isocurvatrue fluctuations. A general formalism to evaluate isocurvature perturbations in the extra radiation is provided in the mixed inflaton-curvaton system, where the extra radiation is produced by the decay of both scalar fields. We also derive constraints on the abundance of the extra radiation and the amount of its isocurvature perturbation. Current observational data favors the existence of an extra radiation component, but does not indicate its having isocurvature perturbation. These constraints are applied to some particle physics motivated models. If future observations detect isocurvature perturbations in the extra radiation, it will give us a hint to the origin of the extra radiation.
2. Dust-ion-acoustic solitons with transverse perturbation
SciTech Connect
Moslem, Waleed M.; El-Taibany, W.F.; El-Shewy, E.K.; El-Shamy, E.F.
2005-05-15
The ionization source model is considered, for the first time, to study the combined effects of trapped electrons, transverse perturbation, ion streaming velocity, and dust charge fluctuations on the propagation of dust-ion-acoustic solitons in dusty plasmas. The solitary waves are investigated through the derivation of the damped modified Kadomtsev-Petviashivili equation using the reductive perturbation method. Conditions for the formation of solitons as well as their properties are clearly explained. The relevance of our investigation to supernovae shells is also discussed.
3. Next-to-leading resummations in cosmological perturbation theory
SciTech Connect
Anselmi, Stefano; Matarrese, Sabino; Pietroni, Massimo E-mail: [email protected]
2011-06-01
One of the nicest results in cosmological perturbation theory is the analytical resummaton of the leading corrections at large momentum, which was obtained by Crocce and Scoccimarro for the propagator in Crocce (2005). Using an exact evolution equation, we generalize this result, by showing that a class of next-to-leading corrections can also be resummed at all orders in perturbation theory. The new corrections modify the propagator by a few percent in the Baryonic Acoustic Oscillation range of scales, and therefore cannot be neglected in resummation schemes aiming at an accuracy compatible with future generation galaxy surveys. Similar tools can be employed to derive improved approximations for the Power Spectrum.
4. Non-adiabatic perturbations in Ricci dark energy model
SciTech Connect
Karwan, Khamphee; Thitapura, Thiti E-mail: [email protected]
2012-01-01
We show that the non-adiabatic perturbations between Ricci dark energy and matter can grow both on superhorizon and subhorizon scales, and these non-adiabatic perturbations on subhorizon scales can lead to instability in this dark energy model. The rapidly growing non-adiabatic modes on subhorizon scales always occur when the equation of state parameter of dark energy starts to drop towards -1 near the end of matter era, except that the parameter α of Ricci dark energy equals to 1/2. In the case where α = 1/2, the rapidly growing non-adiabatic modes disappear when the perturbations in dark energy and matter are adiabatic initially. However, an adiabaticity between dark energy and matter perturbations at early time implies a non-adiabaticity between matter and radiation, this can influence the ordinary Sachs-Wolfe (OSW) effect. Since the amount of Ricci dark energy is not small during matter domination, the integrated Sachs-Wolfe (ISW) effect is greatly modified by density perturbations of dark energy, leading to a wrong shape of CMB power spectrum. The instability in Ricci dark energy is difficult to be alleviated if the effects of coupling between baryon and photon on dark energy perturbations are included.
5. Robust stability under additive perturbations
NASA Technical Reports Server (NTRS)
Bhaya, A.; Desoer, C. A.
1985-01-01
A MIMO linear time-invariant feedback system 1S(P,C) is considered which is assumed to be U-stable. The plant P is subjected to an additive perturbation Delta P which is proper but not necessarily stable. It is proved that the perturbed system is U-stable if and only if Delta P(I + Q x Delta P) exp -1 is U-stable.
6. Modified cyanobacteria
DOEpatents
Vermaas, Willem F J.
2014-06-17
Disclosed is a modified photoautotrophic bacterium comprising genes of interest that are modified in terms of their expression and/or coding region sequence, wherein modification of the genes of interest increases production of a desired product in the bacterium relative to the amount of the desired product production in a photoautotrophic bacterium that is not modified with respect to the genes of interest.
7. The Nervous System Uses Nonspecific Motor Learning in Response to Random Perturbations of Varying Nature
PubMed Central
2010-01-01
We constantly make small errors during movement and use them to adapt our future movements. Movement experiments often probe this error-driven learning by perturbing movements and analyzing the after-effects. Past studies have applied perturbations of varying nature such as visual disturbances, position- or velocity-dependent forces and modified inertia properties of the limb. However, little is known about how the specific nature of a perturbation influences subsequent movements. For a single perturbation trial, the nature of a perturbation may be highly uncertain to the nervous system, given that it receives only noisy information. One hypothesis is that the nervous system can use this rough estimate to partially correct for the perturbation on the next trial. Alternatively, the nervous system could ignore uncertain information about the nature of the perturbation and resort to a nonspecific adaptation. To study how the brain estimates and responds to incomplete sensory information, we test these two hypotheses using a trial-by-trial adaptation experiment. On each trial, the nature of the perturbation was chosen from six distinct types, including a visuomotor rotation and different force fields. We observed that corrective forces aiming to oppose the perturbation in the following trial were independent of the nature of the perturbation. Our results suggest that the nervous system uses a nonspecific strategy when it has high uncertainty about the nature of perturbations during trial-by-trial learning. PMID:20861427
8. Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method
Sarwar, S.; Rashidi, M. M.
2016-07-01
This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems.
9. Optimal Homotopy Asymptotic Method for Flow and Heat Transfer of a Viscoelastic Fluid in an Axisymmetric Channel with a Porous Wall
PubMed Central
Mabood, Fazle; Khan, Waqar A.; Ismail, Ahmad Izani
2013-01-01
In this article, an approximate analytical solution of flow and heat transfer for a viscoelastic fluid in an axisymmetric channel with porous wall is presented. The solution is obtained through the use of a powerful method known as Optimal Homotopy Asymptotic Method (OHAM). We obtained the approximate analytical solution for dimensionless velocity and temperature for various parameters. The influence and effect of different parameters on dimensionless velocity, temperature, friction factor, and rate of heat transfer are presented graphically. We also compared our solution with those obtained by other methods and it is found that OHAM solution is better than the other methods considered. This shows that OHAM is reliable for use to solve strongly nonlinear problems in heat transfer phenomena. PMID:24376722
10. Shock wave perturbation decay in granular materials
SciTech Connect
Vogler, Tracy J.
2015-11-05
A technique in which the evolution of a perturbation in a shock wave front is monitored as it travels through a sample is applied to granular materials. Although the approach was originally conceived as a way to measure the viscosity of the sample, here it is utilized as a means to probe the deviatoric strength of the material. Initial results for a tungsten carbide powder are presented that demonstrate the approach is viable. Simulations of the experiments using continuum and mesoscale modeling approaches are used to better understand the experiments. The best agreement with the limited experimental data is obtained for the mesoscale model, which has previously been shown to give good agreement with planar impact results. The continuum simulations indicate that the decay of the perturbation is controlled by material strength but is insensitive to the compaction response. Other sensitivities are assessed using the two modeling approaches. The simulations indicate that the configuration used in the preliminary experiments suffers from certain artifacts and should be modified to remove them. As a result, the limitations of the current instrumentation are discussed, and possible approaches to improve it are suggested.
11. Shock wave perturbation decay in granular materials
DOE PAGESBeta
Vogler, Tracy J.
2015-11-05
A technique in which the evolution of a perturbation in a shock wave front is monitored as it travels through a sample is applied to granular materials. Although the approach was originally conceived as a way to measure the viscosity of the sample, here it is utilized as a means to probe the deviatoric strength of the material. Initial results for a tungsten carbide powder are presented that demonstrate the approach is viable. Simulations of the experiments using continuum and mesoscale modeling approaches are used to better understand the experiments. The best agreement with the limited experimental data is obtainedmore » for the mesoscale model, which has previously been shown to give good agreement with planar impact results. The continuum simulations indicate that the decay of the perturbation is controlled by material strength but is insensitive to the compaction response. Other sensitivities are assessed using the two modeling approaches. The simulations indicate that the configuration used in the preliminary experiments suffers from certain artifacts and should be modified to remove them. As a result, the limitations of the current instrumentation are discussed, and possible approaches to improve it are suggested.« less
12. Perturbative stability of SFT-based cosmological models
Galli, Federico; Koshelev, Alexey S.
2011-05-01
We review the appearance of multiple scalar fields in linearized SFT based cosmological models with a single non-local scalar field. Some of these local fields are canonical real scalar fields and some are complex fields with unusual coupling. These systems only admit numerical or approximate analysis. We introduce a modified potential for multiple scalar fields that makes the system exactly solvable in the cosmological context of Friedmann equations and at the same time preserves the asymptotic behavior expected from SFT. The main part of the paper consists of the analysis of inhomogeneous cosmological perturbations in this system. We show numerically that perturbations corresponding to the new type of complex fields always vanish. As an example of application of this model we consider an explicit construction of the phantom divide crossing and prove the perturbative stability of this process at the linear order. The issue of ghosts and ways to resolve it are briefly discussed.
13. Orbit Averaging in Perturbed Planetary Rings
Stewart, Glen R.
2015-11-01
The orbital period is typically much shorter than the time scale for dynamical evolution of large-scale structures in planetary rings. This large separation in time scales motivates the derivation of reduced models by averaging the equations of motion over the local orbit period (Borderies et al. 1985, Shu et al. 1985). A more systematic procedure for carrying out the orbit averaging is to use Lie transform perturbation theory to remove the dependence on the fast angle variable from the problem order-by-order in epsilon, where the small parameter epsilon is proportional to the fractional radial distance from exact resonance. This powerful technique has been developed and refined over the past thirty years in the context of gyrokinetic theory in plasma physics (Brizard and Hahm, Rev. Mod. Phys. 79, 2007). When the Lie transform method is applied to resonantly forced rings near a mean motion resonance with a satellite, the resulting orbit-averaged equations contain the nonlinear terms found previously, but also contain additional nonlinear self-gravity terms of the same order that were missed by Borderies et al. and by Shu et al. The additional terms result from the fact that the self-consistent gravitational potential of the perturbed rings modifies the orbit-averaging transformation at nonlinear order. These additional terms are the gravitational analog of electrostatic ponderomotive forces caused by large amplitude waves in plasma physics. The revised orbit-averaged equations are shown to modify the behavior of nonlinear density waves in planetary rings compared to the previously published theory. This reserach was supported by NASA's Outer Planets Reserach program.
14. Gravitational waves from perturbed stars
Ferrari, V.
2011-12-01
Non radial oscillations of neutron stars are associated with the emission of gravitational waves. The characteristic frequencies of these oscillations can be computed using the theory of stellar perturbations, and they are shown to carry detailed information on the internal structure of the emitting source. Moreover, they appear to be encoded in various radiative processes, as for instance, in the tail of the giant flares of Soft Gamma Repeaters. Thus, their determination is central to the theory of stellar perturbation. A viable approach to the problem consists in formulating this theory as a problem of resonant scattering of gravitational waves incident on the potential barrier generated by the spacetime curvature. This approach discloses some unexpected correspondences between the theory of stellar perturbations and the theory of quantum mechanics, and allows us to predict new relativistic effects.
15. Gravitational waves from perturbed stars
Ferrari, V.
2011-03-01
Non radial oscillations of neutron stars are associated with the emission of gravitational waves. The characteristic frequencies of these oscillations can be computed using the theory of stellar perturbations, and they are shown to carry detailed information on the internal structure of the emitting source. Moreover, they appear to be encoded in various radiative processes, as for instance in the tail of the giant flares of Soft Gamma Repeaters. Thus, their determination is central to the theory of stellar perturbation. A viable approach to the problem consists in formulating this theory as a problem of resonant scattering of gravitational waves incident on the potential barrier generated by the spacetime curvature. This approach discloses some unexpected correspondences between the theory of stellar perturbations and the theory of quantum mechanics, and allows us to predict new relativistic effects.
16. Jet Perturbation by HE target
SciTech Connect
Poulsen, P; Kuklo, R M
2001-03-01
We have previously reported the degree of attenuation and perturbation by a Cu jet passing through Comp B explosive. Similar tests have now been performed with high explosive (HE) targets having CJ pressures higher than and lower than the CJ pressure of Comp B. The explosives were LX-14 and TNT, respectively. We found that the measured exit velocity of the jet where it transitions from perturbed to solid did not vary significantly as a function of HE type for each HE thickness. The radial momentum imparted to the perturbed jet segment did vary as a function of HE type, however, and we report the radial spreading of the jet and the penetration of a downstream target as a function of HE type and thickness.
17. Perturbed motion at small eccentricities
Emel'yanov, N. V.
2015-09-01
In the study of the motion of planets and moons, it is often necessary to have a simple approximate analytical motion model, which takes into account major perturbations and preserves almost the same accuracy at long time intervals. A precessing ellipse model is used for this purpose. In this paper, it is shown that for small eccentricities this model of the perturbed orbit does not correspond to body motion characteristics. There is perturbed circular motion with a constant zero mean anomaly. The corresponding solution satisfies the Lagrange equations with respect to Keplerian orbital elements. There are two families of solutions with libration and circulation changes in the mean anomaly close to this particular solution. The paper shows how the eccentricity and mean anomaly change in these solutions. Simple analytical models of the motion of the four closest moons of Jupiter consistent with available ephemerides are proposed, which in turn are obtained by the numerical integration of motion equations and are refined by observations.
18. Thermal perturbation of the Sun
NASA Technical Reports Server (NTRS)
Twigg, L. W.; Endal, A. S.
1981-01-01
An investigation of thermal perturbations of the solar convective zone via changes in the mixing length parameter were carried out, with a view toward understanding the possible solar radius and luminosity changes cited in the literature. The results show that: (a) a single perturbation of alpha is probably not the cause of the solar radius change and (b) the parameter W = d lambda nR./d lambda nL. can not be characterized by a single value, as implied in recent work.
19. Constraining dark sector perturbations II: ISW and CMB lensing tomography
Soergel, B.; Giannantonio, T.; Weller, J.; Battye, R. A.
2015-02-01
Any Dark Energy (DE) or Modified Gravity (MG) model that deviates from a cosmological constant requires a consistent treatment of its perturbations, which can be described in terms of an effective entropy perturbation and an anisotropic stress. We have considered a recently proposed generic parameterisation of DE/MG perturbations and compared it to data from the Planck satellite and six galaxy catalogues, including temperature-galaxy (Tg), CMB lensing-galaxy (varphi g) and galaxy-galaxy (gg) correlations. Combining these observables of structure formation with tests of the background expansion allows us to investigate the properties of DE/MG both at the background and the perturbative level. Our constraints on DE/MG are mostly in agreement with the cosmological constant paradigm, while we also find that the constraint on the equation of state w (assumed to be constant) depends on the model assumed for the perturbation evolution. We obtain w=-0.92+0.20-0.16 (95% CL; CMB+gg+Tg) in the entropy perturbation scenario; in the anisotropic stress case the result is w=-0.86+0.17-0.16. Including the lensing correlations shifts the results towards higher values of w. If we include a prior on the expansion history from recent Baryon Acoustic Oscillations (BAO) measurements, we find that the constraints tighten closely around w=-1, making it impossible to measure any DE/MG perturbation evolution parameters. If, however, upcoming observations from surveys like DES, Euclid or LSST show indications for a deviation from a cosmological constant, our formalism will be a useful tool towards model selection in the dark sector.
20. AGK Rules in Perturbative QCD
Bartels, Jochen
2006-06-01
I summarize the present status of the AGK cutting rules in perturbative QCD. Particular attention is given to the application of the AGK analysis to diffraction and multiple scattering in DIS at HERA and to pp collisions at the LHC. I also discuss the bootstrap conditions which appear in pQCD.
1. VHS Movies: Perturbations for Morphogenesis.
ERIC Educational Resources Information Center
Holmes, Danny L.
This paper discusses the concept of a family system in terms of an interactive system of interrelated, interdependent parts and suggests that VHS movies can act as perturbations, i.e., change promoting agents, for certain dysfunctional family systems. Several distinct characteristics of a family system are defined with particular emphasis on…
2. Adaptation Strategies in Perturbed /s/
ERIC Educational Resources Information Center
Brunner, Jana; Hoole, Phil; Perrier, Pascal
2011-01-01
The purpose of this work is to investigate the role of three articulatory parameters (tongue position, jaw position and tongue grooving) in the production of /s/. Six normal speakers' speech was perturbed by a palatal prosthesis. The fricative was recorded acoustically and through electromagnetic articulography in four conditions: (1) unperturbed,…
3. Basics of QCD perturbation theory
SciTech Connect
Soper, D.E.
1997-06-01
This is an introduction to the use of QCD perturbation theory, emphasizing generic features of the theory that enable one to separate short-time and long-time effects. The author also covers some important classes of applications: electron-positron annihilation to hadrons, deeply inelastic scattering, and hard processes in hadron-hadron collisions. 31 refs., 38 figs.
4. Generalized perturbations in neutrino mixing
Liao, Jiajun; Marfatia, D.; Whisnant, K.
2015-10-01
We derive expressions for the neutrino mixing parameters that result from complex perturbations on (1) the Majorana neutrino mass matrix (in the basis of charged lepton mass eigenstates) and on (2) the charged lepton mass matrix, for arbitrary initial (unperturbed) mixing matrices. In the first case, we find that the phases of the elements of the perturbation matrix, and the initial values of the Dirac and Majorana phases, strongly impact the leading-order corrections to the neutrino mixing parameters and phases. For experimentally compatible scenarios wherein the initial neutrino mass matrix has μ -τ symmetry, we find that the Dirac phase can take any value under small perturbations. Similarly, in the second case, perturbations to the charged lepton mass matrix can generate large corrections to the mixing angles and phases of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. As an illustration of our generalized procedure, we apply it to a situation in which nonstandard scalar and nonstandard vector interactions simultaneously affect neutrino oscillations.
5. Seven topics in perturbative QCD
SciTech Connect
Buras, A.J.
1980-09-01
The following topics of perturbative QCD are discussed: (1) deep inelastic scattering; (2) higher order corrections to e/sup +/e/sup -/ annihilation, to photon structure functions and to quarkonia decays; (3) higher order corrections to fragmentation functions and to various semi-inclusive processes; (4) higher twist contributions; (5) exclusive processes; (6) transverse momentum effects; (7) jet and photon physics.
6. Vector perturbations of galaxy number counts
Durrer, Ruth; Tansella, Vittorio
2016-07-01
We derive the contribution to relativistic galaxy number count fluctuations from vector and tensor perturbations within linear perturbation theory. Our result is consistent with the the relativistic corrections to number counts due to scalar perturbation, where the Bardeen potentials are replaced with line-of-sight projection of vector and tensor quantities. Since vector and tensor perturbations do not lead to density fluctuations the standard density term in the number counts is absent. We apply our results to vector perturbations which are induced from scalar perturbations at second order and give numerical estimates of their contributions to the power spectrum of relativistic galaxy number counts.
7. Neptune's story. [Triton's orbit perturbation
NASA Technical Reports Server (NTRS)
Goldreich, P.; Murray, N.; Longaretti, P. Y.; Banfield, D.
1989-01-01
It is conjectured that Triton was captured from a heliocentric orbit as the result of a collision with what was then one of Neptune's regular satellites. The immediate post-capture orbit was highly eccentric. Dissipation due to tides raised by Neptune in Triton caused Triton's orbit to evolve to its present state in less than one billion years. For much of this time Triton was almost entirely molten. While its orbit was evolving, Triton cannibalized most of the regular satellites of Neptune and also perturbed Nereid, thus accounting for that satellite's highly eccentric and inclined orbit. The only regular satellites of Neptune that survived were those that formed well within 5 Neptune radii, and they move on inclined orbits as the result of chaotic perturbations forced by Triton.
8. Identifying Network Perturbation in Cancer.
PubMed
Grechkin, Maxim; Logsdon, Benjamin A; Gentles, Andrew J; Lee, Su-In
2016-05-01
We present a computational framework, called DISCERN (DIfferential SparsE Regulatory Network), to identify informative topological changes in gene-regulator dependence networks inferred on the basis of mRNA expression datasets within distinct biological states. DISCERN takes two expression datasets as input: an expression dataset of diseased tissues from patients with a disease of interest and another expression dataset from matching normal tissues. DISCERN estimates the extent to which each gene is perturbed-having distinct regulator connectivity in the inferred gene-regulator dependencies between the disease and normal conditions. This approach has distinct advantages over existing methods. First, DISCERN infers conditional dependencies between candidate regulators and genes, where conditional dependence relationships discriminate the evidence for direct interactions from indirect interactions more precisely than pairwise correlation. Second, DISCERN uses a new likelihood-based scoring function to alleviate concerns about accuracy of the specific edges inferred in a particular network. DISCERN identifies perturbed genes more accurately in synthetic data than existing methods to identify perturbed genes between distinct states. In expression datasets from patients with acute myeloid leukemia (AML), breast cancer and lung cancer, genes with high DISCERN scores in each cancer are enriched for known tumor drivers, genes associated with the biological processes known to be important in the disease, and genes associated with patient prognosis, in the respective cancer. Finally, we show that DISCERN can uncover potential mechanisms underlying network perturbation by explaining observed epigenomic activity patterns in cancer and normal tissue types more accurately than alternative methods, based on the available epigenomic data from the ENCODE project. PMID:27145341
9. Adiabatic perturbations in pre-big bang models: Matching conditions and scale invariance
Durrer, Ruth; Vernizzi, Filippo
2002-10-01
At low energy, the four-dimensional effective action of the ekpyrotic model of the universe is equivalent to a slightly modified version of the pre-big bang model. We discuss cosmological perturbations in these models. In particular we address the issue of matching the perturbations from a collapsing to an expanding phase. We show that, under certain physically motivated and quite generic assumptions on the high energy corrections, one obtains n=0 for the spectrum of scalar perturbations in the original pre-big bang model (with a vanishing potential). With the same assumptions, when an exponential potential for the dilaton is included, a scale invariant spectrum (n=1) of adiabatic scalar perturbations is produced under very generic matching conditions, both in a modified pre-big bang and ekpyrotic scenario. We also derive the resulting spectrum for arbitrary power law scale factors matched to a radiation-dominated era.
10. R evolution: Improving perturbative QCD
SciTech Connect
Hoang, Andre H.; Jain, Ambar; Stewart, Iain W.; Scimemi, Ignazio
2010-07-01
Perturbative QCD results in the MS scheme can be dramatically improved by switching to a scheme that accounts for the dominant power law dependence on the factorization scale in the operator product expansion. We introduce the ''MSR scheme'' which achieves this in a Lorentz and gauge invariant way and has a very simple relation to MS. Results in MSR depend on a cutoff parameter R, in addition to the {mu} of MS. R variations can be used to independently estimate (i.) the size of power corrections, and (ii.) higher-order perturbative corrections (much like {mu} in MS). We give two examples at three-loop order, the ratio of mass splittings in the B*-B and D*-D systems, and the Ellis-Jaffe sum rule as a function of momentum transfer Q in deep inelastic scattering. Comparing to data, the perturbative MSR results work well even for Q{approx}1 GeV, and power corrections are reduced compared to MS.
11. Path integral for inflationary perturbations
Prokopec, Tomislav; Rigopoulos, Gerasimos
2010-07-01
The quantum theory of cosmological perturbations in single-field inflation is formulated in terms of a path integral. Starting from a canonical formulation, we show how the free propagators can be obtained from the well-known gauge-invariant quadratic action for scalar and tensor perturbations, and determine the interactions to arbitrary order. This approach does not require the explicit solution of the energy and momentum constraints, a novel feature which simplifies the determination of the interaction vertices. The constraints and the necessary imposition of gauge conditions is reflected in the appearance of various commuting and anticommuting auxiliary fields in the action. These auxiliary fields are not propagating physical degrees of freedom but need to be included in internal lines and loops in a diagrammatic expansion. To illustrate the formalism we discuss the tree-level three-point and four-point functions of the inflaton perturbations, reproducing the results already obtained by the methods used in the current literature. Loop calculations are left for future work.
12. R evolution: Improving perturbative QCD
Hoang, André H.; Jain, Ambar; Scimemi, Ignazio; Stewart, Iain W.
2010-07-01
Perturbative QCD results in the MS¯ scheme can be dramatically improved by switching to a scheme that accounts for the dominant power law dependence on the factorization scale in the operator product expansion. We introduce the “MSR scheme” which achieves this in a Lorentz and gauge invariant way and has a very simple relation to MS¯. Results in MSR depend on a cutoff parameter R, in addition to the μ of MS¯. R variations can be used to independently estimate (i.) the size of power corrections, and (ii.) higher-order perturbative corrections (much like μ in MS¯). We give two examples at three-loop order, the ratio of mass splittings in the B*-B and D*-D systems, and the Ellis-Jaffe sum rule as a function of momentum transfer Q in deep inelastic scattering. Comparing to data, the perturbative MSR results work well even for Q˜1GeV, and power corrections are reduced compared to MS¯.
13. Perturbation growth in accreting filaments
Clarke, S. D.; Whitworth, A. P.; Hubber, D. A.
2016-05-01
We use smoothed particle hydrodynamic simulations to investigate the growth of perturbations in infinitely long filaments as they form and grow by accretion. The growth of these perturbations leads to filament fragmentation and the formation of cores. Most previous work on this subject has been confined to the growth and fragmentation of equilibrium filaments and has found that there exists a preferential fragmentation length-scale which is roughly four times the filament's diameter. Our results show a more complicated dispersion relation with a series of peaks linking perturbation wavelength and growth rate. These are due to gravo-acoustic oscillations along the longitudinal axis during the sub-critical phase of growth. The positions of the peaks in growth rate have a strong dependence on both the mass accretion rate onto the filament and the temperature of the gas. When seeded with a multiwavelength density power spectrum, there exists a clear preferred core separation equal to the largest peak in the dispersion relation. Our results allow one to estimate a minimum age for a filament which is breaking up into regularly spaced fragments, as well as an average accretion rate. We apply the model to observations of filaments in Taurus by Tafalla & Hacar and find accretion rates consistent with those estimated by Palmeirim et al.
14. Cosmological perturbations: Vorticity, isocurvature and magnetic fields
2014-10-01
In this paper, I review some recent, interlinked, work undertaken using cosmological perturbation theory — a powerful technique for modeling inhomogeneities in the universe. The common theme which underpins these pieces of work is the presence of nonadiabatic pressure, or entropy, perturbations. After a brief introduction covering the standard techniques of describing inhomogeneities in both Newtonian and relativistic cosmology, I discuss the generation of vorticity. As in classical fluid mechanics, vorticity is not present in linearized perturbation theory (unless included as an initial condition). Allowing for entropy perturbations, and working to second order in perturbation theory, I show that vorticity is generated, even in the absence of vector perturbations, by purely scalar perturbations, the source term being quadratic in the gradients of first order energy density and isocurvature, or nonadiabatic pressure perturbations. This generalizes Crocco's theorem to a cosmological setting. I then introduce isocurvature perturbations in different models, focusing on the entropy perturbation in standard, concordance cosmology, and in inflationary models involving two scalar fields. As the final topic, I investigate magnetic fields, which are a potential observational consequence of vorticity in the early universe. I briefly review some recent work on including magnetic fields in perturbation theory in a consistent way. I show, using solely analytical techniques, that magnetic fields can be generated by higher order perturbations, albeit too small to provide the entire primordial seed field, in agreement with some numerical studies. I close this paper with a summary and some potential extensions of this work.
15. Superconvergent perturbation method in quantum mechanics
SciTech Connect
Scherer, W. )
1995-02-27
An analog of Kolmogorov's superconvergent perturbation theory in classical mechanics is constructed for self-adjoint operators. It is different from the usual Rayleigh-Schroedinger perturbation theory and yields expansions for eigenvalues and eigenvectors in terms of functions of the perturbation parameter.
16. Geometric Hamiltonian structures and perturbation theory
SciTech Connect
Omohundro, S.
1984-08-01
We have been engaged in a program of investigating the Hamiltonian structure of the various perturbation theories used in practice. We describe the geometry of a Hamiltonian structure for non-singular perturbation theory applied to Hamiltonian systems on symplectic manifolds and the connection with singular perturbation techniques based on the method of averaging.
17. Cosmological perturbations and quasistatic assumption in f (R ) theories
Chiu, Mu-Chen; Taylor, Andy; Shu, Chenggang; Tu, Hong
2015-11-01
f (R ) gravity is one of the simplest theories of modified gravity to explain the accelerated cosmic expansion. Although it is usually assumed that the quasi-Newtonian approach (a combination of the quasistatic approximation and sub-Hubble limit) for cosmic perturbations is good enough to describe the evolution of large scale structure in f (R ) models, some studies have suggested that this method is not valid for all f (R ) models. Here, we show that in the matter-dominated era, the pressure and shear equations alone, which can be recast into four first-order equations to solve for cosmological perturbations exactly, are sufficient to solve for the Newtonian potential, Ψ , and the curvature potential, Φ . Based on these two equations, we are able to clarify how the exact linear perturbations fit into different limits. We find that the Compton length controls the quasistatic behaviors in f (R ) gravity. In addition, regardless the validity of quasistatic approximation, a strong version of the sub-Hubble limit alone is sufficient to reduce the exact linear perturbations in any viable f (R ) gravity to second order. Our findings disagree with some previous studies where we find little difference between our exact and quasi-Newtonian solutions even up to k =10 c-1H0.
18. A new approach to cosmological perturbations in f(R) models
SciTech Connect
Bertacca, Daniele; Bartolo, Nicola; Matarrese, Sabino E-mail: [email protected]
2012-08-01
We propose an analytic procedure that allows to determine quantitatively the deviation in the behavior of cosmological perturbations between a given f(R) modified gravity model and a ΛCDM reference model. Our method allows to study structure formation in these models from the largest scales, of the order of the Hubble horizon, down to scales deeply inside the Hubble radius, without employing the so-called 'quasi-static' approximation. Although we restrict our analysis here to linear perturbations, our technique is completely general and can be extended to any perturbative order.
19. Preliminary Orbit Determination System (PODS) for Tracking and Data Relay Satellite System (TDRSS)-tracked target Spacecraft using the homotopy continuation method
NASA Technical Reports Server (NTRS)
Kirschner, S. M.; Samii, M. V.; Broaddus, S. R.; Doll, C. E.
1988-01-01
The Preliminary Orbit Determination System (PODS) provides early orbit determination capability in the Trajectory Computation and Orbital Products System (TCOPS) for a Tracking and Data Relay Satellite System (TDRSS)-tracked spacecraft. PODS computes a set of orbit states from an a priori estimate and six tracking measurements, consisting of any combination of TDRSS range and Doppler tracking measurements. PODS uses the homotopy continuation method to solve a set of nonlinear equations, and it is particularly effective for the case when the a priori estimate is not well known. Since range and Doppler measurements produce multiple states in PODS, a screening technique selects the desired state. PODS is executed in the TCOPS environment and can directly access all operational data sets. At the completion of the preliminary orbit determination, the PODS-generated state, along with additional tracking measurements, can be directly input to the differential correction (DC) process to generate an improved state. To validate the computational and operational capabilities of PODS, tests were performed using simulated TDRSS tracking measurements for the Cosmic Background Explorer (COBE) satellite and using real TDRSS measurements for the Earth Radiation Budget Satellite (ERBS) and the Solar Mesosphere Explorer (SME) spacecraft. The effects of various measurement combinations, varying arc lengths, and levels of degradation of the a priori state vector on the PODS solutions were considered.
20. Identifying Network Perturbation in Cancer
PubMed Central
Logsdon, Benjamin A.; Gentles, Andrew J.; Lee, Su-In
2016-01-01
We present a computational framework, called DISCERN (DIfferential SparsE Regulatory Network), to identify informative topological changes in gene-regulator dependence networks inferred on the basis of mRNA expression datasets within distinct biological states. DISCERN takes two expression datasets as input: an expression dataset of diseased tissues from patients with a disease of interest and another expression dataset from matching normal tissues. DISCERN estimates the extent to which each gene is perturbed—having distinct regulator connectivity in the inferred gene-regulator dependencies between the disease and normal conditions. This approach has distinct advantages over existing methods. First, DISCERN infers conditional dependencies between candidate regulators and genes, where conditional dependence relationships discriminate the evidence for direct interactions from indirect interactions more precisely than pairwise correlation. Second, DISCERN uses a new likelihood-based scoring function to alleviate concerns about accuracy of the specific edges inferred in a particular network. DISCERN identifies perturbed genes more accurately in synthetic data than existing methods to identify perturbed genes between distinct states. In expression datasets from patients with acute myeloid leukemia (AML), breast cancer and lung cancer, genes with high DISCERN scores in each cancer are enriched for known tumor drivers, genes associated with the biological processes known to be important in the disease, and genes associated with patient prognosis, in the respective cancer. Finally, we show that DISCERN can uncover potential mechanisms underlying network perturbation by explaining observed epigenomic activity patterns in cancer and normal tissue types more accurately than alternative methods, based on the available epigenomic data from the ENCODE project. PMID:27145341
1. Hadronic Structure from Perturbative Dressing
Arash, Firooz
2005-09-01
Perturbative dressing of a valence quark in QCD produces the internal structure of an extended object, the so-called Valon. The valon structure is universal and independent of the hosting hadron. Polarized and unpolarized proton and pion structure functions are calculated in the valon representation. One finds that although all the available data on g1p,n,d are easily reproduced, a sizable orbital angular momentum associated with the partonic structure of the valon is required in order to have a spin 1/2 valon.
2. Scaled Energy Spectroscopy of Collisionally Perturbed Potassium Rydberg States
Keeler, Matthew Len; Setzer, William
2010-03-01
We will present preliminary results on the recurrence spectroscopy (or scaled energy spectroscopy) of highly-excited potassium in the presence of collisional perturbations. Recurrence spectroscopy, with the aid of closed orbit theory, has produced useful insights into the semi-classical description of non-hydrogenic spectral features of excited atoms in external fields. We demonstrate how to apply recurrence spectroscopy to the Stark spectrum of potassium subject to collisional line-shift and line-broadening. When krypton gas is added to the system the absorption spectrum experiences line broadening, differential line shifts, and state mixing. With an appropriately modified energy scale, perturbations of the absorption spectrum become meaningful features within the scaled-energy spectrum. New features found within the recurrence spectra can then, with semi-classical closed orbit theory, be interpreted in terms of classical decoherence, elastic and inelastic collisions.
3. Dark matter perturbations and viscosity: A causal approach
Acquaviva, Giovanni; John, Anslyn; Pénin, Aurélie
2016-08-01
The inclusion of dissipative effects in cosmic fluids modifies their clustering properties and could have observable effects on the formation of large-scale structures. We analyze the evolution of density perturbations of cold dark matter endowed with causal bulk viscosity. The perturbative analysis is carried out in the Newtonian approximation and the bulk viscosity is described by the causal Israel-Stewart (IS) theory. In contrast to the noncausal Eckart theory, we obtain a third-order evolution equation for the density contrast that depends on three free parameters. For certain parameter values, the density contrast and growth factor in IS mimic their behavior in Λ CDM when z ≥1 . Interestingly, and contrary to intuition, certain sets of parameters lead to an increase of the clustering.
4. A Schwarz alternating procedure for singular perturbation problems
SciTech Connect
Garbey, M.; Kaper, H.G.
1994-12-31
The authors show that the Schwarz alternating procedure offers a good algorithm for the numerical solution of singular perturbation problems, provided the domain decomposition is properly designed to resolve the boundary and transition layers. They give sharp estimates for the optimal position of the domain boundaries and present convergence rates of the algorithm for various second-order singular perturbation problems. The splitting of the operator is domain-dependent, and the iterative solution of each subproblem is based on a modified asymptotic expansion of the operator. They show that this asymptotic-induced method leads to a family of efficient massively parallel algorithms and report on implementation results for a turning-point problem and a combustion problem.
5. Stochastic Perturbations in Type I Planetary Migraiton
Adams, Fred C.; Bloch, A. M.
2009-05-01
This talk presents a generalized treatment of Type I planetary migration in the presence of stochastic perturbations. In many planet-forming disks, the Type I migration mechanism, driven by asymmetric torques, can compromise planet formation. If the disk also supports MHD instabilities, however, the corresponding turbulent fluctuations produce additional stochastic torques that modify the steady inward migration scenario. This work studies the migration of planetary cores in the presence of stochastic fluctuations using complementary methods, including iterative maps and a Fokker-Planck approach. Stochastic torques have two main effects: [1] Through outward diffusion, a small fraction of the planetary cores can survive in the face of Type I inward migration. [2] For a given starting condition, the result of any particular realization of migration is uncertain, so that results must be described in terms of the distributions of outcomes. In addition to exploring different regimes of parameter space, this paper considers the effects of the outer disk boundary condition and time-dependence of the torque parameters. For disks with finite radii, the fraction of surviving planets decreases exponentially with time. We find the survival fractions and decay rates for a range of disk models, and find the expected distribution of locations for surviving planets. The survival fraction is expected to lie in the range 0.01 < p_S < 0.1.
6. Phase perturbation measurements through a heated ionosphere
NASA Technical Reports Server (NTRS)
Frey, A.; Gordon, W. E.
1982-01-01
High frequency radiowaves incident on an overdense (i.e., HF-frequency penetration frequency) ionosphere produce electron density irregularities. The effect of such ionospheric irregularities on the phase of UHF-radiowaves was determined. For that purpose the phase of radiowaves originating from celestial radio sources was observed with two antennas. The radiosources were chosen such that the line of sight to at least one of the antennas (usually both) passed through the modified volume of the ionosphere. Observations at 430 MHz and at 2380 MHz indicate that natural irregularities have a much stronger effect on the UHF phase fluctuations than the HF-induced irregularities for presently achieved HF-power densities of 20-80 uW/sq m. It is not clear whether some of the effects observed are the result of HF-modification of the ionosphere. Upper limits on the phase perturbations produced by HF-modification are 10 deg at 2380 MHz and 80 deg at 430 MHz.
7. The stability of the dust acoustic waves under transverse perturbations in a magnetized and collisionless dusty plasma
Gao, Dong-Ning; Qi, Xin; Hong, Xue-Ren; Yang, Xue; Duan, Wen-Shan; Yang, Lei; Yang
2014-06-01
Numerical and theoretical investigations are carried out for the stability of the dust acoustic waves (DAWs) under the transverse perturbation in a two-ion temperature magnetized and collisionless dusty plasma. The Zakharov-Kuznetsov (ZK) equation, modified ZK equation, and Extended ZK (EZK) equation of the DAWs are given by using the reductive perturbation technique. The cut-off frequency is obtained by applying higher-order transverse perturbations to the soliton solution of the EZK equation. The propagation velocity of solitary waves, the real cut-off frequency, as well as the growth rate of the higher-order perturbation to the solitary wave are obtained.
8. "Phonon" scattering beyond perturbation theory
Qiu, WuJie; Ke, XueZhi; Xi, LiLi; Wu, LiHua; Yang, Jiong; Zhang, WenQing
2016-02-01
Searching and designing materials with intrinsically low lattice thermal conductivity (LTC) have attracted extensive consideration in thermoelectrics and thermal management community. The concept of part-crystalline part-liquid state, or even part-crystalline part-amorphous state, has recently been proposed to describe the exotic structure of materials with chemical- bond hierarchy, in which a set of atoms is weakly bonded to the rest species while the other sublattices retain relatively strong rigidity. The whole system inherently manifests the coexistence of rigid crystalline sublattices and fluctuating noncrystalline substructures. Representative materials in the unusual state can be classified into two categories, i.e., caged and non-caged ones. LTCs in both systems deviate from the traditional T -1 relationship ( T, the absolute temperature), which can hardly be described by small-parameter-based perturbation approaches. Beyond the classical perturbation theory, an extra rattling-like scattering should be considered to interpret the liquid-like and sublattice-amorphization-induced heat transport. Such a kind of compounds could be promising high-performance thermoelectric materials, due to the extremely low LTCs. Other physical properties for these part-crystalline substances should also exhibit certain novelty and deserve further exploration.
9. Phytochemicals Perturb Membranes and Promiscuously Alter Protein Function
PubMed Central
2015-01-01
A wide variety of phytochemicals are consumed for their perceived health benefits. Many of these phytochemicals have been found to alter numerous cell functions, but the mechanisms underlying their biological activity tend to be poorly understood. Phenolic phytochemicals are particularly promiscuous modifiers of membrane protein function, suggesting that some of their actions may be due to a common, membrane bilayer-mediated mechanism. To test whether bilayer perturbation may underlie this diversity of actions, we examined five bioactive phenols reported to have medicinal value: capsaicin from chili peppers, curcumin from turmeric, EGCG from green tea, genistein from soybeans, and resveratrol from grapes. We find that each of these widely consumed phytochemicals alters lipid bilayer properties and the function of diverse membrane proteins. Molecular dynamics simulations show that these phytochemicals modify bilayer properties by localizing to the bilayer/solution interface. Bilayer-modifying propensity was verified using a gramicidin-based assay, and indiscriminate modulation of membrane protein function was demonstrated using four proteins: membrane-anchored metalloproteases, mechanosensitive ion channels, and voltage-dependent potassium and sodium channels. Each protein exhibited similar responses to multiple phytochemicals, consistent with a common, bilayer-mediated mechanism. Our results suggest that many effects of amphiphilic phytochemicals are due to cell membrane perturbations, rather than specific protein binding. PMID:24901212
10. Phytochemicals perturb membranes and promiscuously alter protein function.
PubMed
Ingólfsson, Helgi I; Thakur, Pratima; Herold, Karl F; Hobart, E Ashley; Ramsey, Nicole B; Periole, Xavier; de Jong, Djurre H; Zwama, Martijn; Yilmaz, Duygu; Hall, Katherine; Maretzky, Thorsten; Hemmings, Hugh C; Blobel, Carl; Marrink, Siewert J; Koçer, Armağan; Sack, Jon T; Andersen, Olaf S
2014-08-15
A wide variety of phytochemicals are consumed for their perceived health benefits. Many of these phytochemicals have been found to alter numerous cell functions, but the mechanisms underlying their biological activity tend to be poorly understood. Phenolic phytochemicals are particularly promiscuous modifiers of membrane protein function, suggesting that some of their actions may be due to a common, membrane bilayer-mediated mechanism. To test whether bilayer perturbation may underlie this diversity of actions, we examined five bioactive phenols reported to have medicinal value: capsaicin from chili peppers, curcumin from turmeric, EGCG from green tea, genistein from soybeans, and resveratrol from grapes. We find that each of these widely consumed phytochemicals alters lipid bilayer properties and the function of diverse membrane proteins. Molecular dynamics simulations show that these phytochemicals modify bilayer properties by localizing to the bilayer/solution interface. Bilayer-modifying propensity was verified using a gramicidin-based assay, and indiscriminate modulation of membrane protein function was demonstrated using four proteins: membrane-anchored metalloproteases, mechanosensitive ion channels, and voltage-dependent potassium and sodium channels. Each protein exhibited similar responses to multiple phytochemicals, consistent with a common, bilayer-mediated mechanism. Our results suggest that many effects of amphiphilic phytochemicals are due to cell membrane perturbations, rather than specific protein binding. PMID:24901212
11. Perturbations i have Known and Loved
Field, Robert W.
2011-06-01
A spectroscopic perturbation is a disruption of a ^1Σ-^1Σ-like regular pattern that can embody level-shifts, extra lines, and intensity anomalies. Once upon a time, when a band was labeled perturbed,'' it was considered worthless because it could at best yield molecular constants unsuited for archival tables. Nevertheless, a few brave spectroscopists, notably Albin Lagerqvist and Richard Barrow, collected perturbations because they knew that the pattern of multiple perturbations formed an intricate puzzle that would eventually reveal the presence and electronic symmetry of otherwise unobservable electronic states. There are many kinds of patterns of broken patterns. In my PhD thesis I showed how to determine absolute vibrational assignments for the perturber from patterns among the observed values of perturbation matrix elements. When a ^3Π state is perturbed, its six (Ω, parity) components capture a pattern of level shifts and intensity anomalies that reveals more about the nature of the perturber than a simple perturbation of the single component of a ^1Σ state. In perturbation-facilitated OODR, a perturbed singlet level acts as a spectroscopic doorway through which the entire triplet manifold may be systematically explored. For polyatomic molecule vibrations, a vibrational polyad (a group of mutually perturbing vibrational levels, among which the perturbation matrix elements are expected to follow harmonic oscillator scaling rules) can contain more components than a ^3Π state and intrapolyad patterns can be exquisitely sensitive not merely to the nature of an interloper within the polyad but also to the eigenvector character of the vibronic state from which the polyad is viewed. Variation of scaled polyad interaction parameters from one polyad to the next, a pattern of patterns, can signal proximity to an isomerization barrier. Everything in Rydberg-land seems to scale as N⋆-3, yet a trespassing valence state causes all scaling and propensity rules go
12. Speech Compensation for Time-Scale-Modified Auditory Feedback
ERIC Educational Resources Information Center
Ogane, Rintaro; Honda, Masaaki
2014-01-01
Purpose: The purpose of this study was to examine speech compensation in response to time-scale-modified auditory feedback during the transition of the semivowel for a target utterance of /ija/. Method: Each utterance session consisted of 10 control trials in the normal feedback condition followed by 20 perturbed trials in the modified auditory…
13. Entropy mode loops and cosmological correlations during perturbative reheating
SciTech Connect
Kaya, Ali; Kutluk, Emine Seyma E-mail: [email protected]
2015-01-01
Recently, it has been shown that during preheating the entropy modes circulating in the loops, which correspond to the inflaton decay products, meaningfully modify the cosmological correlation functions at superhorizon scales. In this paper, we determine the significance of the same effect when reheating occurs in the perturbative regime. In a typical two scalar field model, the magnitude of the loop corrections are shown to depend on several parameters like the background inflaton amplitude in the beginning of reheating, the inflaton decay rate and the inflaton mass. Although the loop contributions turn out to be small as compared to the preheating case, they still come out larger than the loop effects during inflation.
14. Matter perturbations in scaling cosmology
Fuño, A. Romero; Hipólito-Ricaldi, W. S.; Zimdahl, W.
2016-04-01
A suitable nonlinear interaction between dark matter with an energy density ρM and dark energy with an energy density ρX is known to give rise to a non-canonical scaling ρM ∝ ρXa-ξ, where ξ is a parameter which generally deviates from ξ = 3. Here, we present a covariant generalization of this class of models and investigate the corresponding perturbation dynamics. The resulting matter power spectrum for the special case of a time-varying Lambda model is compared with data from the Sloan Digital Sky Survey (SDSS) DR9 catalogue (Ahn et al.). We find a best-fitting value of ξ = 3.25 which corresponds to a decay of dark matter into the cosmological term. Our results are compatible with the Lambda Cold Dark Matter model at the 2σ confidence level.
15. Perturbativity in the seesaw mechanism
Asaka, Takehiko; Tsuyuki, Takanao
2016-02-01
We consider the Standard Model extended by right-handed neutrinos to explain massive neutrinos through the seesaw mechanism. The new fermion can be observed when it has a sufficiently small mass and large mixings to left-handed neutrinos. If such a particle is the lightest right-handed neutrino, its contribution to the mass matrix of active neutrinos needs to be canceled by that of a heavier one. Yukawa couplings of the heavier one are then larger than those of the lightest one. We show that the perturbativity condition gives a severe upper bound on the mixing of the lightest right-handed neutrino, depending on the masses of heavier ones. Models of high energy phenomena, such as leptogenesis, can be constrained by low energy experiments.
16. Perturbations of vortex ring pairs
Gubser, Steven S.; Horn, Bart; Parikh, Sarthak
2016-02-01
We study pairs of coaxial vortex rings starting from the action for a classical bosonic string in a three-form background. We complete earlier work on the phase diagram of classical orbits by explicitly considering the case where the circulations of the two vortex rings are equal and opposite. We then go on to study perturbations, focusing on cases where the relevant four-dimensional transfer matrix splits into two-dimensional blocks. When the circulations of the rings have the same sign, instabilities are mostly limited to wavelengths smaller than a dynamically generated length scale at which single-ring instabilities occur. When the circulations have the opposite sign, larger wavelength instabilities can occur.
17. Sudakov safety in perturbative QCD
Larkoski, Andrew J.; Marzani, Simone; Thaler, Jesse
2015-06-01
Traditional calculations in perturbative quantum chromodynamics (pQCD) are based on an order-by-order expansion in the strong coupling αs. Observables that are calculable in this way are known as "safe." Recently, a class of unsafe observables was discovered that do not have a valid αs expansion but are nevertheless calculable in pQCD using all-orders resummation. These observables are called "Sudakov safe" since singularities at each αs order are regulated by an all-orders Sudakov form factor. In this paper, we give a concrete definition of Sudakov safety based on conditional probability distributions, and we study a one-parameter family of momentum sharing observables that interpolate between the safe and unsafe regimes. The boundary between these regimes is particularly interesting, as the resulting distribution can be understood as the ultraviolet fixed point of a generalized fragmentation function, yielding a leading behavior that is independent of αs.
18. Robust control with structured perturbations
NASA Technical Reports Server (NTRS)
Keel, Leehyun
1988-01-01
Two important problems in the area of control systems design and analysis are discussed. The first is the robust stability using characteristic polynomial, which is treated first in characteristic polynomial coefficient space with respect to perturbations in the coefficients of the characteristic polynomial, and then for a control system containing perturbed parameters in the transfer function description of the plant. In coefficient space, a simple expression is first given for the l(sup 2) stability margin for both monic and non-monic cases. Following this, a method is extended to reveal much larger stability region. This result has been extended to the parameter space so that one can determine the stability margin, in terms of ranges of parameter variations, of the closed loop system when the nominal stabilizing controller is given. The stability margin can be enlarged by a choice of better stabilizing controller. The second problem describes the lower order stabilization problem, the motivation of the problem is as follows. Even though the wide range of stabilizing controller design methodologies is available in both the state space and transfer function domains, all of these methods produce unnecessarily high order controllers. In practice, the stabilization is only one of many requirements to be satisfied. Therefore, if the order of a stabilizing controller is excessively high, one can normally expect to have a even higher order controller on the completion of design such as inclusion of dynamic response requirements, etc. Therefore, it is reasonable to have a lowest possible order stabilizing controller first and then adjust the controller to meet additional requirements. The algorithm for designing a lower order stabilizing controller is given. The algorithm does not necessarily produce the minimum order controller; however, the algorithm is theoretically logical and some simulation results show that the algorithm works in general.
19. Cosmological perturbations in f(T) gravity
SciTech Connect
Chen, Shih-Hung; Dent, James B.; Dutta, Sourish; Saridakis, Emmanuel N.
2011-01-15
We investigate the cosmological perturbations in f(T) gravity. Examining the pure gravitational perturbations in the scalar sector using a diagonal vierbein, we extract the corresponding dispersion relation, which provides a constraint on the f(T) Ansaetze that lead to a theory free of instabilities. Additionally, upon inclusion of the matter perturbations, we derive the fully perturbed equations of motion, and we study the growth of matter overdensities. We show that f(T) gravity with f(T) constant coincides with General Relativity, both at the background as well as at the first-order perturbation level. Applying our formalism to the power-law model we find that on large subhorizon scales (O(100 Mpc) or larger), the evolution of matter overdensity will differ from {Lambda}CDM cosmology. Finally, examining the linear perturbations of the vector and tensor sectors, we find that (for the standard choice of vierbein) f(T) gravity is free of massive gravitons.
20. Observational tests of modified gravity
SciTech Connect
Jain, Bhuvnesh; Zhang Pengjie
2008-09-15
Modifications of general relativity provide an alternative explanation to dark energy for the observed acceleration of the Universe. Modified gravity theories have richer observational consequences for large-scale structures than conventional dark energy models, in that different observables are not described by a single growth factor even in the linear regime. We examine the relationships between perturbations in the metric potentials, density and velocity fields, and discuss strategies for measuring them using gravitational lensing, galaxy cluster abundances, galaxy clustering/dynamics, and the integrated Sachs-Wolfe effect. We show how a broad class of gravity theories can be tested by combining these probes. A robust way to interpret observations is by constraining two key functions: the ratio of the two metric potentials, and the ratio of the gravitational 'constant' in the Poisson equation to Newton's constant. We also discuss quasilinear effects that carry signatures of gravity, such as through induced three-point correlations. Clustering of dark energy can mimic features of modified gravity theories and thus confuse the search for distinct signatures of such theories. It can produce pressure perturbations and anisotropic stresses, which break the equality between the two metric potentials even in general relativity. With these two extra degrees of freedom, can a clustered dark energy model mimic modified gravity models in all observational tests? We show with specific examples that observational constraints on both the metric potentials and density perturbations can in principle distinguish modifications of gravity from dark energy models. We compare our result with other recent studies that have slightly different assumptions (and apparently contradictory conclusions)
1. Singular perturbation applications in neutron transport
SciTech Connect
Losey, D.C.; Lee, J.C.
1996-09-01
This is a paper on singular perturbation applications in neutron transport for submission at the next ANS conference. A singular perturbation technique was developed for neutron transport analysis by postulating expansion in terms of a small ordering parameter {eta}. Our perturbation analysis is carried, without approximation, through {Omicron}({eta}{sup 2}) to derive a material interface correction for diffusion theory. Here we present results from an analytical application of the perturbation technique to a fixed source problem and then describe and implementation of the technique in a computational scheme.
2. Kato expansion in quantum canonical perturbation theory
Nikolaev, Andrey
2016-06-01
This work establishes a connection between canonical perturbation series in quantum mechanics and a Kato expansion for the resolvent of the Liouville superoperator. Our approach leads to an explicit expression for a generator of a block-diagonalizing Dyson's ordered exponential in arbitrary perturbation order. Unitary intertwining of perturbed and unperturbed averaging superprojectors allows for a description of ambiguities in the generator and block-diagonalized Hamiltonian. We compare the efficiency of the corresponding computational algorithm with the efficiencies of the Van Vleck and Magnus methods for high perturbative orders.
3. Singularity perturbed zero dynamics of nonlinear systems
NASA Technical Reports Server (NTRS)
Isidori, A.; Sastry, S. S.; Kokotovic, P. V.; Byrnes, C. I.
1992-01-01
Stability properties of zero dynamics are among the crucial input-output properties of both linear and nonlinear systems. Unstable, or 'nonminimum phase', zero dynamics are a major obstacle to input-output linearization and high-gain designs. An analysis of the effects of regular perturbations in system equations on zero dynamics shows that whenever a perturbation decreases the system's relative degree, it manifests itself as a singular perturbation of zero dynamics. Conditions are given under which the zero dynamics evolve in two timescales characteristic of a standard singular perturbation form that allows a separate analysis of slow and fast parts of the zero dynamics.
4. Stellar oscillations in modified gravity
Sakstein, Jeremy
2013-12-01
Starting from the equations of modified gravity hydrodynamics, we derive the equations of motion governing linear, adiabatic, radial perturbations of stars in scalar-tensor theories. There are two new features: first, the eigenvalue equation for the period of stellar oscillations is modified such that the eigenfrequencies are always larger than predicted by general relativity. Second, the general relativity condition for stellar instability is altered so that the adiabatic index can fall below 4/3 before unstable modes appear. Stars are more stable in modified gravity theories. Specializing to the case of chameleonlike theories, we investigate these effects numerically using both polytropic Lane-Emden stars and models coming from modified gravity stellar structure simulations. We find that the change in the oscillation period of Cepheid star models can be as large as 30% for order-one matter couplings and the change in the inferred distance using the period-luminosity relation can be up to three times larger than if one had only considered the modified equilibrium structure. We discuss the implications of these results for recent and upcoming astrophysical tests and estimate that previous methods can produce new constraints such that the modifications are screened in regions of Newtonian potential of O(10-8).
5. System-reservoir theory with anharmonic baths: a perturbative approach
2016-04-01
In this paper we develop the formalism of a general system coupled to a reservoir (the words ‘bath’ and ‘reservoir’ will be used interchangeably) consisting of nonlinear oscillators, based on perturbation theory at the classical level, by extending the standard Zwanzig approach of elimination of bath degrees of freedom order by order in perturbation. We observe that the fluctuation dissipation relation (FDR) of the second kind in its standard form for harmonic baths gets modified due to the nonlinearity and this is manifested through higher powers of {{k}\\text{B}}T in the expression for two-time noise correlation. On the flip side, this very modification allows us to define a dressed (renormalized) system-bath coupling that depends on the temperature and the nonlinear parameters of the bath in such a way that the structure of the FDR (of the second kind) is maintained. As an aside, we also observe that the first moment of the noise arising from a nonlinear bath can be non-zero, even in the absence of any external drive, if the reservoir potential is asymmetric with respect to one of its minima, about which one builds up the perturbation theory.
6. Perturbation theory in light-cone quantization
SciTech Connect
Langnau, A.
1992-01-01
A thorough investigation of light-cone properties which are characteristic for higher dimensions is very important. The easiest way of addressing these issues is by analyzing the perturbative structure of light-cone field theories first. Perturbative studies cannot be substituted for an analysis of problems related to a nonperturbative approach. However, in order to lay down groundwork for upcoming nonperturbative studies, it is indispensable to validate the renormalization methods at the perturbative level, i.e., to gain control over the perturbative treatment first. A clear understanding of divergences in perturbation theory, as well as their numerical treatment, is a necessary first step towards formulating such a program. The first objective of this dissertation is to clarify this issue, at least in second and fourth-order in perturbation theory. The work in this dissertation can provide guidance for the choice of counterterms in Discrete Light-Cone Quantization or the Tamm-Dancoff approach. A second objective of this work is the study of light-cone perturbation theory as a competitive tool for conducting perturbative Feynman diagram calculations. Feynman perturbation theory has become the most practical tool for computing cross sections in high energy physics and other physical properties of field theory. Although this standard covariant method has been applied to a great range of problems, computations beyond one-loop corrections are very difficult. Because of the algebraic complexity of the Feynman calculations in higher-order perturbation theory, it is desirable to automatize Feynman diagram calculations so that algebraic manipulation programs can carry out almost the entire calculation. This thesis presents a step in this direction. The technique we are elaborating on here is known as light-cone perturbation theory.
7. Non-perturbative approach for curvature perturbations in stochastic δ N formalism
SciTech Connect
Fujita, Tomohiro; Kawasaki, Masahiro; Tada, Yuichiro E-mail: [email protected]
2014-10-01
In our previous paper [1], we have proposed a new algorithm to calculate the power spectrum of the curvature perturbations generated in inflationary universe with use of the stochastic approach. Since this algorithm does not need the perturbative expansion with respect to the inflaton fields on super-horizon scale, it works even in highly stochastic cases. For example, when the curvature perturbations are very large or the non-Gaussianities of the curvature perturbations are sizable, the perturbative expansion may break down but our algorithm enables to calculate the curvature perturbations. We apply it to two well-known inflation models, chaotic and hybrid inflation, in this paper. Especially for hybrid inflation, while the potential is very flat around the critical point and the standard perturbative computation is problematic, we successfully calculate the curvature perturbations.
8. Importance of Plasma Response to Non-axisymmetric Perturbations in Tokamaks
SciTech Connect
Jong-kyu Park, Allen H. Boozer, Jonathan E. Menard, Andrea M. Garofalo, Michael J. Schaffer, Richard J. Hawryluk, Stanley M. Kaye, Stefan P. Gerhardt, Steve A. Sabbagh, and the NSTX Team
2009-04-22
Tokamaks are sensitive to deviations from axisymmetry as small as δB=B0 ~ 10-4. These non-axisymmetric perturbations greatly modify plasma confinement and performance by either destroying magnetic surfaces with subsequent locking or deforming magnetic surfaces with associated non-ambipolar transport. The Ideal Perturbed Equilibrium Code (IPEC) calculates ideal perturbed equilibria and provides important basis for understanding the sensitivity of tokamak plasmas to perturbations. IPEC calculations indicate that the ideal plasma response, or equiva- lently the effect by ideally perturbed plasma currents, is essential to explain locking experiments on National Spherical Torus eXperiment (NSTX) and DIII-D. The ideal plasma response is also important for Neoclassical Toroidal Viscosity (NTV) in non-ambipolar transport. The consistency between NTV theory and magnetic braking experiments on NSTX and DIII-D can be improved when the variation in the field strength in IPEC is coupled with generalized NTV theory. These plasma response effects will be compared with the previous vacuum superpositions to illustrate the importance. However, plasma response based on ideal perturbed equilibria is still not suffciently accurate to predict the details of NTV transport, and can be inconsistent when currents associated with a toroidal torque become comparable to ideal perturbed currents.
9. Quantum field perturbation theory revisited
Matone, Marco
2016-03-01
Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in D dimension with exponential interactions, such as μDexp (α ϕ ). In particular, we use the relation exp (α δ/δ J (x ) )exp (-Z0[J ])=exp (-Z0[J +αx]) with J the external source, and αx(y )=α δ (y -x ). Such a shift is strictly related to the normal ordering of exp (α ϕ ) and to a scaling relation which follows by renormalizing μ . Next, we derive a new formulation of perturbation theory for the potentials V (ϕ )=λ/n ! :ϕn: , using the generating functional associated to :exp (α ϕ ):. The Δ (0 )-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to J to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.
10. Non-Perturbative Field Theories.
Stephenson, David
Available from UMI in association with The British Library. Requires signed TDF. Some non-perturbative aspects of field theories are studied by applying lattice gauge theory techniques. The low-lying hadronic mass spectrum is calculated numerically using quenched lattice quantum chromodynamics. The results of large numerical simulations performed on a distributed array processor are presented and analysed. Particular emphasis is stressed upon the understanding of systematic and statistical errors in the calculation. In addition, the pion decay constant and the chiral condensate are evaluated. An attempt is made to relate the numerical findings to the experimentally measured quantities. A pioneering attempt to understand Yukawa couplings is discussed. A toy Fermion-Higgs system is studied numerically on a transputer array. Dynamical fermions are included in the investigation of the behavior of the system over a wide range of Yukawa couplings. A phase diagram is found for the model which shows evidence of spontaneous chiral symmetry breaking transitions. Extensions of the model are discussed together some speculations concerning the behaviour of Yukawa couplings in general. The possibility of using the lattice as a model for space-time is investigated by studying the propagation of particles on a fractal lattice. In addition, the use of truncated fractals as novel regulators is studied numerically in the hope that the problem of fermion doubling will be alleviated.
11. Scalar Quantum Electrodynamics: Perturbation Theory and Beyond
SciTech Connect
Bashir, A.; Gutierrez-Guerrero, L. X.; Concha-Sanchez, Y.
2006-09-25
In this article, we calculate scalar propagator in arbitrary dimensions and gauge and the three-point scalar-photon vertex in arbitrary dimensions and Feynman gauge, both at the one loop level. We also discuss constraints on their non perturbative structure imposed by requirements of gauge invariance and perturbation theory.
12. Degenerate Open Shell Density Perturbation Theory
Palenik, Mark; Dunlap, Brett
The density perturbation theory (DPT) methodology we have developed applies the Hohenberg-Kohn theorem to perturbations in density functional theory. At each order, the energy is directly minimized with respect to the density at all lower orders. The difference between the perturbed and unperturbed densities is expanded in terms of a finite number of basis functions, and a single matrix inversion in this space reduces the complexity of the problem to that of non-interacting perturbation theory. For open-shell systems with symmetry, however, the situation becomes more complex. Typically, the perturbation will break the symmetry leading to a zeroth-order shift in the Kohn-Sham potential. Because the symmetry breaking is independent of the strength of the perturbation, the mapping from the initial to the perturbed KS potential is discontinuous and techniques from perturbation theory for noninteracting particles fail. We describe a rigorous formulation of DPT for use in systems that display an initial degeneracy, such as atoms and Fe55Cp*12 clusters and present initial calculations on these systems.
13. Intelligent perturbation algorithms for space scheduling optimization
NASA Technical Reports Server (NTRS)
Kurtzman, Clifford R.
1991-01-01
Intelligent perturbation algorithms for space scheduling optimization are presented in the form of the viewgraphs. The following subject areas are covered: optimization of planning, scheduling, and manifesting; searching a discrete configuration space; heuristic algorithms used for optimization; use of heuristic methods on a sample scheduling problem; intelligent perturbation algorithms are iterative refinement techniques; properties of a good iterative search operator; dispatching examples of intelligent perturbation algorithm and perturbation operator attributes; scheduling implementations using intelligent perturbation algorithms; major advances in scheduling capabilities; the prototype ISF (industrial Space Facility) experiment scheduler; optimized schedule (max revenue); multi-variable optimization; Space Station design reference mission scheduling; ISF-TDRSS command scheduling demonstration; and example task - communications check.
14. Covariant generalization of cosmological perturbation theory
SciTech Connect
Enqvist, Kari; Hoegdahl, Janne; Nurmi, Sami; Vernizzi, Filippo
2007-01-15
We present an approach to cosmological perturbations based on a covariant perturbative expansion between two worldlines in the real inhomogeneous universe. As an application, at an arbitrary order we define an exact scalar quantity which describes the inhomogeneities in the number of e-folds on uniform density hypersurfaces and which is conserved on all scales for a barotropic ideal fluid. We derive a compact form for its conservation equation at all orders and assign it a simple physical interpretation. To make a comparison with the standard perturbation theory, we develop a method to construct gauge-invariant quantities in a coordinate system at arbitrary order, which we apply to derive the form of the nth order perturbation in the number of e-folds on uniform density hypersurfaces and its exact evolution equation. On large scales, this provides the gauge-invariant expression for the curvature perturbation on uniform density hypersurfaces and its evolution equation at any order.
15. On the perturbation of the luminosity distance by peculiar motions
Kaiser, Nick; Hudson, Michael J.
2015-06-01
We consider some aspects of the perturbation to the luminosity distance d(z) that are of relevance for SN1a cosmology and for future peculiar velocity surveys at non-negligible redshifts. (1) Previous work has shown that the correction to the lowest order perturbation δd/d = -δv/cz has the peculiar characteristic that it appears to depend on the absolute state of motion of sources, rather than on their motion relative to that of the observer. The resolution of this apparent violation of the equivalence principle is that it is necessary to allow for evolution of the velocities with time, and also, when considering perturbations on the scale of the observer-source separation, to include the gravitational redshift effect. We provide an expression for δd/d that provides a physically consistent way to measure peculiar velocities and determine their impact for SN1a cosmology. (2) We then calculate the perturbation to the redshift as a function of source flux density, which has been proposed as an alternative probe of large-scale motions. We show how the inclusion of surface brightness modulation modifies the relation between δz(m) and the peculiar velocity, and that, while the noise properties of this method might appear promising, the velocity signal is swamped by the effect of galaxy clustering for most scales of interest. (3) We show how, in linear theory, peculiar velocity measurements are biased downwards by the effect of smaller scale motions or by measurement errors (such as in photometric redshifts). Our results nicely explain the effects seen in simulations by Koda et al. We critically examine the prospects for extending peculiar velocity studies to larger scales with near-term future surveys.
16. Tubulin-perturbing naphthoquinone spiroketals.
PubMed
Balachandran, Raghavan; Hopkins, Tamara D; Thomas, Catherine A; Wipf, Peter; Day, Billy W
2008-02-01
Several natural and synthetic naphthoquinone spiroketals are potent inhibitors of the thioredoxin-thioredoxin reductase redox system. Based on the antimitotic and weak antitubulin actions noted for SR-7 ([8-(furan-3-ylmethoxy)-1-oxo-1,4-dihydronaphthalene-4-spiro-2'-naphtho[1'',8''-de][1',3'][dioxin]), a library of related compounds was screened for tubulin-perturbing properties. Two compounds, TH-169 (5'-hydroxy-4'H-spiro[1,3-dioxolane-2,1'-naphthalen]-4'-one) and TH-223 (5'-methoxy-4'H-spiro[1,3-dioxane-2,1'-naphthalen]-4'-one), had substantial effects on tubulin assembly and were antiproliferative at low micromolar concentrations. TH-169 was the most potent at blocking GTP-dependent polymerization of 10 mum tubulin in vitro with a remarkable 50% inhibitory concentration of ca. 400 nm. It had no effect on paclitaxel-induced microtubule assembly and did not cause microtubule hypernucleation. TH-169 failed to compete with colchicine for binding to beta-tubulin. The 50% antiproliferative concentration of TH-169 against human cancer cells was at or slightly below 1 mum. Flow cytometry showed that 1 mum TH-169 caused an increase in G(2)/M and hypodiploid cells. TH-169 eliminated the PC-3 cells' polyploid population and increased their expression of p21(WAF1) and Hsp70 in a concentration-dependent manner. The antiproliferative effect of TH-169 was irreversible and independent of changes in caspases, actin, tubulin, glyceraldehyde phosphate dehydrogenase or Bcl-x(S/L). This structurally simple naphthoquinone spiroketal represents a small molecule, tubulin-interactive agent with a novel apoptotic pathway and attractive biological function. PMID:18194192
17. Transient dynamics of perturbations in astrophysical disks
Razdoburdin, D. N.; Zhuravlev, V. V.
2015-11-01
We review some aspects of a major unsolved problem in understanding astrophysical (in particular, accretion) disks: whether the disk interiors can be effectively viscous in spite of the absence of magnetorotational instability. A rotational homogeneous inviscid flow with a Keplerian angular velocity profile is spectrally stable, making the transient growth of perturbations a candidate mechanism for energy transfer from regular motion to perturbations. Transient perturbations differ qualitatively from perturbation modes and can grow substantially in shear flows due to the nonnormality of their dynamical evolution operator. Because the eigenvectors of this operator, also known as perturbation modes, are not pairwise orthogonal, they can mutually interfere, resulting in the transient growth of their linear combinations. Physically, a growing transient perturbation is a leading spiral whose branches are shrunk as a result of the differential rotation of the flow. We discuss in detail the transient growth of vortex shearing harmonics in the spatially local limit, as well as methods for identifying the optimal (fastest growth) perturbations. Special attention is given to obtaining such solutions variationally by integrating the respective direct and adjoint equations forward and backward in time. The presentation is intended for experts new to the subject.
18. Expansion of Perturbation Theory Applied to Shim Rotation Automation of the Advanced Test Reactor
Peterson, Joshua Loren
In 2007, the Department of Energy (DOE) declared the Advanced Test Reactor (ATR) a National Scientific User Facility (NSUF). This declaration expanded the focus of the ATR to include diversified classes of academic and industrial experiments. An essential part of the new suite of more accurate and flexible codes being deployed to support the NSUF is their ability to predict reactor behavior at startup, particularly the position of the outer shim control cylinders (OSCC). The current method used for calculating the OSCC positions during a cycle startup utilizes a heuristic trial and error approach that is impractical with the computationally intensive reactor physics tools, such as NEWT. It is therefore desirable that shim rotation prediction for startup be automated. Shim rotation prediction with perturbation theory was chosen to be investigated as one method for use with startup calculation automation. A modified form of first order perturbation theory, called phase space interpolated perturbation theory, was developed to more accurately model shim rotation prediction. Shim rotation prediction is just one application for this new modified form of perturbation theory. Phase space interpolated perturbation theory can be used on any application where the range of change to the system is known a priori, but the magnitude of change is not known. A cubic regression method was also developed to automate shim rotation prediction by using only forward solutions to the transport equation.
19. Calculating nonadiabatic pressure perturbations during multifield inflation
2012-03-01
Isocurvature perturbations naturally occur in models of inflation consisting of more than one scalar field. In this paper, we calculate the spectrum of isocurvature perturbations generated at the end of inflation for three different inflationary models consisting of two canonical scalar fields. The amount of nonadiabatic pressure present at the end of inflation can have observational consequences through the generation of vorticity and subsequently the sourcing of B-mode polarization. We compare two different definitions of isocurvature perturbations and show how these quantities evolve in different ways during inflation. Our results are calculated using the open source Pyflation numerical package which is available to download.
20. Vector perturbations in a contracting Universe
SciTech Connect
Battefeld, T.J.; Brandenberger, R.
2004-12-15
In this note we show that vector perturbations exhibit growing mode solutions in a contracting Universe, such as the contracting phase of the pre big bang or the cyclic/ekpyrotic models of the Universe. This is not a gauge artifact and will in general lead to the breakdown of perturbation theory--a severe problem that has to be addressed in any bouncing model. We also comment on the possibility of explaining, by means of primordial vector perturbations, the existence of the observed large-scale magnetic fields. This is possible since they can be seeded by vorticity.
1. Cosmological perturbations and the Weinberg theorem
2015-12-01
The celebrated Weinberg theorem in cosmological perturbation theory states that there always exist two adiabatic scalar modes in which the comoving curvature perturbation is conserved on super-horizon scales. In particular, when the perturbations are generated from a single source, such as in single field models of inflation, both of the two allowed independent solutions are adiabatic and conserved on super-horizon scales. There are few known examples in literature which violate this theorem. We revisit the theorem and specify the loopholes in some technical assumptions which violate the theorem in models of non-attractor inflation, fluid inflation, solid inflation and in the model of pseudo conformal universe.
2. Evolution of non-spherical perturbations.
Boschan, P.
1995-06-01
In this paper I investigate the evolution of primordial non-spherical positive and negative fluctuations. They can be calculated by second order of perturbation theory. I solved analytically the second order equation for arbitrary density parameter {OMEGA}_M0_ and cosmological constant {LAMBDA} using the approximation introduced by Martell & Freundling (???). The second order solution is compared with the exact one in the spherical case. I find that the initial deformation grows rapidly for positive perturbations, while the negative perturbations (voids) are stable against deformations.
3. Perturbation calculation of thermodynamic density of states
SciTech Connect
Brown, Greg; Schulthess, Thomas C; Nicholson, Don M; Eisenbach, Markus; Stocks, George Malcolm
2011-01-01
The density of states g( ) is frequently used to calculate the temperature-dependent properties of a thermodynamic system. Here a derivation is given for calculating the warped density of states g ( ) resulting from the addition of a perturbation. The method is validated for a classical Heisenberg model of bcc Fe and the errors in the free energy are shown to be second order in the perturbation. Taking the perturbation to be the difference between a first-principles quantum-mechanical energy and a corresponding classical energy, this method can significantly reduce the computational effort required to calculate g( ) for quantum systems using the Wang-Landau approach.
4. HIV-associated memory B cell perturbations
PubMed Central
Hu, Zhiliang; Luo, Zhenwu; Wan, Zhuang; Wu, Hao; Li, Wei; Zhang, Tong; Jiang, Wei
2015-01-01
Memory B-cell depletion, hyperimmunoglobulinemia, and impaired vaccine responses are the hallmark of B cell perturbations inhuman immunodeficiency virus (HIV) disease. Although B cells are not the targets for HIV infection, there is evidence for B cell, especially memory B cell dysfunction in HIV disease mediated by other cells or HIV itself. This review will focus on HIV-associated phenotypic and functional alterations in memory B cells. Additionally, we will discuss the mechanism underlying these perturbations and the effect of anti-retroviral therapy (ART) on these perturbations. PMID:25887082
5. Perturbing macroscopic magnetohydrodynamic stability for toroidal plasmas
Comer, Kathryn J.
We have introduced a new perturbative technique to rapidly explore the dependence of long wavelength ideal magnetohydrodynamic (MHD) instabilities on equilibrium profiles, shaping properties, and wall parameters. Traditionally, these relations are studied with numerical parameter scans using computationally intensive stability codes. Our perturbative technique first finds the equilibrium and stability using traditional methods. Subsequent small changes in the original equilibrium parameters change the stability. We quickly find the new stability with an expansion of the energy principle, rather than with another run of the stability codes. We first semi-analytically apply the technique to the screw pinch after eliminating compressional Alfven wave effects. The screw pinch results validate the approach, but also indicate that allowable perturbations to equilibria with certain features may be restricted. Next, we extend the approach to toroidal geometry using experimental equilibria and a simple constructed equilibrium, with the ideal MHD stability code GATO. Stability properties are successfully predicted from perturbed toroidal equilibria when only the vacuum beyond the plasma is perturbed (through wall parameter variations), rather than the plasma itself. Small plasma equilibrium perturbations to both experimental and simple equilibria result in very large errors to the predicted stability, and valid results are found only over a narrow range of most perturbations. Despite the large errors produced when changing plasma parameters, the wall perturbations revealed two useful applications of this technique. Because the calculations are non-iterative matrix multiplications, the convergence issues that can disrupt a full MHD stability code are absent. Marginal stability, therefore, is much easier to find with the perturbative technique. Also, the perturbed results can be input as the initial guess for the eigenvalue for a full stability code, and improve subsequent
6. A Model for Gaussian Perturbations of Graphene
Dodson, C. T. J.
2015-11-01
Graphene consists nominally of a regular planar hexagonal carbon lattice monolayer. However, its structure experiences perturbations in the presence of external influences, whether from substrate properties, thermal or electromagnetic fields, or ambient fluid movement. Here we give an information geometric model to represent the state space of perturbations as a Riemannian pseudosphere with scalar curvature close to -1/2. This would allow the representation of a trajectory of states under a given ambient or process change, so opening the possibility for geometrically formulated dynamical models to link structural perturbations to the physics.
7. Perturbations of black p-branes
SciTech Connect
Abdalla, Elcio; Fernandez Piedra, Owen Pavel; Oliveira, Jeferson de; Molina, C.
2010-03-15
We consider black p-brane solutions of the low-energy string action, computing scalar perturbations. Using standard methods, we derive the wave equations obeyed by the perturbations and treat them analytically and numerically. We have found that tensorial perturbations obtained via a gauge-invariant formalism leads to the same results as scalar perturbations. No instability has been found. Asymptotically, these solutions typically reduce to a AdS{sub (p+2)}xS{sup (8-p)} space which, in the framework of Maldacena's conjecture, can be regarded as a gravitational dual to a conformal field theory defined in a (p+1)-dimensional flat space-time. The results presented open the possibility of a better understanding the AdS/CFT correspondence, as originally formulated in terms of the relation among brane structures and gauge theories.
8. The Perturbational MO Method for Saturated Systems.
ERIC Educational Resources Information Center
Herndon, William C.
1979-01-01
Summarizes a theoretical approach using nonbonding MO's and perturbation theory to correlate properties of saturated hydrocarbons. Discussion is limited to correctly predicted using this method. Suggests calculations can be carried out quickly in organic chemistry. (Author/SA)
9. Controlling roll perturbations in fruit flies.
PubMed
Beatus, Tsevi; Guckenheimer, John M; Cohen, Itai
2015-04-01
Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 100° within 30 ± 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional-integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom. PMID:25762650
10. SHARP ENTRYWISE PERTURBATION BOUNDS FOR MARKOV CHAINS
PubMed Central
THIEDE, ERIK; VAN KOTEN, BRIAN; WEARE, JONATHAN
2015-01-01
For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem in computational statistical physics. We have derived perturbation bounds on the relative error of the invariant distribution that reveal these variations in sensitivity. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature. Moreover, our bounds have a simple interpretation in terms of hitting times, which can be used to draw intuitive but rigorous conclusions about the sensitivity of a chain to various types of perturbations. PMID:26491218
11. Cosmological perturbations in mimetic Horndeski gravity
Arroja, Frederico; Bartolo, Nicola; Karmakar, Purnendu; Matarrese, Sabino
2016-04-01
We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic G3 theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. We find solutions for the Newtonian potential evolution equation in two simple models. Finally we show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom.
12. Casimir energy for perturbed surfaces of revolution
Morales-Almazan, Pedro
2016-03-01
In this paper, we explore the zeta function arising from a small perturbation on a surface of revolution and the effect of this on the functional determinant and on the change of the Casimir energy associated with the surface.
13. Controlling roll perturbations in fruit flies
PubMed Central
Beatus, Tsevi; Guckenheimer, John M.; Cohen, Itai
2015-01-01
Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 100° within 30 ± 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional–integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom. PMID:25762650
14. Cosmological perturbations in theories with non-minimal coupling between curvature and matter
SciTech Connect
Bertolami, Orfeu; Frazão, Pedro; Páramos, Jorge E-mail: [email protected]
2013-05-01
In this work, we examine how the presence of a non-minimal coupling between spacetime curvature and matter affects the evolution of cosmological perturbations on a homogeneous and isotropic Universe, and hence the formation of large-scale structure. This framework places constraints on the terms which arise due to the coupling with matter and, in particular, on the modified growth of matter density perturbations. We derive approximate analytical solutions for the evolution of matter overdensities during the matter dominated era and discuss the compatibility of the obtained results with the hypothesis that the late time acceleration of the Universe is driven by a non-minimal coupling.
15. Constructing perturbation theory kernels for large-scale structure in generalized cosmologies
Taruya, Atsushi
2016-07-01
We present a simple numerical scheme for perturbation theory (PT) calculations of large-scale structure. Solving the evolution equations for perturbations numerically, we construct the PT kernels as building blocks of statistical calculations, from which the power spectrum and/or correlation function can be systematically computed. The scheme is especially applicable to the generalized structure formation including modified gravity, in which the analytic construction of PT kernels is intractable. As an illustration, we show several examples for power spectrum calculations in f (R ) gravity and Λ CDM models.
16. The relationship between the human state and external perturbations of atmospheric, geomagnetic and solar origin
Gavryuseva, E.; Kroussanova, N.
2002-12-01
The relationship between the state of human body and the external factors such as the different phenomena of solar activity, geomagnetic perturbations and local atmospheric characteristics is studied. The monitoring of blood pressure and electro-conductivity of human body in acupuncture points for a group fo 28 people over the period of 1.5 year has been performed daily from February 2001 to August 2002 in Capodimonte Observatory in Naples, Italy. The modified Voll method of electropuncture diagnostics was used. The strong correlation between the human body state and meteo conditions is found and the probable correlation with geomagnetic perturbations is discussed.
17. Covariant perturbations of f(R) black holes: the Weyl terms
Pratten, Geraint
2015-08-01
In this paper we revisit non-spherical perturbations of the Schwarzschild black hole in the context of f(R) gravity. Previous studies were able to demonstrate the stability of the f(R) Schwarzschild black hole against gravitational perturbations in both the even and odd parity sectors. In particular, it was seen that the Regge-Wheeler (RW) and Zerilli equations in f(R) gravity obey the same equations as their general relativity (GR) counterparts. More recently, the 1+1+2 semi-tetrad formalism has been used to derive a set of two wave equations: one for transverse, trace-free (tensor) perturbations and one for the additional scalar modes that characterize fourth-order theories of gravitation. The master variable governing tensor perturbations was shown to be a modified RW tensor obeying the same equation as in GR. However, it is well known that there is a non-uniqueness in the definition of the master variable. In this paper we derive a set of two perturbation variables and their concomitant wave equations that describe gravitational perturbations in a covariant and gauge invariant manner. These variables can be related to the Newman-Penrose (NP) Weyl scalars as well as the master variables from the 2+2 formalism. As a byproduct of this study, we also derive a set of useful results relating the NP formalism to the 1+1+2 formalism valid for LRS-II spacetimes.
18. On perturbations of a quintom bounce
SciTech Connect
Cai Yifu; Qiu Taotao; Zhang Xinmin; Brandenberger, Robert; Piao Yunsong E-mail: [email protected] E-mail: [email protected]
2008-03-15
A quintom universe with an equation of state crossing the cosmological constant boundary can provide a bouncing solution dubbed the quintom bounce and thus resolve the big bang singularity. In this paper, we investigate the cosmological perturbations of the quintom bounce both analytically and numerically. We find that the fluctuations in the dominant mode in the post-bounce expanding phase couple to the growing mode of the perturbations in the pre-bounce contracting phase.
19. Non Perturbative Aspects of Field Theory
SciTech Connect
Bashir, A.
2009-04-20
For any quantum field theory (QFT), there exists a set of Schwinger-Dyson equations (SDE) for all its Green functions. However, it is not always straight forward to extract quantitatively exact physical information from this set of equations, especially in the non perturbative regime. The situation becomes increasingly complex with growing number of external legs. I give a qualitative account of the hunt for the non perturbative Green functions in gauge theories.
20. Thermally unstable perturbations in stratified conducting atmospheres
Reale, Fabio; Serio, Salvatore; Peres, Giovanni
1994-10-01
We investigate the thermal stability of isobaric perturbations in a stratified isothermal background atmosphere with solar abundances, as resulting from the competition of optically thin plasma radiative cooling and of heating conducted from the surrounding atmosphere. We have analyzed the threshold line between stable and unstable perturbations, in the plane of the two important control parameters: the initial size of the perturbation and the temperature of the unperturbed medium; this line changes with the pressure of the unperturbed atmosphere. We have extended the results of linear perturbation analysis by means of numerical calculations of the evolution of spherical isobaric perturbations, using a two-dimensional hydrodynamic code including Spitzer heat conduction. We explore a wide range of the parameters appropriate to the solar and stellar upper atmospheres: the background uniform temperature is between 105 K and 107 K, the initial pressure betweeen 0.1 and 10 dyn/sq cm, and the perturbation size between 105 and 1010 cm. The numerical results are in substantial agreement with the linear analysis. We discuss possible implications of our results also in terms of observable effects, especially concerning plasma downflows, and propose thermal instability as a possible candidate to explain the observed redshifts in solar and stellar transition region lines.
1. Near horizon extremal geometry perturbations: dynamical field perturbations vs. parametric variations
Hajian, K.; Seraj, A.; Sheikh-Jabbari, M. M.
2014-10-01
In [1] we formulated and derived the three universal laws governing Near Horizon Extremal Geometries (NHEG). In this work we focus on the Entropy Perturbation Law (EPL) which, similarly to the first law of black hole thermodynamics, relates perturbations of the charges labeling perturbations around a given NHEG to the corresponding entropy perturbation. We show that field perturbations governed by the linearized equations of motion and symmetry conditions which we carefully specify, satisfy the EPL. We also show that these perturbations are limited to those coming from difference of two NHEG solutions (i.e. variations on the NHEG solution parameter space). Our analysis and discussions shed light on the "no-dynamics" statements of [2, 3].
2. Perturbations of the Robertson-Walker space
Hwang, Jai Chan
This dissertation contains three parts consisting of thirteen chapters. Each chapter is self-contained, and can be read independently. In chapter 1, we have presented a complete set of cosmological perturbation equations using the covariant equations. We also present an explicit solution for the evolution of large scale cosmological density perturbations assuming a perfect fluid. In chapter 2, two independent gauge-invariant variables are derived which are continuous at any transition where there is a discontinuous change in pressure. In chapter 3, we present a Newtonian counterpart to the general relativistic covariant approach to cosmological perturbations. In chapter 4, we present a simple way of deriving cosmological perturbation equations in generalized gravity theories which accounts for metric perturbations in gauge-invariant way. We apply this approach to the f(phi,R)-omega(phi)phi, cphi;c Lagrangian. In chapter 5, we have derived second order differential equations for cosmological perturbations in a Robertson-Walker space, for each of the following gravity theories: f(R) gravity, generalized scalar-tensor gravity, gravity with non-minimally coupled scalar field, and induced gravity. Asymptotic solutions are derived for the large and small scale limits. In chapter 6, classical evolution of density perturbations in the large scale limit is clarified in the generalized gravity theories. In chapter 7, we apply our method to a theory with the Lagrangian L approximately f(R) + gamma RR;c;c. In chapter 8, T(M)ab;b equals 0 is shown in a general ground. In chapter 9, the origin of the Friedmann-like behavior of the perturbed model in the large scale limit is clarified in a comoving gauge. Thus, when the imperfect fluid contributions are negligible, the large scale perturbations in a nearly flat background evolve like separate Friedmann models. In chapter 10, we generalize the perturbation equations applicable to a class of generalized gravity theories with multi
3. Non-hard sphere thermodynamic perturbation theory
Zhou, Shiqi
2011-08-01
A non-hard sphere (HS) perturbation scheme, recently advanced by the present author, is elaborated for several technical matters, which are key mathematical details for implementation of the non-HS perturbation scheme in a coupling parameter expansion (CPE) thermodynamic perturbation framework. NVT-Monte Carlo simulation is carried out for a generalized Lennard-Jones (LJ) 2n-n potential to obtain routine thermodynamic quantities such as excess internal energy, pressure, excess chemical potential, excess Helmholtz free energy, and excess constant volume heat capacity. Then, these new simulation data, and available simulation data in literatures about a hard core attractive Yukawa fluid and a Sutherland fluid, are used to test the non-HS CPE 3rd-order thermodynamic perturbation theory (TPT) and give a comparison between the non-HS CPE 3rd-order TPT and other theoretical approaches. It is indicated that the non-HS CPE 3rd-order TPT is superior to other traditional TPT such as van der Waals/HS (vdW/HS), perturbation theory 2 (PT2)/HS, and vdW/Yukawa (vdW/Y) theory or analytical equation of state such as mean spherical approximation (MSA)-equation of state and is at least comparable to several currently the most accurate Ornstein-Zernike integral equation theories. It is discovered that three technical issues, i.e., opening up new bridge function approximation for the reference potential, choosing proper reference potential, and/or using proper thermodynamic route for calculation of fex - ref, chiefly decide the quality of the non-HS CPE TPT. Considering that the non-HS perturbation scheme applies for a wide variety of model fluids, and its implementation in the CPE thermodynamic perturbation framework is amenable to high-order truncation, the non-HS CPE 3rd-order or higher order TPT will be more promising once the above-mentioned three technological advances are established.
4. Perturbed Energy Metabolism and Neuronal Circuit Dysfunction in Cognitive Impairment
PubMed Central
Kapogiannis, Dimitrios; Mattson, Mark P.
2010-01-01
Summary Epidemiological, neuropathological and functional neuroimaging evidence implicates global and regional derangements in brain metabolism and energetics in the pathogenesis of cognitive impairment. Nerve cell microcircuits are modified adaptively by excitatory and inhibitory synaptic activity and neurotrophic factors. Aging and Alzheimer’s disease (AD) cause perturbations in cellular energy metabolism, level of excitation/inhibition and neurotrophic factor release that overwhelm compensatory mechanisms and result in neuronal microcircuit and brain network dysfunction. A prolonged positive energy balance impairs the ability of neurons to respond adaptively to oxidative and metabolic stress. Experimental studies in animals demonstrate how derangements related to chronic positive energy balance, such as diabetes, set the stage for accelerated cognitive aging and AD. Therapeutic interventions to allay cognitive dysfunction that target energy metabolism and adaptive stress responses (such as neurotrophin signaling) have shown efficacy in animal models and preliminary studies in humans. PMID:21147038
5. Non-parametric reconstruction of cosmological matter perturbations
González, J. E.; Alcaniz, J. S.; Carvalho, J. C.
2016-04-01
Perturbative quantities, such as the growth rate (f) and index (γ), are powerful tools to distinguish different dark energy models or modified gravity theories even if they produce the same cosmic expansion history. In this work, without any assumption about the dynamics of the Universe, we apply a non-parametric method to current measurements of the expansion rate H(z) from cosmic chronometers and high-z quasar data and reconstruct the growth factor and rate of linearised density perturbations in the non-relativistic matter component. Assuming realistic values for the matter density parameter Ωm0, as provided by current CMB experiments, we also reconstruct the evolution of the growth index γ with redshift. We show that the reconstruction of current H(z) data constrains the growth index to γ=0.56 ± 0.12 (2σ) at z = 0.09, which is in full agreement with the prediction of the ΛCDM model and some of its extensions.
6. Cosmological perturbations during the Bose-Einstein condensation of dark matter
SciTech Connect
Freitas, R.C.; Gonçalves, S.V.B. E-mail: [email protected]
2013-04-01
In the present work, we analyze the evolution of the scalar and tensorial perturbations and the quantities relevant for the physical description of the Universe, as the density contrast of the scalar perturbations and the gravitational waves energy density during the Bose-Einstein condensation of dark matter. The behavior of these parameters during the Bose-Einstein phase transition of dark matter is analyzed in details. To study the cosmological dynamics and evolution of scalar and tensorial perturbations in a Universe with and without cosmological constant we use both analytical and numerical methods. The Bose-Einstein phase transition modifies the evolution of gravitational waves of cosmological origin, as well as the process of large-scale structure formation.
7. Application of Perturbation Method in Investigating the Interaction of thin Shock with Turbulence
SciTech Connect
Ao, X.; Zank, G. P.; Pogorelov, N. V.; Shaikh, D.
2006-09-26
A 2D hydrodynamical model describing the interaction of thin shock with turbulence is developed by adopting a multi-scale perturbation analysis. This is extended to a 2D MHD model. The interaction is found to be governed by a two-dimentional Burger's equation involving ''perturbation terms''. Different perturbation profiles are tested with numerical simulations to show how the shock front is modified by turbulence. The results indicate that while turbulence can balance the nonlinear steepening of shock waves at some regions, it also helps to create a higher jump in physical quantities at other regions. The plasma medium in these regions can therefore experience higher compression, which will result in a downstream state that differs from the usual Rankine-Hugoniot state.
8. Nonlocal Symmetry and its Applications in Perturbed mKdV Equation
Ren, Bo; Lin, Ji
2016-06-01
Based on the modified direct method, the variable-coefficient perturbed mKdV equation is changed to the constant-coefficient perturbed mKdV equation. The truncated Painlevé method is applied to obtain the nonlocal symmetry of the constant-coefficient perturbed mKdV equation. By introducing one new dependent variable, the nonlocal symmetry can be localized to the Lie point symmetry. Thanks to the localization procedure, the finite symmetry transformation is presented by solving the initial value problem of the prolonged systems. Furthermore, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, and Painlevé II solutions are obtained using the symmetry reduction method to the enlarged systems. Two special concrete soliton-cnoidal interaction solutions are studied in both analytical and graphical ways.
9. Implementation of uniform perturbation method for potential flow past axisymmetric and two-dimensional bodies
NASA Technical Reports Server (NTRS)
Wong, T. C.; Tiwari, S. N.
1984-01-01
The aerodynamic characteristics of potential flow past an axisymmetric slender body and a thin airfoil are calculated using a uniform perturbation analysis method. The method is based on the superposition of potentials of point singularities distributed inside the body. The strength distribution satisfies a linear integral equation by enforcing the flow tangency condition on the surface of the body. The complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio by modifying and adapting an existing technique. Results calculated by the perturbation analysis method are compared with the existing surface singularity panel method and some available analytical solutions for a number of cases under identical conditions. From these comparisons, it is found that the perturbation analysis method can provide quite accurate results for bodies with small slenderness ratio. The present method is much simpler and requires less memory and computation time than existing surface singularity panel methods of comparable accuracy.
10. Analyzing magneto-hydrodynamic squeezing flow between two parallel disks with suction or injection by a new hybrid method based on the Tau method and the homotopy analysis method
Shaban, M.; Shivanian, E.; Abbasbandy, S.
2013-11-01
In this paper an algorithm based on the homotopy analysis method (HAM) is introduced to study the magneto-hydrodynamic (MHD) squeeze flow between two parallel infinite disks where one disk is impermeable and the other is porous with either suction or injection of the fluid in the presence of an applied magnetic field. The continuity and momentum equations governing the squeeze flow are reduced to a single, nonlinear, ordinary differential equation via similarity transformations. In addition, by using the Tau method the problem converts to the algebraic equations to obtain the solution iteratively. The combined effect of inertia, electromagnetic forces for both suction and blowing cases is discussed. Additionally, the convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. The applicability, accuracy and efficiency of this new Tau modification of the HAM is demonstrated via the accomplished comparison.
11. NONLINEAR DYNAMICAL FRICTION OF A CIRCULAR-ORBIT PERTURBER IN A GASEOUS MEDIUM
SciTech Connect
Kim, Woong-Tae
2010-12-10
We use three-dimensional hydrodynamic simulations to investigate the nonlinear gravitational responses of gas to, and the resulting drag forces on, very massive perturbers moving in circular orbits. This work extends our previous studies that explored the cases of low-mass perturbers in circular orbits and massive perturbers on straight-line trajectories. The background medium is assumed to be non-rotating, adiabatic with index 5/3, and uniform with density {rho}{sub 0} and sound speed a{sub 0}. We model the gravitating perturber using a Plummer sphere with mass M{sub p} and softening radius r{sub s} in a uniform circular motion at speed V{sub p} and orbital radius R{sub p} , and run various models with differing R{identical_to}r{sub s}/R{sub p}, M{identical_to}V{sub p}/a{sub 0}, and B{identical_to}GM{sub p}/(a{sub 0}{sup 2}R{sub p}). A quasi-steady density wake of a supersonic model consists of a hydrostatic envelope surrounding the perturber, an upstream bow shock, and a trailing low-density region. The continuous change in the direction of the perturber motion reduces the detached shock distance compared to the linear-trajectory cases, while the orbit-averaged gravity of the perturber gathers the gas toward the center of the orbit, modifying the background preshock density to {rho}{sub 1}{approx}(1+0.46B{sup 1.1}){rho}{sub 0} depending weakly on M. For sufficiently massive perturbers, the presence of a hydrostatic envelope makes the drag force smaller than the prediction of the linear perturbation theory, resulting in F=4{pi}{rho}{sub 1}(GM{sub p})?2/V{sub p}?2 x (0.7{eta}{sub B}?-?1) for {eta}{sub B{identical_to}}B/(M?2-1)>0.1; the drag force for low-mass perturbers with {eta}{sub B}<0.1 agrees well with the linear prediction. The nonlinear drag force becomes independent of R as long as R<{eta}{sub B}/2, which places an upper limit on the perturber size for accurate evaluation of the drag force in numerical simulations.
12. Mode coupling of Schwarzschild perturbations: Ringdown frequencies
SciTech Connect
Pazos, Enrique; Brizuela, David; Martin-Garcia, Jose M.; Tiglio, Manuel
2010-11-15
Within linearized perturbation theory, black holes decay to their final stationary state through the well-known spectrum of quasinormal modes. Here we numerically study whether nonlinearities change this picture. For that purpose we study the ringdown frequencies of gauge-invariant second-order gravitational perturbations induced by self-coupling of linearized perturbations of Schwarzschild black holes. We do so through high-accuracy simulations in the time domain of first and second-order Regge-Wheeler-Zerilli type equations, for a variety of initial data sets. We consider first-order even-parity (l=2, m={+-}2) perturbations and odd-parity (l=2, m=0) ones, and all the multipoles that they generate through self-coupling. For all of them and all the initial data sets considered we find that--in contrast to previous predictions in the literature--the numerical decay frequencies of second-order perturbations are the same ones of linearized theory, and we explain the observed behavior. This would indicate, in particular, that when modeling or searching for ringdown gravitational waves, appropriately including the standard quasinormal modes already takes into account nonlinear effects.
13. Cohomology Methods in Causal Perturbation Theory
Grigore, D. R.
2010-01-01
Various problems in perturbation theory of (quantum) gauge models can be rephrased in the language of cohomology theory. This was already noticed in the functional formulation of perturbative gauge theories. Causal perturbation theory is a fully quantum approach: is works only with the chronological products which are defined as operator-valued distributions in the Fock space of the model. The use of causal perturbation theory leads to similar cohomology problems; the main difference with respect to the functional methods comes from the fact that the gauge transformation of the causal approach is, essentially, the linear part of the non-linear BRST transformation. Using these methods it is possible to give a nice determination of the interaction Lagrangians for gauge models (Yang-Mills and gravitation in the linear approximation); one obtains with this method the unicity of the interaction Lagrangian up to trivial terms. The case of quantum gravity is highly non-trivial and can be generalized with this method to the massive graviton case. Going to higher orders of perturbation theory one finds quantum anomalies. Again the cohomological methods can be used to determine the generic form of these anomalies. Finally, one can investigate the arbitrariness of the chronological products in higher orders and reduce this problem to cohomology methods also.
14. Supersymmetry restoration in superstring perturbation theory
Sen, Ashoke
2015-12-01
Superstring perturbation theory based on the 1PI effective theory approach has been useful for addressing the problem of mass renormalization and vacuum shift. We derive Ward identities associated with space-time supersymmetry transformation in this approach. This leads to a proof of the equality of renormalized masses of bosons and fermions and identities relating fermionic amplitudes to bosonic amplitudes after taking into account the effect of mass renormalization. This also relates unbroken supersymmetry to a given order in perturbation theory to absence of tadpoles of massless scalars to higher order. The results are valid at the perturbative vacuum as well as in the shifted vacuum when the latter describes the correct ground state of the theory. We apply this to SO(32) heterotic string theory on Calabi-Yau 3-folds where a one loop Fayet-Iliopoulos term apparently breaks supersymmetry at one loop, but analysis of the low energy effective field theory indicates that there is a nearby vacuum where supersymmetry is restored. We explicitly prove that the perturbative amplitudes of this theory around the shifted vacuum indeed satisfy the Ward identities associated with unbroken supersymmetry. We also test the general arguments by explicitly verifying the equality of bosonic and fermionic masses at one loop order in the shifted vacuum, and the appearance of two loop dilaton tadpole in the perturbative vacuum where supersymmetry is expected to be broken.
15. Cosmological perturbations in teleparallel Loop Quantum Cosmology
Haro, Jaime
2013-11-01
Cosmological perturbations in Loop Quantum Cosmology (LQC) are usually studied incorporating either holonomy corrections, where the Ashtekar connection is replaced by a suitable sinus function in order to have a well-defined quantum analogue, or inverse-volume corrections coming from the eigenvalues of the inverse-volume operator. In this paper we will develop an alternative approach to calculate cosmological perturbations in LQC based on the fact that, holonomy corrected LQC in the flat Friedmann-Lemaître-Robertson-Walker (FLRW) geometry could be also obtained as a particular case of teleparallel F(T) gravity (teleparallel LQC). The main idea of our approach is to mix the simple bounce provided by holonomy corrections in LQC with the non-singular perturbation equations given by F(T) gravity, in order to obtain a matter bounce scenario as a viable alternative to slow-roll inflation. In our study, we have obtained an scale invariant power spectrum of cosmological perturbations. However, the ratio of tensor to scalar perturbations is of order 1, which does not agree with the current observations. For this reason, we suggest a model where a transition from the matter domination to a quasi de Sitter phase is produced in order to enhance the scalar power spectrum.
16. Local perturbations perturb—exponentially–locally
SciTech Connect
De Roeck, W. Schütz, M.
2015-06-15
We elaborate on the principle that for gapped quantum spin systems with local interaction, “local perturbations [in the Hamiltonian] perturb locally [the groundstate].” This principle was established by Bachmann et al. [Commun. Math. Phys. 309, 835–871 (2012)], relying on the “spectral flow technique” or “quasi-adiabatic continuation” [M. B. Hastings, Phys. Rev. B 69, 104431 (2004)] to obtain locality estimates with sub-exponential decay in the distance to the spatial support of the perturbation. We use ideas of Hamza et al. [J. Math. Phys. 50, 095213 (2009)] to obtain similarly a transformation between gapped eigenvectors and their perturbations that is local with exponential decay. This allows to improve locality bounds on the effect of perturbations on the low lying states in certain gapped models with a unique “bulk ground state” or “topological quantum order.” We also give some estimate on the exponential decay of correlations in models with impurities where some relevant correlations decay faster than one would naively infer from the global gap of the system, as one also expects in disordered systems with a localized groundstate.
17. Cosmological perturbations on the phantom brane
Bag, Satadru; Viznyuk, Alexander; Shtanov, Yuri; Sahni, Varun
2016-07-01
We obtain a closed system of equations for scalar perturbations in a multi-component braneworld. Our braneworld possesses a phantom-like equation of state at late times, weff < ‑1, but no big-rip future singularity. In addition to matter and radiation, the braneworld possesses a new effective degree of freedom—the Weyl fluid' or dark radiation'. Setting initial conditions on super-Hubble spatial scales at the epoch of radiation domination, we evolve perturbations of radiation, pressureless matter and the Weyl fluid until the present epoch. We observe a gradual decrease in the amplitude of the Weyl-fluid perturbations after Hubble-radius crossing, which results in a negligible effect of the Weyl fluid on the evolution of matter perturbations on spatial scales relevant for structure formation. Consequently, the quasi-static approximation of Koyama and Maartens provides a good fit to the exact results during the matter-dominated epoch. We find that the late-time growth of density perturbations on the brane proceeds at a faster rate than in ΛCDM. Additionally, the gravitational potentials Φ and Ψ evolve differently on the brane than in ΛCDM, for which Φ = Ψ. On the brane, by contrast, the ratio Φ/Ψ exceeds unity during the late matter-dominated epoch (z lesssim 50). These features emerge as smoking gun tests of phantom brane cosmology and allow predictions of this scenario to be tested against observations of galaxy clustering and large-scale structure.
18. Improved WKB analysis of cosmological perturbations
SciTech Connect
Casadio, Roberto; Luzzi, Mattia; Venturi, Giovanni; Finelli, Fabio
2005-02-15
Improved Wentzel-Kramers-Brillouin (WKB)-type approximations are presented in order to study cosmological perturbations beyond the lowest order. Our methods are based on functions which approximate the true perturbation modes over the complete range of the independent (Langer) variable, from subhorizon to superhorizon scales, and include the region near the turning point. We employ both a perturbative Green's function technique and an adiabatic (or semiclassical) expansion (for a linear turning point) in order to compute higher order corrections. Improved general expressions for the WKB scalar and tensor power spectra are derived for both techniques. We test our methods on the benchmark of power-law inflation, which allows comparison with exact expressions for the perturbations, and find that the next-to-leading order adiabatic expansion yields the amplitude of the power spectra with excellent accuracy, whereas the next-to-leading order with the perturbative Green's function method does not improve the leading order result significantly. However, in more general cases, either or both methods may be useful.
19. Rolling axions during inflation: perturbativity and signatures
Peloso, Marco; Sorbo, Lorenzo; Unal, Caner
2016-09-01
The motion of a pseudo-scalar field X during inflation naturally induces a significant amplification of the gauge fields to which it is coupled. The amplified gauge fields can source characteristic scalar and tensor primordial perturbations. Several phenomenological implications have been discussed in the cases in which (i) X is the inflaton, and (ii) X is a field different from the inflaton, that experiences a temporary speed up during inflation. In this second case, visible sourced gravitational waves (GW) can be produced at the CMB scales without affecting the scalar perturbations, even if the scale of inflation is several orders of magnitude below what is required to produce a visible vacuum GW signal. Perturbativity considerations can be used to limit the regime in which these results are under perturbative control. We revised limits recently claimed for the case (i), and we extend these considerations to the case (ii). We show that, in both cases, these limits are satisfied by the applications that generate signals at CMB scales. Applications that generate gravitational waves and primordial black holes at much smaller scales are at the limit of the validity of this perturbativity analysis, so we expect those results to be valid up to possibly order one corrections.
20. Nonderivative modified gravity: a classification
SciTech Connect
Comelli, D.; Nesti, F.; Pilo, L. E-mail: [email protected]
2014-11-01
We analyze the theories of gravity modified by a generic nonderivative potential built from the metric, under the minimal requirement of unbroken spatial rotations. Using the canonical analysis, we classify the potentials V according to the number of degrees of freedom (DoF) that propagate at the nonperturbative level. We then compare the nonperturbative results with the perturbative DoF propagating around Minkowski and FRW backgrounds. A generic V implies 6 propagating DoF at the non-perturbative level, with a ghost on Minkowski background. There exist potentials which propagate 5 DoF, as already studied in previous works. Here, no V with unbroken rotational invariance admitting 4 DoF is found. Theories with 3 DoF turn out to be strongly coupled on Minkowski background. Finally, potentials with only the 2 DoF of a massive graviton exist. Their effect on cosmology is simply equivalent to a cosmological constant. Potentials with 2 or 5 DoF and explicit time dependence appear to be a further viable possibility.
1. Non-gravitational perturbations and satellite geodesy
SciTech Connect
Milani, A.; Nobill, A.M.; Farinella, P.
1987-01-01
This book presents the basic ideas of the physics of non-gravitational perturbations and the mathematics required to compute their orbital effects. It conveys the relevance of the different problems that must be solved to achieve a given level of accuracy in orbit determination and in recovery of geophysically significant parameters. Selected Contents are: Orders of Magnitude of the Perturbing Forces, Tides and Apparent Forces, Tools from Celestial Mechanics, Solar Radiation Pressure-Direct Effects: Satellite-Solar Radiation Interaction, Long-Term Effects on Semi-Major Axis, Radiation Pressure-Indirect Effects: Earth-Reflected Radiation Pressure, Anisotropic Thermal Emission, Drag: Orbital Perturbations by a Drag-Like Force, and Charged Particle Drag.
2. Perturbations in a regular bouncing universe
SciTech Connect
Battefeld, T.J.; Geshnizjani, G.
2006-03-15
We consider a simple toy model of a regular bouncing universe. The bounce is caused by an extra timelike dimension, which leads to a sign flip of the {rho}{sup 2} term in the effective four dimensional Randall Sundrum-like description. We find a wide class of possible bounces: big bang avoiding ones for regular matter content, and big rip avoiding ones for phantom matter. Focusing on radiation as the matter content, we discuss the evolution of scalar, vector and tensor perturbations. We compute a spectral index of n{sub s}=-1 for scalar perturbations and a deep blue index for tensor perturbations after invoking vacuum initial conditions, ruling out such a model as a realistic one. We also find that the spectrum (evaluated at Hubble crossing) is sensitive to the bounce. We conclude that it is challenging, but not impossible, for cyclic/ekpyrotic models to succeed, if one can find a regularized version.
3. Non-perturbative quantum geometry III
Krefl, Daniel
2016-08-01
The Nekrasov-Shatashvili limit of the refined topological string on toric Calabi-Yau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kähler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local ℙ1 + ℙ1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local ℙ2 near the conifold point in moduli space is also provided.
4. Interactions of Blast Waves with Perturbed Interfaces
Henry de Frahan, Marc; Johnsen, Eric
2015-11-01
Richtmyer-Meshkov and Rayleigh-Taylor instabilities induce hydrodynamic mixing in many important physical systems such as inertial confinement fusion, supernova collapse, and scramjet combustion. Blast waves interacting with perturbed interfaces are prevelant in such applications and dictate the mixing dynamics. This study increases our understanding of blast-driven hydrodynamic instabilities by providing models for the time-dependent perturbation growth and vorticity production mechanisms. The strength and length of the blast wave determine the different growth regimes and the importance of the Richtmyer-Meshkov or Rayleigh-Taylor growth. Our analysis is based on simulations of a 2D planar blast wave, modeled by a shock (instantaneous acceleration) followed by a rarefaction (time-dependent deceleration), interacting with a sinusoidal perturbation at an interface between two fluids. A high-order accurate Discontinuous Galerkin method is used to solve the multifluid Euler equations.
5. Hypersurface-invariant approach to cosmological perturbations
Salopek, D. S.; Stewart, J. M.
1995-01-01
Using Hamilton-Jacobi theory, we develop a formalism for solving semiclassical cosmological perturbations which does not require an explicit choice of time hypersurface. The Hamilton-Jacobi equation for gravity interacting with matter (either a scalar or dust field) is solved by making an ansatz which includes all terms quadratic in the spatial curvature. Gravitational radiation and scalar perturbations are treated on an equal footing. Our technique encompasses linear perturbation theory and it also describes some mild nonlinear effects. As a concrete example of the method, we compute the galaxy-galaxy correlation function as well as large-angle microwave background fluctuations for power-law inflation, and we compare with recent observations.
6. Elementary theorems regarding blue isocurvature perturbations
Chung, Daniel J. H.; Yoo, Hojin
2015-04-01
Blue CDM-photon isocurvature perturbations are attractive in terms of observability and may be typical from the perspective of generic mass relations in supergravity. We present and apply three theorems useful for blue isocurvature perturbations arising from linear spectator scalar fields. In the process, we give a more precise formula for the blue spectrum associated with the axion model of Kasuya and Kawasaki [Axion Isocurvature Fluctuations with Extremely Blue Spectrum, Phys. Rev. D 80, 023516 (2009).], which can in a parametric corner give a factor of O (10 ) correction. We explain how a conserved current associated with Peccei-Quinn symmetry plays a crucial role and explicitly plot several example spectra including the breaks in the spectra. We also resolve a little puzzle arising from a naive multiplication of isocurvature expression that sheds light on the gravitational imprint of the adiabatic perturbations on the fields responsible for blue isocurvature fluctuations.
7. Perturbation measurement of waveguides for acoustic thermometry
Lin, H.; Feng, X. J.; Zhang, J. T.
2013-09-01
Acoustic thermometers normally embed small acoustic transducers in the wall bounding a gas-filled cavity resonator. At high temperature, insulators of transducers loss electrical insulation and degrade the signal-to-noise ratio. One essential solution to this technical trouble is to couple sound by acoustic waveguides between resonator and transducers. But waveguide will break the ideal acoustic surface and bring perturbations(Δf+ig) to the ideal resonance frequency. The perturbation model for waveguides was developed based on the first-order acoustic theory in this paper. The frequency shift Δf and half-width change g caused by the position, length and radius of waveguides were analyzed using this model. Six different length of waveguides (52˜1763 mm) were settled on the cylinder resonator and the perturbation (Δf+ig) were measured at T=332 K and p=250˜500 kPa. The experiment results agreed with the theoretical prediction very well.
8. Perturbation analysis of electromagnetic geodesic acoustic modes
SciTech Connect
Ren, Haijun
2014-06-15
Lagrangian displacement and magnetic field perturbation response to the geodesic acoustic mode is analyzed by using the ideal magnetohydrodynamic equations in a large-aspect-ratio tokamak. δB{sub θ}, the poloidal component of magnetic field perturbation, has poloidal wave number m = 2 created by the poloidal displacement ξ{sub θ}. The parallel perturbation of magnetic field, δB{sub ∥}, has a poloidally asymmetric structure with m = 1 and is on the same order of magnitude with δB{sub θ} to the leading order. The radial displacement ξ{sub r} is of order O(βϵξ{sub θ}) but plays a significant role in determining δB{sub ∥}, where β is the plasma/magnetic pressure ratio and ϵ is the inverse aspect ratio.
9. Instability of charged Lovelock black holes: Vector perturbations and scalar perturbations
Takahashi, Tomohiro
2013-01-01
We examine the stability of charged Lovelock black hole solutions under vector-type and scalar-type perturbations. We find suitable master variables for the stability analysis; the equations for these variables are Schrödinger-type equations with two components, and these Schrödinger operators are symmetric. By these master equations, we show that charged Lovelock black holes are stable under vector-type perturbations. For scalar-type perturbations, we show the criteria for instability and check these numerically. In our previous paper [T. Takahashi, Prog. Theor. Phys. 125, 1289 (2011)], we have shown that nearly extreme black holes show instability under tensor-type perturbations. In this paper, we find that black holes with a small charge show instability under scalar-type perturbations even if they have a relatively large mass.
10. CHARACTERIZATION OF THE RESONANT CAUSTIC PERTURBATION
SciTech Connect
Chung, Sun-Ju
2009-11-01
Four of nine exoplanets found by microlensing were detected by the resonant caustic, which represents the merging of the planetary and central caustics at the position when the projected separation of a host star and a bounded planet is s approx 1. One of the resonant caustic lensing events, OGLE-2005-BLG-169, was a caustic-crossing high-magnification event with A {sub max}approx 800 and the source star was much smaller than the caustic, nevertheless the perturbation was not obviously apparent on the light curve of the event. In this paper, we investigate the perturbation pattern of the resonant caustic to understand why the perturbations induced by the caustic do not leave strong traces on the light curves of high-magnification events despite a small source/caustic size ratio. From this study, we find that the regions with small magnification excess around the center of the resonant caustic are rather widely formed, and the event passing the small-excess region produces a high-magnification event with a weak perturbation that is small relative to the amplification caused by the star and thus does not noticeably appear on the light curve of the event. We also find that the positive excess of the inside edge of the resonant caustic and the negative excess inside the caustic become stronger and wider as q increases, and thus the resonant caustic-crossing high-magnification events with the weak perturbation occur in the range of q <= 10{sup -4}. We determine the probability of the occurrence of events with the small excess |epsilon| <= 3% in high-magnification events induced by a resonant caustic. As a result, we find that for Earth-mass planets with a separation of approx2.5 AU the resonant caustic high-magnification events with the weak perturbation can occur with a significant frequency.
11. Perturbative Lagrangian approach to gravitational instability.
Bouchet, F. R.; Colombi, S.; Hivon, E.; Juszkiewicz, R.
1995-04-01
This paper deals with the time evolution in the matter era of perturbations in Friedman-Lemaitre models with arbitrary density parameter {OMEGA}, with either a zero cosmological constant, {LAMBDA}=0, or with a non-zero cosmological constant in a spatially flat Universe. Unlike the classical Eulerian approach where the density contrast is expanded in a perturbative series, this analysis relies instead on a perturbative expansion of particles trajectories in Lagrangian coordinates. This brings a number of advantages over the classical analysis. In particular, it enables the description of stronger density contrasts. Indeed the linear term in the Lagrangian perturbative series is the famous Zeldovich approximate solution (1970). The idea to consider the higher order terms was introduced by Moutarde et al. (1991), generalized by Bouchet et al. (1992), and further developed by many others. We present here a systematic and detailed account of this approach. We give analytical results (or fits to numerical results) up to the third order (which is necessary to compute, for instance, the four point spatial correlation function or the corrections to the linear evolution of the two-point correlation function, as well as the secondary temperature anisotropies of the Cosmic Microwave Background). We then proceed to explore the link between the Lagrangian description and statistical measures. We show in particular that Lagrangian perturbation theory provides a natural framework to compute the effect of redshift distortions, using the skewness of the density distribution function as an example. Finally, we show how well the second order theory does as compared to other approximations in the case of spherically symmetric perturbations. We also compare this second order approximation and Zeldovich solution to N-body simulations with scale-free (n=-2) Gaussian initial conditions. We find that second order theory is both simple and powerful.
12. Nonlinear Growth of Singular Vector Based Perturbations
Reynolds, C. A.
2002-12-01
The nonlinearity of singular vector-based perturbation growth is examined within the context of a global atmospheric forecast model. The characteristics of these nonlinearities and their impact on the utility of SV-based diagnostics are assessed both qualitatively and quantitatively. Nonlinearities are quantified by examining the symmetry of evolving positive and negative "twin" perturbations. Perturbations initially scaled to be consistent with estimates of analysis uncertainty become significantly nonlinear by 12 hours. However, the relative magnitude of the nonlinearities is a strong function of scale and metric. Small scales become nonlinear very quickly while synoptic scales can remain significantly linear out to three day. Small shifts between positive and negative perturbations can result in significant nonlinearities even when the basic anomaly patterns are quite similar. Thus, singular vectors may be qualitatively useful even when nonlinearities are large. Post-time pseudo-inverse experiments show that despite significant nonlinear perturbation growth, the nonlinear forecast corrections are similar to the expected linear corrections, even at 72 hours. When the nonlinear correction does differ significantly from the expected linear correction, the nonlinear correction is usually better, indicating that in some cases the pseudo-inverse correction effectively suppresses error growth outside the subspace defined by the leading (dry) singular vectors. Because a significant portion of the nonlinear growth occurs outside of the dry singular vector subspace, an a priori nonlinearity index based on the full perturbations is not a good predictor of when pseudo-inverse based corrections will be ineffective. However, one can construct a reasonable predictor of pseudo-inverse ineffectiveness by focusing on nonlinearities in the synoptic scales or in the singular vector subspace only.
13. Tensor-vector-scalar cosmology: Covariant formalism for the background evolution and linear perturbation theory
Skordis, Constantinos
2006-11-01
A relativistic theory of gravity has recently been proposed by Bekenstein, where gravity is mediated by a tensor, a vector, and a scalar field, thus called TeVeS. The theory aims at modifying gravity in such a way as to reproduce Milgrom’s modified Newtonian dynamics (MOND) in the weak field, nonrelativistic limit, which provides a framework to solve the missing mass problem in galaxies without invoking dark matter. In this paper I apply a covariant approach to formulate the cosmological equations for this theory, for both the background and linear perturbations. I derive the necessary perturbed equations for scalar, vector, and tensor modes without adhering to a particular gauge. Special gauges are considered in the appendixes.
14. Modification of plasma rotation with resonant magnetic perturbations in the STOR-M tokamak
Elgriw, S.; Liu, Y.; Hirose, A.; Xiao, C.
2016-04-01
The toroidal plasma flow velocity of impurity ions has been significantly modified in the Saskatchewan Torus-Modified (STOR-M) tokamak by means of resonant magnetic perturbations (RMP). It has been found that the toroidal flow velocities of OV and CVI impurity ions change towards the co-current direction after the application of a current through a set of (l = 2, n = 1) RMP field coils. It has been observed that the reduction of the toroidal flow velocity is closely correlated to the reduction of the magnetohydrodynamic (MHD) fluctuation frequency measured by Mirnov coils. Modulation of the flow velocity has been achieved by switching the RMP current pulses. Non-resonant magnetic perturbations have also induced a much smaller change in the toroidal plasma flow. A theoretical model has been adopted to assess the contributions of different drift mechanisms to magnetic islands rotation in STOR-M.
15. Perturbative approach to Markovian open quantum systems
PubMed Central
Li, Andy C. Y.; Petruccione, F.; Koch, Jens
2014-01-01
The exact treatment of Markovian open quantum systems, when based on numerical diagonalization of the Liouville super-operator or averaging over quantum trajectories, is severely limited by Hilbert space size. Perturbation theory, standard in the investigation of closed quantum systems, has remained much less developed for open quantum systems where a direct application to the Lindblad master equation is desirable. We present such a perturbative treatment which will be useful for an analytical understanding of open quantum systems and for numerical calculation of system observables which would otherwise be impractical. PMID:24811607
16. Evolution of perturbations in an inflationary universe
NASA Technical Reports Server (NTRS)
Frieman, J. A.; Will, C. M.
1982-01-01
The evolution of inhomogeneous density perturbations in a model of the very early universe that is dominated for a time by a constant energy density of a false quantum-mechanical vacuum is analyzed. During this period, the universe inflates exponentially and supercools exponentially, until a phase transition back to the true vacuum reheats the matter and radiation. Focus is on the physically measurable, coordinate-independent modes of inhomogeneous perturbations of this model and it is found that all modes either are constant or are exponentially damped during the inflationary era.
17. Death to perturbative QCD in exclusive processes?
SciTech Connect
Eckardt, R.; Hansper, J.; Gari, M.F.
1994-04-01
The authors discuss the question of whether perturbative QCD is applicable in calculations of exclusive processes at available momentum transfers. They show that the currently used method of determining hadronic quark distribution amplitudes from QCD sum rules yields wave functions which are completely undetermined because the polynomial expansion diverges. Because of the indeterminacy of the wave functions no statement can be made at present as to whether perturbative QCD is valid. The authors emphasize the necessity of a rigorous discussion of the subject and the importance of experimental data in the range of interest.
18. Perturbations in bouncing and cyclic models
Biswas, Tirthabir; Mayes, Riley; Lattyak, Colleen
2016-03-01
Being able to reliably track perturbations across bounces and turnarounds in cyclic and bouncing cosmology lies at the heart of being able to compare the predictions of these models with the cosmic microwave background observations. This has been a challenging task due to the unknown nature of the physics involved during the bounce as well as the technical challenge of matching perturbations precisely between the expansion and contraction phases. In this paper, we present some general techniques (analytical and numerical) that can be applied to understand the physics of the fluctuations, especially those with "long" wavelengths, and apply our techniques to nonsingular cosmological models such as the bounce inflation and cyclic inflation.
19. Non-Gaussianity from isocurvature perturbations
SciTech Connect
Kawasaki, Masahiro; Nakayama, Kazunori; Sekiguchi, Toyokazu; Suyama, Teruaki; Takahashi, Fuminobu E-mail: [email protected] E-mail: [email protected]
2008-11-15
We develop a formalism for studying non-Gaussianity in both curvature and isocurvature perturbations. It is shown that non-Gaussianity in the isocurvature perturbation between dark matter and photons leaves distinct signatures in the cosmic microwave background temperature fluctuations, which may be confirmed in future experiments, or possibly even in the currently available observational data. As an explicit example, we consider the quantum chromodynamics axion and show that it can actually induce sizable non-Gaussianity for the inflationary scale, H{sub inf} = O(10{sup 9}-10{sup 11}) GeV.
20. Continuum methods in lattice perturbation theory
SciTech Connect
Becher, Thomas G
2002-11-15
We show how methods of continuum perturbation theory can be used to simplify perturbative lattice calculations. We use the technique of asymptotic expansions to expand lattice loop integrals around the continuum limit. After the expansion, all nontrivial dependence on momenta and masses is encoded in continuum loop integrals and the only genuine lattice integrals left are tadpole integrals. Using integration-by-parts relations all of these can be expressed in terms of a small number of master integrals. Four master integrals are needed for bosonic one loop integrals, sixteen in QCD with Wilson or staggered fermions.
1. Robustness of braneworld scenarios against tensorial perturbations
Bazeia, D.; Losano, L.; Menezes, R.; Olmo, Gonzalo J.; Rubiera-Garcia, D.
2015-11-01
Inspired by the peculiarities of the effective geometry of crystalline structures, we reconsider thick brane scenarios from a metric-affine perspective. We show that for a rather general family of theories of gravity, whose Lagrangian is an arbitrary function of the metric and the Ricci tensor, the background and scalar field equations can be written in first-order form, and tensorial perturbations have a non negative definite spectrum, which makes them stable under linear perturbations regardless of the form of the gravity Lagrangian. We find, in particular, that the tensorial zero modes are exactly the same as predicted by Einstein’s theory regardless of the scalar field and gravitational Lagrangians.
2. Perturbative renormalization of the electric field correlator
Christensen, C.; Laine, M.
2016-04-01
The momentum diffusion coefficient of a heavy quark in a hot QCD plasma can be extracted as a transport coefficient related to the correlator of two colour-electric fields dressing a Polyakov loop. We determine the perturbative renormalization factor for a particular lattice discretization of this correlator within Wilson's SU(3) gauge theory, finding a ∼ 12% NLO correction for values of the bare coupling used in the current generation of simulations. The impact of this result on existing lattice determinations is commented upon, and a possibility for non-perturbative renormalization through the gradient flow is pointed out.
3. Conservative perturbation theory for nonconservative systems
Shah, Tirth; Chattopadhyay, Rohitashwa; Vaidya, Kedar; Chakraborty, Sagar
2015-12-01
In this paper, we show how to use canonical perturbation theory for dissipative dynamical systems capable of showing limit-cycle oscillations. Thus, our work surmounts the hitherto perceived barrier for canonical perturbation theory that it can be applied only to a class of conservative systems, viz., Hamiltonian systems. In the process, we also find Hamiltonian structure for an important subset of Liénard system—a paradigmatic system for modeling isolated and asymptotic oscillatory state. We discuss the possibility of extending our method to encompass an even wider range of nonconservative systems.
4. Spontaneous breaking of Lorentz symmetry by ghost condensation in perturbative quantum gravity
Faizal, Mir
2011-10-01
In this paper, we will study the spontaneous breakdown of the Lorentz symmetry by ghost condensation in perturbative quantum gravity. Our analysis will be done in the Curci-Ferrari gauge. We will also analyse the modification of the BRST and anti-BRST transformations by the formation of this ghost condensate. It will be shown that even though the modified BRST and anti-BRST transformations are not nilpotent, their nilpotency is restored on-shell.
5. Perturbed particle orbits and kinetic plasma response in non-axisymmetric tokamaks
Kim, Kimin; Park, J.-K.; Boozer, A. H.; Logan, N. C.; Wang, Z. R.; Menard, J. E.
2014-10-01
Non-axisymmetric magnetic fields interact with the drift trajectories of ions and electrons to create an anisotropic plasma pressure. The force produced by the gradient of this anisotropic pressure produces a torque, the Neoclassical Toroidal Viscosity (NTV), which tends to relax the plasma rotation to a specific offset rotation, and modifies the energy required to perturb the plasma. Complexities, such as resonances of the ExB drift with particle bounce frequencies, finite orbit width, and full collisional effects, require full numerical simulation to determine the NTV and the perturbation energy. The POCA delta-f drift kinetic particle code has been used to: (1) demonstrate the existence of the bounce resonances with the ExB drift and show that they often dominate the magnitude of the NTV, (2) show the NTV of perturbations with different toroidal mode numbers are generally decoupled, and (3) verify a quadratic NTV dependence on the asymmetric magnetic field. Such results imply the pressure anisotropy is linear in the magnetic perturbation and can produce a significant change in the applied non-axisymmetric field. Progress on integrating this pressure anisotropy into a perturbed equilibrium solver to obtain self-consistent solutions is presented. This work was supported by US DOE Contract DE-AC02-09CH11466.
6. Gauge-invariant cosmological perturbation theory with seeds
SciTech Connect
Durrer, R. )
1990-10-15
Gauge-invariant cosmological perturbation theory is extended to handle perturbations induced by seeds. A calculation of the Sachs-Wolfe effect is presented. A second-order differential equation for the growth of density perturbations is derived and the perturbation of Liouville's equation for collisionless particles is also given. The results are illustrated by a simple analytic example of a single texture knot, where we calculate the induced perturbations of the energy of microwave photons, of baryonic matter, and of collisionless particles.
7. Modified gravity as dark energy
Sawicki, Ignacy
2007-08-01
We study the effects of introducing modifications to general relativity ("GR") at large scales as an alternative to exotic forms of matter required to replicate the observed cosmic acceleration. We survey the effects on cosmology and solar-system tests of Dvali-Gabadadze-Porrati ("DGP") gravity, f ( R ) he changes to the background expansion history of the universe, these modifications have substantial impact on structure formation and its observable predictions. For DGP, we develop a scaling approximation for the behaviour of perturbations off the brane, for which the predicted integrated Sachs-Wolf ("ISW") effect is much stronger than observed, requiring new physics at around horizon scale to bring it into agreement with data. We develop a test based on cross-correlating galaxies and the ISW effect which is independent of the initial power spectrum for perturbations and is a smoking-gun test for DGP gravity. For f ( R ) models, we find that, for the expansion history to resemble that of Lambda-CDM, it is required that the second derivative of f with respect to R be non-negative. We then find the conditions on f ( R ) which allow this subset of models to pass solar-system tests. Provided that gravity behave like GR in the galaxy, these constraints are weak. However, for a model to allow large deviations from GR in the cosmology, the galactic halo must differ significantly from that predicted by structure evolution in GR. We then discuss the effect that these models have on structure formation, and find that even in the most conservative of models, percent-level deviations in the matter power spectrum will exist and should be detectable in the future. Finally, for MSG, we investigate the cosmology of a theory of gravity with a modified constraint structure. The acceleration era can be replicated in these models; however, linear perturbations become unstable as the universe begins to accelerate. Once the perturbations become non-linear, the model reverts to GR
8. Circumstellar Debris Disks: Diagnosing the Unseen Perturber
Nesvold, Erika R.; Naoz, Smadar; Vican, Laura; Farr, Will M.
2016-07-01
The first indication of the presence of a circumstellar debris disk is usually the detection of excess infrared emission from the population of small dust grains orbiting the star. This dust is short-lived, requiring continual replenishment, and indicating that the disk must be excited by an unseen perturber. Previous theoretical studies have demonstrated that an eccentric planet orbiting interior to the disk will stir the larger bodies in the belt and produce dust via interparticle collisions. However, motivated by recent observations, we explore another possible mechanism for heating a debris disk: a stellar-mass perturber orbiting exterior to and inclined to the disk and exciting the disk particles’ eccentricities and inclinations via the Kozai–Lidov mechanism. We explore the consequences of an exterior perturber on the evolution of a debris disk using secular analysis and collisional N-body simulations. We demonstrate that a Kozai–Lidov excited disk can generate a dust disk via collisions and we compare the results of the Kozai–Lidov excited disk with a simulated disk perturbed by an interior eccentric planet. Finally, we propose two observational tests of a dust disk that can distinguish whether the dust was produced by an exterior brown dwarf or stellar companion or an interior eccentric planet.
9. Do cosmological perturbations have zero mean?
SciTech Connect
Armendariz-Picon, Cristian
2011-03-01
A central assumption in our analysis of cosmic structure is that cosmological perturbations have a constant ensemble mean, which can be set to zero by appropriate choice of the background. This property is one of the consequences of statistical homogeneity, the invariance of correlation functions under spatial translations. In this article we explore whether cosmological perturbations indeed have zero mean, and thus test one aspect of statistical homogeneity. We carry out a classical test of the zero mean hypothesis against a class of alternatives in which primordial perturbations have inhomogeneous non-vanishing means, but homogeneous and isotropic covariances. Apart from Gaussianity, our test does not make any additional assumptions about the nature of the perturbations and is thus rather generic and model-independent. The test statistic we employ is essentially Student's t statistic, applied to appropriately masked, foreground-cleaned cosmic microwave background anisotropy maps produced by the WMAP mission. We find evidence for a non-zero mean in a particular range of multipoles, but the evidence against the zero mean hypothesis goes away when we correct for multiple testing. We also place constraints on the mean of the temperature multipoles as a function of angular scale. On angular scales smaller than four degrees, a non-zero mean has to be at least an order of magnitude smaller than the standard deviation of the temperature anisotropies.
10. Degenerate adiabatic perturbation theory: Foundations and applications
Rigolin, Gustavo; Ortiz, Gerardo
2014-08-01
11. Aharonov-Bohm Effect in Perturbation Theory.
ERIC Educational Resources Information Center
Purcell, Kay M.; Henneberger, Walter C.
1978-01-01
The Aharonov-Bohn effect is obtained in first-order perturbation theory. It is shown that the effect occurs only when the initial state is a superposition of eigenstates of Lz corresponding to eigenvalues having opposite sign. (Author/GA)
12. On the divergences of inflationary superhorizon perturbations
SciTech Connect
Enqvist, K; Nurmi, S; Podolsky, D; Rigopoulos, G I E-mail: [email protected] E-mail: [email protected]
2008-04-15
We discuss the infrared divergences that appear to plague cosmological perturbation theory. We show that, within the stochastic framework, they are regulated by eternal inflation so that the theory predicts finite fluctuations. Using the {Delta}N formalism to one loop, we demonstrate that the infrared modes can be absorbed into additive constants and the coefficients of the diagrammatic expansion for the connected parts of two-and three-point functions of the curvature perturbation. As a result, the use of any infrared cutoff below the scale of eternal inflation is permitted, provided that the background fields are appropriately redefined. The natural choice for the infrared cutoff would, of course, be the present horizon; other choices manifest themselves in the running of the correlators. We also demonstrate that it is possible to define observables that are renormalization-group-invariant. As an example, we derive a non-perturbative, infrared finite and renormalization point-independent relation between the two-point correlators of the curvature perturbation for the case of the free single field.
13. What Perturbs the ggrdgr Rings of Uranus?
PubMed
French, R G; Kangas, J A; Elliot, J L
1986-01-31
The gamma and delta rings have by far the largest radial perturbations of any of the nine known Uranian rings. These two rings deviate from Keplerian orbits, having typical root-mean-square residuals of about 3 kilometers (compared to a few hundred meters for the other seven known rings). Possible causes for the perturbations include nearby shepherd satellites and Lindblad resonances. If shepherd satellites are responsible, they could be as large as several tens of kilometers in diameter. The perturbation patterns of the gamma and delta rings have been examined for evidence of Lindblad resonances of azimuthal wave number m = 0, 1, 2, 3, and 4. The beta ring radial residuals are well matched by a 2:1 Lindblad resonance. If this represents a real physical phenomenon and is not an artifact of undersampling, then the most plausible interpretation is that there is an undiscovered satellite orbiting 76,522 +/- 8 kilometers from Uranus, with an orbital period of 15.3595 +/- 0.0001 hours and a radius of 75 to 100 kilometers. Such a satellite would be easily detected by the Voyager spacecraft when it encounters Uranus. The 2:1 resonance location is 41 +/- 9 kilometers inside the delta ring, which makes it unlikely that the resonance is due to a viscous instability within the ring. In contrast, no low-order Lindblad resonance matches the gamma ring perturbations, which are probably caused by one or more shepherd satellites large enough to be clearly visible in Voyager images. PMID:17776019
14. On-Shell Methods in Perturbative QCD
SciTech Connect
Bern, Zvi; Dixon, Lance J.; Kosower, David A.
2007-04-25
We review on-shell methods for computing multi-parton scattering amplitudes in perturbative QCD, utilizing their unitarity and factorization properties. We focus on aspects which are useful for the construction of one-loop amplitudes needed for phenomenological studies at the Large Hadron Collider.
15. Magnetic perturbation inspection of inner bearing races
NASA Technical Reports Server (NTRS)
Barton, J. R.; Lankford, J.
1972-01-01
Approximately 100 inner race bearings were inspected nondestructively prior to endurance testing. Two of the bearings which failed during testing spalled at the sites of subsurface inclusions previously detected by using magnetic field perturbation. At other sites initially judged to be suspect, subsurface inclusion-nucleated cracking was observed. Inspection records and metallurgical sectioning results are presented and discussed.
16. Partially Quenched Chiral Perturbation Theory to NNLO
SciTech Connect
Laehde, Timo; Bijnens, Johan; Danielsson, Niclas
2006-07-11
This paper summarizes the recent calculations of the masses and decay constants of the pseudoscalar mesons at the two-loop level, or NNLO, in Partially Quenched Chiral Perturbation theory (PQ{chi}PT). Possible applications include chiral extrapolations of Lattice QCD, as well as the determination of the low-energy constants (LEC:s) of QCD.
17. Staggered heavy baryon chiral perturbation theory
Bailey, Jon A.
2008-03-01
Although taste violations significantly affect the results of staggered calculations of pseudoscalar and heavy-light mesonic quantities, those entering staggered calculations of baryonic quantities have not been quantified. Here I develop staggered chiral perturbation theory in the light-quark baryon sector by mapping the Symanzik action into heavy baryon chiral perturbation theory. For 2+1 dynamical quark flavors, the masses of flavor-symmetric nucleons are calculated to third order in partially quenched and fully dynamical staggered chiral perturbation theory. To this order the expansion includes the leading chiral logarithms, which come from loops with virtual decuplet-like states, as well as terms of O(mπ3), which come from loops with virtual octet-like states. Taste violations enter through the meson propagators in loops and tree-level terms of O(a2). The pattern of taste symmetry breaking and the resulting degeneracies and mixings are discussed in detail. The resulting chiral forms are appropriate to lattice results obtained with operators already in use and could be used to study the restoration of taste symmetry in the continuum limit. I assume that the fourth root of the fermion determinant can be incorporated in staggered chiral perturbation theory using the replica method.
18. Cell cycle population effects in perturbation studies
PubMed Central
O'Duibhir, Eoghan; Lijnzaad, Philip; Benschop, Joris J; Lenstra, Tineke L; van Leenen, Dik; Groot Koerkamp, Marian JA; Margaritis, Thanasis; Brok, Mariel O; Kemmeren, Patrick; Holstege, Frank CP
2014-01-01
Growth condition perturbation or gene function disruption are commonly used strategies to study cellular systems. Although it is widely appreciated that such experiments may involve indirect effects, these frequently remain uncharacterized. Here, analysis of functionally unrelated Saccharyomyces cerevisiae deletion strains reveals a common gene expression signature. One property shared by these strains is slower growth, with increased presence of the signature in more slowly growing strains. The slow growth signature is highly similar to the environmental stress response (ESR), an expression response common to diverse environmental perturbations. Both environmental and genetic perturbations result in growth rate changes. These are accompanied by a change in the distribution of cells over different cell cycle phases. Rather than representing a direct expression response in single cells, both the slow growth signature and ESR mainly reflect a redistribution of cells over different cell cycle phases, primarily characterized by an increase in the G1 population. The findings have implications for any study of perturbation that is accompanied by growth rate changes. Strategies to counter these effects are presented and discussed. PMID:24952590
19. Theoretical priors on modified growth parametrisations
SciTech Connect
Song, Yong-Seon; Hollenstein, Lukas; Caldera-Cabral, Gabriela; Koyama, Kazuya E-mail: [email protected] E-mail: [email protected]
2010-04-01
Next generation surveys will observe the large-scale structure of the Universe with unprecedented accuracy. This will enable us to test the relationships between matter over-densities, the curvature perturbation and the Newtonian potential. Any large-distance modification of gravity or exotic nature of dark energy modifies these relationships as compared to those predicted in the standard smooth dark energy model based on General Relativity. In linear theory of structure growth such modifications are often parameterised by virtue of two functions of space and time that enter the relation of the curvature perturbation to, first, the matter over- density, and second, the Newtonian potential. We investigate the predictions for these functions in Brans-Dicke theory, clustering dark energy models and interacting dark energy models. We find that each theory has a distinct path in the parameter space of modified growth. Understanding these theoretical priors on the parameterisations of modified growth is essential to reveal the nature of cosmic acceleration with the help of upcoming observations of structure formation.
20. Non-adiabatic perturbations in multi-component perfect fluids
SciTech Connect
Koshelev, N.A.
2011-04-01
The evolution of non-adiabatic perturbations in models with multiple coupled perfect fluids with non-adiabatic sound speed is considered. Instead of splitting the entropy perturbation into relative and intrinsic parts, we introduce a set of symmetric quantities, which also govern the non-adiabatic pressure perturbation in models with energy transfer. We write the gauge invariant equations for the variables that determine on a large scale the non-adiabatic pressure perturbation and the rate of changes of the comoving curvature perturbation. The analysis of evolution of the non-adiabatic pressure perturbation has been made for several particular models.
1. Evolution of the curvature perturbations during warm inflation
SciTech Connect
Matsuda, Tomohiro
2009-06-15
This paper considers warm inflation as an interesting application of multi-field inflation. Delta-N formalism is used for the calculation of the evolution of the curvature perturbations during warm inflation. Although the perturbations considered in this paper are decaying after the horizon exit, the corrections to the curvature perturbations sourced by these perturbations can remain and dominate the curvature perturbations at large scales. In addition to the typical evolution of the curvature perturbations, inhomogeneous diffusion rate is considered for warm inflation, which may lead to significant non-Gaussianity of the spectrum.
2. Cosmological tests of modified gravity
Koyama, Kazuya
2016-04-01
We review recent progress in the construction of modified gravity models as alternatives to dark energy as well as the development of cosmological tests of gravity. Einstein’s theory of general relativity (GR) has been tested accurately within the local universe i.e. the Solar System, but this leaves the possibility open that it is not a good description of gravity at the largest scales in the Universe. This being said, the standard model of cosmology assumes GR on all scales. In 1998, astronomers made the surprising discovery that the expansion of the Universe is accelerating, not slowing down. This late-time acceleration of the Universe has become the most challenging problem in theoretical physics. Within the framework of GR, the acceleration would originate from an unknown dark energy. Alternatively, it could be that there is no dark energy and GR itself is in error on cosmological scales. In this review, we first give an overview of recent developments in modified gravity theories including f(R) gravity, braneworld gravity, Horndeski theory and massive/bigravity theory. We then focus on common properties these models share, such as screening mechanisms they use to evade the stringent Solar System tests. Once armed with a theoretical knowledge of modified gravity models, we move on to discuss how we can test modifications of gravity on cosmological scales. We present tests of gravity using linear cosmological perturbations and review the latest constraints on deviations from the standard Λ CDM model. Since screening mechanisms leave distinct signatures in the non-linear structure formation, we also review novel astrophysical tests of gravity using clusters, dwarf galaxies and stars. The last decade has seen a number of new constraints placed on gravity from astrophysical to cosmological scales. Thanks to on-going and future surveys, cosmological tests of gravity will enjoy another, possibly even more, exciting ten years.
3. Cosmological tests of modified gravity.
PubMed
Koyama, Kazuya
2016-04-01
We review recent progress in the construction of modified gravity models as alternatives to dark energy as well as the development of cosmological tests of gravity. Einstein's theory of general relativity (GR) has been tested accurately within the local universe i.e. the Solar System, but this leaves the possibility open that it is not a good description of gravity at the largest scales in the Universe. This being said, the standard model of cosmology assumes GR on all scales. In 1998, astronomers made the surprising discovery that the expansion of the Universe is accelerating, not slowing down. This late-time acceleration of the Universe has become the most challenging problem in theoretical physics. Within the framework of GR, the acceleration would originate from an unknown dark energy. Alternatively, it could be that there is no dark energy and GR itself is in error on cosmological scales. In this review, we first give an overview of recent developments in modified gravity theories including f(R) gravity, braneworld gravity, Horndeski theory and massive/bigravity theory. We then focus on common properties these models share, such as screening mechanisms they use to evade the stringent Solar System tests. Once armed with a theoretical knowledge of modified gravity models, we move on to discuss how we can test modifications of gravity on cosmological scales. We present tests of gravity using linear cosmological perturbations and review the latest constraints on deviations from the standard [Formula: see text]CDM model. Since screening mechanisms leave distinct signatures in the non-linear structure formation, we also review novel astrophysical tests of gravity using clusters, dwarf galaxies and stars. The last decade has seen a number of new constraints placed on gravity from astrophysical to cosmological scales. Thanks to on-going and future surveys, cosmological tests of gravity will enjoy another, possibly even more, exciting ten years. PMID:27007681
4. Thermostat-Like Perturbations of an Oscillator
Freidlin, Mark
2016-07-01
We consider an oscillator with one degree of freedom perturbed by a deterministic thermostat-like perturbation and another system, in particular, another oscillator, coupled with the first one. If the Hamiltonian of the first system has saddle points, the whole system has, in a sense, a stochastic behavior on long time intervals. Under certain conditions, one can introduce the relative entropy and describe metastability and other large deviation effects in this deterministic system. If the coupled system is also an oscillator, the long time evolution of the energy of this oscillator has a diffusion approximation. To get these results one has to regularize the system. But the results are, to some extent, independent of the regularization: the stochasticity is due to instabilities at saddle points of the original system.
5. Resummation Approach in QCD Analytic Perturbation Theory
Bakulev, Alexander P.; Potapova, Irina V.
2011-10-01
We discuss the resummation approach in QCD Analytic Perturbation Theory (APT). We start we a simple example of asymptotic ower series for a zero-dimensional analog of the scalar g φ model. Then we give a short historic preamble of APT and show that renormgroup improvement of the QCD perturbation theory dictates to use the Fractional APT (FAPT). After that we discuss the (F)PT resummation of nonpower series and provide the one-, two-, and three-loop resummation recipes. We show the results of applications of these recipes to the estimation of the Adler function D(Q) in the N=4 region of Q and of the Higgs-boson-decay width Γ(mH2) for M=100-180 GeV.
6. Perturbative type II amplitudes for BPS interactions
Basu, Anirban
2016-02-01
We consider the perturbative contributions to the {{ R }}4, {D}4{{ R }}4 and {D}6{{ R }}4 interactions in toroidally compactified type II string theory. These BPS interactions do not receive perturbative contributions beyond genus three. We derive Poisson equations satisfied by these moduli dependent string amplitudes. These T-duality invariant equations have eigenvalues that are completely determined by the structure of the integrands of the multi-loop amplitudes. The source terms are given by boundary terms of the moduli space of Riemann surfaces corresponding to both separating and non-separating nodes. These are determined directly from the string amplitudes, as well as from U-duality constraints and logarithmic divergences of maximal supergravity. We explicitly solve these Poisson equations in nine and eight-dimensions.
7. A Renormalisation Group Method. III. Perturbative Analysis
Bauerschmidt, Roland; Brydges, David C.; Slade, Gordon
2015-05-01
This paper is the third in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. In this paper, we motivate and present a general approach towards second-order perturbative renormalisation, and apply it to a specific supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on . Our focus is on the critical dimension . The results include the derivation of the perturbative flow of the coupling constants, with accompanying estimates on the coefficients in the flow. These are essential results for subsequent application to the 4-dimensional weakly self-avoiding walk, including a proof of existence of logarithmic corrections to their critical scaling. With minor modifications, our results also apply to the 4-dimensional -component spin model.
8. Inflationary tensor perturbations after BICEP2.
PubMed
Caligiuri, Jerod; Kosowsky, Arthur
2014-05-16
The measurement of B-mode polarization of the cosmic microwave background at large angular scales by the BICEP experiment suggests a stochastic gravitational wave background from early-Universe inflation with a surprisingly large amplitude. The power spectrum of these tensor perturbations can be probed both with further measurements of the microwave background polarization at smaller scales and also directly via interferometry in space. We show that sufficiently sensitive high-resolution B-mode measurements will ultimately have the ability to test the inflationary consistency relation between the amplitude and spectrum of the tensor perturbations, confirming their inflationary origin. Additionally, a precise B-mode measurement of the tensor spectrum will predict the tensor amplitude on solar system scales to 20% accuracy for an exact power-law tensor spectrum, so a direct detection will then measure the running of the tensor spectral index to high precision. PMID:24877926
9. RECONSTRUCTING COSMOLOGICAL MATTER PERTURBATIONS USING STANDARD CANDLES AND RULERS
SciTech Connect
Alam, Ujjaini; Sahni, Varun; Starobinsky, Alexei A. E-mail: [email protected]
2009-10-20
For a large class of dark energy (DE) models, for which the effective gravitational constant is a constant and there is no direct exchange of energy between DE and dark matter (DM), knowledge of the expansion history suffices to reconstruct the growth factor of linearized density perturbations in the non-relativistic matter component on scales much smaller than the Hubble distance. In this paper, we develop a non-parametric method for extracting information about the perturbative growth factor from data pertaining to the luminosity or angular size distances. A comparison of the reconstructed density contrast with observations of large-scale structure and gravitational lensing can help distinguish DE models such as the cosmological constant and quintessence from models based on modified gravity theories as well as models in which DE and DM are either unified or interact directly. We show that for current supernovae (SNe) data, the linear growth factor at z = 0.3 can be constrained to 5% and the linear growth rate to 6%. With future SNe data, such as expected from the Joint Dark Energy Mission, we may be able to constrain the growth factor to 2%-3% and the growth rate to 3%-4% at z = 0.3 with this unbiased, model-independent reconstruction method. For future baryon acoustic oscillation data which would deliver measurements of both the angular diameter distance and the Hubble parameter, it should be possible to constrain the growth factor at z = 2.5%-9%. These constraints grow tighter with the errors on the data sets. With a large quantity of data expected in the next few years, this method can emerge as a competitive tool for distinguishing between different models of dark energy.
10. Reconstructing cosmological matter perturbations using standard candles and rulers
SciTech Connect
Alam, Ujjaini; Sahni, Varun; Starobinsky, Alexei A
2008-01-01
For a large class of dark energy (DE) models, for which the effective gravitational constant is a constant and there is no direct exchange of energy between DE and dark matter (DM), knowledge of the expansion history suffices to reconstruct the growth factor of linearized density perturbations in the non-relativistic matter component on scales much smaller than the Hubble distance. In this paper, we develop a non-parametric method for extracting information about the perturbative growth factor from data pertaining to the luminosity or angular size distances. A comparison of the reconstructed density contrast with observations of large-scale structure and gravitational lensing can help distinguish DE models such as the cosmological constant and quintessence from models based on modified gravity theories as well as models in which DE and DM are either unified or interact directly. We show that for current supernovae (SNe) data, the linear growth factor at z = 0.3 can be constrained to 5% and the linear growth rate to 6%. With future SNe data, such as expected from the Joint Dark Energy Mission, we may be able to constrain the growth factor to 2%-3% and the growth rate to 3%-4% at z = 0.3 with this unbiased, model-independent reconstruction method. For future baryon acoustic oscillation data which would deliver measurements of both the angular diameter distance and the Hubble parameter, it should be possible to constrain the growth factor at z = 2.5%-9%. These constraints grow tighter with the errors on the data sets. With a large quantity of data expected in the next few years, this method can emerge as a competitive tool for distinguishing between different models of dark energy.
11. Study of the spectrum of inflaton perturbations
SciTech Connect
Glenz, Matthew M.; Parker, Leonard
2009-09-15
We examine the spectrum of inflaton fluctuations resulting from any given long period of exponential inflation. Infrared and ultraviolet divergences in the inflaton dispersion summed over all modes do not appear in our approach. We show how the scale invariance of the perturbation spectrum arises. We also examine the spectrum of scalar perturbations of the metric that is created by the inflaton fluctuations that have left the Hubble sphere during inflation and the spectrum of density perturbations that they produce at reentry after inflation has ended. When the inflaton dispersion spectrum is renormalized during the expansion, we show (for the case of the quadratic inflaton potential) that the density perturbation spectrum approaches a mass-independent limit as the inflaton mass approaches zero, and remains near that limiting value for masses less than about 1/4 of the inflationary Hubble constant. We show that this limiting behavior does not occur if one only makes the Minkowski space subtraction, without the further adiabatic subtractions that involve time derivatives of the expansion scale factor a(t). We also find a parametrized expression for the energy density produced by the change in a(t) as inflation ends. If the end of inflation were sufficiently abrupt, then the temperature corresponding to this energy density could be very significant. We also show that fluctuations of the inflaton field that are present before inflation starts are not dissipated during inflation and could have a significant observational effect today. The mechanism for this is caused by the initial fluctuations through stimulated emission from the vacuum.
12. Tests of Chiral Perturbation Theory with COMPASS
SciTech Connect
Friedrich, Jan
2010-12-28
The COMPASS experiment at CERN studies with high precision pion-photon induced reactions on nuclear targets via the Primakoff effect. This offers the possibility to test chiral perturbation theory (ChPT) in various channels: Pion Compton scattering allows to clarify the longstanding question of the pion polarisabilities, single neutral pion production is related to the chiral anomaly, and for the two-pion production cross sections exist as yet untested ChPT predictions.
13. Geometric perturbation theory and plasma physics
SciTech Connect
Omohundro, S.M.
1985-04-04
Modern differential geometric techniques are used to unify the physical asymptotics underlying mechanics, wave theory and statistical mechanics. The approach gives new insights into the structure of physical theories and is suited to the needs of modern large-scale computer simulation and symbol manipulation systems. A coordinate-free formulation of non-singular perturbation theory is given, from which a new Hamiltonian perturbation structure is derived and related to the unperturbed structure. The theory of perturbations in the presence of symmetry is developed, and the method of averaging is related to reduction by a circle group action. The pseudo-forces and magnetic Poisson bracket terms due to reduction are given a natural asymptotic interpretation. Similar terms due to changing reference frames are related to the method of variation of parameters, which is also given a Hamiltonian formulation. These methods are used to answer a question about nearly periodic systems. The answer leads to a new secular perturbation theory that contains no ad hoc elements. Eikonal wave theory is given a Hamiltonian formulation that generalizes Whitham's Lagrangian approach. The evolution of wave action density on ray phase space is given a Hamiltonian structure using a Lie-Poisson bracket. The relationship between dissipative and Hamiltonian systems is discussed. A new type of attractor is defined which attracts both forward and backward in time and is shown to occur in infinite-dimensional Hamiltonian systems with dissipative behavior. The theory of Smale horseshoes is applied to gyromotion in the neighborhood of a magnetic field reversal and the phenomenon of reinsertion in area-preserving horseshoes is introduced. The central limit theorem is proved by renormalization group techniques. A natural symplectic structure for thermodynamics is shown to arise asymptotically from the maximum entropy formalism.
14. Stability of SIRS system with random perturbations
Lu, Qiuying
2009-09-01
Epidemiological models with bilinear incidence rate λSI usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable non-trivial equilibrium (i.e. interior equilibrium) corresponding to the endemic state. In this paper, we consider an epidemiological model, which is an SIRS model with or without distributed time delay influenced by random perturbations. We present the stability conditions of the disease-free equilibrium of the associated stochastic SIRS system.
15. Perturbation Theory for Superfluid in Nonuniform Potential
Koshida, Shinji; Kato, Yusuke
2016-05-01
Perturbation theory of superfluid fraction in terms of nonuniform potential is constructed. We find that the coefficient of the leading term is determined by the dynamical structure factor or density fluctuation of the system. The results for the ideal Bose gas and the interacting Bose system with linear dispersion are consistent to implications from Landau's criterion. We also find that the superfluidity of Tomonaga-Luttinger liquid with K>2 is shown to be stable against nonuniform potential.
16. Intelligent perturbation algorithms to space scheduling optimization
NASA Technical Reports Server (NTRS)
Kurtzman, Clifford R.
1991-01-01
The limited availability and high cost of crew time and scarce resources make optimization of space operations critical. Advances in computer technology coupled with new iterative search techniques permit the near optimization of complex scheduling problems that were previously considered computationally intractable. Described here is a class of search techniques called Intelligent Perturbation Algorithms. Several scheduling systems which use these algorithms to optimize the scheduling of space crew, payload, and resource operations are also discussed.
17. (Perturbed angular correlations in zirconia ceramics)
SciTech Connect
Not Available
1990-01-01
This is the progress report for the first year of the currently-approved three year funding cycle. We have carried on a vigorous program of experimental and theoretical research on microscopic properties of zirconia and ceria using the Perturbed Angular Correlation (PAC) experimental technique. The experimental method was described in the original proposal and in a number of references as well as several of the technical reports that accompany this progress report.
18. Perturbations of nested branes with induced gravity
SciTech Connect
Sbisà, Fulvio; Koyama, Kazuya E-mail: [email protected]
2014-06-01
We study the behaviour of weak gravitational fields in models where a 4D brane is embedded inside a 5D brane equipped with induced gravity, which in turn is embedded in a 6D spacetime. We consider a specific regularization of the branes internal structures where the 5D brane can be considered thin with respect to the 4D one. We find exact solutions corresponding to pure tension source configurations on the thick 4D brane, and study perturbations at first order around these background solutions. To perform the perturbative analysis, we adopt a bulk-based approach and we express the equations in terms of gauge invariant and master variables using a 4D scalar-vector-tensor decomposition. We then propose an ansatz on the behaviour of the perturbation fields when the thickness of the 4D brane goes to zero, which corresponds to configurations where gravity remains finite everywhere in the thin limit of the 4D brane. We study the equations of motion using this ansatz, and show that they give rise to a consistent set of differential equations in the thin limit, from which the details of the internal structure of the 4D brane disappear. We conclude that the thin limit of the ''ribbon'' 4D brane inside the (already thin) 5D brane is well defined (at least when considering first order perturbations around pure tension configurations), and that the gravitational field on the 4D brane remains finite in the thin limit. We comment on the crucial role of the induced gravity term on the 5D brane.
19. Neutron stars in a perturbative f(R) gravity model with strong magnetic fields
SciTech Connect
Cheoun, Myung-Ki; Deliduman, Cemsinan; Güngör, Can; Keleş, Vildan; Ryu, C.Y.; Kajino, Toshitaka; Mathews, Grant J. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]
2013-10-01
In Kaluza-Klein electromagnetism it is natural to associate modified gravity with strong electromagnetic fields. Hence, in this paper we investigate the combined effects of a strong magnetic field and perturbative f(R) gravity on the structure of neutron stars. The effect of an interior strong magnetic field of about 10{sup 17−18} G on the equation of state is derived in the context of a quantum hadrodynamics (QHD) equation of state (EoS) including effects of the magnetic pressure and energy along with occupied Landau levels. Adopting a random orientation of interior field domains, we solve the modified spherically symmetric hydrostatic equilibrium equations derived for a gravity model with f(R) = R+αR{sup 2}. Effects of both the finite magnetic field and the modified gravity are detailed for various values of the magnetic field and the perturbation parameter α along with a discussion of their physical implications. We show that there exists a parameter space of the modified gravity and the magnetic field strength, in which even a soft equation of state can accommodate a large ( > 2 M{sub s}un) maximum neutron star mass.
20. Cosmic perturbations through the cyclic ages
SciTech Connect
Erickson, Joel K.; Gratton, Steven; Steinhardt, Paul J.; Turok, Neil
2007-06-15
We analyze the evolution of cosmological perturbations in the cyclic model, paying particular attention to their behavior and interplay over multiple cycles. Our key results are: (1) galaxies and large scale structure present in one cycle are generated by the quantum fluctuations in the preceding cycle without interference from perturbations or structure generated in earlier cycles and without interfering with structure generated in later cycles; (2) the ekpyrotic phase, an epoch of gentle contraction with equation of state w>>1 preceding the hot big bang, makes the universe homogeneous, isotropic and flat within any given observer's horizon; and (3) although the universe is uniform within each observer's horizon, the structure of the cyclic universe on very large scales is more complex, owing to the effects of superhorizon length perturbations, and cannot be described globally as a Friedmann-Robertson-Walker cosmology. In particular, we show that the ekpyrotic contraction phase is so effective in smoothing, flattening and isotropizing the universe within the horizon that this phase alone suffices to solve the horizon and flatness problems even without an extended period of dark energy domination (a kind of low energy inflation). Instead, the cyclic model rests on a genuinely novel, noninflationary mechanism (ekpyrotic contraction) for resolving the classic cosmological conundrums.
1. Non-perturbative effects in spin glasses.
PubMed
Castellana, Michele; Parisi, Giorgio
2015-01-01
We present a numerical study of an Ising spin glass with hierarchical interactions--the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the -expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects. PMID:25733337
2. Noninflationary model with scale invariant cosmological perturbations
SciTech Connect
Peter, Patrick; Pinho, Emanuel J. C.; Pinto-Neto, Nelson
2007-01-15
We show that a contracting universe which bounces due to quantum cosmological effects and connects to the hot big-bang expansion phase, can produce an almost scale invariant spectrum of perturbations provided the perturbations are produced during an almost matter dominated era in the contraction phase. This is achieved using Bohmian solutions of the canonical Wheeler-DeWitt equation, thus treating both the background and the perturbations in a fully quantum manner. We find a very slightly blue spectrum (n{sub S}-1>0). Taking into account the spectral index constraint as well as the cosmic microwave background normalization measure yields an equation of state that should be less than {omega} < or approx. 8x10{sup -4}, implying n{sub S}-1{approx}O(10{sup -4}), and that the characteristic curvature scale of the Universe at the bounce is L{sub 0}{approx}10{sup 3}l{sub Pl}, a region where one expects that the Wheeler-DeWitt equation should be valid without being spoiled by string or loop quantum gravity effects. We have also obtained a consistency relation between the tensor-to-scalar ratio T/S and the scalar spectral index as T/S{approx}4.6x10{sup -2}{radical}(n{sub S}-1), leading to potentially measurable differences with inflationary predictions.
3. Relativistic Positioning System in perturbed spacetime
Kostić, Uroš; Horvat, Martin; Gomboc, Andreja
2015-11-01
We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a spacetime around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid tide, ocean tide, and Kerr rotation effect. The exchange of signals between the satellites and a user was calculated using a ray-tracing method in the Schwarzschild spacetime. We find that positioning in a perturbed spacetime is feasible and is highly accurate already with standard numerical procedures: the positioning algorithms used to transform between the emission and the Schwarzschild coordinates of the user are very accurate and time efficient—on a laptop it takes 0.04 s to determine the user’s spatial and time coordinates with a relative accuracy of {10}-28-{10}-26 and {10}-32-{10}-30, respectively.
4. Dark matter dispersion tensor in perturbation theory
Aviles, Alejandro
2016-03-01
We compute the dark matter velocity dispersion tensor up to third order in perturbation theory using the Lagrangian formalism, revealing growing solutions at the third and higher orders. Our results are general and can be used for any other perturbative formalism. As an application, corrections to the matter power spectrum are calculated, and we find that some of them have the same structure as those in the effective field theory of large-scale structure, with "EFT-like" coefficients that grow quadratically with the linear growth function and are further suppressed by powers of the logarithmic linear growth factor f ; other corrections present additional k dependence. Due to the velocity dispersions, there exists a free-streaming scale that suppresses the whole 1-loop power spectrum. Furthermore, we find that as a consequence of the nonlinear evolution, the free-streaming length is shifted towards larger scales, wiping out more structure than that expected in linear theory. Therefore, we argue that the formalism developed here is better suited for a perturbation treatment of warm dark matter or neutrino clustering, where the velocity dispersion effects are well known to be important. We discuss implications related to the nature of dark matter.
5. Using Lagrangian perturbation theory for precision cosmology
SciTech Connect
Sugiyama, Naonori S.
2014-06-10
We explore the Lagrangian perturbation theory (LPT) at one-loop order with Gaussian initial conditions. We present an expansion method to approximately compute the power spectrum LPT. Our approximate solution has good convergence in the series expansion and enables us to compute the power spectrum in LPT accurately and quickly. Non-linear corrections in this theory naturally satisfy the law of conservation of mass because the relation between matter density and the displacement vector of dark matter corresponds to the conservation of mass. By matching the one-loop solution in LPT to the two-loop solution in standard perturbation theory, we present an approximate solution of the power spectrum which has higher order corrections than the two-loop order in standard perturbation theory with the conservation of mass satisfied. With this approximation, we can use LPT to compute a non-linear power spectrum without any free parameters, and this solution agrees with numerical simulations at k = 0.2 h Mpc{sup –1} and z = 0.35 to better than 2%.
6. Predicting pathway perturbations in Down syndrome.
PubMed
Gardiner, K
2003-01-01
Comparative annotation of human chromosome 21 genomic sequence with homologous regions of mouse chromosomes 16, 17 and 10 has identified 170 orthologous gene pairs. Functional annotation of these genes, based on literature reports and computationally-derived predictions, shows that a broad range of cellular processes are represented. A goal of Down syndrome research is to determine which of these processes are perturbed by overexpression of chromosome 21 genes, and which may, therefore, contribute to the cognitive deficits that characterize Down syndrome. Eleven chromosome 21 genes are annotated to interact with or be affected by components of the MAP Kinase pathway and eight are involved in Ca2+/calcineurin signaling. Both pathways are critical for normal neurological function, and consequently their perturbations are proposed as candidates for phenotypic relevance. We present evidence suggesting that the MAP Kinase pathway is perturbed in the Ts65Dn mouse model of Down syndrome at 4-6 months of age. Analysis is complicated by the observation that overexpression of chromosome 21 genes in trisomy may be affected by method of detection, organism, tissue or brain region, and/or developmental age. PMID:15068236
7. Perturbative gravity in the causal approach
Grigore, D. R.
2010-01-01
Quantum theory of the gravitation in the causal approach is studied up to the second order of perturbation theory in the causal approach. We emphasize the use of cohomology methods in this framework. After describing in detail the mathematical structure of the cohomology method we apply it in three different situations: (a) the determination of the most general expression of the interaction Lagrangian; (b) the proof of gauge invariance in the second order of perturbation theory for the pure gravity system—massless and massive; (c) the investigation of the arbitrariness of the second-order chronological products compatible with renormalization principles and gauge invariance (i.e. the renormalization problem in the second order of perturbation theory). In case (a) we investigate pure gravity systems and the interaction of massless gravity with matter (described by scalars and spinors) and massless Yang-Mills fields. We obtain a difference with respect to the classical field theory due to the fact that in quantum field theory one cannot enforce the divergenceless property on the vector potential and this spoils the divergenceless property of the usual energy-momentum tensor. To correct this one needs a supplementary ghost term in the interaction Lagrangian. In all three case, the computations are more simple than by the usual methods.
8. Perturbative Critical Behavior from Spacetime Dependent Couplings
SciTech Connect
Dong, Xi; Horn, Bart; Silverstein, Eva; Torroba, Gonzalo
2012-08-03
We find novel perturbative fixed points by introducing mildly spacetime-dependent couplings into otherwise marginal terms. In four-dimensional QFT, these are physical analogues of the small-{epsilon} Wilson-Fisher fixed point. Rather than considering 4-{epsilon} dimensions, we stay in four dimensions but introduce couplings whose leading spacetime dependence is of the form {lambda}x{sup {kappa}}{mu}{sup {kappa}}, with a small parameter {kappa} playing a role analogous to {epsilon}. We show, in {phi}{sup 4} theory and in QED and QCD with massless flavors, that this leads to a critical theory under perturbative control over an exponentially wide window of spacetime positions x. The exact fixed point coupling {lambda}{sub *}(x) in our theory is identical to the running coupling of the translationally invariant theory, with the scale replaced by 1/x. Similar statements hold for three-dimensional {phi}{sup 6} theories and two-dimensional sigma models with curved target spaces. We also describe strongly coupled examples using conformal perturbation theory.
9. Topological quantum order: Stability under local perturbations
SciTech Connect
Bravyi, Sergey; Hastings, Matthew B.; Michalakis, Spyridon
2010-09-15
We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of geometrically local commuting projectors on a D-dimensional lattice with certain topological order conditions. Given such a Hamiltonian H{sub 0}, we prove that there exists a constant threshold {epsilon}>0 such that for any perturbation V representable as a sum of short-range bounded-norm interactions, the perturbed Hamiltonian H=H{sub 0}+{epsilon}V has well-defined spectral bands originating from low-lying eigenvalues of H{sub 0}. These bands are separated from the rest of the spectra and from each other by a constant gap. The band originating from the smallest eigenvalue of H{sub 0} has exponentially small width (as a function of the lattice size). Our proof exploits a discrete version of Hamiltonian flow equations, the theory of relatively bounded operators, and the Lieb-Robinson bound.
10. Combining coupled cluster and perturbation theory
Nooijen, Marcel
1999-12-01
Single reference coupled cluster (CC) singles and doubles theory is combined with low-order perturbation theory (PT) to treat ground state electron correlation. Two variants of the general scheme are discussed that differ in the type of amplitudes that are approximated perturbatively and which are treated to infinite order. The combined CC/PT methods to include ground state correlation are merged with equation-of-motion (EOM) and similarity transformed EOM methods to describe excitation spectra of the highly correlated s-tetrazine, MnO4- and Ni(CO)4 systems. It is shown that the computationally efficient CC/PT schemes can reproduce full CCSD results even if perturbation theory by itself is a very poor approximation, as is the case for many transition metal compounds. In a second test CC/PT is applied to determine ground state equilibrium molecular structures and harmonic vibrational frequencies for a set of small molecules. Using either variant of CC/PT, full CCSD geometries are easily recovered, while vibrational frequencies can be more sensitive to details of the approximation.
11. Baryonic matter perturbations in decaying vacuum cosmology
SciTech Connect
Marttens, R.F. vom; Zimdahl, W.; Hipólito-Ricaldi, W.S. E-mail: [email protected]
2014-08-01
We consider the perturbation dynamics for the cosmic baryon fluid and determine the corresponding power spectrum for a Λ(t)CDM model in which a cosmological term decays into dark matter linearly with the Hubble rate. The model is tested by a joint analysis of data from supernovae of type Ia (SNIa) (Constitution and Union 2.1), baryonic acoustic oscillations (BAO), the position of the first peak of the anisotropy spectrum of the cosmic microwave background (CMB) and large-scale-structure (LSS) data (SDSS DR7). While the homogeneous and isotropic background dynamics is only marginally influenced by the baryons, there are modifications on the perturbative level if a separately conserved baryon fluid is included. Considering the present baryon fraction as a free parameter, we reproduce the observed abundance of the order of 5% independently of the dark-matter abundance which is of the order of 32% for this model. Generally, the concordance between background and perturbation dynamics is improved if baryons are explicitly taken into account.
12. Multifrequency perturbations in matter-wave interferometry
Günther, A.; Rembold, A.; Schütz, G.; Stibor, A.
2015-11-01
High-contrast matter-wave interferometry is essential in various fundamental quantum mechanical experiments as well as for technical applications. Thereby, contrast and sensitivity are typically reduced by decoherence and dephasing effects. While decoherence accounts for a general loss of quantum information in a system due to entanglement with the environment, dephasing is due to collective time-dependent external phase shifts, which can be related to temperature drifts, mechanical vibrations, and electromagnetic oscillations. In contrast to decoherence, dephasing can, in principle, be reversed. Here, we demonstrate in experiment and theory a method for the analysis and reduction of the influence of dephasing noise and perturbations consisting of several external frequencies in an electron interferometer. This technique uses the high spatial and temporal resolution of a delay-line detector to reveal and remove dephasing perturbations by second-order correlation analysis. It allows matter-wave experiments under perturbing laboratory conditions and can be applied, in principle, to electron, atom, ion, neutron, and molecule interferometers.
13. Fluvial response to environmental perturbations: a perspective from physical experiments
Savi, Sara; Tofelde, Stefanie; Wickert, Andrew; Schildgen, Taylor; Paola, Chris; Strecker, Manfred
2016-04-01
Fluvial terraces and alluvial fans that are perched above the modern base level testify to environmental conditions that were different from today. Sedimentological studies combined with chronological constraints can be used to reconstruct the evolution of these landforms in the context of past changes in regional to global forcing. Despite the improvements in the most commonly used dating techniques (e.g. cosmogenic nuclides, 14C, and OSL), field data from fluvial and alluvial archives often represent only a brief glimpse into the evolution of that particular landscape. As such, the challenge of interpreting landscape development and its relationship to external forcing in the remaining time gaps is often unclear. To gain more insight, we performed physical experiments to test how a fluvial system responds to changes in the boundary conditions. This approach allows us to continuously record the evolution of the fluvial system and to observe, step by step, the response of the fluvial system and the development of the landscape. Additionally, we can directly link the geomorphic modifications to a specific environmental perturbation. Starting with a simple model and a single channel, we changed the amount of discharge (Qw) and sediment supply (Qs) in the system. The most prominent response results from a sudden increase in water discharge. In general, changes in the Qs/Qw ratio control the fluvial morphology (particularly the height/width ratio), the channel's profile, the dynamics of the river, and its ability to modify the surrounding landscape. Responses get more complex with the introduction of a lateral tributary, which changes the dynamics of the main stem and creates feed-back mechanisms between the two systems. For example, a change in the main stem can influence the fluvial morphology and the steepness of the tributary (even with no perturbations in the tributary) and vice-versa, illustrating the potential for non-unique interpretations of fluvial landforms
14. A Lattice Non-Perturbative Definition of an SO(10) Chiral Gauge Theory and Its Induced Standard Model
Wen, Xiao-Gang
2013-11-01
The standard model is a chiral gauge theory where the gauge fields couple to the right-hand and the left-hand fermions differently. The standard model is defined perturbatively and describes all elementary particles (except gravitons) very well. However, for a long time, we do not know if we can have a non-perturbative definition of the standard model as a Hamiltonian quantum mechanical theory. Here we propose a way to give a modified standard model (with 48 two-component Weyl fermions) a non-perturbative definition by embedding the modified standard model into an SO (10) chiral gauge theory. We show that the SO (10) chiral gauge theory can be put on a lattice (a 3D spatial lattice with a continuous time) if we allow fermions to interact. Such a non-perturbatively defined standard model is a Hamiltonian quantum theory with a finite-dimensional Hilbert space for a finite space volume. More generally, using the defining connection between gauge anomalies and the symmetry-protected topological orders, one can show that any truly anomaly-free chiral gauge theory can be non-perturbatively defined by putting it on a lattice in the same dimension.
15. Quantum-to-classical transition for ekpyrotic perturbations
Battarra, Lorenzo; Lehners, Jean-Luc
2014-03-01
We examine the processes of quantum squeezing and decoherence of density perturbations produced during a slowly contracting ekpyrotic phase in which entropic perturbations are converted to curvature perturbations before the bounce to an expanding phase. During the generation phase, the entropic fluctuations evolve into a highly squeezed quantum state, analogous to the evolution of inflationary perturbations. Subsequently, during the conversion phase, quantum coherence is lost very efficiently due to the interactions of entropy and adiabatic modes. Moreover, while decoherence occurs, the adiabatic curvature perturbations inherit their semiclassicality from the entropic perturbations. Our results confirm that, just as for inflation, an ekpyrotic phase can generate nearly scale-invariant curvature perturbations which may be treated as a statistical ensemble of classical density perturbations, in agreement with observations of the cosmic background radiation.
16. The collision singularity in a perturbed n-body problem.
NASA Technical Reports Server (NTRS)
Sperling, H. J.
1972-01-01
Collision of all bodies in a perturbed n-body problem is analyzed by an extension of the author's results for a perturbed two-body problem (1969). A procedure is set forth to prove that the absolute value of energy in a perturbed n-body system remains bounded until the moment of collision. It is shown that the characteristics of motion in both perturbed problems are basically the same.
17. CDM/baryon isocurvature perturbations in a sneutrino curvaton model
SciTech Connect
Harigaya, Keisuke; Kawasaki, Masahiro; Hayakawa, Taku; Yokoyama, Shuichiro E-mail: [email protected] E-mail: [email protected]
2014-10-01
Matter isocurvature perturbations are strictly constrained from cosmic microwave background observations. We study a sneutrino curvaton model where both cold dark matter (CDM)/baryon isocurvature perturbations are generated. In our model, total matter isocurvature perturbations are reduced since the CDM/baryon isocurvature perturbations compensate for each other. We show that this model can not only avoid the stringent observational constraints but also suppress temperature anisotropies on large scales, which leads to improved agreement with observations.
18. Nonlinear electromagnetic perturbations in a degenerate ultrarelativistic electron-positron plasma.
PubMed
El-Taibany, W F; Mamun, A A
2012-02-01
Nonlinear propagation of fast and slow magnetosonic perturbation modes in an ultrarelativistic, ultracold, degenerate (extremely dense) electron positron (EP) plasma (containing ultrarelativistic, ultracold, degenerate electron and positron fluids) has been investigated by the reductive perturbation method. The Alfvén wave velocity is modified due to the presence of the enthalpy correction in the fluid equations of motion. The degenerate EP plasma system (under consideration) supports the Korteweg-de Vries (KdV) solitons, which are associated with either fast or slow magnetosonic perturbation modes. It is found that the ultrarelativistic model leads to compressive (rarefactive) electromagnetic solitons corresponding to the fast (slow) wave mode. There are certain critical angles, θ(c), at which no soliton solution is found corresponding to the fast wave mode. For the slow mode, the magnetic-field intensity affects both the soliton amplitude and width. It is also illustrated that the basic features of the electromagnetic solitary structures, which are found to exist in such a degenerate EP plasma, are significantly modified by the effects of enthalpy correction, electron and positron degeneracy, magnetic-field strength, and the relativistic effect. The applications of the results in a pair-plasma medium, which occurs in many astrophysical objects (e.g., pulsars, white dwarfs, and neutron stars) are briefly discussed. PMID:22463336
19. Exact Controllability and Perturbation Analysis for Elastic Beams
SciTech Connect
Moreles, Miguel Angel
2004-05-15
The Rayleigh beam is a perturbation of the Bernoulli-Euler beam. We establish convergence of the solution of the Exact Controllability Problem for the Rayleigh beam to the corresponding solution of the Bernoulli-Euler beam. Convergence is related to a Singular Perturbation Problem. The main tool in solving this perturbation problem is a weak version of a lower bound for hyperbolic polynomials.
20. Covariant Perturbation Expansion of Off-Diagonal Heat Kernel
Gou, Yu-Zi; Li, Wen-Du; Zhang, Ping; Dai, Wu-Sheng
2016-07-01
Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.
1. On spectral perturbation caused by bounded variation of potential
SciTech Connect
Ismagilov, R S
2014-01-31
The harmonic oscillator operator is perturbed by an arbitrary bounded continuous term. This results in the perturbation of the spectrum. The map sending the first of these perturbations into the second is examined. Its approximation by a linear map is studied. Bibliography: 2 titles.
2. Perturbations of matter fields in the second-order gauge-invariant cosmological perturbation theory
Nakamura, Kouji
2009-12-01
To show that the general framework of the second-order gauge-invariant perturbation theory developed by K. Nakamura [Prog. Theor. Phys. 110, 723 (2003)PTPKAV0033-068X10.1143/PTP.110.723; Prog. Theor. Phys. 113, 481 (2005)PTPKAV0033-068X10.1143/PTP.113.481] is applicable to a wide class of cosmological situations, some formulas for the perturbations of the matter fields are summarized within the framework of the second-order gauge-invariant cosmological perturbation theory in a four-dimensional homogeneous isotropic universe, which is developed in Prog. Theor. Phys. 117, 17 (2007)PTPKAV0033-068X10.1143/PTP.117.17. We derive the formulas for the perturbations of the energy-momentum tensors and equations of motion for a perfect fluid, an imperfect fluid, and a single scalar field, and show that all equations are derived in terms of gauge-invariant variables without any gauge fixing. Through these formulas, we may say that the decomposition formulas for the perturbations of any tensor field into gauge-invariant and gauge-variant parts, which are proposed in the above papers, are universal.
3. Application of functional analysis to perturbation theory of differential equations. [nonlinear perturbation of the harmonic oscillator
NASA Technical Reports Server (NTRS)
Bogdan, V. M.; Bond, V. B.
1980-01-01
The deviation of the solution of the differential equation y' = f(t, y), y(O) = y sub O from the solution of the perturbed system z' = f(t, z) + g(t, z), z(O) = z sub O was investigated for the case where f and g are continuous functions on I x R sup n into R sup n, where I = (o, a) or I = (o, infinity). These functions are assumed to satisfy the Lipschitz condition in the variable z. The space Lip(I) of all such functions with suitable norms forms a Banach space. By introducing a suitable norm in the space of continuous functions C(I), introducing the problem can be reduced to an equivalent problem in terminology of operators in such spaces. A theorem on existence and uniqueness of the solution is presented by means of Banach space technique. Norm estimates on the rate of growth of such solutions are found. As a consequence, estimates of deviation of a solution due to perturbation are obtained. Continuity of the solution on the initial data and on the perturbation is established. A nonlinear perturbation of the harmonic oscillator is considered a perturbation of equations of the restricted three body problem linearized at libration point.
4. Generating scale-invariant tensor perturbations in the non-inflationary universe
Li, Mingzhe
2014-09-01
It is believed that the recent detection of large tensor perturbations strongly favors the inflation scenario in the early universe. This common sense depends on the assumption that Einstein's general relativity is valid at the early universe. In this paper we show that nearly scale-invariant primordial tensor perturbations can be generated during a contracting phase before the radiation dominated epoch if the theory of gravity is modified by the scalar-tensor theory at that time. The scale-invariance protects the tensor perturbations from suppressing at large scales and they may have significant amplitudes to fit BICEP2's result. We construct a model to achieve this purpose and show that the universe can bounce to the hot big bang after long time contraction, and at almost the same time the theory of gravity approaches to general relativity through stabilizing the scalar field. Theoretically, such models are dual to inflation models if we change to the frame in which the theory of gravity is general relativity. Dual models are related by the conformal transformations. With this study we reinforce the point that only the conformal invariant quantities such as the scalar and tensor perturbations are physical. How did the background evolve before the radiation time depends on the frame and has no physical meaning. It is impossible to distinguish different pictures by later time cosmological probes.
5. A non-perturbative approach to freezing of superfuid 4He in density functional theory
Minoguchi, T.; Galli, De; Rossi, M.; Yoshimori, A.
2012-12-01
Freezing of various classical liquids is successfully described by density functional theory (DFT). On the other hand, so far no report has been published that DFT describes the freezing of superfuid 4He correctly. In fact, DFT gives too stable solid phase and the superfuid phase does not exist at finite positive pressures within a second order perturbation. In this paper we try a non-perturbative version of DFT, that is modified weighted density approximation (MWDA) to go beyond second order perturbation for the freezing of superfuid 4He. Via an exact zero temperature quantum Monte-Carlo (QMC) method we have computed the equation of state and the compressibility of superfuid 4He. By utilizing a recently introduced analytic continuation method (the GIFT method), we have obtained also density response functions at different densities from QMC imaginary time correlation functions. Contrary to second order perturbation, by employing these QMC data as DFT input we find a too stable superfuid phase, preventing freezing around the experimentally observed freezing pressure. We find the same pathological behavior by using another model energy functional of superfuid 4He (Orsay-Trento model). We conclude that the straightforward MWDA calculation gives such a poor result when liquid-gas transition is present.
6. Investigation of the spatial structure and developmental dynamics of near-Earth plasma perturbations under the action of powerful HF radio waves
SciTech Connect
Belov, A. S.
2015-10-15
Results of numerical simulations of the near-Earth plasma perturbations induced by powerful HF radio waves from the SURA heating facility are presented. The simulations were performed using a modified version of the SAMI2 ionospheric model for the input parameters corresponding to the series of in-situ SURA–DEMETER experiments. The spatial structure and developmental dynamics of large-scale plasma temperature and density perturbations have been investigated. The characteristic formation and relaxation times of the induced large-scale plasma perturbations at the altitudes of the Earth’s outer ionosphere have been determined.
7. Effect of Ambipolar Plasma Flow on the Penetration of Resonant Magnetic Perturbations in a Quasi-axisymmetric Stellarator
SciTech Connect
A. Reiman; M. Zarnstorff; D. Mikkelsen; L. Owen; H. Mynick; S. Hudson; D. Monticello
2005-04-19
A reference equilibrium for the U.S. National Compact Stellarator Experiment is predicted to be sufficiently close to quasi-symmetry to allow the plasma to flow in the toroidal direction with little viscous damping, yet to have sufficiently large deviations from quasi-symmetry that nonambipolarity significantly affects the physics of the shielding of resonant magnetic perturbations by plasma flow. The unperturbed velocity profile is modified by the presence of an ambipolar potential, which produces a broad velocity profile. In the presence of a resonant magnetic field perturbation, nonambipolar transport produces a radial current, and the resulting j x B force resists departures from the ambipolar velocity and enhances the shielding.
8. Compensated isocurvature perturbations in the curvaton model
He, Chen; Grin, Daniel; Hu, Wayne
2015-09-01
Primordial fluctuations in the relative number densities of particles, or isocurvature perturbations, are generally well constrained by cosmic microwave background (CMB) data. A less probed mode is the compensated isocurvature perturbation (CIP), a fluctuation in the relative number densities of cold dark matter and baryons. In the curvaton model, a subdominant field during inflation later sets the primordial curvature fluctuation ζ . In some curvaton-decay scenarios, the baryon and cold dark matter isocurvature fluctuations nearly cancel, leaving a large CIP correlated with ζ . This correlation can be used to probe these CIPs more sensitively than the uncorrelated CIPs considered in past work, essentially by measuring the squeezed bispectrum of the CMB for triangles whose shortest side is limited by the sound horizon. Here, the sensitivity of existing and future CMB experiments to correlated CIPs is assessed, with an eye towards testing specific curvaton-decay scenarios. The planned CMB Stage 4 experiment could detect the largest CIPs attainable in curvaton scenarios with more than 3 σ significance. The significance could improve if small-scale CMB polarization foregrounds can be effectively subtracted. As a result, future CMB observations could discriminate between some curvaton-decay scenarios in which baryon number and dark matter are produced during different epochs relative to curvaton decay. Independent of the specific motivation for the origin of a correlated CIP perturbation, cross-correlation of CIP reconstructions with the primary CMB can improve the signal-to-noise ratio of a CIP detection. For fully correlated CIPs the improvement is a factor of ˜2 - 3 .
9. Exponentially modified QCD coupling
SciTech Connect
Cvetic, Gorazd; Valenzuela, Cristian
2008-04-01
We present a specific class of models for an infrared-finite analytic QCD coupling, such that at large spacelike energy scales the coupling differs from the perturbative one by less than any inverse power of the energy scale. This condition is motivated by the Institute for Theoretical and Experimental Physics operator product expansion philosophy. Allowed by the ambiguity in the analytization of the perturbative coupling, the proposed class of couplings has three parameters. In the intermediate energy region, the proposed coupling has low loop-level and renormalization scheme dependence. The present modification of perturbative QCD must be considered as a phenomenological attempt, with the aim of enlarging the applicability range of the theory of the strong interactions at low energies.
10. Tests of Chiral perturbation theory with COMPASS
Friedrich, Jan M.
2014-06-01
The COMPASS experiment at CERN accesses pion-photon reactions via the Primakoff effect., where high-energetic pions react with the quasi-real photon field surrounding the target nuclei. When a single real photon is produced, pion Compton scattering is accessed and from the measured cross-section shape, the pion polarisability is determined. The COMPASS measurement is in contradiction to the earlier dedicated measurements, and rather in agreement with the theoretical expectation from ChPT. In the same experimental data taking, reactions with neutral and charged pions in the final state are measured and analyzed in the context of chiral perturbation theory.
11. Optimized Perturbation Theory:. Finite Temperature Applications
Pinto, Marcus Benghi
2001-09-01
We review the optimized perturbation theory (or linear δ-expansion) illustrating with an application to the anharmonic oscillator. We then apply the method to multi-field O(N1) × O(N2) scalar theories at high temperatures to investigate the possibility of inverse symmetry breaking (or symmetry non restoration). Our results support inverse symmetry breaking and reveal the possibility of other high temperature symmetry breaking patterns for which the last term in the breaking sequence is O(N1 - 1) × O(N2 - 1).
12. Exciton dynamics in perturbed vibronic molecular aggregates
PubMed Central
Brüning, C.; Wehner, J.; Hausner, J.; Wenzel, M.; Engel, V.
2015-01-01
A site specific perturbation of a photo-excited molecular aggregate can lead to a localization of excitonic energy. We investigate this localization dynamics for laser-prepared excited states. Changing the parameters of the electric field significantly influences the exciton localization which offers the possibility for a selective control of this process. This is demonstrated for aggregates possessing a single vibrational degree of freedom per monomer unit. It is shown that the effects identified for the molecular dimer can be generalized to larger aggregates with a high density of vibronic states. PMID:26798840
13. Anisotropic perturbations due to dark energy
2006-08-01
A variety of observational tests seem to suggest that the Universe is anisotropic. This is incompatible with the standard dogma based on adiabatic, rotationally invariant perturbations. We point out that this is a consequence of the standard decomposition of the stress-energy tensor for the cosmological fluids, and that rotational invariance need not be assumed, if there is elastic rigidity in the dark energy. The dark energy required to achieve this might be provided by point symmetric domain wall network with P/ρ=-2/3, although the concept is more general. We illustrate this with reference to a model with cubic symmetry and discuss various aspects of the model.
14. Light-Front Perturbation Without Spurious Singularities
Przeszowski, Jerzy A.; Dzimida-Chmielewska, Elżbieta; Żochowski, Jan
2016-07-01
A new form of the light front Feynman propagators is proposed. It contains no energy denominators. Instead the dependence on the longitudinal subinterval x^2_L = 2 x+ x- is explicit and a new formalism for doing the perturbative calculations is invented. These novel propagators are implemented for the one-loop effective potential and various 1-loop 2-point functions for a massive scalar field. The consistency with results for the standard covariant Feynman diagrams is obtained and no spurious singularities are encountered at all. Some remarks on the calculations with fermion and gauge fields in QED and QCD are added.
15. Cosmological explosions from cold dark matter perturbations
NASA Technical Reports Server (NTRS)
Scherrer, Robert J.
1992-01-01
The cosmological-explosion model is examined for a universe dominated by cold dark matter in which explosion seeds are produced from the growth of initial density perturbations of a given form. Fragmentation of the exploding shells is dominated by the dark-matter potential wells rather than the self-gravity of the shells, and particular conditions are required for the explosions to bootstrap up to very large scales. The final distribution of dark matter is strongly correlated with the baryons on small scales, but uncorrelated on large scales.
16. Light-Front Perturbation Without Spurious Singularities
Przeszowski, Jerzy A.; Dzimida-Chmielewska, Elżbieta; Żochowski, Jan
2016-03-01
A new form of the light front Feynman propagators is proposed. It contains no energy denominators. Instead the dependence on the longitudinal subinterval x^2_L = 2 x+ x- is explicit and a new formalism for doing the perturbative calculations is invented. These novel propagators are implemented for the one-loop effective potential and various 1-loop 2-point functions for a massive scalar field. The consistency with results for the standard covariant Feynman diagrams is obtained and no spurious singularities are encountered at all. Some remarks on the calculations with fermion and gauge fields in QED and QCD are added.
17. Amplification of curvature perturbations in cyclic cosmology
SciTech Connect
Zhang Jun; Liu Zhiguo; Piao Yunsong
2010-12-15
We analytically and numerically show that through the cycles with nonsingular bounce, the amplitude of curvature perturbation on a large scale will be amplified and the power spectrum will redden. In some sense, this amplification will eventually destroy the homogeneity of the background, which will lead to the ultimate end of cycles of the global universe. We argue that for the model with increasing cycles, it might be possible that a fissiparous multiverse will emerge after one or several cycles, in which the cycles will continue only at corresponding local regions.
18. Gravitational perturbation and Kerr/CFT correspondence
Ghezelbash, A. M.
2016-07-01
We find the explicit form of two-point function for the conformal spin-2 energy momentum operators on the near horizon of a near extremal Kerr black hole by variation of a proper boundary action. In this regard, we consider an appropriate boundary action for the gravitational perturbation of the Kerr black hole. We show that the variation of the boundary action with respect to the boundary fields yields the two-point function for the energy momentum tensor of a conformal field theory. We find agreement between the two-point function and the correlators of the dual conformal field theory to the Kerr black hole.
19. Exciton dynamics in perturbed vibronic molecular aggregates.
PubMed
Brüning, C; Wehner, J; Hausner, J; Wenzel, M; Engel, V
2016-07-01
A site specific perturbation of a photo-excited molecular aggregate can lead to a localization of excitonic energy. We investigate this localization dynamics for laser-prepared excited states. Changing the parameters of the electric field significantly influences the exciton localization which offers the possibility for a selective control of this process. This is demonstrated for aggregates possessing a single vibrational degree of freedom per monomer unit. It is shown that the effects identified for the molecular dimer can be generalized to larger aggregates with a high density of vibronic states. PMID:26798840
20. Numerical simulation of small perturbation transonic flows
NASA Technical Reports Server (NTRS)
Seebass, A. R.; Yu, N. J.
1976-01-01
The results of a systematic study of small perturbation transonic flows are presented. Both the flow over thin airfoils and the flow over wedges were investigated. Various numerical schemes were employed in the study. The prime goal of the research was to determine the efficiency of various numerical procedures by accurately evaluating the wave drag, both by computing the pressure integral around the body and by integrating the momentum loss across the shock. Numerical errors involved in the computations that affect the accuracy of drag evaluations were analyzed. The factors that effect numerical stability and the rate of convergence of the iterative schemes were also systematically studied.
1. Approximate conservation laws in perturbed integrable lattice models
Mierzejewski, Marcin; Prosen, Tomaž; Prelovšek, Peter
2015-11-01
We develop a numerical algorithm for identifying approximately conserved quantities in models perturbed away from integrability. In the long-time regime, these quantities fully determine correlation functions of local observables. Applying the algorithm to the perturbed XXZ model, we find that the main effect of perturbation consists in expanding the support of conserved quantities. This expansion follows quadratic dependence on the strength of perturbation. The latter result, together with correlation functions of conserved quantities obtained from the memory function analysis, confirms the feasibility of the perturbation theory.
2. Perturbative charged rotating 5D Einstein-Maxwell black holes
Navarro-Lérida, Francisco
2010-12-01
We present perturbative charged rotating 5D Einstein-Maxwell black holes with spherical horizon topology. The electric charge Q is the perturbative parameter, the perturbations being performed up to 4th order. The expressions for the relevant physical properties of these black holes are given. The gyromagnetic ratio g, in particular, is explicitly shown to be non-constant in higher order, and thus to deviate from its lowest order value, g = 3. Comparison of the perturbative analytical solutions with their non-perturbative numerical counterparts shows remarkable agreement.
3. On the generation of a non-gaussian curvature perturbation during preheating
SciTech Connect
Kohri, Kazunori; Lyth, David H.; Valenzuela-Toledo, Cesar A. E-mail: [email protected]
2010-02-01
The perturbation of a light field might affect preheating and hence generate a contribution to the spectrum and non-gaussianity of the curvature perturbation ζ. The field might appear directly in the preheating model (curvaton-type preheating) or indirectly through its effect on a mass or coupling (modulated preheating). We give general expressions for ζ based on the δN formula, and apply them to the cases of quadratic and quartic chaotic inflation. For the quadratic case, curvaton-type preheating is ineffective in contributing to ζ, but modulated preheating can be effective. For quartic inflation, curvaton-type preheating may be effective but the usual δN formalism has to be modified. We see under what circumstances the recent numerical simulation of Bond et al. [0903.3407] may be enough to provide a rough estimate for this case.
4. Analysis of a Leslie-Gower-type prey-predator model with periodic impulsive perturbations
Chen, Yiping; Liu, Zhijun; Haque, Mainul
2009-08-01
A modified Leslie-Gower-type prey-predator model with periodic impulsive perturbations is proposed and investigated. It is proved that there exists an asymptotically stable prey-free periodic solution when the impulsive period is less than some critical value. Otherwise, the above system can be permanent. And then the numerical simulations are carried out to study the effects of the impulsive varying parameters of the system. The results of simulations show that the model we consider, under the effects of impulsive perturbations for biologically feasible parametric values, has more complex dynamics including cycle, period adding, 3 T -period oscillation, chaos, period-doubling bifurcation, period-halving bifurcation, period windows, symmetry-breaking pitchfork bifurcation, and non-unique dynamics, meaning that several attractors coexist.
5. An Experimental Device for Generating High Frequency Perturbations in Supersonic Wind Tunnels
NASA Technical Reports Server (NTRS)
Melcher, Kevin J.; Ibrahim, Mounir B.
1996-01-01
This paper describes the analytical study of a device that has been proposed as a mechanism for generating gust-like perturbations in supersonic wind tunnels. The device is envisioned as a means to experimentally validate dynamic models and control systems designed for high-speed inlets. The proposed gust generator is composed of two flat trapezoidal plates that modify the properties of the flow ingested by the inlet. One plate may be oscillated to generate small perturbations in the flow. The other plate is held stationary to maintain a constant angle-of-attack. Using an idealized approach, design equations and performance maps for the new device were developed from the compressible flow relations. A two-dimensional CFD code was used to confirm the correctness of these results. The idealized approach was then used to design and evaluate a new gust generator for a 3.05-meter by 3.05-meter (10-foot by 10-foot) supersonic wind tunnel.
6. Ideal plasma response to vacuum magnetic fields with resonant magnetic perturbations in non-axisymmetric tokamaks
SciTech Connect
Kim, Kimin; Ahn, J-W; Scotti, F.; Park, J-K; Menard, J. E.
2015-09-03
Ideal plasma shielding and amplification of resonant magnetic perturbations in non-axisymmetric tokamak is presented by field line tracing simulation with full ideal plasma response, compared to measurements of divertor lobe structures. Magnetic field line tracing simulations in NSTX with toroidal non-axisymmetry indicate the ideal plasma response can significantly shield/amplify and phase shift the vacuum resonant magnetic perturbations. Ideal plasma shielding for n = 3 mode is found to prevent magnetic islands from opening as consistently shown in the field line connection length profile and magnetic footprints on the divertor target. It is also found that the ideal plasma shielding modifies the degree of stochasticity but does not change the overall helical lobe structures of the vacuum field for n = 3. Amplification of vacuum fields by the ideal plasma response is predicted for low toroidal mode n = 1, better reproducing measurements of strong striation of the field lines on the divertor plate in NSTX.
7. Ideal plasma response to vacuum magnetic fields with resonant magnetic perturbations in non-axisymmetric tokamaks
SciTech Connect
Kim, Kimin; Ahn, J. -W.; Scotti, F.; Park, J. -K.; Menard, J. E.
2015-09-03
Ideal plasma shielding and amplification of resonant magnetic perturbations in non-axisymmetric tokamak is presented by field line tracing simulation with full ideal plasma response, compared to measurements of divertor lobe structures. Magnetic field line tracing simulations in NSTX with toroidal non-axisymmetry indicate the ideal plasma response can significantly shield/amplify and phase shift the vacuum resonant magnetic perturbations. Ideal plasma shielding for n = 3 mode is found to prevent magnetic islands from opening as consistently shown in the field line connection length profile and magnetic footprints on the divertor target. It is also found that the ideal plasma shielding modifies the degree of stochasticity but does not change the overall helical lobe structures of the vacuum field for n = 3. Furthermore, amplification of vacuum fields by the ideal plasma response is predicted for low toroidal mode n = 1, better reproducing measurements of strong striation of the field lines on the divertor plate in NSTX.
8. General Analysis of Type I Planetary Migration with Stochastic Perturbations
Adams, Fred C.; Bloch, Anthony M.
2009-08-01
This paper presents a generalized treatment of Type I planetary migration in the presence of stochastic perturbations. In many planet-forming disks, the Type I migration mechanism, driven by asymmetric torques, acts on a short timescale and compromises planet formation. If the disk also supports magnetohydrodynamics instabilities, however, the corresponding turbulent fluctuations produce additional stochastic torques that modify the steady inward migration scenario. This work studies the migration of planetary cores in the presence of stochastic fluctuations using complementary methods, including a Fokker-Planck approach and iterative maps. Stochastic torques have two main effects. (1) Through outward diffusion, a small fraction of the planetary cores can survive in the face of Type I inward migration. (2) For a given starting condition, the result of any particular realization of migration is uncertain, so that results must be described in terms of the distributions of outcomes. In addition to exploring different regimes of parameter space, this paper considers the effects of the outer disk boundary condition, varying initial conditions, and time dependence of the torque parameters. For disks with finite radii, the fraction of surviving planets decreases exponentially with time. We find the survival fractions and decay rates for a range of disk models, and find the expected distribution of locations for surviving planets. For expected disk properties, the survival fraction lies in the range 0.01 < pS < 0.1.
9. GATO Code Modification to Compute Plasma Response to External Perturbations
Turnbull, A. D.; Chu, M. S.; Ng, E.; Li, X. S.; James, A.
2006-10-01
It has become increasingly clear that the plasma response to an external nonaxiymmetric magnetic perturbation cannot be neglected in many situations of interest. This response can be described as a linear combination of the eigenmodes of the ideal MHD operator. The eigenmodes of the system can be obtained numerically with the GATO ideal MHD stability code, which has been modified for this purpose. A key requirement is the removal of inadmissible continuum modes. For Finite Hybrid Element codes such as GATO, a prerequisite for this is their numerical restabilization by addition of small numerical terms to δ,to cancel the analytic numerical destabilization. In addition, robustness of the code was improved and the solution method speeded up by use of the SuperLU package to facilitate calculation of the full set of eigenmodes in a reasonable time. To treat resonant plasma responses, the finite element basis has been extended to include eigenfunctions with finite jumps at rational surfaces. Some preliminary numerical results for DIII-D equilibria will be given.
10. Perturbation Theory for Parent Hamiltonians of Matrix Product States
Szehr, Oleg; Wolf, Michael M.
2015-05-01
This article investigates the stability of the ground state subspace of a canonical parent Hamiltonian of a Matrix product state against local perturbations. We prove that the spectral gap of such a Hamiltonian remains stable under weak local perturbations even in the thermodynamic limit, where the entire perturbation might not be bounded. Our discussion is based on preceding work by Yarotsky that develops a perturbation theory for relatively bounded quantum perturbations of classical Hamiltonians. We exploit a renormalization procedure, which on large scale transforms the parent Hamiltonian of a Matrix product state into a classical Hamiltonian plus some perturbation. We can thus extend Yarotsky's results to provide a perturbation theory for parent Hamiltonians of Matrix product states and recover some of the findings of the independent contributions (Cirac et al in Phys Rev B 8(11):115108, 2013) and (Michalakis and Pytel in Comm Math Phys 322(2):277-302, 2013).
11. Molecular cluster perturbation theory. I. Formalism
Byrd, Jason N.; Jindal, Nakul; Molt, Robert W., Jr.; Bartlett, Rodney J.; Sanders, Beverly A.; Lotrich, Victor F.
2015-11-01
We present second-order molecular cluster perturbation theory (MCPT(2)), a linear scaling methodology to calculate arbitrarily large systems with explicit calculation of individual wave functions in a coupled-cluster framework. This new MCPT(2) framework uses coupled-cluster perturbation theory and an expansion in terms of molecular dimer interactions to obtain molecular wave functions that are infinite order in both the electronic fluctuation operator and all possible dimer (and products of dimers) interactions. The MCPT(2) framework has been implemented in the new SIA/Aces4 parallel architecture, making use of the advanced dynamic memory control and fine-grained parallelism to perform very large explicit molecular cluster calculations. To illustrate the power of this method, we have computed energy shifts, lattice site dipole moments, and harmonic vibrational frequencies via explicit calculation of the bulk system for the polar and non-polar polymorphs of solid hydrogen fluoride. The explicit lattice size (without using any periodic boundary conditions) was expanded up to 1000 HF molecules, with 32,000 basis functions and 10,000 electrons. Our obtained HF lattice site dipole moments and harmonic vibrational frequencies agree well with the existing literature.
12. Aircraft Range Optimization Using Singular Perturbations
NASA Technical Reports Server (NTRS)
Oconnor, Joseph Taffe
1973-01-01
An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.
13. Hormonal Perturbations in Occupationally Exposed Nickel Workers
PubMed Central
Beshir, Safia; Ibrahim, Khadiga Salah; Shaheen, Weam; Shahy, Eman M.
2016-01-01
BACKGROUND: Nickel exposure is recognized as an endocrine disruptor because of its adverse effects on reproduction. AIM: This study was designed to investigate the possible testiculo-hormonal perturbations on workers occupationally exposed to nickel and to assess its effects on human male sexual function. METHODS: Cross-sectional comparative study, comprising 105 electroplating male non-smoker, non-alcoholic workers exposed to soluble nickel and 60 controls was done. Serum luteinizing hormone, follicle stimulating hormone, testosterone levels and urinary nickel concentrations were determined for the studied groups. RESULTS: Serum luteinizing hormone, follicle stimulating hormone, urinary nickel and the simultaneous incidence of more than one sexual disorder were significantly higher in the exposed workers compared to controls. The occurrence of various types of sexual disorders (decreased libido, impotence and premature ejaculation) in the exposed workers was 9.5, 5.1 and 4.4 folds respectively than the controls. CONCLUSIONS: Exposure to nickel produces possible testiculo-hormonal perturbations in those exposed workers. PMID:27335607
14. Perturbed cholesterol homeostasis in aging spinal cord.
PubMed
Parkinson, Gemma M; Dayas, Christopher V; Smith, Doug W
2016-09-01
The spinal cord is vital for the processing of sensorimotor information and for its propagation to and from both the brain and the periphery. Spinal cord function is affected by aging, however, the mechanisms involved are not well-understood. To characterize molecular mechanisms of spinal cord aging, microarray analyses of gene expression were performed on cervical spinal cords of aging rats. Of the metabolic and signaling pathways affected, cholesterol-associated pathways were the most comprehensively altered, including significant downregulation of cholesterol synthesis-related genes and upregulation of cholesterol transport and metabolism genes. Paradoxically, a significant increase in total cholesterol content was observed-likely associated with cholesterol ester accumulation. To investigate potential mechanisms for the perturbed cholesterol homeostasis, we quantified the expression of myelin and neuroinflammation-associated genes and proteins. Although there was minimal change in myelin-related expression, there was an increase in phagocytic microglial and astrogliosis markers, particularly in the white matter. Together, these results suggest that perturbed cholesterol homeostasis, possibly as a result of increased inflammatory activation in spinal cord white matter, may contribute to impaired spinal cord function with aging. PMID:27459933
15. Articulatory and acoustic adaptation to palatal perturbation.
PubMed
Thibeault, Mélanie; Ménard, Lucie; Baum, Shari R; Richard, Gabrielle; McFarland, David H
2011-04-01
Previous work has established that speakers have difficulty making rapid compensatory adjustments in consonant production (especially in fricatives) for structural perturbations of the vocal tract induced by artificial palates with thicker-than-normal alveolar regions. The present study used electromagnetic articulography and simultaneous acoustic recordings to estimate tongue configurations during production of [s š t k] in the presence of a thin and a thick palate, before and after a practice period. Ten native speakers of English participated in the study. In keeping with previous acoustic studies, fricatives were more affected by the palate than were the stops. The thick palate lowered the center of gravity and the jaw was lower and the tongue moved further backwards and downwards. Center of gravity measures revealed complete adaptation after training, and with practice, subjects' decreased interlabial distance. The fact that adaptation effects were found for [k], which are produced with an articulatory gesture not directly impeded by the palatal perturbation, suggests a more global sensorimotor recalibration that extends beyond the specific articulatory target. PMID:21476667
16. Quasi-periodic oscillations of perturbed tori
Parthasarathy, Varadarajan; Manousakis, Antonios; Kluźniak, Włodzimierz
2016-05-01
We performed axisymmetric hydrodynamical simulations of oscillating tori orbiting a non-rotating black hole. The tori in equilibrium were constructed with a constant distribution of angular momentum in a pseudo-Newtonian potential (Kluźniak-Lee). Motions of the torus were triggered by adding subsonic velocity fields: radial, vertical and diagonal to the tori in equilibrium. As the perturbed tori evolved in time, we measured L2 norm of density and obtained the power spectrum of L2 norm which manifested eigenfrequencies of tori modes. The most prominent modes of oscillation excited in the torus by a quasi-random perturbation are the breathing mode and the radial and vertical epicyclic modes. The radial and the plus modes, as well as the vertical and the breathing modes will have frequencies in an approximate 3:2 ratio if the torus is several Schwarzschild radii away from the innermost stable circular orbit. Results of our simulations may be of interest in the context of high-frequency quasi-periodic oscillations observed in stellar-mass black hole binaries, as well as in supermassive black holes.
17. BICEP2, the curvature perturbation and supersymmetry
Lyth, David H.
2014-11-01
The tensor fraction r simeq 0.16 found by BICEP2 corresponds to a Hubble parameter H simeq 1.0 × 1014 GeV during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than 10% of the curvature perturbation ζ, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating ζ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed r as well as the observed spectral index ns. The shift symmetry does not require supersymmetry. Also, the big H may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC.
18. Coupled oscillator model for nonlinear gravitational perturbations
Yang, Huan; Zhang, Fan; Green, Stephen R.; Lehner, Luis
2015-04-01
Motivated by the gravity-fluid correspondence, we introduce a new method for characterizing nonlinear gravitational interactions. Namely we map the nonlinear perturbative form of the Einstein equation to the equations of motion of a collection of nonlinearly coupled harmonic oscillators. These oscillators correspond to the quasinormal or normal modes of the background spacetime. We demonstrate the mechanics and the utility of this formalism within the context of perturbed asymptotically anti-de Sitter black brane spacetimes. We confirm in this case that the boundary fluid dynamics are equivalent to those of the hydrodynamic quasinormal modes of the bulk spacetime. We expect this formalism to remain valid in more general spacetimes, including those without a fluid dual. In other words, although born out of the gravity-fluid correspondence, the formalism is fully independent and it has a much wider range of applicability. In particular, as this formalism inspires an especially transparent physical intuition, we expect its introduction to simplify the often highly technical analytical exploration of nonlinear gravitational dynamics.
19. Perturbative unitarity of Higgs derivative interactions
Kikuta, Yohei; Yamamoto, Yasuhiro
2013-05-01
We study the perturbative unitarity bound given by dimension six derivative interactions consisting of Higgs doublets. These interactions emerge from kinetic terms of composite Higgs models or integrating out heavy particles that interact with Higgs doublets. They lead to new phenomena beyond the Standard Model. One of characteristic contributions from derivative interactions appear in vector boson scattering processes. Longitudinal modes of massive vector bosons can be regarded as Nambu Goldstone bosons eaten by each vector field. Since their effects become larger and larger as the collision energy of vector bosons increases, vector boson scattering processes become important in the high energy region around the TeV scale. On the other hand, in such a high energy region, we have to take into account the unitarity of amplitudes. We have obtained the unitarity condition in terms of the parameter included in the effective Lagrangian for one Higgs doublet models. Applying it to some models, we have found that contributions of derivative interactions are not so large enough to clearly discriminate them from the Standard Model ones. We also study the unitarity bound in two Higgs doublet models. Because they are too complex to obtain it in the general effective Lagrangian, we have calculated it in explicit models. These analyses tell that the perturbative unitarity bounds are highly model dependent.
20. Perturbative Corrections for Staggered Fermion Bilinears
SciTech Connect
Patel, Apoorva; Sharpe, Stephen
1992-01-01
We calculate the perturbative corrections to fermion bilinears that are used in numerical simulations when extracting weak matric elements using staggered fermions.This extends previous calculations of Golterman and Smit, and Daniel and Sheard.In particular, we calculate the corrections for non-local bilinears defined in Landau gauge with gauge links excluded.We do this for the simplest operators, i.e. those defined on a 2^4 hypercube, and for tree level improved operators which live on 4^4 hypercubes.We also consider gauge invariant operators in which the "tadpole" contributions are suppressed by projecting the sums of products of gauge links back in to the gauge group.In all cases, we find that the variation in the size of the perturbative corrections is smaller than those with the gauge invariant unimproved operators.This is most strikingly true for the smeared operators.We investigate the efficacy of the mean-field method of Lepage and Mackenzie at summing up tadpole
1. Gradient expansion, curvature perturbations, and magnetized plasmas
SciTech Connect
Giovannini, Massimo; Rezaei, Zahra
2011-04-15
The properties of magnetized plasmas are always investigated under the hypothesis that the relativistic inhomogeneities stemming from the fluid sources and from the geometry itself are sufficiently small to allow for a perturbative description prior to photon decoupling. The latter assumption is hereby relaxed and predecoupling plasmas are described within a suitable expansion where the inhomogeneities are treated to a given order in the spatial gradients. It is argued that the (general relativistic) gradient expansion shares the same features of the drift approximation, customarily employed in the description of cold plasmas, so that the two schemes are physically complementary in the large-scale limit and for the low-frequency branch of the spectrum of plasma modes. The two-fluid description, as well as the magnetohydrodynamical reduction, is derived and studied in the presence of the spatial gradients of the geometry. Various solutions of the coupled system of evolution equations in the anti-Newtonian regime and in the quasi-isotropic approximation are presented. The relation of this analysis to the so-called separate universe paradigm is outlined. The evolution of the magnetized curvature perturbations in the nonlinear regime is addressed for the magnetized adiabatic mode in the plasma frame.
2. Quantum cosmological perturbations of multiple fluids
Peter, Patrick; Pinto-Neto, N.; Vitenti, Sandro D. P.
2016-01-01
The formalism to treat quantization and evolution of cosmological perturbations of multiple fluids is described. We first construct the Lagrangian for both the gravitational and matter parts, providing the necessary relevant variables and momenta leading to the quadratic Hamiltonian describing linear perturbations. The final Hamiltonian is obtained without assuming any equations of motions for the background variables. This general formalism is applied to the special case of two fluids, having in mind the usual radiation and matter mix which made most of our current Universe history. Quantization is achieved using an adiabatic expansion of the basis functions. This allows for an unambiguous definition of a vacuum state up to the given adiabatic order. Using this basis, we show that particle creation is well defined for a suitable choice of vacuum and canonical variables, so that the time evolution of the corresponding quantum fields is unitary. This provides constraints for setting initial conditions for an arbitrary number of fluids and background time evolution. We also show that the common choice of variables for quantization can lead to an ill-defined vacuum definition. Our formalism is not restricted to the case where the coupling between fields is small, but is only required to vary adiabatically with respect to the ultraviolet modes, thus paving the way to consistent descriptions of general models not restricted to single-field (or fluid).
3. Perturbative quantum gravity in double field theory
Boels, Rutger H.; Horst, Christoph
2016-04-01
We study perturbative general relativity with a two-form and a dilaton using the double field theory formulation which features explicit index factorisation at the Lagrangian level. Explicit checks to known tree level results are performed. In a natural covariant gauge a ghost-like scalar which contributes even at tree level is shown to decouple consistently as required by perturbative unitarity. In addition, a lightcone gauge is explored which bypasses the problem altogether. Using this gauge to study BCFW on-shell recursion, we can show that most of the D-dimensional tree level S-matrix of the theory, including all pure graviton scattering amplitudes, is reproduced by the double field theory. More generally, we argue that the integrand may be reconstructed from its single cuts and provide limited evidence for off-shell cancellations in the Feynman graphs. As a straightforward application of the developed technology double field theory-like expressions for four field string corrections are derived.
4. Parallel magnetic field perturbations in gyrokinetic simulations
SciTech Connect
Joiner, N.; Hirose, A.; Dorland, W.
2010-07-15
At low beta it is common to neglect parallel magnetic field perturbations on the basis that they are of order beta{sup 2}. This is only true if effects of order beta are canceled by a term in the nablaB drift also of order beta[H. L. Berk and R. R. Dominguez, J. Plasma Phys. 18, 31 (1977)]. To our knowledge this has not been rigorously tested with modern gyrokinetic codes. In this work we use the gyrokinetic code GS2[Kotschenreuther et al., Comput. Phys. Commun. 88, 128 (1995)] to investigate whether the compressional magnetic field perturbation B{sub ||} is required for accurate gyrokinetic simulations at low beta for microinstabilities commonly found in tokamaks. The kinetic ballooning mode (KBM) demonstrates the principle described by Berk and Dominguez strongly, as does the trapped electron mode, in a less dramatic way. The ion and electron temperature gradient (ETG) driven modes do not typically exhibit this behavior; the effects of B{sub ||} are found to depend on the pressure gradients. The terms which are seen to cancel at long wavelength in KBM calculations can be cumulative in the ion temperature gradient case and increase with eta{sub e}. The effect of B{sub ||} on the ETG instability is shown to depend on the normalized pressure gradient beta{sup '} at constant beta.
5. Perturbation analysis for patch occupancy dynamics
USGS Publications Warehouse
Martin, Julien; Nichols, James D.; McIntyre, Carol L.; Ferraz, Goncalo; Hines, James E.
2009-01-01
Perturbation analysis is a powerful tool to study population and community dynamics. This article describes expressions for sensitivity metrics reflecting changes in equilibrium occupancy resulting from small changes in the vital rates of patch occupancy dynamics (i.e., probabilities of local patch colonization and extinction). We illustrate our approach with a case study of occupancy dynamics of Golden Eagle (Aquila chrysaetos) nesting territories. Examination of the hypothesis of system equilibrium suggests that the system satisfies equilibrium conditions. Estimates of vital rates obtained using patch occupancy models are used to estimate equilibrium patch occupancy of eagles. We then compute estimates of sensitivity metrics and discuss their implications for eagle population ecology and management. Finally, we discuss the intuition underlying our sensitivity metrics and then provide examples of ecological questions that can be addressed using perturbation analyses. For instance, the sensitivity metrics lead to predictions about the relative importance of local colonization and local extinction probabilities in influencing equilibrium occupancy for rare and common species.
6. BICEP2, the curvature perturbation and supersymmetry
SciTech Connect
Lyth, David H.
2014-11-01
The tensor fraction r ≅ 0.16 found by BICEP2 corresponds to a Hubble parameter H ≅ 1.0 × 10{sup 14} GeV during inflation. This has two implications for the (single-field) slow-roll inflation hypothesis. First, the inflaton perturbation must account for much more than 10% of the curvature perturbation ζ, which barring fine-tuning means that it accounts for practically all of it. It follows that a curvaton-like mechanism for generating ζ requires an alternative to slow roll such as k-inflation. Second, accepting slow-roll inflation, the excursion of the inflaton field is at least of order Planck scale. As a result, the flatness of the inflaton presumably requires a shift symmetry. I point out that if such is the case, the resulting potential is likely to have at least approximately the quadratic form suggested in 1983 by Linde, which is known to be compatible with the observed r as well as the observed spectral index n{sub s}. The shift symmetry does not require supersymmetry. Also, the big H may rule out a GUT by restoring the symmetry and producing fatal cosmic strings. The absence of a GUT would correspond to the absence of superpartners for the Standard Model particles, which indeed have yet to be found at the LHC.
7. Perturbation solutions of combustion instability problems
NASA Technical Reports Server (NTRS)
Googerdy, A.; Peddieson, J., Jr.; Ventrice, M.
1979-01-01
A method involving approximate modal analysis using the Galerkin method followed by an approximate solution of the resulting modal-amplitude equations by the two-variable perturbation method (method of multiple scales) is applied to two problems of pressure-sensitive nonlinear combustion instability in liquid-fuel rocket motors. One problem exhibits self-coupled instability while the other exhibits mode-coupled instability. In both cases it is possible to carry out the entire linear stability analysis and significant portions of the nonlinear stability analysis in closed form. In the problem of self-coupled instability the nonlinear stability boundary and approximate forms of the limit-cycle amplitudes and growth and decay rates are determined in closed form while the exact limit-cycle amplitudes and growth and decay rates are found numerically. In the problem of mode-coupled instability the limit-cycle amplitudes are found in closed form while the growth and decay rates are found numerically. The behavior of the solutions found by the perturbation method are in agreement with solutions obtained using complex numerical methods.
8. Harmonically Perturbed Gas-Solid Fluidized System
Nix, S. T.; Muller, M. R.
1996-11-01
Experiments were performed on a harmonically perturbed gas-solid fluidized system, to determine the extent to which the total system behaved as a liquid in regards to the aspects of resonant frequency, wave shapes, and damping effects. The fluidized system consists of a cylindrical alumina oxide/air fluidized bed which is vertically perturbed in a symmetrical fashion by externally vibrating the entire vessel. The external vibrations were carried out over various flow rates, amplitudes, and frequencies. The results obtained could then be compared to the natural frequencies of water for the same governing parameters by analytical means. The effects of excitations on the formation of voidage disturbances or "gas bubbles" was also investigated. Data acquisition enabled the determination of both the amplitude and frequency of the waves generated in the bath. Results indicate that external vertical vibration caused the mean surface level of the bed to drop. This can be attributed to a closer packing of the particles along with a decrease in the number and size of gas bubbles in the bed.
9. Cosmological perturbations in a mimetic matter model
Matsumoto, Jiro; Odintsov, Sergei D.; Sushkov, Sergey V.
2015-03-01
We investigate the cosmological evolution of a mimetic matter model with arbitrary scalar potential. The cosmological reconstruction—which is the method for constructing a model for an arbitrary evolution of the scale factor—is explicitly performed for different choices of potential. The cases where the mimetic matter model shows the evolution as cold dark matter (CDM), the w CDM model, dark matter and dark energy with a dynamical O m (z ) [where O m (z )≡[(H (z )/H0)2-1 ]/[(1 +z )3-1 ] ], and phantom dark energy with a phantom-nonphantom crossing are presented in detail. The cosmological perturbations for such evolutions are studied in the mimetic matter model. For instance, the evolution behavior of the matter density contrast (which is different than the usual one, i.e., δ ¨+2 H δ ˙-κ2ρ δ /2 =0 ) is investigated. The possibility of a peculiar evolution of δ in the model under consideration is shown. Special attention is paid to the behavior of the matter density contrast near the future singularity, where the decay of perturbations may occur much earlier than the singularity.
10. Dynamic Epistasis under Varying Environmental Perturbations
PubMed Central
Barker, Brandon; Xu, Lin; Gu, Zhenglong
2015-01-01
Epistasis describes the phenomenon that mutations at different loci do not have independent effects with regard to certain phenotypes. Understanding the global epistatic landscape is vital for many genetic and evolutionary theories. Current knowledge for epistatic dynamics under multiple conditions is limited by the technological difficulties in experimentally screening epistatic relations among genes. We explored this issue by applying flux balance analysis to simulate epistatic landscapes under various environmental perturbations. Specifically, we looked at gene-gene epistatic interactions, where the mutations were assumed to occur in different genes. We predicted that epistasis tends to become more positive from glucose-abundant to nutrient-limiting conditions, indicating that selection might be less effective in removing deleterious mutations in the latter. We also observed a stable core of epistatic interactions in all tested conditions, as well as many epistatic interactions unique to each condition. Interestingly, genes in the stable epistatic interaction network are directly linked to most other genes whereas genes with condition-specific epistasis form a scale-free network. Furthermore, genes with stable epistasis tend to have similar evolutionary rates, whereas this co-evolving relationship does not hold for genes with condition-specific epistasis. Our findings provide a novel genome-wide picture about epistatic dynamics under environmental perturbations. PMID:25625594
11. Fast ion loss associated with perturbed field by resonant magnetic perturbation coils in KSTAR
Kim, Jun Young; Kim, Junghee; Rhee, Tongnyeol; Yoon, S. W.; Park, G. Y.; Jeon, Y. M.; Isobe, M.; Shimizu, A.; Ogawa, K.; Park, J.-K.; Garcia-Munoz, M.
2013-10-01
Resonant magnetic perturbation (RMP) is the most promising strategies for ELM mitigation/suppression. However, it has been found through the modeling and the experiments that RMP for the ELM mitigation can enhance the toroidally localized fast ion loss. During KSTAR experimental campaigns in 2011 and 2012, sudden increase or decrease of the fast ion loss has been observed by the scintillator-based fast ion loss detector (FILD) when the RMP is applied. Three-dimensional perturbed magnetic field by RMP coil in vacuum is calculated by Biot-Savart's law embedded in the Lorentz orbit code (LORBIT). The LORBIT code which is based on gyro-orbit following motion has been used for the simulation of the three-dimensional fast ion trajectories in presence of non-axisymmetric magnetic perturbation. It seems the measured fast ion loss rate at the localized position depends on not only the RMP field configuration but also the plasma profile such as safety factor and so on, varying the ratio between radial drift and stochastization of the fat-ion orbits. The simulation results of fast ion orbit under magnetic perturbation w/ and w/o plasma responses will be presented and compared with KSTAR FILD measurement results in various cases.
12. Wave propagation in modified gravity
Lindroos, Jan Ø.; Llinares, Claudio; Mota, David F.
2016-02-01
We investigate the propagation of scalar waves induced by matter sources in the context of scalar-tensor theories of gravity which include screening mechanisms for the scalar degree of freedom. The usual approach when studying these theories in the nonlinear regime of cosmological perturbations is based on the assumption that scalar waves travel at the speed of light. Within general relativity this approximation is valid and leads to no loss of accuracy in the estimation of observables. We find, however, that mass terms and nonlinearities in the equations of motion lead to propagation and dispersion velocities significantly different from the speed of light. As the group velocity is the one associated with the propagation of signals, a reduction of its value has direct impact on the behavior and dynamics of nonlinear structures within modified gravity theories with screening. For instance, the internal dynamics of galaxies and satellites submerged in large dark matter halos could be affected by the fact that the group velocity is smaller than the speed of light. It is therefore important, within such a framework, to take into account the fact that different parts of a galaxy will see changes in the environment at different times. A full nonstatic analysis may be necessary under those conditions.
13. Dynamics of free surface perturbations along an annular viscous film
Smolka, Linda B.; North, Justin; Guerra, Bree K.
2008-03-01
It is known that the free surface of an axisymmetric viscous film flowing down the outside of a thin vertical fiber under the influence of gravity becomes unstable to interfacial perturbations. We present an experimental study using fluids with different densities, surface tensions, and viscosities to investigate the growth and dynamics of these interfacial perturbations and to test the assumptions made by previous authors. We find that the initial perturbation growth is exponential, followed by a slower phase as the amplitude and wavelength saturate in size. Measurements of the perturbation growth for experiments conducted at low and moderate Reynolds numbers are compared to theoretical predictions developed from linear stability theory. Excellent agreement is found between predictions from a long-wave Stokes flow model [Craster and Matar, J. Fluid Mech. 553, 85 (2006)] and data, while fair to excellent agreement (depending on fiber size) is found between predictions from a moderate-Reynolds-number model [Sisoev , Chem. Eng. Sci. 61, 7279 (2006)] and data. Furthermore, we find that a known transition in the longer-time perturbation dynamics from unsteady to steady behavior at a critical flow rate Qc is correlated with a transition in the rate at which perturbations naturally form along the fiber. For Qperturbation formation is constant. As a result, the position along the fiber where perturbations form is nearly fixed, and the spacing between consecutive perturbations remains constant as they travel 2 m down the fiber. For Q>Qc (unsteady case), the rate of perturbation formation is modulated. As a result, the position along the fiber where perturbations form oscillates irregularly, and the initial speed and spacing between perturbations varies, resulting in the coalescence of neighboring perturbations further down the fiber.
14. Perturbative analysis of sheared flow Kelvin-Helmholtz instability in a weakly relativistic magnetized electron fluid
SciTech Connect
Sundar, Sita; Das, Amita; Kaw, Predhiman
2012-05-15
In the interaction of intense lasers with matter/plasma, energetic electrons having relativistic energies get created. These energetic electrons can often have sheared flow profiles as they propagate through the plasma medium. In an earlier study [Phys. Plasmas 17, 022101 (2010)], it was shown that a relativistic sheared electron flow modifies the growth rate and threshold condition of the conventional Kelvin-Helmholtz instability. A perturbative analytic treatment for the case of weakly relativistic regime has been provided here. It provides good agreement with the numerical results obtained earlier.
15. Study on perturbation schemes for achieving the real PMNS matrix from various symmetric textures
Wang, Bin; Tang, Jian; Li, Xue-Qian
2013-10-01
The Pontecorvo-Maki-Nakawaga-Sakata matrix displays an obvious symmetry but not exact. There are several textures proposed in the literature which possess various symmetry patterns and seem to originate from different physics scenarios at high energy scales. To be consistent with the experimental measurement, however, the symmetry must be broken. Following the schemes given in the literature, we modify the matrices (ten in total) to gain the real Pontecorvo-Maki-Nakawaga-Sakata matrix by perturbative rotations. The results may be useful for future model builders.
16. Perturbative approach to open circuit QED systems
Li, Andy C. Y.; Petruccione, Francesco; Koch, Jens
2014-03-01
Perturbation theory (PT) is a powerful and commonly used tool in the investigation of closed quantum systems. In the context of open quantum systems, PT based on the Markovian quantum master equation is much less developed. The investigation of open systems mostly relies on exact diagonalization of the Liouville superoperator or quantum trajectories. In this approach, the system size is rather limited by current computational capabilities. Analogous to closed-system PT, we develop a PT suitable for open quantum systems. The proposed method is useful in the analytical understanding of open systems as well as in the numerical calculation of system observables, which would otherwise be impractical. This enables us to investigate a variety of open circuit QED systems, including the open Jaynes-Cummings lattice model.
17. Robustness of topological quantum codes: Ising perturbation
2015-02-01
We study the phase transition from two different topological phases to the ferromagnetic phase by focusing on points of the phase transition. To this end, we present a detailed mapping from such models to the Ising model in a transverse field. Such a mapping is derived by rewriting the initial Hamiltonian in a new basis so that the final model in such a basis has a well-known approximated phase transition point. Specifically, we consider the toric codes and the color codes on various lattices with Ising perturbation. Our results provide a useful table to compare the robustness of the topological codes and to explicitly show that the robustness of the topological codes depends on triangulation of their underlying lattices.
18. The onion method for multiple perturbation theory
Cross, R. J.
1988-04-01
We develop a method of successive approximations for molecular scattering theory. This consists of a recipe for removing from the Schrödinger equation, one by one, the wave functions of a set of approximate solutions. The radial wave function is expressed as a linear combination of the well-behaved and singular solutions of the first approximation, and a set of coupled differential equations is obtained for the coefficients of the approximate solutions. A similar set of coefficients is obtained for the next approximation, and the exact coefficients are expressed in terms of the approximate coefficients to yield a set of second-level coefficients. The process can be continued like pealing off the layers of an onion. At each stage the coupled differential equations for the coefficients is equivalent to the Schrödinger equation. Finally, one can either ignore the remaining coefficients or approximate the coupled equations by a simple perturbation theory.
19. Orbit Perturbations Due to Solar Radiation Pressure
NASA Technical Reports Server (NTRS)
Sawyer, G. A.
1972-01-01
This disturbing force will be important for satellites with a large area to mass ratio and also for those whose orbits are high enough that atmospheric drag is not the more dominate force. The procedure for the analysis is to represent the radiation force as the gradient of a scalar function to be compatible with existing procedures for studying perturbations due to earth's oblateness. From this analysis, solar radiation pressure appears not to be responsible for any secular or long-periodic variations in the semi-major axis of the orbit nor does it provide any secular changes in the eccentricity of the orbit or the angle of inclination of the osculating plane. Solar radiation pressure does produce secular effects in the other orbital elements, but these are in the opposite sense of secularities caused by the gravitational attraction of the sun and tend to reduce the total secularity.
20. Applications of partially quenched chiral perturbation theory
SciTech Connect
Golterman, M.F.; Leung, K.C.
1998-05-01
Partially quenched theories are theories in which the valence- and sea-quark masses are different. In this paper we calculate the nonanalytic one-loop corrections of some physical quantities: the chiral condensate, weak decay constants, Goldstone boson masses, B{sub K}, and the K{sup +}{r_arrow}{pi}{sup +}{pi}{sup 0} decay amplitude, using partially quenched chiral perturbation theory. Our results for weak decay constants and masses agree with, and generalize, results of previous work by Sharpe. We compare B{sub K} and the K{sup +} decay amplitude with their real-world values in some examples. For the latter quantity, two other systematic effects that plague lattice computations, namely, finite-volume effects and unphysical values of the quark masses and pion external momenta, are also considered. We find that typical one-loop corrections can be substantial. {copyright} {ital 1998} {ital The American Physical Society}
1. Systematic analysis of endocytosis by cellular perturbations.
PubMed
Kühling, Lena; Schelhaas, Mario
2014-01-01
Endocytosis is an essential process of eukaryotic cells that facilitates numerous cellular and organismal functions. The formation of vesicles from the plasma membrane serves the internalization of ligands and receptors and leads to their degradation or recycling. A number of distinct mechanisms have been described over the years, several of which are only partially characterized in terms of mechanism and function. These are often referred to as novel endocytic pathways. The pathways differ in their mode of uptake and in their intracellular destination. Here, an overview of the set of cellular proteins that facilitate the different pathways is provided. Further, the approaches to distinguish between the pathways by different modes of perturbation are critically discussed, emphasizing the use of genetic tools such as dominant negative mutant proteins. PMID:24947372
2. Perturbative double field theory on general backgrounds
Hohm, Olaf; Marques, Diego
2016-01-01
We develop the perturbation theory of double field theory around arbitrary solutions of its field equations. The exact gauge transformations are written in a manifestly background covariant way and contain at most quadratic terms in the field fluctuations. We expand the generalized curvature scalar to cubic order in fluctuations and thereby determine the cubic action in a manifestly background covariant form. As a first application we specialize this theory to group manifold backgrounds, such as S U (2 )≃S3 with H -flux. In the full string theory this corresponds to a Wess-Zumino-Witten background CFT. Starting from closed string field theory, the cubic action around such backgrounds has been computed before by Blumenhagen, Hassler, and Lüst. We establish precise agreement with the cubic action derived from double field theory. This result confirms that double field theory is applicable to arbitrary curved background solutions, disproving assertions in the literature to the contrary.
3. New Representations of the Perturbative S Matrix.
PubMed
Baadsgaard, Christian; Bjerrum-Bohr, N E J; Bourjaily, Jacob L; Caron-Huot, Simon; Damgaard, Poul H; Feng, Bo
2016-02-12
We propose a new framework to represent the perturbative S matrix which is well defined for all quantum field theories of massless particles, constructed from tree-level amplitudes and integrable term by term. This representation is derived from the Feynman expansion through a series of partial fraction identities, discarding terms that vanish upon integration. Loop integrands are expressed in terms of "Q-cuts" that involve both off-shell and on-shell loop momenta, defined with a precise contour prescription that can be evaluated by ordinary methods. This framework implies recent results found in the scattering equation formalism at one loop, and it has a natural extension to all orders--even nonplanar theories without well-defined forward limits or good ultraviolet behavior. PMID:26918978
4. Planck constraints on neutrino isocurvature density perturbations
Di Valentino, Eleonora; Melchiorri, Alessandro
2014-10-01
The recent cosmic microwave background data from the Planck satellite experiment, when combined with Hubble Space Telescope determinations of the Hubble constant, are compatible with a larger, nonstandard number of relativistic degrees of freedom at recombination, parametrized by the neutrino effective number Neff . In the curvaton scenario, a larger value for Neff could arise from a nonzero neutrino chemical potential connected to residual neutrino isocurvature density (NID) perturbations after the decay of the curvaton field, the component of which is parametrized by the amplitude αNID . Here we present new constraints on Neff and αNID from an analysis of recent cosmological data. We find that the Planck+WMAP polarization data set does not show any indication of a NID component (severely constraining its amplitude), and that current indications for a nonstandard Neff are further relaxed.
5. Perturbation approach applied to modal diffraction methods.
PubMed
Bischoff, Joerg; Hehl, Karl
2011-05-01
Eigenvalue computation is an important part of many modal diffraction methods, including the rigorous coupled wave approach (RCWA) and the Chandezon method. This procedure is known to be computationally intensive, accounting for a large proportion of the overall run time. However, in many cases, eigenvalue information is already available from previous calculations. Some of the examples include adjacent slices in the RCWA, spectral- or angle-resolved scans in optical scatterometry and parameter derivatives in optimization. In this paper, we present a new technique that provides accurate and highly reliable solutions with significant improvements in computational time. The proposed method takes advantage of known eigensolution information and is based on perturbation method. PMID:21532698
6. Eigenvector dynamics under perturbation of modular networks
Sarkar, Somwrita; Chawla, Sanjay; Robinson, P. A.; Fortunato, Santo
2016-06-01
Rotation dynamics of eigenvectors of modular network adjacency matrices under random perturbations are presented. In the presence of q communities, the number of eigenvectors corresponding to the q largest eigenvalues form a "community" eigenspace and rotate together, but separately from that of the "bulk" eigenspace spanned by all the other eigenvectors. Using this property, the number of modules or clusters in a network can be estimated in an algorithm-independent way. A general argument and derivation for the theoretical detectability limit for sparse modular networks with q communities is presented, beyond which modularity persists in the system but cannot be detected. It is shown that for detecting the clusters or modules using the adjacency matrix, there is a "band" in which it is hard to detect the clusters even before the theoretical detectability limit is reached, and for which the theoretically predicted detectability limit forms the sufficient upper bound. Analytic estimations of these bounds are presented and empirically demonstrated.
7. Degenerate R-S perturbation theory
NASA Technical Reports Server (NTRS)
Hirschfelder, J. O.; Certain, P. R.
1973-01-01
A concise, systematic procedure is given for determining the Rayleigh-Schrodinger energies and wave functions of degenerate states to arbitrarily high orders even when the degeneracies of the various states are resolved in arbitrary orders. The procedure is expressed in terms of an iterative cycle in which the energy through the (2n+1)st order is expressed in terms of the partially determined wave function through the n-th order. Both a direct and an operator derivation are given. The two approaches are equivalent and can be transcribed into each other. The direct approach deals with the wave functions (without the use of formal operators) and has the advantage that it resembles the usual treatment of nondegenerate perturbations and maintains close contact with the basic physics. In the operator approach, the wave functions are expressed in terms of infinite order operators which are determined by the successive resolution of the space of the zeroth order functions.
8. NEXCADE: perturbation analysis for complex networks.
PubMed
2012-01-01
9. Relativistic Lagrangian displacement field and tensor perturbations
Rampf, Cornelius; Wiegand, Alexander
2014-12-01
We investigate the purely spatial Lagrangian coordinate transformation from the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for extracting the relativistic displacement field from a given solution in the Lagrangian frame. These techniques are (a) from defining a local set of Eulerian coordinates embedded into the Lagrangian frame; (b) from performing a specific gauge transformation; and (c) from a fully nonperturbative approach based on the Arnowitt-Deser-Misner (ADM) split. The latter approach shows that this decomposition is not tied to a specific perturbative formulation for the solution of the Einstein equations. Rather, it can be defined at the level of the nonperturbative coordinate change from the Lagrangian to the Eulerian description. Studying such different techniques is useful because it allows us to compare and develop further the various approximation techniques available in the Lagrangian formulation. We find that one has to solve the gravitational wave equation in the relativistic analysis, otherwise the corresponding Newtonian limit will necessarily contain spurious nonpropagating tensor artifacts at second order in the Eulerian frame. We also derive the magnetic part of the Weyl tensor in the Lagrangian frame, and find that it is not only excited by gravitational waves but also by tensor perturbations which are induced through the nonlinear frame dragging. We apply our findings to calculate for the first time the relativistic displacement field, up to second order, for a Λ CDM Universe in the presence of a local primordial non-Gaussian component. Finally, we also comment on recent claims about whether mass conservation in the Lagrangian frame is violated.
10. NEXCADE: Perturbation Analysis for Complex Networks
PubMed Central
2012-01-01
11. Multiloop calculations in perturbative quantum field theory
Blokland, Ian Richard
This thesis deals with high-precision calculations in perturbative quantum field theory. In conjunction with detailed experimental measurements, perturbative quantum field theory provides the quantitative framework with which much of modern particle physics is understood. The results of three new theoretical calculations are presented. The first is a definitive resolution of a recent controversy involving the interaction of a muon with a magnetic field. Specifically, the light-by-light scattering contribution to the anomalous magnetic moment of the muon is shown to be of positive sign, thereby decreasing the discrepancy between theory and experiment. Despite this adjustment to the theoretical prediction, the remaining discrepancy might be a subtle signature of new kinds of particles. The second calculation involves the energy levels of a bound state formed from two charged particles of arbitrary masses. By employing recently developed mass expansion techniques, new classes of solutions are obtained for problems in a field of particle physics with a very rich history. The third calculation provides an improved prediction for the decay of a top quark. In order to obtain this result, a large class of multiloop integrals has been solved for the first time. Top quark decay is just one member of a family of interesting physical processes to which these new results apply. Since specialized calculational techniques are essential ingredients in all three calculations, they are motivated and explained carefully in this thesis. These techniques, once automated with symbolic computational software, have recently opened avenues of solution to a wide variety of important problems in particle physics.
12. Adiabatic and isocurvature perturbation projections in multi-field inflation
Gordon, Chris; Saffin, Paul M.
2013-08-01
Current data are in good agreement with the predictions of single field inflation. However, the hemispherical asymmetry, seen in the cosmic microwave background data, may hint at a potential problem. Generalizing to multi-field models may provide one possible explanation. A useful way of modeling perturbations in multi-field inflation is to investigate the projection of the perturbation along and perpendicular to the background fields' trajectory. These correspond to the adiabatic and isocurvature perturbations. However, it is important to note that in general there are no corresponding adiabatic and isocurvature fields. The purpose of this article is to highlight the distinction between a field redefinition and a perturbation projection. We provide a detailed derivation of the evolution of the isocurvature perturbation to show that no assumption of an adiabatic or isocurvature field is needed. We also show how this evolution equation is consistent with the field covariant evolution equations for the adiabatic perturbation in the flat field space limit.
13. Non-perturbative String Theory from Water Waves
SciTech Connect
Iyer, Ramakrishnan; Johnson, Clifford V.; Pennington, Jeffrey S.; /SLAC
2012-06-14
We use a combination of a 't Hooft limit and numerical methods to find non-perturbative solutions of exactly solvable string theories, showing that perturbative solutions in different asymptotic regimes are connected by smooth interpolating functions. Our earlier perturbative work showed that a large class of minimal string theories arise as special limits of a Painleve IV hierarchy of string equations that can be derived by a similarity reduction of the dispersive water wave hierarchy of differential equations. The hierarchy of string equations contains new perturbative solutions, some of which were conjectured to be the type IIA and IIB string theories coupled to (4, 4k ? 2) superconformal minimal models of type (A, D). Our present paper shows that these new theories have smooth non-perturbative extensions. We also find evidence for putative new string theories that were not apparent in the perturbative analysis.
14. Perturbative Expansion of τ Hadronic Spectral Function Moments
Boito, Diogo
2014-12-01
In the extraction of αs from hadronic τ decay data several moments of the spectral functions have been employed. Furthermore, different renormalization group improvement (RGI) frameworks have been advocated, leading to conflicting values of αs. Recently, we performed a systematic study of the perturbative behavior of these moments in the context of the two main-stream RGI frameworks: Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT). The yet unknown higher order coefficients of the perturbative series were modelled using the available knowledge of the renormalon singularities of the QCD Adler function. We were able to show that within these RGI frameworks some of the commonly employed moments should be avoided due to their poor perturbative behavior. Furthermore, under reasonable assumptions about the higher order behavior of the perturbative series FOPT provides the preferred RGI framework.
15. Non-Gaussian isocurvature perturbations in dark radiation
SciTech Connect
Kawakami, Etsuko; Kawasaki, Masahiro; Miyamoto, Koichi; Nakayama, Kazunori; Sekiguchi, Toyokazu E-mail: [email protected] E-mail: [email protected]
2012-07-01
We study non-Gaussian properties of the isocurvature perturbations in the dark radiation, which consists of the active neutrinos and extra light species, if exist. We first derive expressions for the bispectra of primordial perturbations which are mixtures of curvature and dark radiation isocurvature perturbations. We also discuss CMB bispectra produced in our model and forecast CMB constraints on the non-linearity parameters based on the Fisher matrix analysis. Some concrete particle physics motivated models are presented in which large isocurvature perturbations in extra light species and/or the neutrino density isocurvature perturbations as well as their non-Gaussianities may be generated. Thus detections of non-Gaussianity in the dark radiation isocurvature perturbation will give us an opportunity to identify the origin of extra light species and lepton asymmetry.
16. Wild worm embryogenesis harbors ubiquitous polygenic modifier variation
PubMed Central
Paaby, Annalise B; White, Amelia G; Riccardi, David D; Gunsalus, Kristin C; Piano, Fabio; Rockman, Matthew V
2015-01-01
Embryogenesis is an essential and stereotypic process that nevertheless evolves among species. Its essentiality may favor the accumulation of cryptic genetic variation (CGV) that has no effect in the wild-type but that enhances or suppresses the effects of rare disruptions to gene function. Here, we adapted a classical modifier screen to interrogate the alleles segregating in natural populations of Caenorhabditis elegans: we induced gene knockdowns and used quantitative genetic methodology to examine how segregating variants modify the penetrance of embryonic lethality. Each perturbation revealed CGV, indicating that wild-type genomes harbor myriad genetic modifiers that may have little effect individually but which in aggregate can dramatically influence penetrance. Phenotypes were mediated by many modifiers, indicating high polygenicity, but the alleles tend to act very specifically, indicating low pleiotropy. Our findings demonstrate the extent of conditional functionality in complex trait architecture. DOI: http://dx.doi.org/10.7554/eLife.09178.001 PMID:26297805
17. Investigations of Probe Induced Perturbations in a Hall Thruster
SciTech Connect
D. Staack; Y. Raitses; N.J. Fisch
2002-08-12
An electrostatic probe used to measure spatial plasma parameters in a Hall thruster generates perturbations of the plasma. These perturbations are examined by varying the probe material, penetration distance, residence time, and the nominal thruster conditions. The study leads us to recommendations for probe design and thruster operating conditions to reduce discharge perturbations, including metal shielding of the probe insulator and operation of the thruster at lower densities.
18. Constraints on primordial density perturbations from induced gravitational waves
SciTech Connect
2010-01-15
We consider the stochastic background of gravitational waves produced during the radiation-dominated hot big bang as a constraint on the primordial density perturbation on comoving length scales much smaller than those directly probed by the cosmic microwave background or large-scale structure. We place weak upper bounds on the primordial density perturbation from current data. Future detectors such as BBO and DECIGO will place much stronger constraints on the primordial density perturbation on small scales.
19. Equation-of-motion coupled cluster perturbation theory revisited
SciTech Connect
Eriksen, Janus J. Jørgensen, Poul; Olsen, Jeppe; Gauss, Jürgen
2014-05-07
The equation-of-motion coupled cluster (EOM-CC) framework has been used for deriving a novel series of perturbative corrections to the coupled cluster singles and doubles energy that formally converges towards the full configuration interaction energy limit. The series is based on a Møller-Plesset partitioning of the Hamiltonian and thus size extensive at any order in the perturbation, thereby remedying the major deficiency inherent to previous perturbation series based on the EOM-CC ansatz.
20. Multifield cosmological perturbations at third order and the ekpyrotic trispectrum
SciTech Connect
Lehners, Jean-Luc; Renaux-Petel, Sebastien
2009-09-15
Using the covariant formalism, we derive the equations of motion for adiabatic and entropy perturbations at third order in perturbation theory for cosmological models involving two scalar fields. We use these equations to calculate the trispectrum of ekpyrotic and cyclic models in which the density perturbations are generated via the entropic mechanism. In these models, the conversion of entropy into curvature perturbations occurs just before the big bang, either during the ekpyrotic phase or during the subsequent kinetic energy dominated phase. In both cases, we find that the nonlinearity parameters f{sub NL} and g{sub NL} combine to leave a very distinct observational imprint.
1. Resonant magnetic perturbations and edge ergodization on the COMPASS tokamak
SciTech Connect
Cahyna, P.; Fuchs, V.; Krlin, L.
2008-09-15
Results of calculations of resonant magnetic perturbation spectra on the COMPASS tokamak are presented. Spectra of the perturbations are calculated from the vacuum field of the perturbation coils. Ergodization is then estimated by applying the criterion of overlap of the resulting islands and verified by field line tracing. Results show that for the chosen configuration of perturbation coils an ergodic layer appears in the pedestal region. The ability to form an ergodic layer is similar to the theoretical results for the ELM suppression experiment at DIII-D; thus, a comparable effect on ELMs can be expected.
2. Perturbations of generic Kasner spacetimes and their stability
SciTech Connect
Kofman, Lev; Uzan, Jean-Philippe; Pitrou, Cyril E-mail: [email protected]
2011-05-01
This article investigates the stability of a generic Kasner spacetime to linear perturbations, both at late and early times. It demonstrates that the perturbation of the Weyl tensor diverges at late time in all cases but in the particular one in which the Kasner spacetime is the product of a two-dimensional Milne spacetime and a two-dimensional Euclidean space. At early times, the perturbation of the Weyl tensor also diverges unless one imposes a condition on the perturbations so as to avoid the most divergent modes to be excited.
3. Non-linear isocurvature perturbations and non-Gaussianities
SciTech Connect
Langlois, David; Vernizzi, Filippo; Wands, David E-mail: [email protected]
2008-12-15
We study non-linear primordial adiabatic and isocurvature perturbations and their non-Gaussianity. After giving a general formulation in the context of an extended {delta}N formalism, we analyse in detail two illustrative examples. The first is a mixed curvaton-inflaton scenario in which fluctuations of both the inflaton and a curvaton (a light isocurvature field during inflation) contribute to the primordial density perturbation. The second example is that of double inflation involving two decoupled massive scalar fields during inflation. In the mixed curvaton-inflaton scenario we find that the bispectrum of primordial isocurvature perturbations may be large and comparable to the bispectrum of adiabatic curvature perturbations.
4. A new method to compute lunisolar perturbations in satellite motions
NASA Technical Reports Server (NTRS)
Kozai, Y.
1973-01-01
A new method to compute lunisolar perturbations in satellite motion is proposed. The disturbing function is expressed by the orbital elements of the satellite and the geocentric polar coordinates of the moon and the sun. The secular and long periodic perturbations are derived by numerical integrations, and the short periodic perturbations are derived analytically. The perturbations due to the tides can be included in the same way. In the Appendix, the motion of the orbital plane for a synchronous satellite is discussed; it is concluded that the inclination cannot stay below 7 deg.
5. Bursting process of large- and small-scale structures in turbulent boundary layer perturbed by a cylinder roughness element
Tang, Zhanqi; Jiang, Nan; Zheng, Xiaobo; Wu, Yanhua
2016-05-01
Hot-wire measurements on a turbulent boundary layer flow perturbed by a wall-mounted cylinder roughness element (CRE) are carried out in this study. The cylindrical element protrudes into the logarithmic layer, which is similar to those employed in turbulent boundary layers by Ryan et al. (AIAA J 49:2210-2220, 2011. doi: 10.2514/1.j051012) and Zheng and Longmire (J Fluid Mech 748:368-398, 2014. doi: 10.1017/jfm.2014.185) and in turbulent channel flow by Pathikonda and Christensen (AIAA J 53:1-10, 2014. doi: 10.2514/1.j053407). The similar effects on both the mean velocity and Reynolds stress are observed downstream of the CRE perturbation. The series of hot-wire data are decomposed into large- and small-scale fluctuations, and the characteristics of large- and small-scale bursting process are observed, by comparing the bursting duration, period and frequency between CRE-perturbed case and unperturbed case. It is indicated that the CRE perturbation performs the significant impact on the large- and small-scale structures, but within the different impact scenario. Moreover, the large-scale bursting process imposes a modulation on the bursting events of small-scale fluctuations and the overall trend of modulation is not essentially sensitive to the present CRE perturbation, even the modulation extent is modified. The conditionally averaging fluctuations are also plotted, which further confirms the robustness of the bursting modulation in the present experiments.
6. Invariant exchange perturbation theory for multicenter systems: Time-dependent perturbations
SciTech Connect
Orlenko, E. V. Evstafev, A. V.; Orlenko, F. E.
2015-02-15
A formalism of exchange perturbation theory (EPT) is developed for the case of interactions that explicitly depend on time. Corrections to the wave function obtained in any order of perturbation theory and represented in an invariant form include exchange contributions due to intercenter electron permutations in complex multicenter systems. For collisions of atomic systems with an arbitrary type of interaction, general expressions are obtained for the transfer (T) and scattering (S) matrices in which intercenter electron permutations between overlapping nonorthogonal states belonging to different centers (atoms) are consistently taken into account. The problem of collision of alpha particles with lithium atoms accompanied by the redistribution of electrons between centers is considered. The differential and total charge-exchange cross sections of lithium are calculated.
7. A perturbative approach to the redshift space power spectrum: beyond the Standard Model
Bose, Benjamin; Koyama, Kazuya
2016-08-01
We develop a code to produce the power spectrum in redshift space based on standard perturbation theory (SPT) at 1-loop order. The code can be applied to a wide range of modified gravity and dark energy models using a recently proposed numerical method by A.Taruya to find the SPT kernels. This includes Horndeski's theory with a general potential, which accommodates both chameleon and Vainshtein screening mechanisms and provides a non-linear extension of the effective theory of dark energy up to the third order. Focus is on a recent non-linear model of the redshift space power spectrum which has been shown to model the anisotropy very well at relevant scales for the SPT framework, as well as capturing relevant non-linear effects typical of modified gravity theories. We provide consistency checks of the code against established results and elucidate its application within the light of upcoming high precision RSD data.
8. Imposing Neumann boundary condition on cosmological perturbation equations and trajectories of particles
Shenavar, Hossein
2016-03-01
We impose Neumann boundary condition to solve cosmological perturbation equations and we derive a modified Friedmann equation and a new lensing equation. To check the new lensing equation and the value of Neumann constant, a sample that contains ten strong lensing systems is surveyed. Except for one lens, masses of the other lenses are found to be within the constrains of the observational data. Furthermore, we argue that by using the concept of geometrodynamic clocks it is possible to modify the equation of motion of massive particles too. Also, a sample that includes 101 HSB and LSB galaxies is used to re-estimate the value of the Neumann constant and we found that this value is consistent with the prior evaluation from Friedmann and lensing equations. Finally, the growth of structure is studied by a Newtonian approach which resulted in a more rapid rate of the structure formation in matter dominated era.
9. Perturbative and non-perturbative aspects of the two-dimensional string/Yang-Mills correspondence
Lelli, Simone; Maggiore, Michele; Rissone, Anna
2003-04-01
It is known that YM 2 with gauge group SU( N) is equivalent to a string theory with coupling gs=1/ N, order by order in the 1/ N expansion. We show how this result can be obtained from the bosonization of the fermionic formulation of YM 2, improving on results in the literature, and we examine a number of non-perturbative aspects of this string/YM correspondence. We find contributions to the YM 2 partition function of order exp{- kA/( πα' gs)} with k an integer and A the area of the target space, which would correspond, in the string interpretation, to D1-branes. Effects which could be interpreted as D0-branes are instead strictly absent, suggesting a non-perturbative structure typical of type 0B string theories. We discuss effects from the YM side that are interpreted in terms of the stringy exclusion principle of Maldacena and Strominger. We also find numerically an interesting phase structure, with a region where YM 2 is described by a perturbative string theory separated from a region where it is described by a topological string theory.
10. Enhanced diamagnetic perturbations and electric currents observed downstream of the high power helicon
Roberson, B. Race; Winglee, Robert; Prager, James
2011-05-01
The high power helicon (HPH) is capable of producing a high density plasma (1017-1018 m-3) and directed ion energies greater than 20 eV that continue to increase tens of centimeters downstream of the thruster. In order to understand the coupling mechanism between the helicon antenna and the plasma outside the immediate source region, measurements were made in the plasma plume downstream from the thruster of the propagating wave magnetic field and the perturbation of the axial bulk field using a type `R' helicon antenna. This magnetic field perturbation (ΔB) peaks at more than 15 G in strength downstream of the plasma source, and is 3-5 times larger than those previously reported from HPH. Taking the curl of this measured magnetic perturbation and assuming azimuthal symmetry suggests that this magnetic field is generated by a (predominantly) azimuthal current ring with a current density on the order of tens of kA m-2. At this current density the diamagnetic field is intense enough to cancel out the B0 axial magnetic field near the source region. The presence of the diamagnetic current is important as it demonstrates modification of the vacuum fields well beyond the source region and signifies the presence of a high density, collimated plasma stream. This diamagnetic current also modifies the propagation of the helicon wave, which facilitates a better understanding of coupling between the helicon wave and the resultant plasma acceleration.
11. Evidence of Dayside Asymmetries in High-Latitude Magnetic Perturbations Measured By ST5 and DMSP
Knipp, D. J.; Kilcommons, L. M.; Slavin, J. A.; Le, G.; Gjerloev, J. W.; Redmon, R. J.
2014-12-01
Magnetometer data from the three-spacecraft Space Technology 5 (ST5) mission and from two Defense Meteorological Spacecraft Program (DMSP) spacecraft are available during the 90-day ST5 mission (late March - late June, 2006). The ST5 spacecraft were in a 300 km x 4500 km orbit while the DMSP spacecraft were in near circular orbit at ~ 850 km. We removed gain jumps from the ST5 data and improved a high-latitude background removal method for DMSP. All data were processed through the Modified Apex Coordinate transformation and mapped to 110 km. Agreement between different spacecraft observations was verified by computing magnetic conjunctions and extracting data from a small region around each ST5-DMSP conjunction. We found excellent agreement between the magnetic perturbations at 277 conjunctions between DMSP and ST5. With these data we show: 1) a global view of storm-time field aligned current (FAC) perturbations; 2) a global view of calm-before-the-storm FAC events; and 3) clear evidence of hemispheric asymmetries in cusp region magnetic perturbation strength during the equinox season. Our methods provide a 'path forward' in terms of comparing on-orbit electric and magnetic field data from multiple spacecraft.
12. Engineering of perturbation effects in onion-like heteronanocrystal quantum dot-quantum well
SalmanOgli, A.; Rostami, R.
2013-10-01
In this article, the perturbation influences on optical characterization of quantum dot and quantum dot-quantum well (modified quantum dot) heteronanocrystal is investigated. The original aim of this article is to investigate the quantum dot-quantum well heteronanocrystal advantages and disadvantages, when used as a functionalized particle in biomedical applications. Therefore, all of the critical features of quantum dots are fundamentally studied and their influences on optical properties are simulated. For the first time, the perturbation effects on optical characteristics are observed in the quantum dot-quantum well heteronanocrystals by 8-band K.P theory. The impact of perturbation on optical features such as photoluminescence and shifting of wavelength is studied. The photoluminescence and operation wavelength of quantum dots play a vital role in biomedical applications, where their absorption and emission in biological assays are altered by shifting of wavelength. Furthermore, in biomedical applications, by tuning the emission wavelengths of the quantum dot into far-red and near-infrared ranges, non-invasive in-vivo imaging techniques have been easily developed. In this wavelength window, tissue absorption, scattering and auto-fluorescence intensities have minimum quantities; thus fixing or minimizing of wavelength shifting can be regarded as an important goal which is investigated in this work.
13. Enhanced diamagnetic perturbations and electric currents observed downstream of the high power helicon
SciTech Connect
Roberson, B. Race; Winglee, Robert; Prager, James
2011-05-15
The high power helicon (HPH) is capable of producing a high density plasma (10{sup 17}-10{sup 18} m{sup -3}) and directed ion energies greater than 20 eV that continue to increase tens of centimeters downstream of the thruster. In order to understand the coupling mechanism between the helicon antenna and the plasma outside the immediate source region, measurements were made in the plasma plume downstream from the thruster of the propagating wave magnetic field and the perturbation of the axial bulk field using a type 'R' helicon antenna. This magnetic field perturbation ({Delta}B) peaks at more than 15 G in strength downstream of the plasma source, and is 3-5 times larger than those previously reported from HPH. Taking the curl of this measured magnetic perturbation and assuming azimuthal symmetry suggests that this magnetic field is generated by a (predominantly) azimuthal current ring with a current density on the order of tens of kA m{sup -2}. At this current density the diamagnetic field is intense enough to cancel out the B{sub 0} axial magnetic field near the source region. The presence of the diamagnetic current is important as it demonstrates modification of the vacuum fields well beyond the source region and signifies the presence of a high density, collimated plasma stream. This diamagnetic current also modifies the propagation of the helicon wave, which facilitates a better understanding of coupling between the helicon wave and the resultant plasma acceleration.
14. Applications of the Kustaanheimo-Stieffel transformation of the perturbed two-body problem
NASA Technical Reports Server (NTRS)
Bond, V. R.
1973-01-01
The Newtonian differential equations of motion for the two-body problem can be transformed into four linear harmonic-oscillator equations by simultaneously applying the regularization step dt/ds = r and the Kustaanheimo-Stieffel (KS) transformation. The regularization step changes the independent variable from time to a new variable s, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. A derivation of a uniform, regular solution for the perturbed two-body problem in the four-dimensional space is presented. The variation-of-parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. This ten-element solution has mixed secular terms that degrade the long-term accuracy during numerical integration. Therefore, to eliminate these terms, the solution is modified by introducing two additional elements.
15. Field-Aligned and Ionospheric Current Contributions to Ground Magnetic Perturbations
Connors, M. G.; McPherron, R. L.; Anderson, B. J.; Korth, H.; Russell, C. T.; Chu, X.
2014-12-01
AMPERE data provides global space-derived radial electric currents on temporal and spatial scales suited to studying magnetic fields at ULF frequencies. It responds little to ionspheric currents, which dominate ground-based measurements, so that AMPERE and ground datasets complement each other to give a comprehensive view of near-Earth electric currents. Connors et al. (GRL, 2014) found that a three-dimensional current system slightly modified from the original substorm current wedge (SCW) concept of McPherron et al. (JGR, 1973) represented substorm midnight sector perturbations well both in the auroral and subauroral regions, if a current equivalent to that found by integrating AMPERE downward current was used, located where clear SCW signatures were indicated by AMPERE, and featuring an ionospheric electrojet. The AMPERE upward current was found to exceed that in the SCW, at least in part since the evening sector electrojet fed into it. We extend these results with a more detailed accounting of field-aligned and ionospheric currents throughout the active period (including growth phase). Ionospheric currents for the study are obtained from ground perturbations through optimization of a simple forward model over regions or on a meridian chain. We also investigate the degree to which subauroral perturbations may be directly calculated from AMPERE results. We further find that auroral zone currents may be very localized, to the extent that the entire SCW ionospheric current flows in a very restricted latitudinal range near onset, possibly corresponding to a single auroral arc.
16. Dose perturbation due to the polysulfone cap surrounding a Fletcher-Williamson colpostat.
PubMed
Price, Michael J; Kry, Stephen F; Eifel, Patricia J; Salehpour, Mohammad; Mourtada, Firas
2010-01-01
We conducted a metrological evaluation of the dosimetric impact due to the polysulfone cap used with the Fletcher-Williamson (FW) colpostat for 192Ir high-dose rate and pulsed-dose rate intracavitary brachytherapy using Monte Carlo simulations. Polysulfone caps with diameter of 30 mm, 25 mm, 20 mm, and 16 mm (mini-ovoid) were simulated and the absorbed dose rate in the surrounding water was calculated and compared to the dose rate for a bare 192Ir source in water. The dose perturbation depended on the cap diameter, distance away from the cap surface, and angular position around the cap. The largest dose rate reductions were found to be in the direction of the tumor bed where the cap is thickest. The range of perturbation over all depths and cap diameters was +2.8% (dose enhancement) to -6.8% (dose reduction). The FW colpostat cap's material composition should be modified to reduce this dosimetric effect or brachytherapy treatment planning dose algorithms should be improved to account for this perturbation. PMID:20160700
17. Stabilization of perturbed Boolean network attractors through compensatory interactions
PubMed Central
2014-01-01
Background Understanding and ameliorating the effects of network damage are of significant interest, due in part to the variety of applications in which network damage is relevant. For example, the effects of genetic mutations can cascade through within-cell signaling and regulatory networks and alter the behavior of cells, possibly leading to a wide variety of diseases. The typical approach to mitigating network perturbations is to consider the compensatory activation or deactivation of system components. Here, we propose a complementary approach wherein interactions are instead modified to alter key regulatory functions and prevent the network damage from triggering a deregulatory cascade. Results We implement this approach in a Boolean dynamic framework, which has been shown to effectively model the behavior of biological regulatory and signaling networks. We show that the method can stabilize any single state (e.g., fixed point attractors or time-averaged representations of multi-state attractors) to be an attractor of the repaired network. We show that the approach is minimalistic in that few modifications are required to provide stability to a chosen attractor and specific in that interventions do not have undesired effects on the attractor. We apply the approach to random Boolean networks, and further show that the method can in some cases successfully repair synchronous limit cycles. We also apply the methodology to case studies from drought-induced signaling in plants and T-LGL leukemia and find that it is successful in both stabilizing desired behavior and in eliminating undesired outcomes. Code is made freely available through the software package BooleanNet. Conclusions The methodology introduced in this report offers a complementary way to manipulating node expression levels. A comprehensive approach to evaluating network manipulation should take an "all of the above" perspective; we anticipate that theoretical studies of interaction modification
18. Carbon content on perturbed wetlands of Yucatan
Morales Ojeda, S. M.; Orellana, R.; Herrera Silveira, J.
2013-05-01
The north coast of Yucatan Peninsula is a karstic scenario where the water flows mainly underground through the so called "cenotes"-ring system ("sink holes") toward the coast. This underground water system enhances the connection between watershed condition and coastal ecosystem health. Inland activities such as livestock, agriculture and urban development produce changes in the landscape, hydrological connectivity and in the water quality that can decrease wetland coverage specially mangroves and seagrasses. We conducted studies on the description of structure, biomass and carbon content of the soil, above and below ground of four different types of wetland in a perturbed region. The wetland ecological types were freshwater (Typha domingensis), dwarf mangroves (Avicenia germinans), grassland (Cyperacea) and Seagrasses. Due to the area is mainly covered by mangroves, they represent the most important carbon storage nevertheless the condition of the structure determine the carbon content in soil. Through GIS tools we explore the relationships between land use and costal condition in order to determine priority areas for conservation within the watershed that could be efficient to preserve the carbon storage of this area.
19. Waterhole: An auroral-ionosphere perturbation experiment
Whalen, B. A.; Yau, A. W.; Creutzberg, F.; Pongratz, M. B.
A sounding rocket carrying 100 kg of high explosives and plasma diagnostic instrumentation was launched from Churchill Research Range on 6 April 1980 over a premidnight auroral arc. The object of the experiment was to produce an ionospheric hole or plasma density depletion at about 300 km altitude on field lines connected to an auroral arc. The plasma depletion is produced when the explosive by-products (mostly water) charge-exchange with the ambient O+ ions and then rapidly recombine. It was speculated that the presence of the "hole" would interfere with the field-aligned current systems associated with the arc and would in turn perturb the auroral source mechanism. The release occurred about 10 km poleward of the auroral arc fieldlines. As expected, a large ionospheric hole was detected by rocket-borne plasma sensors. Within a few seconds following the release (a) the energetic electron precipitation observed in the hole dropped to background levels, (b) the luminosity of the auroral arc observed by a ground-based auroral scanning photometer decreased by a factor of two, and (c) the ionospheric E region density below the hole decayed at a rate consistent with a sudden reduction in particle precipitation. The simultaneous onset of these gross changes in electron precipitation coincident with the release strongly suggests a cause and effect relationship. In particular, these results suggest that the ionospheric plasma and the field-aligned current systems play a crucial role in the auroral acceleration process.
20. Cerebral perturbations during exercise in hypoxia.
PubMed
Verges, Samuel; Rupp, Thomas; Jubeau, Marc; Wuyam, Bernard; Esteve, François; Levy, Patrick; Perrey, Stéphane; Millet, Guillaume Y
2012-04-15
Reduction of aerobic exercise performance observed under hypoxic conditions is mainly attributed to altered muscle metabolism due to impaired O(2) delivery. It has been recently proposed that hypoxia-induced cerebral perturbations may also contribute to exercise performance limitation. A significant reduction in cerebral oxygenation during whole body exercise has been reported in hypoxia compared with normoxia, while changes in cerebral perfusion may depend on the brain region, the level of arterial oxygenation and hyperventilation induced alterations in arterial CO(2). With the use of transcranial magnetic stimulation, inconsistent changes in cortical excitability have been reported in hypoxia, whereas a greater impairment in maximal voluntary activation following a fatiguing exercise has been suggested when arterial O(2) content is reduced. Electromyographic recordings during exercise showed an accelerated rise in central motor drive in hypoxia, probably to compensate for greater muscle contractile fatigue. This accelerated development of muscle fatigue in moderate hypoxia may be responsible for increased inhibitory afferent signals to the central nervous system leading to impaired central drive. In severe hypoxia (arterial O(2) saturation <70-75%), cerebral hypoxia per se may become an important contributor to impaired performance and reduced motor drive during prolonged exercise. This review examines the effects of acute and chronic reduction in arterial O(2) (and CO(2)) on cerebral blood flow and cerebral oxygenation, neuronal function, and central drive to the muscles. Direct and indirect influences of arterial deoxygenation on central command are separated. Methodological concerns as well as future research avenues are also considered. PMID:22319046
1. Thermospheric density perturbations in response to substorms
Clausen, L. B. N.; Milan, S. E.; Grocott, A.
2014-06-01
We use 5 years (2001-2005) of CHAMP (Challenging Minisatellite Payload) satellite data to study average spatial and temporal mass density perturbations caused by magnetospheric substorms in the thermosphere. Using statistics from 2306 substorms to construct superposed epoch time series, we find that the largest average increase in mass density of about 6% occurs about 90 min after substorm expansion phase onset about 3 h of magnetic local time east of the onset region. Averaged over the entire polar auroral region, a mass density increase of about 4% is observed. Using a simple model to estimate the mass density increase at the satellite altitude, we find that an energy deposition rate of 30 GW applied for half an hour predominantly at an altitude of 110 km is able to produce mass density enhancements of the same magnitude. When taking into account previous work that has shown that 80% of the total energy input is due to Joule heating, i.e., enhanced electric fields, whereas 20% is due to precipitation of mainly electrons, our results suggest that the average substorm deposits about 6 GW in the polar thermosphere through particle precipitation. Our result is in good agreement with simultaneous measurements of the NOAA Polar-orbiting Operational Environmental Satellite (POES) Hemispheric Power Index; however, it is about 1 order of magnitude less than reported previously.
2. Neutron star solutions in perturbative quadratic gravity
SciTech Connect
Deliduman, Cemsinan; Ekşi, K.Y.; Keleş, Vildan E-mail: [email protected]
2012-05-01
We study the structure of neutron stars in R+βR{sup μν}R{sub μν} gravity model with perturbative method. We obtain mass-radius relations for six representative equations of state (EoSs). We find that, for |β| ∼ 10{sup 11} cm{sup 2}, the results differ substantially from the results of general relativity. Some of the soft EoSs that are excluded within the framework of general relativity can be reconciled for certain values of β of this order with the 2 solar mass neutron star recently observed. For values of β greater than a few 10{sup 11} cm{sup 2} we find a new solution branch allowing highly massive neutron stars. By referring some recent observational constraints on the mass–radius relation we try to constrain the value of β for each EoS. The associated length scale (β){sup 1/2} ∼ 10{sup 6} cm is of the order of the the typical radius of neutron stars, the probe used in this test. This implies that the true value of β is most likely much smaller than 10{sup 11} cm{sup 2}.
3. Previrialization: Perturbative and N-Body Results
Lokas, E. L.; Juszkiewicz, R.; Bouchet, F. R.; Hivon, E.
1996-08-01
We present a series of N-body experiments which confirm the reality of the previrialization effect. We use also a weakly nonlinear perturbative approach to study the phenomenon. These two approaches agree when the rms density contrast, σ, is small; more surprisingly, they remain in agreement when σ ~ 1. When the slope of the initial power spectrum is n > -1, nonlinear tidal interactions slow down the growth of density fluctuations, and the magnitude of the suppression increases when n (i.e., the relative amount of small-scale power) is increased. For n < -1, we see an opposite effect: the fluctuations grow more rapidly than in linear theory. The transition occurs at n = -1 when the weakly nonlinear correction to σ is close to zero and the growth rate is close to linear. Our results resolve recent controversy between two N- body studies of previrialization. Peebles assumed n = 0 and found strong evidence in support of previrialization, while Evrard & Crone, who assumed n = -1, reached opposite conclusions. As we show here, the initial conditions with n = -1 are rather special because the nonlinear effects nearly cancel out for that particular spectrum. In addition to our calculations for scale-free initial spectra, we show results for a more realistic spectrum of Peacock & Dodds. Its slope near the scale usually adopted for normalization is close to -1, so σ is close to linear. Our results retroactively justify linear normalization at 8 h^-1^ Mpc while also demonstrating the danger and limitations of this practice.
4. Cosmological perturbations across an S-brane
SciTech Connect
Brandenberger, Robert H.; Kounnas, Costas; Partouche, Hervé; Patil, Subodh P.; Toumbas, Nicolaos E-mail: [email protected] E-mail: [email protected]
2014-03-01
Space-filling S-branes can mediate a transition between a contracting and an expanding universe in the Einstein frame. Following up on previous work that uncovered such bouncing solutions in the context of weakly coupled thermal configurations of a certain class of type II superstrings, we set up here the formalism in which we can study the evolution of metric fluctuations across such an S-brane. Our work shows that the specific nature of the S-brane, which is sourced by non-trivial massless thermal string states and appears when the universe reaches a maximal critical temperature, allows for a scale invariant spectrum of curvature fluctuations to manifest at late times via a stringy realization of the matter bounce scenario. The finite energy density at the transition from contraction to expansion provides calculational control over the propagation of the curvature perturbations through the bounce, furnishing a working proof of concept that such a stringy universe can result in viable late time cosmology.
5. Red density perturbations and inflationary gravitational waves
SciTech Connect
Pagano, Luca; Melchiorri, Alessandro; Cooray, Asantha; Kamionkowski, Marc E-mail: [email protected] E-mail: [email protected]
2008-04-15
We study the implications of recent indications from the Wilkinson Microwave Anisotropy Probe (WMAP) and other cosmological data for a red spectrum of primordial density perturbations for the detection of inflationary gravitational waves (IGWs) with forthcoming cosmic microwave background experiments. We consider a variety of single-field power-law, chaotic, spontaneous symmetry-breaking and Coleman-Weinberg inflationary potentials which are expected to provide a sizable tensor component and quantify the expected tensor-to-scalar ratio given existing constraints from WMAP on the tensor-to-scalar ratio and the power spectrum tilt. We discuss the ability of the near-future Planck satellite to detect the IGW background in the framework of those models. We find that the proposed satellite missions of the Cosmic Vision and Inflation Probe programs will be able to detect IGWs from all the models we have surveyed at better than 5{sigma} confidence level. We also provide an example of what is required if the IGW background is to remain undetected even by these latter experiments.
6. Relativistically Covariant Many-Body Perturbation Procedure
Lindgren, Ingvar; Salomonson, Sten; Hedendahl, Daniel
A covariant evolution operator (CEO) can be constructed, representing the time evolution of the relativistic wave unction or state vector. Like the nonrelativistic version, it contains (quasi-)singularities. The regular part is referred to as the Green’s operator (GO), which is the operator analogue of the Green’s function (GF). This operator, which is a field-theoretical concept, is closely related to the many-body wave operator and effective Hamiltonian, and it is the basic tool for our unified theory. The GO leads, when the perturbation is carried to all orders, to the Bethe-Salpeter equation (BSE) in the equal-time or effective-potential approximation. When relaxing the equal-time restriction, the procedure is fully compatible with the exact BSE. The calculations are performed in the photonic Fock space, where the number of photons is no longer constant. The procedure has been applied to helium-like ions, and the results agree well with S-matrix results in cases when comparison can be performed. In addition, evaluation of higher-order quantum-electrodynamical (QED) correlational effects has been performed, and the effects are found to be quite significant for light and medium-heavy ions.
7. Coupled Oscillator Model for Nonlinear Gravitational Perturbations
Yang, Huan; Zhang, Fan; Green, Stephen; Lehner, Luis
2015-04-01
Motivated by the fluid/gravity correspondence, we introduce a new method for characterizing nonlinear gravitational interactions. Namely we map the nonlinear perturbative form of the Einstein's equation to the equations of motion of a series of nonlinearly-coupled harmonic oscillators. These oscillators correspond to the quasinormal modes of the background spacetime. We demonstrate the mechanics and the utility of this formalism with an asymptotically AdS black-brane spacetime, where the equations of motion for the oscillators are shown to be equivalent to the Navier-Stokes equation for the boundary fluid in the mode-expansion picture. We thereby expand on the explicit correspondence connecting the fluid and gravity sides for this particular physical set-up. Perhaps more importantly, we expect this formalism to remain valid in more general spacetimes, including those without a fluid/gravity correspondence. In other words, although born out of the correspondence, the formalism survives independently of it and has a much wider range of applicability.
8. Parallelization of a multiconfigurational perturbation theory.
PubMed
Vancoillie, Steven; Delcey, Mickaël G; Lindh, Roland; Vysotskiy, Victor; Malmqvist, Per-Åke; Veryazov, Valera
2013-08-15
In this work, we present a parallel approach to complete and restricted active space second-order perturbation theory, (CASPT2/RASPT2). We also make an assessment of the performance characteristics of its particular implementation in the Molcas quantum chemistry programming package. Parallel scaling is limited by memory and I/O bandwidth instead of available cores. Significant time savings for calculations on large and complex systems can be achieved by increasing the number of processes on a single machine, as long as memory bandwidth allows, or by using multiple nodes with a fast, low-latency interconnect. We found that parallel efficiency drops below 50% when using 8-16 cores on the shared-memory architecture, or 16-32 nodes on the distributed-memory architecture, depending on the calculation. This limits the scalability of the implementation to a moderate amount of processes. Nonetheless, calculations that took more than 3 days on a serial machine could be performed in less than 5 h on an InfiniBand cluster, where the individual nodes were not even capable of running the calculation because of memory and I/O requirements. This ensures the continuing study of larger molecular systems by means of CASPT2/RASPT2 through the use of the aggregated computational resources offered by distributed computing systems. PMID:23749386
9. Perturbative string thermodynamics near black hole horizons
Mertens, Thomas G.; Verschelde, Henri; Zakharov, Valentin I.
2015-06-01
We provide further computations and ideas to the problem of near-Hagedorn string thermodynamics near (uncharged) black hole horizons, building upon our earlier work [1]. The relevance of long strings to one-loop black hole thermodynamics is emphasized. We then provide an argument in favor of the absence of α'-corrections for the (quadratic) heterotic thermal scalar action in Rindler space. We also compute the large k limit of the cigar orbifold partition functions (for both bosonic and type II superstrings) which allows a better comparison between the flat cones and the cigar cones. A discussion is made on the general McClain-Roth-O'Brien-Tan theorem and on the fact that different torus embeddings lead to different aspects of string thermodynamics. The black hole/string correspondence principle for the 2d black hole is discussed in terms of the thermal scalar. Finally, we present an argument to deal with arbitrary higher genus partition functions, suggesting the breakdown of string perturbation theory (in g s ) to compute thermodynam-ical quantities in black hole spacetimes.
10. Optimized perturbation theory: the pion form factor
SciTech Connect
Gupta, R.
1981-10-01
The order ..cap alpha../sup 2//sub s/(Q/sup 2/) corrections to the pion form-factor F/sub ..pi../(Q/sup 2/) are calculated using perturbative QCD and dimensional regularization. The result is compared in the MS and MOM subtraction schemes and plotted as a function of Q/sup 2//Q/sup 2/ where Q is the subtraction point. There is a large dependence on the scheme, the definition of the running coupling constant ..cap alpha../sub s/(Q/sup 2/) and the subtraction point Q. We find it best to invert the ..beta..-function equation for the definition of ..cap alpha../sub s/ rather than make an expansion in powers of log(Q/sup 2//..lambda../sup 2/). We study two methods to optimize the result with respect to Q: Stevenson's prescription and putting the 0(..cap alpha../sup 2//sub s/) term to zero. Both methods give almost the same value for Q/sup 2/F/sub ..pi../ and this value is scheme independent.
11. MCNP Perturbation Capability for Monte Carlo Criticality Calculations
SciTech Connect
Hendricks, J.S.; Carter, L.L.; McKinney, G.W.
1999-09-20
The differential operator perturbation capability in MCNP4B has been extended to automatically calculate perturbation estimates for the track length estimate of k{sub eff} in MCNP4B. The additional corrections required in certain cases for MCNP4B are no longer needed. Calculating the effect of small design changes on the criticality of nuclear systems with MCNP is now straightforward.
12. Reconfigurable Optical Spectra from Perturbations on Elliptical Whispering Gallery Resonances
NASA Technical Reports Server (NTRS)
Mohageg, Makan; Maleki, Lute
2008-01-01
Elastic strain, electrical bias, and localized geometric deformations were applied to elliptical whispering-gallery-mode resonators fabricated with lithium niobate. The resultant perturbation of the mode spectrum is highly dependant on the modal indices, resulting in a discretely reconfigurable optical spectrum. Breaking of the spatial degeneracy of the whispering-gallery modes due to perturbation is also observed.
13. Variational Perturbation Treatment of the Confined Hydrogen Atom
ERIC Educational Resources Information Center
Montgomery, H. E., Jr.
2011-01-01
The Schrodinger equation for the ground state of a hydrogen atom confined at the centre of an impenetrable cavity is treated using variational perturbation theory. Energies calculated from variational perturbation theory are comparable in accuracy to the results from a direct numerical solution. The goal of this exercise is to introduce the…
14. Reconfigurable optical spectra from perturbations on elliptical whispering gallery resonances.
PubMed
Mohageg, Makan; Maleki, Lute
2008-02-01
Elastic strain, electrical bias, and localized geometric deformations were applied to elliptical whispering-gallery-mode resonators fabricated with lithium niobate. The resultant perturbation of the mode spectrum is highly dependant on the modal indices, resulting in a discretely reconfigurable optical spectrum. Breaking of the spatial degeneracy of the whispering-gallery modes due to perturbation is also observed. PMID:18542283
15. An analytic solution for the J2 perturbed equatorial orbit
NASA Technical Reports Server (NTRS)
Jezewski, D. J.
1983-01-01
An analytic solution for the J2 perturbed equatorial orbit is obtained in terms of elliptic functions and integrals. The necessary equations for computing the position and elocity vectors, and the time are given in terms of known functions. The perturbed periapsis and apoapsis distances are determined from the roots of a characteristic cubic.
16. A Unified Approach for Solving Nonlinear Regular Perturbation Problems
ERIC Educational Resources Information Center
Khuri, S. A.
2008-01-01
This article describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified…
17. Biological response modifiers
SciTech Connect
Weller, R.E.
1988-10-01
Much of what used to be called immunotherapy is now included in the term biological response modifiers. Biological response modifiers (BRMs) are those agents or approaches that modify the relationship between the tumor and host by modifying the host's biological response to tumor cells with resultant therapeutic effects. Most of the early work with BRMs centered around observations of spontaneous tumor regression and the association of tumor regression with concurrent bacterial infections. The BRM can modify the host response by increasing the host's antitumor responses through augmentation and/or restoration of effector mechanisms or mediators of the host's defense or decrease the deleterious component by the host's reaction, increasing the host's defenses by the administration of natural biologics (or the synthetic derivatives thereof) as effectors or mediators of an antitumor response, augmenting the host's response to modified tumor cells or vaccines, which might stimulate a greater response by the host or increase tumor-cell sensitivity to an existing response, decreasing the transformation and/or increase differentiation (maturation) of tumor cells, or increasing the ability of the host to tolerate damage by cytotoxic modalities of cancer treatment.
18. Biological response modifiers
SciTech Connect
Weller, R.E.
1991-10-01
Much of what used to be called immunotherapy is now included in the term biological response modifiers. Biological response modifiers (BRMs) are defined as those agents or approaches that modify the relationship between the tumor and host by modifying the host's biological response to tumor cells with resultant therapeutic effects.'' Most of the early work with BRMs centered around observations of spontaneous tumor regression and the association of tumor regression with concurrent bacterial infections. The BRM can modify the host response in the following ways: Increase the host's antitumor responses through augmentation and/or restoration of effector mechanisms or mediators of the host's defense or decrease the deleterious component by the host's reaction; Increase the host's defenses by the administration of natural biologics (or the synthetic derivatives thereof) as effectors or mediators of an antitumor response; Augment the host's response to modified tumor cells or vaccines, which might stimulate a greater response by the host or increase tumor-cell sensitivity to an existing response; Decrease the transformation and/or increase differentiation (maturation) of tumor cells; or Increase the ability of the host to tolerate damage by cytotoxic modalities of cancer treatment.
19. Elliptic inflation: generating the curvature perturbation without slow-roll
Matsuda, Tomohiro
2006-09-01
There are many inflationary models in which the inflaton field does not satisfy the slow-roll condition. However, in such models, it is always difficult to generate the curvature perturbation during inflation. Thus, to generate the curvature perturbation, one must introduce another component into the theory. To cite a case, curvatons may generate the dominant part of the curvature perturbation after inflation. However, we question whether it is realistic to consider the generation of the curvature perturbation during inflation without slow-roll. Assuming multifield inflation, we encounter the generation of curvature perturbation during inflation without slow-roll. The potential along the equipotential surface is flat by definition and thus we do not have to worry about symmetry. We also discuss KKLT (Kachru Kallosh Linde Trivedi) models, in which corrections lifting the inflationary direction may not become a serious problem if there is a symmetry enhancement at the tip (not at the moving brane) of the inflationary throat.
20. Finite field-dependent symmetries in perturbative quantum gravity
2014-01-01
In this paper we discuss the absolutely anticommuting nilpotent symmetries for perturbative quantum gravity in general curved spacetime in linear and non-linear gauges. Further, we analyze the finite field-dependent BRST (FFBRST) transformation for perturbative quantum gravity in general curved spacetime. The FFBRST transformation changes the gauge-fixing and ghost parts of the perturbative quantum gravity within functional integration. However, the operation of such symmetry transformation on the generating functional of perturbative quantum gravity does not affect the theory on physical ground. The FFBRST transformation with appropriate choices of finite BRST parameter connects non-linear Curci-Ferrari and Landau gauges of perturbative quantum gravity. The validity of the results is also established at quantum level using Batalin-Vilkovisky (BV) formulation.
1. Quasilinear perturbed equilibria of resistively unstable current carrying plasma
Hu, Di; Zakharov, Leonid E.
2015-12-01
> A formalism for consideration of island formation is presented using a model of a cylindrical resistively unstable plasma. Both current and pressure driven island formation at resonant surfaces are considered. The proposed formalism of perturbed equilibria avoids problems typical for linear analysis of resistive magneto-hydrodynamic instabilities related to extraction of the so-called small solution near the resonant surfaces. The matching technique of this paper is not sensitive to configuration parameters near the resonant surfaces. The comparison of the perturbed equilibrium method with the frequently used quasilinear mode analysis based on a perturbed averaged current density profile shows that the latter is limited in its applicability and underestimates the stability. Presented here for a cylindrical case, the perturbed equilibrium technique can be used in toroidal perturbed equilibrium codes with minor modifications.
2. Duality between QCD perturbative series and power corrections
Narison, S.; Zakharov, V. I.
2009-08-01
We elaborate on the relation between perturbative and power-like corrections to short-distance sensitive QCD observables. We confront theoretical expectations with explicit perturbative calculations existing in literature. As is expected, the quadratic correction is dual to a long perturbative series and one should use one of them but not both. However, this might be true only for very long perturbative series, with number of terms needed in most cases exceeding the number of terms available. What has not been foreseen, the quartic corrections might also be dual to the perturbative series. If confirmed, this would imply a crucial modification of the dogma. We confront this quadratic correction against existing phenomenology (QCD (spectral) sum rules scales, determinations of light quark masses and of αs from τ-decay). We find no contradiction and (to some extent) better agreement with the data and with recent lattice calculations.
3. Reducing Plasma Perturbations with Segmented Metal Shielding on Electrostatic Probes
SciTech Connect
Staack, D.; Raitses, Y.; Fisch N.J.
2002-10-02
Electrostatic probes are widely used to measure spatial plasma parameters in the quasi-neutral plasma created in Hall thrusters and similar E x B electric discharge devices. Significant perturbations of the plasma, induced by such probes, can mask the actual physics involved in operation of these devices. In an attempt to reduce these perturbations in Hall thrusters, the perturbations were examined by varying the component material, penetration distance, and residence time of various probe designs. This study leads us to a conclusion that secondary electron emission from insulator ceramic tubes of the probe can affect local changes of the plasma parameters causing plasma perturbations. A probe design, which consists of a segmented metal shielding of the probe insulator, is suggested to reduce these perturbations. This new probe design can be useful for plasma applications in which the electron temperature is sufficient to produce secondary electron emission by interaction of plasma electrons with dielectric materials.
4. Plasma-satellite interaction driven magnetic field perturbations
SciTech Connect
Saeed-ur-Rehman; Marchand, Richard
2014-09-15
We report the first fully kinetic quantitative estimate of magnetic field perturbations caused by the interaction of a spacecraft with space environment. Such perturbations could affect measurements of geophysical magnetic fields made with very sensitive magnetometers on-board satellites. Our approach is illustrated with a calculation of perturbed magnetic fields near the recently launched Swarm satellites. In this case, magnetic field perturbations do not exceed 20 pT, and they are below the sensitivity threshold of the on-board magnetometers. Anticipating future missions in which satellites and instruments would be subject to more intense solar UV radiation, however, it appears that magnetic field perturbations associated with satellite interaction with space environment, might approach or exceed instruments' sensitivity thresholds.
5. May chaos always be suppressed by parametric perturbations?
PubMed
Schwalger, Tilo; Dzhanoev, Arsen; Loskutov, Alexander
2006-06-01
The problem of chaos suppression by parametric perturbations is considered. Despite the widespread opinion that chaotic behavior may be stabilized by perturbations of any system parameter, we construct a counterexample showing that this is not necessarily the case. In general, chaos suppression means that parametric perturbations should be applied within a set of parameters at which the system has a positive maximal Lyapunov exponent. Analyzing the known Duffing-Holmes model by a Melnikov method, we showed that chaotic dynamics cannot be suppressed by harmonic perturbations of a certain parameter, independently from the other parameter values. Thus, to stabilize the behavior of chaotic systems, the perturbation and parameters should be carefully chosen. PMID:16822012
6. A new model for realistic random perturbations of stochastic oscillators
Dieci, Luca; Li, Wuchen; Zhou, Haomin
2016-08-01
Classical theories predict that solutions of differential equations will leave any neighborhood of a stable limit cycle, if white noise is added to the system. In reality, many engineering systems modeled by second order differential equations, like the van der Pol oscillator, show incredible robustness against noise perturbations, and the perturbed trajectories remain in the neighborhood of a stable limit cycle for all times of practical interest. In this paper, we propose a new model of noise to bridge this apparent discrepancy between theory and practice. Restricting to perturbations from within this new class of noise, we consider stochastic perturbations of second order differential systems that -in the unperturbed case- admit asymptotically stable limit cycles. We show that the perturbed solutions are globally bounded and remain in a tubular neighborhood of the underlying deterministic periodic orbit. We also define stochastic Poincaré map(s), and further derive partial differential equations for the transition density function.
7. Analyzing molecular static linear response properties with perturbed localized orbitals
Autschbach, Jochen; King, Harry F.
2010-07-01
Perturbed localized molecular orbitals (LMOs), correct to first order in an applied static perturbation and consistent with a chosen localization functional, are calculated using analytic derivative techniques. The formalism is outlined for a general static perturbation and variational localization functionals. Iterative and (formally) single-step approaches are compared. The implementation employs an iterative sequence of 2×2 orbital rotations. The procedure is verified by calculations of molecular electric-field perturbations. Boys LMO contributions to the electronic static polarizability and the electric-field perturbation of the ⟨r2⟩ expectation value are calculated and analyzed for ethene, ethyne, and fluoroethene (H2CCHF). For ethene, a comparison is made with results from a Pipek-Mezey localization. The calculations show that a chemically intuitive decomposition of the calculated properties is possible with the help of the LMO contributions and that the polarizability contributions in similar molecules are approximately transferable.
8. Investigating perturbed pathway modules from gene expression data via structural equation models
PubMed Central
2014-01-01
Background It is currently accepted that the perturbation of complex intracellular networks, rather than the dysregulation of a single gene, is the basis for phenotypical diversity. High-throughput gene expression data allow to investigate changes in gene expression profiles among different conditions. Recently, many efforts have been made to individuate which biological pathways are perturbed, given a list of differentially expressed genes (DEGs). In order to understand these mechanisms, it is necessary to unveil the variation of genes in relation to each other, considering the different phenotypes. In this paper, we illustrate a pipeline, based on Structural Equation Modeling (SEM) that allowed to investigate pathway modules, considering not only deregulated genes but also the connections between the perturbed ones. Results The procedure was tested on microarray experiments relative to two neurological diseases: frontotemporal lobar degeneration with ubiquitinated inclusions (FTLD-U) and multiple sclerosis (MS). Starting from DEGs and dysregulated biological pathways, a model for each pathway was generated using databases information biological databases, in order to design how DEGs were connected in a causal structure. Successively, SEM analysis proved if pathways differ globally, between groups, and for specific path relationships. The results confirmed the importance of certain genes in the analyzed diseases, and unveiled which connections are modified among them. Conclusions We propose a framework to perform differential gene expression analysis on microarray data based on SEM, which is able to: 1) find relevant genes and perturbed biological pathways, investigating putative sub-pathway models based on the concept of disease module; 2) test and improve the generated models; 3) detect a differential expression level of one gene, and differential connection between two genes. This could shed light, not only on the mechanisms affecting variations in gene
9. Transmission and dose perturbations with high-Z materials in clinical electron beams.
PubMed
Das, Indra J; Cheng, Chee-Wai; Mitra, Raj K; Kassaee, Alireza; Tochner, Zelig; Solin, Lawrence J
2004-12-01
High density and atomic number (Z) materials used in various prostheses, eye shielding, and beam modifiers produce dose enhancements on the backscatter side in electron beams and is well documented. However, on the transmission side the dose perturbation is given very little clinical importance, which is investigated in this study. A simple and accurate method for dose perturbation at metallic interfaces with soft tissues and transmission through these materials is required for all clinical electron beams. Measurements were taken with thin-window parallel plate ion chambers for various high-Z materials (Al, Ti, Cu, and Pb) on a Varian and a Siemens accelerator in the energy range of 5-20 MeV. The dose enhancement on both sides of the metallic sheet is due to increased electron fluence that is dependent on the beam energy and Z. On the transmission side, the magnitude of dose enhancement depends on the thickness of the high-Z material. With increasing thickness, dose perturbation reduces to the electron transmission. The thickness of material to reduce 100% (range of dose perturbation), 50% and 10% transmission is linear with the beam energy. The slope (mm/MeV) of the transmission curve varies exponentially with Z. A nonlinear regression expression (t=E[alpha+beta exp(-0.1Z)]) is derived to calculate the thickness at a given transmission, namely 100%, 50%, and 10% for electron energy, E, which is simple, accurate and well suited for a quick estimation in clinical use. Caution should be given to clinicians for the selection of thickness of high-Z materials when used to shield critical structures as small thickness increases dose significantly at interfaces. PMID:15651605
10. Transmission and dose perturbations with high-Z materials in clinical electron beams
SciTech Connect
Das, Indra J.; Cheng, C.-W.; Mitra, Raj K.; Kassaee, Alireza; Tochner, Zelig; Solin, Lawrence J.
2004-12-01
High density and atomic number (Z) materials used in various prostheses, eye shielding, and beam modifiers produce dose enhancements on the backscatter side in electron beams and is well documented. However, on the transmission side the dose perturbation is given very little clinical importance, which is investigated in this study. A simple and accurate method for dose perturbation at metallic interfaces with soft tissues and transmission through these materials is required for all clinical electron beams. Measurements were taken with thin-window parallel plate ion chambers for various high-Z materials (Al, Ti, Cu, and Pb) on a Varian and a Siemens accelerator in the energy range of 5-20 MeV. The dose enhancement on both sides of the metallic sheet is due to increased electron fluence that is dependent on the beam energy and Z. On the transmission side, the magnitude of dose enhancement depends on the thickness of the high-Z material. With increasing thickness, dose perturbation reduces to the electron transmission. The thickness of material to reduce 100% (range of dose perturbation), 50% and 10% transmission is linear with the beam energy. The slope (mm/MeV) of the transmission curve varies exponentially with Z. A nonlinear regression expression {l_brace}t=E[{alpha}+{beta} exp(-0.1Z)]{r_brace} is derived to calculate the thickness at a given transmission, namely 100%, 50%, and 10% for electron energy, E, which is simple, accurate and well suited for a quick estimation in clinical use. Caution should be given to clinicians for the selection of thickness of high-Z materials when used to shield critical structures as small thickness increases dose significantly at interfaces.
11. Detection of perturbed quantization class stego images based on possible change modes
Zhang, Yi; Liu, Fenlin; Yang, Chunfang; Luo, Xiangyang; Song, Xiaofeng
2015-11-01
To improve the detection performance for perturbed quantization (PQ) class [PQ, energy-adaptive PQ (PQe), and texture-adaptive PQ (PQt)] stego images, a detection method based on possible change modes is proposed. First, by using the relationship between the changeable coefficients used for carrying secret messages and the second quantization steps, the modes having even second quantization steps are identified as possible change modes. Second, by referencing the existing features, the modified features that can accurately capture the embedding changes based on possible change modes are extracted. Next, feature sensitivity analyses based on the modifications performed before and after the embedding are carried out. These analyses show that the modified features are more sensitive to the original features. Experimental results indicate that detection performance of the modified features is better than that of the corresponding original features for three typical feature models [Cartesian calibrated PEVny (ccPEV), Cartesian calibrated co-occurrence matrix features (CF), and JPEG rich model (JRM)], and the integrated feature consisting of enhanced histogram features (EHF) and the modified JRM outperforms two current state-of-the-art feature models, namely, phase aware projection model (PHARM) and Gabor rich model (GRM).
12. CMB constraint on non-Gaussianity in isocurvature perturbations
SciTech Connect
Hikage, Chiaki; Kawasaki, Masahiro; Sekiguchi, Toyokazu; Takahashi, Tomo E-mail: [email protected] E-mail: [email protected]
2013-07-01
We study the CMB constraints on non-Gaussianity in CDM isocurvature perturbations. Non-Gaussian isocurvature perturbations can be produced in various models at the very early stage of the Universe. Since the isocurvature perturbations little affect the structure formation at late times, CMB is the best probe of isocurvature non-Gaussianity at least in the near future. In this paper, we focus on non-Gaussian curvature and isocurvature perturbations of the local-type, which are uncorrelated and in the form ζ = ζ{sub G}+(3/5)f{sub NL}(ζ{sub G}{sup 2}−(ζ{sub G}{sup 2})) and S = S{sub G}+f{sub NL}{sup (ISO)}(S{sub G}−(S{sub G}{sup 2})), and constrain the non-linearity parameter of isocurvature perturbations, f{sub NL}{sup (ISO)}, as well as the curvature one f{sub NL}. For this purpose, we employ several state-of-art techniques for the analysis of CMB data and simulation. Assuming that isocurvature perturbations are subdominant, we apply our method to the WMAP 7-year data of temperature anisotropy and obtain constraints on a combination α{sup 2}f{sub NL}{sup (ISO)}, where α is the ratio of the power spectrum of isocurvature perturbations to that of the adiabatic ones. When the adiabatic perturbations are assumed to be Gaussian, we obtained a constraint α{sup 2}f{sub NL}{sup (ISO)} = 40±66 assuming the power spectrum of isocurvature perturbations is scale-invariant. When we assume that the adiabatic perturbations can also be non-Gaussian, we obtain f{sub NL} = 38±24 and α{sup 2}f{sub NL}{sup (ISO)} = −8±72. We also discuss implications of our results for the axion CDM isocurvature model.
13. Effect of mechanical perturbation on the biomechanics, primary growth and secondary tissue development of inflorescence stems of Arabidopsis thaliana
PubMed Central
Paul-Victor, Cloé; Rowe, Nick
2011-01-01
Background and Aims Mechanical perturbation is known to inhibit elongation of the inflorescence stem of Arabidopsis thaliana. The phenomenon has been reported widely for both herbaceous and woody plants, and has implications for how plants adjust their size and form to survive in mechanically perturbed environments. While this response is an important aspect of the plant's architecture, little is known about how mechanical properties of the inflorescence stem are modified or how its primary and secondary tissues respond to mechanical perturbation. Methods Plants of the Columbia-0 ecotype were exposed to controlled brushing treatments and then submitted to three-point bending tests to determine stem rigidity and stiffness. Contributions of different tissues to the inflorescence stem geometry were analysed. Key Results Perturbed plants showed little difference in stem diameter, were 50 % shorter, 75 % less rigid and 70 % less stiff than controls. Changes in mechanical properties were linked to significant changes in tissue geometry – size and position of the pith, lignified interfascicular tissue and cortex – as well as a reduction in density of lignified cells. Stem mechanical properties were modified by changes in primary tissues and thus differ from changes observed in most woody plants tested with indeterminate growth – even though a vascular cambium is present in the inflorescence axis. Conclusions The study suggests that delayed development of key primary developmental features of the stem in this ecotype of Arabidopsis results in a ‘short and flexible’ rather than a ‘short and rigid’ strategy for maintaining upright axes in conditions of severe mechanical perturbation. The mechanism is comparable with more general phenomena in plants where changes in developmental rate can significantly affect the overall growth form of the plant in both ecological and evolutionary contexts. PMID:21118840
14. Flexoelectricity from density-functional perturbation theory
Stengel, Massimiliano
2013-11-01
We derive the complete flexoelectric tensor, including electronic and lattice-mediated effects, of an arbitrary insulator in terms of the microscopic linear response of the crystal to atomic displacements. The basic ingredient, which can be readily calculated from first principles in the framework of density-functional perturbation theory, is the quantum-mechanical probability current response to a long-wavelength acoustic phonon. Its second-order Taylor expansion in the wave vector q around the Γ (q=0) point in the Brillouin zone naturally yields the flexoelectric tensor. At order one in q we recover Martin's theory of piezoelectricity [Martin, Phys. Rev. B 5, 1607 (1972)], thus providing an alternative derivation thereof. To put our derivations on firm theoretical grounds, we perform a thorough analysis of the nonanalytic behavior of the dynamical matrix and other response functions in a vicinity of Γ. Based on this analysis, we find that there is an ambiguity in the specification of the “zero macroscopic field” condition in the flexoelectric case; such arbitrariness can be related to an analytic band-structure term, in close analogy to the theory of deformation potentials. As a by-product, we derive a rigorous generalization of the Cochran-Cowley formula [Cochran and Cowley, J. Phys. Chem. Solids 23, 447 (1962)] to higher orders in q. This can be of great utility in building reliable atomistic models of electromechanical phenomena, as well as for improving the accuracy of the calculation of phonon dispersion curves. Finally, we discuss the physical interpretation of the various contributions to the flexoelectric response, either in the static or dynamic regime, and we relate our findings to earlier theoretical works on the subject.
15. AGK Cutting Rules and Perturbative QCD
Treleani, D.
The purpose of this article is to describe a few features of semihard interactions, in high energy nuclear collisions, that are better understood with the help of the AGK cutting rules, and of the probabilistic picture of the interaction which follows. In the first part of the article the cutting rules are discussed for the simplest component of the forward three-body parton amplitude in the large s fixed t limit. The case considered corresponds to the term — at the lowest order in the coupling constant and with vacuum quantum number exchange in both t channels — of the amplitude which describes the interaction of a high energy quark with the two target quarks. The different leading cuts of the amplitude are shown to be proportional to one another with the same weights of the cutting rules derived in the context of multi-Pomeron exchange. The probabilistic picture of the multiple interactions, which originates from the cutting rules, and the self-shadowing cross sections are then discussed. The second part of the article deals with the semihard interactions. The semihard cross section in high energy nucleus-nucleus collisions is represented as a self-shadowing cross section, and a feature which is pointed out is that the single scattering factorized expression of the perturbative QCD parton model holds at any order in the multiparton correlations, the relation being the analog of the AGK cancellation for the average number of soft interactions in high energy hadron-nucleus collisions. Finally, an infrared problem which finds a solution within the self-shadowing representation of the semihard cross section is discussed.
16. The Tidal Perturbations of the Galilean Satellites
Jacobson, Robert A.; Folkner, William M.
2014-11-01
To support the Juno mission currently enroute to Jupiter and preproject studies for the Europa Clipper mission, we developed new ephemerides for the Jovian satellites (the Galileans and four inners). The ephemerides are based on orbits that were determined by fitting a data set that included Earth-based astrometry from 1891 through 2013, Galilean satellite mutual events from 1973 through 2009, Galilean satellite eclipse timings from 1878 to 2013, and data acquired by the Pioneer, Voyager, Ulysses, Cassini, Galileo, and New Horizons spacecraft. As a part of the data fit we also redetermined the Jovian system gravity parameters and the spacecraft trajectories to be consistent with the satellite orbits. The dynamical model for the satellite orbits did not include tidal perturbations. Lainey et al. (2009 Nature 459, 957) determined tidal parameters for Jupiter and Io from a fit of the Galilean satellite orbits to Earth-based astrometry from 1891 to 2007 and mutual events from 1973 to 2003; he estimated only the satellite states and the tidal parameters. Subsequent to our ephemeris development, we activated the tide model and repeated our orbit analysis adding the determination of the tidal parameters. We found that if we omitted the spacecraft data and estimated only satellite states and tidal parameters, we obtained results similar to Lainey. However, when we included the spacecraft data in the fit, the tidal acceleration on Io was smaller but still caused a positive secular acceleration. The remaining task is to discriminate between the effects of the tide raised on Jupiter by Io and that raised on Io by Jupiter.
17. Developing perturbations for Climate Change Impact Assessments
Hewitson, Bruce
Following the 2001 Intergovernmental Panel on Climate Change (IPCC) Third Assessment Report [TAR; IPCC, 2001], and the paucity of climate change impact assessments from developing nations, there has been a significant growth in activities to redress this shortcoming. However, undertaking impact assessments (in relation to malaria, crop stress, regional water supply, etc.) is contingent on available climate-scale scenarios at time and space scales of relevance to the regional issues of importance. These scales are commonly far finer than even the native resolution of the Global Climate Models (GCMs) (the principal tools for climate change research), let alone the skillful resolution (scales of aggregation at which GCM observational error is acceptable for a given application) of GCMs.Consequently, there is a growing demand for regional-scale scenarios, which in turn are reliant on techniques to downscale from GCMs, such as empirical downscaling or nested Regional Climate Models (RCMs). These methods require significant skill, experiential knowledge, and computational infrastructure in order to derive credible regional-scale scenarios. In contrast, it is often the case that impact assessment researchers in developing nations have inadequate resources with limited access to scientists in the broader international scientific community who have the time and expertise to assist. However, where developing effective downscaled scenarios is problematic, it is possible that much useful information can still be obtained for impact assessments by examining the system sensitivity to largerscale climate perturbations. Consequently, one may argue that the early phase of assessing sensitivity and vulnerability should first be characterized by evaluation of the first-order impacts, rather than immediately addressing the finer, secondary factors that are dependant on scenarios derived through downscaling.
18. Statistics and dynamics of the perturbed universe
Lemson, G.
1995-09-01
Wilson discovered the corresponding radiation field, at a temperature of roughly 3K (Penzias & Wilson, 1965). It soon appeared that this microwave background radiation was isotropic to a high degree, which conrmed the assumptions made about the homogeneity of the early Universe. At present however, we see that the Universe is no longer featureless and smooth. Starting from the smallest scales we see matter organized in structures up to very large scales: from planets to stars to stellar systems to galaxies to groups and clusters of galaxies, up to super-clusters, where clusters and galaxies are organized in the largest structures known. Somewhere during the evolution of the Universe, these structures must have developed out of the featureless, uniform sea of matter and radiation. Various different theories have been developed to explain the emergence of structure, but in this thesis I will concentrate exclusively on the most generally accepted theory, that of gravitational instability. In this theory it is assumed that in the early Universe, small fluctuations in the density were present, and these would grow under the influence of gravity towards the presently observed structures. There is actually a rather complete theory of the early stages of this process, that regime where these deviations from homogeneity are small. In that case, the inhomogeneous field may be seen as a small disturbance to the uniform model, and the standard apparatus of perturbation theory may be applied. In this thesis I investigate the later stages of this process of structure formation, where the fluctuations have grown to such a size that this 'linear' perturbation approach breaks down. There is as yet no comprehensive model describing this 'nonlinear' regime as successfully as the linear theory describes the early stages of structure formation. Instead, the problem is approached from many different directions, using different, approximate models for describing the dynamics and other
19. Challenges in the extraction of TMDs from SIDIS data: perturbative vs non-perturbative aspects
SciTech Connect
Boglione, Mariaelena; Gonzalez Hernandez, Jose O.; Melis, Stefano; Prokudin, Alexey
2015-09-01
We present our recent results on the study of the Semi-Inclusive Deep Inelastic Scattering (SIDIS) cross section as a function of the transverse momentum, qT. Using the Collins-Soper-Sterman (CSS) formalism, we study the matching between the region where fixed-order perturbative QCD can successfully be applied and the region where soft gluon resummation is necessary. We find that the commonly used prescription of matching through the so-called Y-factor cannot be applied in the SIDIS kinematical configurations we examine. We comment on the impact that the nonperturbative component has even at relatively high energies.
20. Modified blank ammunition injuries.
PubMed
Ogunc, Gokhan I; Ozer, M Tahir; Coskun, Kagan; Uzar, Ali Ihsan
2009-12-15
Blank firing weapons are designed only for discharging blank ammunition cartridges. Because they are cost-effective, are easily accessible and can be modified to live firearms plus their unclear legal situation in Turkish Law makes them very popular in Turkey. 2004 through 2008, a total of 1115 modified blank weapons were seized in Turkey. Blank firing weapons are easily modified by owners, making them suitable for discharging live firearm ammunition or modified blank ammunitions. Two common methods are used for modification of blank weapons. After the modification, these weapons can discharge the live ammunition. However, due to compositional durability problems with these types of weapons; the main trend is to use the modified blank ammunitions rather than live firearm ammunition fired from modified blank firing weapons. In this study, two types of modified blank weapons and two types of modified blank cartridges were tested on three different target models. Each of the models' shooting side was coated with 1.3+/-2 mm thickness chrome tanned cowhide as a skin simulant. The first model was only coated with skin simulant. The second model was coated with skin simulant and 100% cotton police shirt. The third model was coated with skin simulant and jean denim. After the literature evaluation four high risky anatomic locations (the neck area; the eyes; the thorax area and inguinal area) were pointed out for the steel and lead projectiles are discharged from the modified blank weapons especially in close range (0-50 cm). The target models were designed for these anatomic locations. For the target models six Transparent Ballistic Candle blocks (TCB) were prepared and divided into two test groups. The first group tests were performed with lead projectiles and second group with steel projectile. The shortest penetration depth (lead projectile: 4.358 cm; steel projectile 8.032 cm) was recorded in the skin simulant and jean denim coated block for both groups. In both groups
1. Aminoglycoside Modifying Enzymes
PubMed Central
Ramirez, Maria S.; Tolmasky, Marcelo E.
2010-01-01
Aminoglycosides have been an essential component of the armamentarium in the treatment of life-threatening infections. Unfortunately, their efficacy has been reduced by the surge and dissemination of resistance. In some cases the levels of resistance reached the point that rendered them virtually useless. Among many known mechanisms of resistance to aminoglycosides, enzymatic modification is the most prevalent in the clinical setting. Aminoglycoside modifying enzymes catalyze the modification at different −OH or −NH2 groups of the 2-deoxystreptamine nucleus or the sugar moieties and can be nucleotidyltranferases, phosphotransferases, or acetyltransferases. The number of aminoglycoside modifying enzymes identified to date as well as the genetic environments where the coding genes are located is impressive and there is virtually no bacteria that is unable to support enzymatic resistance to aminoglycosides. Aside from the development of new aminoglycosides refractory to as many as possible modifying enzymes there are currently two main strategies being pursued to overcome the action of aminoglycoside modifying enzymes. Their successful development would extend the useful life of existing antibiotics that have proven effective in the treatment of infections. These strategies consist of the development of inhibitors of the enzymatic action or of the expression of the modifying enzymes. PMID:20833577
2. Spherical dust fluctuations: The exact versus the perturbative approach
Sussman, Roberto A.; Hidalgo, Juan Carlos; Dunsby, Peter K. S.; German, Gabriel
2015-03-01
We examine the relation between the dynamics of Lemaître-Tolman-Bondi (LTB) dust models (with and without Λ ) and the dynamics of dust perturbations in two of the more familiar formalisms used in cosmology: the metric based cosmological perturbation theory (CPT) and the covariant gauge invariant (GIC) perturbations. For this purpose we recast the evolution of LTB models in terms of a covariant and gauge invariant formalism of local and nonlocal "exact fluctuations" on a Friedmann-Lemaître-Robertson-Walker (FLRW) background defined by suitable averages of covariant scalars. We examine the properties of these fluctuations, which can be defined for a confined comoving domain or for an asymptotic domain extending to whole time slices. In particular, the nonlocal density fluctuation provides a covariant and precise definition for the notion of the "density contrast." We show that in their linear regime these LTB exact fluctuations (local and nonlocal) are fully equivalent to the conventional cosmological perturbations in the synchronous-comoving gauge of CPT and to GIC perturbations. As an immediate consequence, we show the time-invariance of the spatial curvature perturbation in a simple form. The present work may provide important theoretical connections between the exact and perturbative (linear or nonlinear) approach to the dynamics of dust sources in general relativity.
3. Transport of energetic ions by low-n magnetic perturbations
SciTech Connect
Mynick, H.E.
1992-10-01
The stochastic transport of MeV ions induced by low-n magnetic perturbations is studied, focussing chiefly on the stochastic mechanism operative for passing particles in low frequency perturbations. Beginning with a single-harmonic form for the perturbing field, it iii first shown numerically and analytically that the stochastic threshold of energetic particles can be much lower than that of the magnetic field, contrary to earlier expectations, so that MHD perturbations could cause appreciable loss of energetic ions without destroying the bulk confinement. The analytic theory is then extended in a number of directions, to darity the relation of the present stochaistic mechanism to instances already found, to allow for more complex perturbations, and to consider the more general relationship between the stochasticity of magnetic fields, and that of particles of differing energies (and pitch angles) moving in those fields. It is shown that the stochastic threshold is in general a nonmonotonic function of energy, whose form can to some extent be tailored to achieve desired goals (e.g., burn control or ash removal) by a judicious choice of the perturbation. Illustrative perturbations are exhibited which are stochastic for low but not for high-energy ions, for high but not for low-energy ions, and for intermediate-energy ions, but not for low or high energy. The second possibility is the behavior needed for burn control; the third provides a possible mechanism for ash removal.
4. Perturbing polynomials with all their roots on the unit circle
Mossinghoff, M. J.; Pinner, C. G.; Vaaler, J. D.
1998-10-01
Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most 4, with 4 achieved only for polynomials of the form x(2n) + cx(n) + 1 with c in [-2, 2]. The problem can also be formulated in terms of perturbing the constant coefficient of a polynomial having all its roots in [-1, 1]. If we restrict to integer coefficients, then the polynomials in question are products of cyclotomics. We show that in this case there are no perturbations of length 3 that do not arise from a perturbation of length 4. We also investigate the connection between slightly perturbed products of cyclotomic polynomials and polynomials with small Mahler measure. We describe an algorithm for searching for polynomials with small Mahler measure by perturbing the middle coefficients of products of cyclotomic polynomials. We show that the complexity of this algorithm is O(C-root d), where d is the degree, and we report on the polynomials found by this algorithm through degree 64.
5. Development of a perturbation generator for vortex stability studies
NASA Technical Reports Server (NTRS)
Riester, J. E.; Ash, Robert L.
1991-01-01
Theory predicts vortex instability when subjected to certain types of disturbances. It was desired to build a device which could introduce controlled velocity perturbations into a trailing line vortex in order to study the effects on stability. A perturbation generator was designed and manufactured which can be attached to the centerbody of an airfoil type vortex generator. Details of design tests and manufacturing of the perturbation generator are presented. The device produced controlled perturbation with frequencies in excess of 250 Hz. Preliminary testing and evaluation of the perturbation generator performance was conducted in a 4 inch cylindrical pipe. Observations of vortex shedding frequencies from a centerbody were measured. Further evaluation with the perturbation generator attached to the vortex generator in a 2 x 3 foot wind tunnel were also conducted. Hot-wire anemometry was used to confirm the perturbation generator's ability to introduce controlled frequency fluctuations. Comparison of the energy levels of the disturbances in the vortex core was made between locations 42 chord lengths and 15 chord lengths downstream.
6. Black hole perturbation theory in a light cone gauge
Preston, Brent
The metric of a Schwarzschild black hole immersed in a uniform magnetic field is studied using black hole perturbation theory in a light crone coordinate system that penetrates the event horizon and possesses a clear geometrical meaning. The magnetic field, which is distorted due to the presence of the black hole, has strength B which is assumed to be small compared to the curvature of the spacetime which allows the perturbed metric to be calculated to order B 2 only. The coordinates allow for an easy identification of the event horizon and the properties of the perturbed black hole are studied. To interpret this perturbed metric, the advanced coordinates are decomposed into irreducible parts which yields the metric of a perturbed black hole in the limit r >> 2 M . Finally we compare our perturbed solution to an exact solution. We show that our perturbed solution is able to match the exact solution but has the freedom to describe a larger class of physically relevant solutions.
7. Boundary Layer Instabilities Generated by Freestream Laser Perturbations
NASA Technical Reports Server (NTRS)
Chou, Amanda; Schneider, Steven P.
2015-01-01
A controlled, laser-generated, freestream perturbation was created in the freestream of the Boeing/AFOSR Mach-6 Quiet Tunnel (BAM6QT). The freestream perturbation convected downstream in the Mach-6 wind tunnel to interact with a flared cone model. The geometry of the flared cone is a body of revolution bounded by a circular arc with a 3-meter radius. Fourteen PCB 132A31 pressure transducers were used to measure a wave packet generated in the cone boundary layer by the freestream perturbation. This wave packet grew large and became nonlinear before experiencing natural transition in quiet flow. Breakdown of this wave packet occurred when the amplitude of the pressure fluctuations was approximately 10% of the surface pressure for a nominally sharp nosetip. The initial amplitude of the second mode instability on the blunt flared cone is estimated to be on the order of 10 -6 times the freestream static pressure. The freestream laser-generated perturbation was positioned upstream of the model in three different configurations: on the centerline, offset from the centerline by 1.5 mm, and offset from the centerline by 3.0 mm. When the perturbation was offset from the centerline of a blunt flared cone, a larger wave packet was generated on the side toward which the perturbation was offset. The offset perturbation did not show as much of an effect on the wave packet on a sharp flared cone as it did on a blunt flared cone.
8. Evolution equation for non-linear cosmological perturbations
SciTech Connect
Brustein, Ram; Riotto, Antonio E-mail: [email protected]
2011-11-01
We present a novel approach, based entirely on the gravitational potential, for studying the evolution of non-linear cosmological matter perturbations. Starting from the perturbed Einstein equations, we integrate out the non-relativistic degrees of freedom of the cosmic fluid and obtain a single closed equation for the gravitational potential. We then verify the validity of the new equation by comparing its approximate solutions to known results in the theory of non-linear cosmological perturbations. First, we show explicitly that the perturbative solution of our equation matches the standard perturbative solutions. Next, using the mean field approximation to the equation, we show that its solution reproduces in a simple way the exponential suppression of the non-linear propagator on small scales due to the velocity dispersion. Our approach can therefore reproduce the main features of the renormalized perturbation theory and (time)-renormalization group approaches to the study of non-linear cosmological perturbations, with some possibly important differences. We conclude by a preliminary discussion of the nature of the full solutions of the equation and their significance.
9. Perturbative and non-perturbative aspects of moments of the thrust distribution in e+e-
Gardi, Einan
2000-04-01
Resummation and power-corrections play a crucial role in the phenomenology of event-shape variables like the thrust T. Previous investigations showed that the perturbative contribution to the average thrust is dominated by gluons of small invariant mass, of the order of 10% of Q, where Q is the center-of-mass energy. The effect of soft gluons is also important, leading to a non-perturbative 1/Q correction. These conclusions are based on renormalon analysis in the single dressed gluon (SDG) approximation. Here we analyze higher moments of the thrust distribution using a similar technique. We find that the characteristic gluon invariant mass contributing to langle(1-T)mrangle increases with m. Yet, for m = 2 this scale is quite low, around 27% of Q, and therefore renormalon resummation is still very important. On the other hand, the power-correction to langle(1-T)2rangle from a single soft gluon emission is found to be highly suppressed: it scales as 1/Q3. In practice, langle(1-T)2rangle and higher moments depend also on soft gluon emission from configurations of three hard partons, which may lead to αs(Q2)/Q power-corrections. This issue is yet to be investigated.
10. Storm Track Response to Perturbations in Climate
Mbengue, Cheikh Oumar
This thesis advances our understanding of midlatitude storm tracks and how they respond to perturbations in the climate system. The midlatitude storm tracks are regions of maximal turbulent kinetic energy in the atmosphere. Through them, the bulk of the atmospheric transport of energy, water vapor, and angular momentum occurs in midlatitudes. Therefore, they are important regulators of climate, controlling basic features such as the distribution of surface temperatures, precipitation, and winds in midlatitudes. Storm tracks are robustly projected to shift poleward in global-warming simulations with current climate models. Yet the reasons for this shift have remained unclear. Here we show that this shift occurs even in extremely idealized (but still three-dimensional) simulations of dry atmospheres. We use these simulations to develop an understanding of the processes responsible for the shift and develop a conceptual model that accounts for it. We demonstrate that changes in the convective static stability in the deep tropics alone can drive remote shifts in the midlatitude storm tracks. Through simulations with a dry idealized general circulation model (GCM), midlatitude storm tracks are shown to be located where the mean available potential energy (MAPE, a measure of the potential energy available to be converted into kinetic energy) is maximal. As the climate varies, even if only driven by tropical static stability changes, the MAPE maximum shifts primarily because of shifts of the maximum of near-surface meridional temperature gradients. The temperature gradients shift in response to changes in the width of the tropical Hadley circulation, whose width is affected by the tropical static stability. Storm tracks generally shift in tandem with shifts of the subtropical terminus of the Hadley circulation. We develop a one-dimensional diffusive energy-balance model that links changes in the Hadley circulation to midlatitude temperature gradients and so to the storm
11. Extreme Value Analysis of Tidal Stream Velocity Perturbations
SciTech Connect
Harding, Samuel; Thomson, Jim; Polagye, Brian; Richmond, Marshall C.; Durgesh, Vibhav; Bryden, Ian
2011-04-26
This paper presents a statistical extreme value analysis of maximum velocity perturbations from the mean flow speed in a tidal stream. This study was performed using tidal velocity data measured using both an Acoustic Doppler Velocimeter (ADV) and an Acoustic Doppler Current Profiler (ADCP) at the same location which allows for direct comparison of predictions. The extreme value analysis implements of a Peak-Over-Threshold method to explore the effect of perturbation length and time scale on the magnitude of a 50-year perturbation.
12. Perturbative analysis of multiple-field cosmological inflation
SciTech Connect
Lahiri, Joydev . E-mail: [email protected]
2006-04-15
We develop a general formalism for analyzing linear perturbations in multiple-field cosmological inflation based on the gauge-ready approach. Our inflationary model consists of an arbitrary number of scalar fields with non-minimal kinetic terms. We solve the equations for scalar- and tensor-type perturbations during inflation to the first order in slow roll, and then obtain the super-horizon solutions for adiabatic and isocurvature perturbations after inflation. Analytic expressions for power-spectra and spectral indices arising from multiple-field inflation are presented.
13. Disentangling perturbative and power corrections in precision tau decay analysis
SciTech Connect
Gorbunov, D.S.; Pivovarov, A.A.
2005-01-01
Hadronic tau decay precision data are analyzed with account of both perturbative and power corrections of high orders within QCD. It is found that contributions of high order power corrections are essential for extracting a numerical value for the strange quark mass from the data on Cabibbo suppressed tau decays. We show that with inclusion of new five-loop perturbative corrections in the analysis the convergence of perturbation theory remains acceptable only for few low order moments. We obtain m{sub s}(M{sub {tau}})=130{+-}27 MeV in agreement with previous estimates.
14. Stability of coflowing capillary jets under nonaxisymmetric perturbations.
PubMed
Montanero, J M; Gañán-Calvo, A M
2008-04-01
In this paper, linear hydrodynamic stability analysis is used to study the response of a capillary jet and a coflowing fluid to both axisymmetric and nonaxisymmetric perturbations. The temporal analysis revealed that nonaxisymmetric perturbations were damped (or overdamped) within the region of parameter space explored, which involved equal velocities for the jet and focusing fluid. It is explained how an extension to a spatiotemporal analysis implies that those perturbations can yield no transition from convective (jetting) to absolute (whipping) instability for that parameter region. This result provides a theoretical explanation for the absence of that kind of transition in most experimental results in the literature. PMID:18517726
15. Charged scalar perturbations around a regular magnetic black hole
Huang, Yang; Liu, Dao-Jun
2016-05-01
We study charged scalar perturbations in the background of a regular magnetic black hole. In this case, the charged scalar perturbation does not result in superradiance. By using a careful time-domain analysis, we show that the charge of the scalar field can change the real part of the quasinormal frequency, but has little impact on the imaginary part of the quasinormal frequency and the behavior of the late-time tail. Therefore, the regular magnetic black hole may be stable under the perturbations of a charged scalar field at the linear level.
16. Secular perturbations of the Uranian satellites - Theory and practice
NASA Technical Reports Server (NTRS)
Malhotra, R.; Nicholson, P. D.; Fox, K.; Murray, C. D.
1989-01-01
A simple revised secular perturbation theory which incorporates the averaged secular effect of first-order near-resonances is derived. By including the effects of these near-resonances, the largest error in the secular frequencies is reduced from 16 percent to less than 3 percent. It is concluded that the revised secular perturbation theory is adequate for the quantitative modeling of the long-term perturbations in the Uranian satellite system. If incorporated within the general theory of Laskar (1986), this theory would lead to completely analytic theory.
17. Current Density and Plasma Displacement Near Perturbed Rational Surface
SciTech Connect
A.H. Boozer and N. Pomphrey
2010-10-10
The current density in the vicinity of a rational surface of a force-free magnetic field subjected to an ideal perturbation is shown to be the sum of both a smooth and a delta-function distribution, which give comparable currents. The maximum perturbation to the smooth current density is comparable to a typical equilibrium current density and the width of the layer in which the current flows is shown to be proportional to the perturbation amplitude. In the standard linearized theory, the plasma displacement has an unphysical jump across the rational surface, but the full theory gives a continuous displacement.
18. Axion inflation with cross-correlated axion isocurvature perturbations
Kadota, Kenji; Kobayashi, Tatsuo; Otsuka, Hajime
2016-01-01
We study the inflation scenarios, in the framework of superstring theory, where the inflaton is an axion producing the adiabatic curvature perturbations while there exists another light axion producing the isocurvature perturbations. We discuss how the non-trivial couplings among string axions can generically arise, and calculate the consequent cross-correlations between the adiabatic and isocurvature modes through concrete examples. Based on the Planck analysis on the generally correlated isocurvature perturbations, we show that there is a preference for the existence of the correlated isocurvature modes for the axion monodromy inflation while the natural inflation disfavors such isocurvature modes.
19. Stochastic systems with delay: Perturbation theory for second order statistics
Frank, T. D.
2016-03-01
Within the framework of delay Fokker-Planck equations, a perturbation theoretical method is developed to determine second-order statistical quantities such as autocorrelation functions for stochastic systems with delay. Two variants of the perturbation theoretical approach are presented. The first variant is based on a non-local Fokker-Planck operator. The second variant requires to solve a Fokker-Planck equation with source term. It is shown that the two variants yield consistent results. The perturbation theoretical approaches are applied to study negative autocorrelations that are induced by feedback delays and mediated by the strength of the fluctuating forces that act on the feedback systems.
20. A general analysis of non-gaussianity from isocurvature perturbations
SciTech Connect
Kawasaki, Masahiro; Nakayama, Kazunori; Sekiguchi, Toyokazu; Suyama, Teruaki; Takahashi, Fuminobu E-mail: [email protected] E-mail: [email protected]
2009-01-15
Light scalars may be ubiquitous in nature, and their quantum fluctuations can produce large non-Gaussianity in the cosmic microwave background temperature anisotropy. The non-Gaussianity may be accompanied with a small admixture of isocurvature perturbations, which often have correlations with the curvature perturbations. We present a general method to calculate the non-Gaussianity in the adiabatic and isocurvature perturbations with and without correlations, and see how it works in several explicit examples. We also show that they leave distinct signatures on the bispectrum of the cosmic microwave background temperature fluctuations. | {"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": 0.8642237186431885, "perplexity": 1420.2583956419855}, "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/1487501171171.24/warc/CC-MAIN-20170219104611-00404-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.clutchprep.com/organic-chemistry/practice-problems/14078/deuterium-d-is-an-isotope-of-hydrogen-in-which-the-nucleus-160-has-one-proton-an | 🤓 Based on our data, we think this question is relevant for Professor Graham's class at SJU.
# Solution: Deuterium (D) is an isotope of hydrogen, in which the nucleus has one proton and one neutron. This nucleus, called a deuteron, behaves very much like a proton, although there are observed differences in the rates of reactions involving either protons or deuterons (an effect called the kinetic isotope effect). Deuterium can be introduced into a compound via the process below:(a) The C—Mg bond in compound 3 can be drawn as ionic. Redraw 3 as an ionic species, with BrMg + as a counterion, and then draw the mechanism for the conversion of 3 to 4.
###### Problem
Deuterium (D) is an isotope of hydrogen, in which the nucleus has one proton and one neutron. This nucleus, called a deuteron, behaves very much like a proton, although there are observed differences in the rates of reactions involving either protons or deuterons (an effect called the kinetic isotope effect). Deuterium can be introduced into a compound via the process below:
(a) The C—Mg bond in compound 3 can be drawn as ionic. Redraw 3 as an ionic species, with BrMg + as a counterion, and then draw the mechanism for the conversion of 3 to 4. | {"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": 0.904444694519043, "perplexity": 2491.448193189367}, "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-2020-16/segments/1585371813538.73/warc/CC-MAIN-20200408104113-20200408134613-00554.warc.gz"} |
https://www.physicsforums.com/threads/mathematical-physics-vs-theoritical-physics.146881/ | # Mathematical Physics vs Theoritical Physics
1. Dec 5, 2006
### Werg22
I don't really understand the difference between the two. Can someone please highlight the differences and resemblances? Thank you in advance.
2. Dec 5, 2006
### pivoxa15
MP emphasises the maths more and may try to rigorously establish the valididty of physical theories on mathematical grounds.
TP does not emphasis the rigorous maths and mostly tries to reconcile theory with experiment. Either finding theory to explain an observation or predict physical phenomenas that can be tested by experiment.
3. Dec 5, 2006 | {"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": 0.981730580329895, "perplexity": 2554.283777648106}, "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/1476988720000.45/warc/CC-MAIN-20161020183840-00523-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathhelpforum.com/differential-geometry/177561-partial-derivation-issue.html | # Thread: Partial derivation issue
1. ## Partial derivation issue
I got this problem with a simple partial derivation that is driving me crazy, since the solution I obtain is different from the one published in the article, in which I believe.
Basically I got this equation:
$P=\frac{RT}{v-b}-\frac{a(T)}{v(v+b)+b(v-b)}$
Where:
P= pressure
R=Gas constant
T= temperature
V= molar volume
And
a(T) = a(Tc)*( 1+ k( 1-Tr^(1/2) ) )^2
a(Tc) = 0.45724 * R^2 * Tc^2 / Pc
Tc & Pc are the critical temperature & pressure respectively
k = 0.37464 + 1.54226*w - 0.26992*w^2
b=0.0778*R * Tc / Pc
Tr = T / Tc
w is the acentric factor.
Now I need to make this partial derivation:
pi = internal pressure = $T*/frac{dP}{dT}v -P$
Guy sorry first time I'm using latex, with this $frac{dP} {dT}v$ (if some one can show me the code please) I mean the partial derivation of P respect to T at constant volume.
The solution on the article is:
$pi=\frac{a(T)}{v(v+b)+b(v-b)}(1+kTr^(^1^/^2^))$
Could you please help me to understand how to arrive there.
Thanks
2. Might be worth posting this one again, seems a little tricky to follow
under the "math" tags a fraction has the code \frac{RT}{v-b} giving, $\frac{RT}{v-b}$
Click on the code to see how!
3. Thanks pickslides,
but i do really do not succede to write the partial derivetion correctely. Do you have any tips for this. | {"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": 0.8845797181129456, "perplexity": 2581.576926529204}, "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-2016-36/segments/1471983577646.93/warc/CC-MAIN-20160823201937-00038-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://philpapers.org/s/Liang%20Yu | ## Works by Liang Yu
16 found
Sort by:
Disambiguations:
Liang Yu [18] Liangzao Yu [2]
Profile: Liangyu Yu
1. Liang Yu (2012). Characterizing Strong Randomness Via Martin-Löf Randomness. Annals of Pure and Applied Logic 163 (3):214-224.
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2. Yun Fan & Liang Yu (2011). Maximal Pairs of Ce Reals in the Computably Lipschitz Degrees. Annals of Pure and Applied Logic 162 (5):357-366.
Computably Lipschitz reducibility , was suggested as a measure of relative randomness. We say α≤clβ if α is Turing reducible to β with oracle use on x bounded by x+c. In this paper, we prove that for any non-computable real, there exists a c.e. real so that no c.e. real can cl-compute both of them. So every non-computable c.e. real is the half of a cl-maximal pair of c.e. reals.
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3. Liang Yu (2011). A New Proof of Friedman's Conjecture. Bulletin of Symbolic Logic 17 (3):455-461.
We give a new proof of Friedman's conjecture that every uncountable Δ11 set of reals has a member of each hyperdegree greater than or equal to the hyperjump.
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4. Bjørn Kjos-Hanssen, André Nies, Frank Stephan & Liang Yu (2010). Higher Kurtz Randomness. Annals of Pure and Applied Logic 161 (10):1280-1290.
A real x is -Kurtz random if it is in no closed null set . We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.
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5. Klaus Ambos-Spies, Decheng Ding, Wei Wang & Liang Yu (2009). Bounding Non- GL ₂ and R.E.A. Journal of Symbolic Logic 74 (3):989-1000.
We prove that every Turing degree a bounding some non-GL₂ degree is recursively enumerable in and above (r.e.a.) some 1-generic degree.
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6. Liang Yu (2008). Zheng Ju Xiang Guan Xing Yan Jiu =. Beijing da Xue Chu Ban She.
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7. C. T. Chong & Liang Yu (2007). Maximal Chains in the Turing Degrees. Journal of Symbolic Logic 72 (4):1219 - 1227.
We study the problem of existence of maximal chains in the Turing degrees. We show that: 1. ZF+DC+"There exists no maximal chain in the Turing degrees" is equiconsistent with ZFC+"There exists an inaccessible cardinal"; 2. For all a ∈ 2ω.(ω₁)L[a] = ω₁ if and only if there exists a \$\Pi _{1}^{1}[a]\$ maximal chain in the Turing degrees. As a corollary, ZFC + "There exists an inaccessible cardinal" is equiconsistent with ZFC + "There is no (bold face) \$\utilde{\Pi}{}_{1}^{1}\$ maximal chain of (...)
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8. Rod Downey, Andre Nies, Rebecca Weber & Liang Yu (2006). Lowness and Π₂⁰ Nullsets. Journal of Symbolic Logic 71 (3):1044-1052.
We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.
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9. Rod Downey & Liang Yu (2006). Arithmetical Sacks Forcing. Archive for Mathematical Logic 45 (6):715-720.
We answer a question of Jockusch by constructing a hyperimmune-free minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented.
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10. Yue Yang & Liang Yu (2006). On Σ₁-Structural Differences Among Finite Levels of the Ershov Hierarchy. Journal of Symbolic Logic 71 (4):1223 - 1236.
We show that the structure R of recursively enumerable degrees is not a Σ₁-elementary substructure of Dn, where Dn (n > 1) is the structure of n-r.e. degrees in the Ershov hierarchy.
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11. Liang Yu (2006). Measure Theory Aspects of Locally Countable Orderings. Journal of Symbolic Logic 71 (3):958 - 968.
We prove that for any locally countable \$\Sigma _{1}^{1}\$ partial order P = 〈2ω,≤P〉, there exists a nonmeasurable antichain in P. Some applications of the result are also presented.
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12. Liang Yu (2006). Lowness for Genericity. Archive for Mathematical Logic 45 (2):233-238.
We study lowness for genericity. We show that there exists no Turing degree which is low for 1-genericity and all of computably traceable degrees are low for weak 1-genericity.
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13. Liang Yu & Yue Yang (2005). On the Definable Ideal Generated by Nonbounding C.E. Degrees. Journal of Symbolic Logic 70 (1):252 - 270.
Let [NB]₁ denote the ideal generated by nonbounding c.e. degrees and NCup the ideal of noncuppable c.e. degrees. We show that both [NB]₁ ∪ NCup and the ideal generated by nonbounding and noncuppable degrees are new, in the sense that they are different from M, [NB]₁ and NCup—the only three known definable ideals so far.
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14. Liang Yu & Decheng Ding (2004). There Is No SW-Complete C.E. Real. Journal of Symbolic Logic 69 (4):1163 - 1170.
We prove that there is no sw-complete c.e. real, negatively answering a question in [6].
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15. Liang Yu, Decheng Ding & Rodney Downey (2004). The Kolmogorov Complexity of Random Reals. Annals of Pure and Applied Logic 129 (1-3):163-180.
We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K (...) | {"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": 0.9592664241790771, "perplexity": 2999.582443279521}, "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-2015-32/segments/1438042987775.70/warc/CC-MAIN-20150728002307-00157-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://thirdspacelearning.com/gcse-maths/geometry-and-measure/angle-rules/ | GCSE Maths Geometry and Measure
Angles
Angle Rules
# Angle Rules
Here we will learn about angle rules including how to solve problems involving angles on a straight line, angles around a point, vertically opposite angles, complementary angles and supplementary angles.
There are also angle rules worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
## What are angle rules?
Angle rules enable us to calculate unknown angles:
• Angles on a straight line equal 180º
Angles on a straight line always add up to 180º.
E.g.
Here the two angles are labelled 30º and 150º . When added together they equal 180º and therefore lie on a straight lie. These angles share a vertex.
However, below we can see an example of where two angles do not equal 180º:
This is because they do not share a vertex and therefore do not lie on the same line segment.
Note: you can try out the above rule by drawing out the above diagrams and measuring the angles using a protractor.
Step-by-step guide: Angles on a straight line
• Angles around a point equal 360o
Angles around a point will always equal 360º.
E.g.
The three angles above share a vertex and, when added together equal 360o.
Step-by-step guide: Angles around a point
• Supplementary angles
Two angles are supplementary when they add up to 180º , they do not have to be next to each other.
E.g.
These two angles are supplementary because when added together they equal 180º.
Step-by-step guide: Supplementary angles
• Complementary angles
Two angles are complementary when they add up to 90º , they do not have to be next to each other.
E.g.
These two angles are supplementary because when added together they equal 90º.
Step-by-step guide: Complementary angles
• Vertically opposite angles
Vertically opposite angles refer to angles that are opposite one another at a specific vertex and are created by two lines crossing.
E.g.
Here the two angles labelled ‘a’ are equal to one another because they are ‘vertically opposite’ at the same vertex.
The same applies to angles labelled as ‘b’.
Step-by-step guide: Vertically opposite angles
When solving problems involving angles sometimes we use more than the above rules. Below you will see a range of problems involving angles with links to lessons that will go into more detail with more complex questions.
### Keywords
It is important we are familiar with some key words, terminology and symbols for this topic:
• Angle: defined as the amount of turn round a common vertex.
• Vertex: the point created by two line segments (plural is vertices).
We normally label angles in two main ways:
1By giving the angle a ‘name’ which is normally a lower case letter such as a, x or y or the greek letter θ (theta). See below for an example:
2By referring to the angle as the three letters that define the angle. The middle letter refers to the vertex at which the angle is e.g. see below for the angle we call ABC:
## How to use angle rules
In order to solve problems involving angles you should follow these steps:
1. Identify which angle you need to find.
2. Identity which angle rule/s apply to the context and write them down.
3. Solve the problem using the above angle rule/s. Give reasons where applicable.
4. Clearly state the answer using angle terminology.
## Angle rules examples
### Example 1: angles on a straight line
Find angles a and b.
1. Identify which angle you need to find (label it if you need to).
You need to find angles labelled a and b.
2Identify which angle rule/s apply to the context and write them down (remember multiple rules may be needed).
Angles on a straight line at the same vertex always add up to 180o.
Notice how angles a and b do not share a vertex.
3Solve the problem using the above angle rule/s. Give reasons where applicable.
\begin{aligned} a+110&=180 \\ a=&70 \end{aligned}
\begin{aligned} b+55&=180 \\ b&=125 \end{aligned}
4Clearly state the answer using angle terminology.
Angle a = 70°
Angle b = 125°
### Example 2: angles around a point
Find the size of θ:
You need to find the angle labelled θ (theta).
All the angles are around a single vertex and we know that angles around a point equal 360o.
\begin{aligned} \theta+120+95+30&=360 \\ \theta+245&=360 \\ \theta&=115 \end{aligned}
$\text{Angle } \theta = 115^{\circ}$
### Example 3: using supplementary angles
Two angles are supplementary and one of them is 127°. What is the size of the other angle?
You need to find the other angle in a pair of supplementary angles where one is 127°. We will call this angle ‘a’.
Supplementary angles add up to 180°.
\begin{aligned} 127 +a &= 180\\ a&=53\\ \end{aligned}
The other angle is 53°.
### Example 4: using complementary angles
If two angles are complementary and one of them is 34°, what is the size of the other angle?
You need to find the other angle in a pair of supplementary angles where one is 34°. We will call this angle ‘b’.
Complementary angles add up to 90°.
\begin{aligned} 34 + b &=90\\ b &= 56\\ \end{aligned}
The other angle is 56°.
### Example 5: vertically opposite angles
Find angle BCD.
Find the angle at vertex C made from lines CD and CB. The diagram shows it labelled as x
Angle x and 80are vertically opposite one another at a vertex which has been created by two lines crossing.
Vertically opposite angles are equal to one another
x = 80 because they are vertically opposite
Angle BCD = 80°
### Example 6: Applying multiple rules to solve a problem
In the diagram below:
• Angle AOB is a right angle.
• AOE and EOD are complementary angles.
• Angle AOE is 50 degrees.
Find angle COD.
Let’s start by labelling the diagram. We have labelled the angle we are trying to find x.
Two angles are complementary when they add up to 90o.
Angles around a point will always equal 360o.
Angles on one part of a straight line always add up to 180o.
Vertically opposite angles are equal.
Note: there are multiple ways we can solve this problem. Below is just one method.
As AOE and EOD are complementary angles then they must equal 90 degrees. Therefore:
\begin{aligned} 50+E O D&=90 \\ E O D&=40 \end{aligned}
EOD and BOC are vertically opposite. Therefore:
\begin{aligned} B O C&=E O D \\ B O C&=40 \end{aligned}
Angles around a point are equal to 360 degrees. Therefore:
\begin{aligned} A O B+A O E+E O D+B O C+x=360 \\ 90+50+40+40+x=360 \\ 220+x=360 \\ x=140 \end{aligned}
Angle COD = 140°
### Common misconceptions
• Incorrectly labelling angles
• Misuse of the ‘straight line’ rule where angles do not share a vertex
• Mixing up the rules for supplementary and complementary angles
• Finding the incorrect angle due to misunderstanding the terminology
### Practice angle rules questions
1. Find angle x
x=30^{\circ}
x=60^{\circ}
x=90^{\circ}
x=180^{\circ}
Using angle on a straight line we have 180 – (90 + 30) = 60^{\circ}
2. Can angles 40^{\circ}, 100^{\circ}, 115^{\circ}, 105^{\circ} lie around a single point?
Yes
No
The sum of these angles is 360 .
3. Find angle a:
a=30^{\circ}
a=60^{\circ}
a=90^{\circ}
a=50^{\circ}
Using angles around a point, we have 360 – (125 + 125 + 50) = 360 – 300 = 60^{\circ}
4. Are angles 60^{\circ} \text{ and } 90^{\circ} supplementary angles?
Yes
No
The sum of these angles is not 180 .
5. Are angles 75^{\circ} \text{ and } 15^{\circ} complementary angles?
Yes
No
The sum of these angles is 90 .
6. Find angle z
z=115^{\circ}
z=60^{\circ}
z=65^{\circ}
z=55^{\circ}
Vertically opposite angles are equal
### Angle rules GCSE questions
1. Work out the size of angle z .
(2 marks)
360-169-83
(1)
108^{\circ}
(1)
2.
(a) Find the size of angle a .
(b) Find the size of angle b .
(3 marks)
a)
49^{\circ}
(1)
b)
180-49
(1)
131^{\circ}
(1)
3. Work out the size of angle x . Give reasons for your answer.
(3 marks)
180-90-57=33
(1)
Angles on a straight line add up to 180^{\circ}.
(1)
x=33^{\circ}
(1)
## Learning checklist
You have now learned how to:
• Use conventional terms and notation for angles
• Apply the properties of angles on a straight line, around a point and on vertically opposite angles
• Apply angle facts and properties (e.g. supplementary and complementary angles) to solve problems
## Still stuck?
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.
Find out more about our GCSE maths revision programme.
x
#### GCSE Maths Scheme of Work Guide
An essential guide for all SLT and subject leaders looking to plan and build a new scheme of work, including how to decide what to teach and for how long
Explores tried and tested approaches and includes examples and templates you can use immediately in your school. | {"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": 0.8510176539421082, "perplexity": 2085.5242442262543}, "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/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00704.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/knowing-sign-work-done-object-crucial-element-understanding-work-positive-work-indicates-o-q1614227 | Knowing the sign of the work done on an object is a crucial element to understanding work. Positive work indicates that an object has been acted on by a force that tranfers energy to the object, thereby increasing the object's energy. Negative work indicates that an object has been acted on by a force that has reduced the energy of the object.
The next few questions will ask you to determine the sign of the work done by the various forces acting on a box that is being pushed across a rough floor. As illustrated in the figure (Intro 1 figure) , the box is being acted on by a normal force , the force of gravity (i.e., the box's weight ), the force of kinetic friction , and the pushing force . The displacement of the box is .
Part A
Which of the following statements accurately describes the sign of the work done on the box by the force of the push?
positive
negative
zero
Part B
Which of the following statements accurately decribes the sign of the work done on the box by the normal force?
positive
negative
zero
Part C
Which of the following statements accurately decribes the sign of the work done on the box by the force of kinetic friction?
positive
negative
zero
Part D
Which of the following statements accurately decribes the sign of the work done on the box by the force of gravity (i.e., the weight)?
positive
negative
zero
Part E
You have just moved into a new apartment and are trying to arrange your bedroom. You would like to move your dresser of weight 3,500 across the carpet to a spot 5 away on the opposite wall. Hoping to just slide your dresser easily across the floor, you do not empty your clothes out of the drawers before trying to move it. You push with all your might but cannot move the dresser before becoming completely exhausted. How much work do you do on the dresser?
Part F
A box of weight is sliding down a frictionless plane that is inclined at an angle above the horizontal, as shown in the figure (Part F figure) . What is the work done on the box by the force of gravity if the box moves a distance ?
None of these
Part G
The planet Earth travels in a circular orbit at constant speed around the Sun. What is the net work done on the Earth by the gravitational attraction between it and the Sun in one complete orbit? Assume that the mass of the Earth is given by , the mass of the Sun is given by , and the Earth-Sun distance is given by .
None of these.
Part H
A block of mass is pushed up against a spring with spring constant until the spring has been compressed a distance from equilibrium. What is the work done on the block by the spring?
None of these.
ContinueSee Score and Provide Feedback | {"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": 0.8003066182136536, "perplexity": 281.9699715862269}, "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-2014-35/segments/1408500832032.48/warc/CC-MAIN-20140820021352-00177-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.aanda.org/articles/aa/full/2008/13/aa9058-07/aa9058-07.right.html | A&A 481, 229-233 (2008)
DOI: 10.1051/0004-6361:20079058
## Chromospheric features of LQ Hydrae from H line profiles
A. Frasca1 - Zs. Kovári2 - K. G. Strassmeier3 - K. Biazzo1
1 - INAF - Catania Astrophysical Observatory, via S. Sofia 78, 95123 Catania, Italy
2 - Konkoly Observatory, 1525 Budapest, PO Box 67, Hungary
3 - Astrophysical Institute Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 13 November 2007 / Accepted 3 January 2008
Abstract
We analyze the H spectral variability of the rapidly-rotating K1-dwarf LQ Hya using high-resolution H spectra recorded during April-May 2000. Chromospheric parameters were computed from the H profile as a function of rotational phase. We find that all these parameters vary in phase, with a higher chromospheric electron density coinciding with the maximum H emission. We find a clear rotational modulation of the H emission that is better emphasized by subtracting a reference photospheric template built up with a spectrum of a non-active star of the same spectral type. A geometrical plage model applied to the H variation curve allows us to derive the location of the active regions that come out to be close in longitude to the most pronounced photospheric spots found with Doppler imaging applied to the photospheric lines in the same spectra. Our analysis suggests that the H features observed in LQ Hya in 2000 are a scaled-up version of the solar plages as regards dimensions and/or flux contrast. No clear indication of chromospheric mass motions emerges.
Key words: stars: activity - stars: late-type - stars: chromospheres - stars: starspots - stars: individual: LQ Hydrae
### 1 Introduction
According to the solar-stellar analogy, chromospheric stellar activity can be established by the presence of emission in the core of the Balmer H line. As in the Sun, H emission intensification has often been observed in surface features (plages) spatially connected with the photospheric starspots (see, e.g., Biazzo et al. 2007; Catalano et al. 2000; Biazzo et al. 2006, and references therein). Thus, the time variability of the H spectral features can be used to estimate the basic properties of the emitting sources, allowing the geometry of the stellar chromosphere to be mapped.
In this paper we analyze the H spectral variability of the young, rapidly-rotating ( d) single K2-dwarf LQ Hydrae (HD 82558 = Gl 355) using 15 high-resolution spectra recorded during April-May 2000. All H spectra were acquired simultaneously with the mapping lines used for the year-2000 Doppler imaging study presented in our previous paper (Kovári et al. 2004, hereafter Paper I). LQ Hya was first recognized as a chromospherically active star through Ca II H&K emission (Bidelman 1981; Heintz 1981). The H absorption filled in by chromospheric emission was reported first by Fekel et al. (1986), and LQ Hya was classified as a BY Dra-type spotted star. Variable H emission-peak asymmetry was investigated by Strassmeier et al. (1993, hereafter Paper II), who attributed it to the presence of chromospheric velocity fields in the H forming layer probably surrounding photospheric spots.
In our spectroscopic study in Paper I, we presented Doppler images using the Fe I-6411, Fe I-6430, and Ca I-6439 lines for both the late April data and the early May 2000 dataset. Doppler imaging was supported by simultaneous photometric measurements in Johnson-Cousins VI bands. Doppler images showed spot activity uniformly at latitudes between -20and +50, sometimes with high-latitude appendages, but without a polar spot. Comparing the respective maps from two weeks apart, rapid spot evolution was detected, which was attributed to strong cross-talks between the neighboring surface features through magnetic reconnections. In Table 1 we give a summary of the stellar parameters as they emerged from Paper I.
Table 1: Astrophysical data for LQ Hya (adopted from Kovári et al. 2004).
The observations are again briefly presented in Sect. 2 and the method used for the H spectral study is described in Sect. 3. The results are presented in Sect. 4. Since the H observations of this paper were included in the spectroscopic data used in Paper I to reconstruct the year-2000 Doppler images, in Sect. 5 we take the opportunity to compare the photospheric features with the contemporaneous H-emitting regions.
### 2 Spectroscopic observations
The series of 15 H spectra was collected, one spectrum per night, during two observing runs (April 4-9, 2000, and April 25-May 3, 2000) at the Kitt Peak National Observatory (KPNO) with the 0.9 m coudé-feed telescope. The 3096 1024 F3KB CCD detector was employed, together with grating A, camera 5, and the long collimator. The spectra were centered at 6500 Å with a wavelength range of 300 Å. The effective resolution was 28 000 (11 km s-1). We achieved signal-to-noise ratios (S/N) of about 250 in 45-min integration time, with the only exception being the spectrum at 2.726 phase for which an S/N of just 50 could be reached due to bad sky conditions.
The mean HJDs of the observations, the phases, as well as the radial velocities, are summarized in Table 2. For phasing the spectra, was used, where the arbitrarily chosen zero point is the same as used for our former Doppler imaging studies (Kovári et al. 2004; Strassmeier et al. 1993). As radial velocity (RV) standard star, Gem ( km s-1) was measured, except for HJD 2 451 643.744 when 16 Vir ( km s-1) was observed (Scarfe et al. 1990). For more details we refer to Paper I.
Table 2: Observing log and radial velocities.
### 3 Data analysis
The Balmer H line is a useful and easily accessible indicator of chromospheric activity in the optical spectrum. It has been proven to be very effective for detecting chromospheric plages both in the Sun and in active stars, due to its high contrast against the surrounding chromosphere (see, e.g., Biazzo et al. 2007; Frasca et al. 1998; Biazzo et al. 2006).
However, the H line is formed in a wide depth range in the stellar atmosphere, ranging from the temperature minimum to the upper chromosphere. To extract the chromospheric contribution from the line core, we apply a method usually called spectral synthesis'', which consists in the subtraction of synthetic spectra or of observed spectra of non-active standard stars (reference spectra) (see, e.g., Barden 1985; Montes et al. 1995; Herbig 1985; Frasca & Catalano 1994). The difference between observed and reference spectra provides the net H emission, which can be integrated to estimate the total radiative chromospheric loss in the line.
We point out that one can obtain the true chromospheric H emission with this method only if the chromosphere is also optically thin inside the active regions or if the plages are not extended enough to appreciably affect the photospheric spectrum. In the opposite case, the subtraction of the underlying photospheric profile would overestimate the H chromospheric emission.
Figure 1: Time series of H profiles for LQ Hya in April-May 2000 sorted in phase order. The non-active template built up with a spectrum of the non-active, slowly-rotating K2 V star HD 3765 is shown by a dash-dotted line at the bottom. The vertical dotted lines mark the integration interval for the raw H equivalent width ( ). Open with DEXTER
Table 3: Equivalent widths of the raw ( ) and the residual spectra ( ), errors of the equivalent widths ( ), and velocity shifts and for the raw and the residual spectra, respectively.
For this reason we have also measured the H equivalent width in the observed spectra ( ), fixing a constant integration window of 5.0 Å, i.e. integrating the emission core inside the absorption wings (see Fig. 1). The net H equivalent width, , was instead measured in the residual spectra obtained from the original ones with the spectral synthesis'' method. The non-active template was built using a high-resolution (R=42 000) spectrum of the K2 V star HD 3765 retrieved from the ELODIE archive. We chose HD 3765 because its spectral type and color index B-V=0.942(Weis 1996) are nearly identical with those of LQ Hya, because its rotation velocity is very low ( km s-1, Strassmeier et al. 2000), and because it has a low level of magnetic activity compared to LQ Hya. The spectrum was first degraded to the resolution of the KPNO spectra by convolving it with a Gaussian kernel and subsequently broadened by convolution with a rotational profile corresponding to the of 28 km s-1 of LQ Hya.
The error of the equivalent width, , was evaluated by multiplying the integration range by the photometric error on each point. This was estimated by the standard deviation of the observed flux values on the difference spectra in two spectral regions near the H line. The values of , , and are reported in Table 3.
We also measured the central wavelength of the H emission both in the raw (observed) and residual spectra by evaluating the centroid of the core emission. This quantity provided us with the radial velocity shifts between the H emission and the photosphere measured both in the raw ( ) and in the residual spectra ( ). The values of and are reported in Table 3 as well.
### 4 Results
In the LQ Hya spectra, the H line is a strongly variable feature displaying absorption wings and a core changing from a completely filled-in configuration to a moderate emission above the continuum (see Fig. 1). A faint single peak just filling in the line core, without reaching the continuum, is observed at the phases of minimum H emission (-0.4), whereas double-peaked emission profiles are observed with a central reversal at the other phases. The blue'' emission peak is often stronger than the red'' one, although at some phases, nearly equal peak heights and even reversed intensity ratios are also observed. The H profiles are shown in Fig. 1 along with the photospheric template built up with an HD 3765 spectrum (see Sect. 3).
Figure 2: From top to bottom. The computed electron density (cm-3), the flux ratio between the red and the blue peaks, and the peak separation in Å against the rotational phase. Open with DEXTER
We estimated the chromospheric electron density in LQ Hya adopting the assumptions from Paper II, i.e. an isothermal chromosphere with K for which the optical depth in the H center can be derived by the formulation of Cram & Mullan (1979):
(1)
where is the wavelength separation of the blue and red emission peaks, and Å is the chromospheric Doppler width. We measured on the observed spectra by fitting two Lorentzian functions to all the double-peaked H profiles. Obviously, this analysis could not be applied to the single-peaked spectra close to the minimum emission at phase 0.2-0.5. Combining Eqs. (8) and (14) in Cram & Mullan (1979), the electron density in cm-3 can be evaluated as
(2)
where is the ratio of the H peak to the continuum flux, as measured on our spectra, and B is the Planck function calculated for the effective temperature of LQ Hya ( K) and at K. The optical depth at the H center changes from about 30 to 4 and the electron density from to cm-3 from phase 0.2 (near the minimum H emission) to the phases of maximum emission (-0.8, see Fig. 2). In the same figure, the flux ratio between the red and the blue peaks and the peak separation are plotted versus the rotational phase. All these quantities appear modulated by the stellar rotation. Symmetric profiles ( ) tend to be observed when the most active regions, judging from the photospheric spot contrast, are in the visible hemisphere, similar to what was found by Strassmeier et al. in Paper II.
The H equivalent width measured in the observed ( ) and in the residual spectra ( ), the radial velocity shifts of the H emission ( and ), and the contemporaneous light curve in the V band (from Paper I) are plotted in Fig. 3 as functions of the rotational phase. A remarkable anti-correlation of V brightness and H emission is apparent.
The highest electron density is observed when the H line is strongest, indicating a denser chromosphere above more extended active regions, the opposite of what was observed in 1991 (Paper II). We also found an average value for the chromospheric electron density that was higher than the one reported in Paper II ( cm-3). Moreover, the rotational modulation of all the quantities deduced by the H line analysis is much more evident in the present data. Presumably, in April-May 2000, LQ Hya was more active than in 1991, and it displayed more and larger active regions.
Figure 3: Top panel: V light curve contemporaneous to the spectra. Middle panels: rotational modulation of the raw H equivalent width ( , open circles) and of the residual H EW ( , filled circles). Bottom panel: velocity shift between H emission and photospheric lines both for observed (open circles) and residual spectra (filled circles). The continuous lines in the middle and bottom panels represent the best fit of our 3-plage model (cf. Fig. 4), while the dotted lines represent the result of our 2-plage model. Open with DEXTER
### 5 Discussion and conclusions
To obtain information about the surface location of the active regions in the chromosphere of LQ Hya, we applied a simple geometric plage model to the rotationally modulated chromospheric emission. This method had been described and successfully applied to the H modulation curves of several other active stars (e.g., Frasca et al. 2000,2008; Biazzo et al. 2006). On LQ Hya, two circular bright spots (plages) are normally sufficient for reproducing the observed variations within the data errors (cf. Frasca et al. 2000; Kovári & Bartus 2001). In our model the flux ratio between plages and the surrounding chromosphere ( ) is a free parameter. Solar values of , as deduced from averaging many plages in H (e.g., Ellison 1952; Ayres et al. 1986; LaBonte 1986), are too low to model the high amplitudes of H curves in very active stars (Frasca et al. 2008; Biazzo et al. 2006). In fact, extremely large plages, covering a significant fraction of the stellar surface, would be required with such a low flux ratio, and they could not reproduce the observed modulation. In order to achieve a good fit, a flux ratio of 5, which is also typical of the brightest parts of solar plages or of flare regions (e.g., Svestka 1976; Zirin 1988), was adopted.
The solutions essentially provide the longitude of the plages and give only rough estimates of their latitude and size. The information about the latitude of surface features can be recovered from the analysis of spectral-line profiles broadened by the stellar rotation, as we did for the spots of LQ Hya with Doppler imaging in Paper I. A similar technique cannot reach the same level of accuracy when applied to the H profile whose broadening is dominated by chromospheric heating effects, which are particularly efficient in the active regions (e.g., Lanzafame et al. 2000).
We searched for the best solution by varying the longitudes, latitudes, and radii of the active regions. The radii are strongly dependent on the assumed flux contrast . Thus, only the combined information between plage dimensions and flux contrast, i.e. some kind of plage luminosity'' in units of the quiet chromosphere ( ), can be deduced as a meaningful parameter. Note also that we cannot estimate the true quiet chromospheric contribution (network), since the Hminimum value, Å, could still be affected by a homogeneous distribution of smaller plages. However, such an approximation seems valid for LQ Hya because our aim was only to compare the spatial distribution of the main surface inhomogeneities at chromospheric and photospheric levels.
We also searched for solutions with three active regions, finding a small but possibly significant increase in the goodness of the fit, with the passing from 5.77 to 4.45 (see Fig. 3). The model with only two plages requires features at rather high latitudes (Table 4) to explain the nearly flat and long-lasting maximum, while a 3-plage model allows placing plages at lower latitudes, in better agreement with the spot locations from Doppler imaging. However, the longitudes of the two largest plages do not change very much (less than 15 degrees).
The resulting plage longitudes are in good agreement with the Doppler results in Paper I; i.e., photospheric minima at 0.6 and 0.9 with the largest/coolest photospheric spots correspond to the most luminous chromospheric phases, and vice versa, the brightest photometric phase at 0.3 overlap with the lowest chromospheric H emission, again supporting the paradigm that photospheric spots are physically connected with chromospheric plages (Fig. 4).
The synthetic H EW curve for the model with three plages is plotted in the middle panel of Fig. 3 as a continuous line superimposed on the data (dots), while the 2-plage solution is displayed by a dotted line. Both of them reproduce the observed rotational modulation quite well. With the same models we were able to calculate the radial velocity shift between the H synthetic emission profile, resulting from the quiet chromosphere plus the visible plages, and the photosphere. It is clear in the bottom panel of Fig. 3 that the amplitude of the theoretical curves, both for a 2-plage and a 3-plage model, is very low, consistent with the values derived from the residual profiles ( ) and in total disagreement with the velocity shifts derived from the raw spectra. This strongly supports the spectral synthesis method we used to evaluate the chromospheric emission in the H line.
We would like to outline the rotational modulation of other features observed in the H profile, such as the peak separation, which is related to the chromospheric electron density (Sect. 4) and the asymmetry of blue/red peak intensity (Fig. 2). The chromosphere of LQ Hya displays a higher electron density above active regions. The blue emission peak is stronger than the red one ( ) in the quiet'' chromosphere of LQ Hya, while nearly equal peaks ( ) tend to be observed when the most active regions are in the visible hemisphere.
Table 4: Plage parameters for LQ Hya in April-May 2000.
Figure 4: Schematic representation of the surface features of LQ Hya in the chromosphere as reconstructed by the 2-plage ( top) and by the 3-plage models ( middle) and in the photosphere ( bottom). In these maps, the star is rotating counterclockwise. Dominant features are found at similar longitudes at both chromospheric and photospheric levels. Open with DEXTER
Asymmetry in the peaks of an emission line with a central reversal has been frequently observed for Ca II K line both in the Sun (e.g., Ding & Schleicher 1998; Zirin 1988, and reference therein) and in cool stars (e.g., Montes et al. 2000, and reference therein). The blue asymmetry is more frequently observed in the solar chromosphere and is commonly attributed to the propagation of acoustic waves (e.g., Cram 1976) with an upward velocity on the order of 10 km s-1 in the layer in which the K2 emission peaks are formed or a downward motion in the layer producing the central reversal K3 (e.g., Durrant et al. 1976).
Oranje (1983) has shown that the average Ca II K emission profile in solar plages has a completely different shape from the surrounding chromosphere, with the red peak slightly stronger than the blue one. This could reflect the different physical conditions and velocity fields in active regions compared to the quiet chromosphere. A similar analysis cannot be made for the Sun as a star in H, because this line is an absorption feature in the quiet chromosphere and only a filling in is observed in plages. However, similar changes of the H line profile can be expected in the plages of any stars that are more active than the Sun. Thus, the rotational modulation of is consistent with the presence of plages in the chromosphere of LQ Hya.
These results suggest scaled-up versions (regarding size and brightness) of solar-type plages for the H features observed in LQ Hya in 2000, without any clear evidence of strong mass motions in its chromosphere.
Acknowledgements
We are grateful to an anonymous referee for helpful comments and suggestions. This work has been partially supported by the Italian Ministero dell'Università e Ricerca (MUR), which is gratefully acknowledged. Zs.K. is a grantee of the Bolyai János Scholarship of the Hungarian Academy of Sciences and is also grateful to the Hungarian Science Research Program (OTKA) for support under grants T-048961 and K 68626. K.G.S. thanks the US Kitt Peak National Observatory for the possibility to record long time series of stellar data with the late Coudé-feed telescope, now retired. | {"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": 0.9411224126815796, "perplexity": 2178.1034507051822}, "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-2020-45/segments/1603107885126.36/warc/CC-MAIN-20201025012538-20201025042538-00056.warc.gz"} |
https://nyuscholars.nyu.edu/en/publications/phase-transition-for-potentials-of-high-dimensional-wells | # Phase transition for potentials of high-dimensional wells
Fanghua Lin, Xing Bin Pan, Changyou Wang
Research output: Contribution to journalArticlepeer-review
## Abstract
For a potential function F : R{double-struck} k → R{double-struck} + that attains its global minimum value at two disjoint compact connected submanifolds N ± in R{double-struck} k, we discuss the asymptotics, as ε{lunate} → 0, of minimizers u ε{lunate} of the singular perturbed functional E ε(u)=∫ ω (|∇u| 2+ 1/∈2 F(u))dx under suitable Dirichlet boundary data g : ∂Ω → R{double-struck} k. In the expansion of E ε{lunate} (u ε{lunate}) with respect to ${1 \over \varepsilon }$, we identify the first-order term by the area of the sharp interface between the two phases, an area-minimizing hypersurface Γ, and the energy c0F of minimal connecting orbits between N + and N -, and the zeroth-order term by the energy of minimizing harmonic maps into N ± both under the Dirichlet boundary condition on ∂Ω and a very interesting partially constrained boundary condition on the sharp interface Γ.
Original language English (US) 833-888 56 Communications on Pure and Applied Mathematics 65 6 https://doi.org/10.1002/cpa.21386 Published - Jun 2012
## ASJC Scopus subject areas
• Mathematics(all)
• Applied Mathematics
## Fingerprint
Dive into the research topics of 'Phase transition for potentials of high-dimensional wells'. Together they form a unique fingerprint. | {"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": 0.8886888027191162, "perplexity": 4335.90369769695}, "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/1664030335396.92/warc/CC-MAIN-20220929225326-20220930015326-00512.warc.gz"} |
https://www.physicsforums.com/threads/i-need-help-understanding-archimedes-principle.701390/ | # I need help understanding Archimedes' principle.
1. Jul 14, 2013
### threy
I want to know why does the buoyant force equal to the weight of fluid displaced and how the weight of water displaced is equal to the weight of object for free floating objects? What's buoyant force by the way?
2. Jul 14, 2013
### jhosamelly
The weight of the object is the downward force. Buoyant force is the upward force exerted by the fluid on the object. (Just like, for example, if you have a book on a table, the table exerts a normal force on the book) . For objects to float the downward force which is the weight should be equal to the upward force which is the buoyant force.
I hope you get the idea.
3. Jul 14, 2013
### threy
Yeah, I get this but I still have no idea why does the buoyant force equal to the weight of fluid displaced?
4. Jul 14, 2013
### jhosamelly
Since I think you want some mathematical explanation.
Buoyant Force = (weight of displaced water ÷ volume of displaced water) * depth * (surface area in contact with water)
depth * (surface area in contact with water) = volume of the object that is under water.
Buoyant Force = (Weight of displaced water ÷ volume of displaced water) * (volume of the object that is under water)
The volume of the object that is under water = volume of displaced water, because the object displaced the water.
Buoyant Force = (Weight of displaced water ÷ volume of displaced water) * (volume of displaced water)
Buoyant Force = Weight of displaced water
5. Jul 14, 2013
### Staff: Mentor
The fluid is displaced upwards - take a bowl of water, float something in it, and the water level will rise slightly. It takes some force to push that water up, and that force has to exactly balance the weight of the object if it's going to float at the surface.
6. Jul 14, 2013
### Staff: Mentor
Because the fluid surrounding the object does not know that the object has replaced the water that was there previously. The surrounding fluid was supporting the weight of the water that was there previously. So now it is applying the very same distribution of forces to the object.
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https://kr.mathworks.com/help/ident/ref/freqresp.html | # freqresp
Evaluate system response over a grid of frequencies
## Syntax
``````[H,wout] = freqresp(sys)``````
``H = freqresp(sys,w)``
``H = freqresp(sys,w,units)``
``````[H,wout,covH] = freqresp(sys,___)``````
## Description
Use `freqresp` to evaluate the system response over a grid of frequencies. To obtain the magnitude and phase data as well as plots of the frequency response, use `bode` instead.
example
``````[H,wout] = freqresp(sys)``` returns the frequency response of the dynamic system model `sys` at frequencies `wout`. `freqresp` automatically determines the frequencies based on the dynamics of `sys`. For more information about frequency response, see Frequency Response.```
example
````H = freqresp(sys,w)` returns the frequency response on the real frequency grid specified by the vector `w`. ```
````H = freqresp(sys,w,units)` explicitly specifies the frequency units of `w` with `units`.```
example
``````[H,wout,covH] = freqresp(sys,___)``` also returns the covariance `covH` of the frequency response. Use this syntax only when `sys` is an identified model of one of the types listed in Identified LTI Models (Control System Toolbox).```
## Examples
collapse all
For this example, consider the following SISO state-space model:
`$A=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}0.5\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=0$`
Create the SISO state-space model defined by the following state-space matrices:
```A = [-1.5,-2;1,0]; B = [0.5;0]; C = [0,1]; D = 0; sys = ss(A,B,C,D);```
Compute the frequency response of the system.
```[H,wout] = freqresp(sys); size(H)```
```ans = 1×3 1 1 56 ```
`H` contains the frequency response at 56 frequencies that are automatically chosen based on the dynamics of `sys`.
`size(wout)`
```ans = 1×2 56 1 ```
`wout` contains the corresponding 56 frequencies.
Create the following 2-input, 2-output system:
`$sys=\left[\begin{array}{cc}0& \frac{1}{s+1}\\ \frac{s-1}{s+2}& 1\end{array}\right]$`
```sys11 = 0; sys22 = 1; sys12 = tf(1,[1 1]); sys21 = tf([1 -1],[1 2]); sys = [sys11,sys12;sys21,sys22];```
Compute the frequency response of the system.
`[H,wout] = freqresp(sys);`
`H` is a 2-by-2-by-45 array. Each entry `H(:,:,k)` in `H` is a 2-by-2 matrix giving the complex frequency response of all input-output pairs of `sys` at the corresponding frequency `wout(k)`. The 45 frequencies in `wout` are automatically selected based on the dynamics of `sys`.
Create the following 2-input, 2-output system:
`$sys=\left[\begin{array}{cc}0& \frac{1}{s+1}\\ \frac{s-1}{s+2}& 1\end{array}\right]$`
```sys11 = 0; sys22 = 1; sys12 = tf(1,[1 1]); sys21 = tf([1 -1],[1 2]); sys = [sys11,sys12;sys21,sys22];```
Create a logarithmically-spaced grid of 200 frequency points between 10 and 100 radians per second.
`w = logspace(1,2,200);`
Compute the frequency response of the system on the specified frequency grid.
`H = freqresp(sys,w);`
`H` is a 2-by-2-by-200 array. Each entry `H(:,:,k)` in `H` is a 2-by-2 matrix giving the complex frequency response of all input-output pairs of `sys` at the corresponding frequency `w(k)`.
Compute the frequency response and associated covariance for an identified process model at its peak response frequency.
Load estimation data `z1`.
`load iddata1 z1`
Estimate a SISO process model using the data.
`model = procest(z1,'P2UZ');`
Compute the frequency at which the model achieves the peak frequency response gain. To get a more accurate result, specify a tolerance value of `1e-6`.
`[gpeak,fpeak] = getPeakGain(model,1e-6);`
Compute the frequency response and associated covariance for `model` at its peak response frequency.
`[H,wout,covH] = freqresp(model,fpeak);`
`H` is the response value at `fpeak` frequency, and `wout` is the same as `fpeak`.
`covH` is a 5-dimensional array that contains the covariance matrix of the response from the input to the output at frequency `fpeak`. Here `covH(1,1,1,1,1)` is the variance of the real part of the response, and `covH(1,1,1,2,2)` is the variance of the imaginary part. The `covH(1,1,1,1,2)` and `covH(1,1,1,2,1)` elements are the covariance between the real and imaginary parts of the response.
## Input Arguments
collapse all
Dynamic system, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can use include:
• LTI models such as `ss` (Control System Toolbox), `tf` (Control System Toolbox), and `zpk` (Control System Toolbox) models.
• Sparse state-space models, such as `sparss` (Control System Toolbox) or `mechss` (Control System Toolbox) models.
• Generalized or uncertain state-space models such as `genss` (Control System Toolbox) or `uss` (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)
• For tunable control design blocks, the function evaluates the model at its current value to evaluate the frequency response.
• For uncertain control design blocks, the function evaluates the frequency response at the nominal value and random samples of the model.
• Identified state-space models, such as `idss` models.
For a complete list of models, see Dynamic System Models.
Frequency values to evaluate system response, specified as either a vector of scalar values or a vector of complex values. Specify frequencies in units of `rad/TimeUnit`, where `TimeUnit` is the time units specified in the `TimeUnit` property of `sys`.
You can specify the frequency in terms of the Laplace variable `s` or `z` based on whether `sys` is a continuous-time or discrete-time model, respectively. For instance, if you want to evaluate the frequency response of a system `sys` at a frequency value of `w` rad/s, then specify the values in terms of
• `s = jw`, if `sys` is in continuous-time.
• `z = ejwT`, if `sys` is in discrete-time. Here, `T` is the sample time.
Units of the frequencies in the input frequency vector `w`, specified as one of the following values:
• `'rad/TimeUnit'` — radians per the time unit specified in the `TimeUnit` property of `sys`
• `'cycles/TimeUnit'` — cycles per the time unit specified in the `TimeUnit` property of `sys`
• `'rad/s'`
• `'Hz'`
• `'kHz'`
• `'MHz'`
• `'GHz'`
• `'rpm'`
## Output Arguments
collapse all
Frequency response values, returned as an array.
When `sys` is
• An individual dynamic system model with `Ny` outputs and `Nu` inputs, `H` is a 3D array with dimensions `Ny`-by-`Nu`-by-`Nw`, where `Nw` is the number of frequency points. Thus, `H(:,:,k)` is the response at the frequency `w(k)` or `wout(k)`.
• A model array of size `[Ny` `Nu` `S1` `...` `Sn]`, `H` is an array with dimensions `Ny`-by-`Nu`-by-`Nw`-by-`S1`-by-...-by-`Sn`] array.
• A frequency response data model (such as `frd`, `genfrd`, or `idfrd`), `freqresp(sys,w)` evaluates to `NaN` for values of `w` falling outside the frequency interval defined by `sys.frequency`. The `freqresp` command can interpolate between frequencies in `sys.frequency`. However, `freqresp` cannot extrapolate beyond the frequency interval defined by `sys.frequency`.
Output frequencies corresponding to the frequency response `H`, returned as a vector. When you omit `w` from the inputs to `freqresp`, the command automatically determines the frequencies of `wout` based on the system dynamics. If you specify `w`, then `wout` = `w`
Covariance of the frequency response, returned as a 5D array. For instance, `covH(i,j,k,:,:)` contains the 2-by-2 covariance matrix of the response from the `i`th input to the `j`th output at frequency `w(k)`. The (1,1) element of this 2-by-2 matrix is the variance of the real part of the response. The (2,2) element is the variance of the imaginary part. The (1,2) and (2,1) elements are the covariance between the real and imaginary parts of the response.
collapse all
### Frequency Response
In continuous time, the frequency response at a frequency ω is the transfer function value at s = . For state-space models, this value is given by
`$H\left(j\omega \right)=D+C{\left(j\omega I-A\right)}^{-1}B$`
In discrete time, the frequency response is the transfer function evaluated at points on the unit circle that correspond to the real frequencies. `freqresp` maps the real frequencies `w(1)`,..., `w(N)` to points on the unit circle using the transformation $z={e}^{j\omega {T}_{s}}$. Ts is the sample time. The function returns the values of the transfer function at the resulting z values. For models with unspecified sample time, `freqresp` uses Ts = 1.
## Algorithms
For transfer functions or zero-pole-gain models, `freqresp` evaluates the numerator(s) and denominator(s) at the specified frequency points. For continuous-time state-space models (A, B, C, D), the frequency response is
`$\begin{array}{cc}D+C{\left(j\omega -A\right)}^{-1}B,& \omega =\end{array}{\omega }_{1},\dots ,{\omega }_{N}$`
For efficiency, A is reduced to upper Hessenberg form and the linear equation (jω − A)X = B is solved at each frequency point, taking advantage of the Hessenberg structure. The reduction to Hessenberg form provides a good compromise between efficiency and reliability. For more details on this technique, see [1] (Control System Toolbox).
## References
[1] Laub, A.J., "Efficient Multivariable Frequency Response Computations," IEEE® Transactions on Automatic Control, AC-26 (1981), pp. 407-408.
## 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": 6, "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": 0.9460858106613159, "perplexity": 1266.9775955811742}, "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/1669446710719.4/warc/CC-MAIN-20221130024541-20221130054541-00577.warc.gz"} |
http://mymathforum.com/probability-statistics/37283-probability-doubts.html | My Math Forum Probability doubts
Probability and Statistics Basic Probability and Statistics Math Forum
July 28th, 2013, 12:55 AM #1 Member Joined: Sep 2011 Posts: 99 Thanks: 1 Probability doubts I have encountered some problem with the probability qns. I have gotten 5/20 x 4/19= 1/19 for b (i) and (1/4 x 15/19) + (1/4 x 4/19) = 1/4 for b (ii). The answer is with the multiplication of 2 to 5/20 x 4/19 and to (1/4 x 15/19) + (1/4 x 4/19) as stated below. I do not understand why 2 must be multiplied to the probability? I need help in the explanation. Your help will be greatly appreciated. Many thanks. Here is the qns: Sam bought a set of 20 past year test papers. The set contains 4 subjects; English, Mathematics, Physics and Chemistry. There are 5 test papers for each subject numbered 1,2,3,4 and 5. (a) A test paper is chosen at random. Find , as a fraction, the probability that it is a mathematics test paper and numbered 5 or an English test paper. (b) The test paper is placed back into the set and now, two test papers are chosen at random. Find as a fraction, in its lowest terms, the probability that (i) they are both chemistry test papers. (ii) at least one is a mathematics test paper. Answer to (a) 1/20 + 1/4 = 3/10 (b) (i) 5/20 x 4/19 x 2 = 2/19 (b) (ii) (1/4 x 15/19 x 2) + (1/4 x 4/19 x 2) = 1/2
July 28th, 2013, 04:57 AM #2 Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Probability doubts The correct answers to b (i) and b(ii) are: (i) (5/20) * (4/19) = 1/19 (You are correct. There is no reason to multiply this by 2.) (ii) 1/4 + (3/4)*(5/19) = 17/38 The answers given are definitely wrong. But, your calculations for b (ii) are not quite right. Her'e's how to think about b (ii): First, there is 5/20 = 1/4 probability that the first paper is maths. The second paper can then be anything. Note that this includes the possibility that both are maths. But, if the first paper is not maths (15/20) = 3/4, then there is still a chance that the second is maths (5/19). So, the answer is the sum of these. (5/20) + (15/20)(5/19) = 17/38. Note that an alternative way to work this out is to calculate separately: prob the first is maths and the second isn't + prob first is not maths and the second is + prob first and second are both maths. This will give you the same answer of 17/38.
July 28th, 2013, 05:22 AM #3 Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Probability doubts One of the things I like to do is think of different ways to do these probability questions and then, hopefully, they all give the same answer. So, here's another solution for b (ii): Calculate the probability that neither is maths. Then subtract this from 1 to get the probability that at least one is maths: Prob neither is maths = (3/4)(14/19) = 21/38. So prob at least one is maths = 1 - 21/38 = 17/38. That settles it! And, another solution for b (i); Calculate the probability that both papers are the same, then divide this by 4, as they are equally like to be any subject. The first paper can be anything, and the second is the same with probablity 4/19. This is the prob that both papers are the same subject. So, there is 1/19 of both chemistry, 1/19 both maths etc.
July 28th, 2013, 08:36 AM #4 Member Joined: Sep 2011 Posts: 99 Thanks: 1 Re: Probability doubts Thank you so much for the help! Here, I have another qns. I have gotten( 1/6*5/6*6/6) + (6/6*5/6*1/6) + (6/6*1/6*1/6) = 11/36 for a(ii). I am not sure if my answer is correct or not as i find that my answer seems logical from the approach used in the previous doubts stated above. Three unbiased dice are thrown. Find the probability that they (a) (i) all show different numbers (ii) at least two show the same numbers. Answer for a(i) is 6/6 * 5/6 * 4/6 = 5/9 a(ii) is ( 6/6*1/6*1/6) + (6/6*1/6*5/6) = 1/6
July 28th, 2013, 08:51 AM #5 Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Probability doubts The quick way is to note that (i) and (ii) are complements. I.e. either all dice are different or at least two are the same. The answer give to (i) is correct: 5/9. So, the answer to (ii) is 1 - 5/9 = 4/9. It is, of course, possible to crunch the answer to (ii) by looking at all the options where at least two are the same, but this is much more complicated than calculating the complement of this and subtracting from one.
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## MoreForJoe 3 years ago minimizing area: A 36-in. piece of wire is cut into two pieces. One Piece is used to form a circle while the other is used to form a square. How should the wire be cut so that the sum of the areas is minimal? What is the minimum value? Delete Cancel Submit
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1. Dalvoron
• 3 years ago
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|dw:1323260051758:dw| The sum of the areas is$A(r)=\frac{1}{2}(2\pi)r^2 + (36-2\pi r)^2$You want this to be a $$\textit{minimum}$$. This will happen when the first derivative of A = 0. (though this could also be a maximum. To figure out which, you'd need the second derivative I think).
2. MoreForJoe
• 3 years ago
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what do i do next?
3. Dalvoron
• 3 years ago
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That should get you a value for r.
4. Dalvoron
• 3 years ago
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Then you work out the length $$2\pi r$$, and the length $$(36 - 2\pi r)$$
5. Dalvoron
• 3 years ago
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Actually, according to my sanity test, that gives you lengths longer than 36, so it can't be right.
6. Dalvoron
• 3 years ago
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Nope, calculation error. Never mind.
7. MoreForJoe
• 3 years ago
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i never did one of these problems and im sure he's going to put it on the test so i have to know how to do it -..-
8. MoreForJoe
• 3 years ago
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i still dont know where to begin...
9. Dalvoron
• 3 years ago
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The total length of the wire is 36 in, right? That length will then turn into two lengths when you cut it in half. One length will become the circumference of your circle ($$2\pi r)$$, and the other will become the perimeter of your square (36 - $$2\pi r$$). So now you have: |dw:1323260762343:dw| I actually made a mistake earlier, I forgot the perimeter of the square and the length of its side are not the same. So the area of the circle is $$\pi r^2$$, and the area of the square is the side length squared. The side length is one quarter of the perimeter, so the area of the square is $$\left(\frac{1}{4}(36-2\pi r)\right)^2$$ Thus, the sum of the two areas is....?
10. MoreForJoe
• 3 years ago
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πr^2+((1/4)(36-2πr))^2?
11. Dalvoron
• 3 years ago
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Precisely. So $A(r)=\pi r^2 + 81-9\pi r+\frac{1}{4}\pi^2r^2$when you simplify, and expand that bracket. What we have is a function, which defines the area in terms of the radius. To give you an idea of what that looks like, go to http://www.wolframalpha.com/input/?i=%5Cpi+r^2%2B%289-%5Cfrac {%5Cpi}{2}r%29^2 We're looking for that lowest point on the graph. Where is that point? Well it's at the point where the slope of the graph is zero. The slope of the graph is given by $\frac{dA}{dr}$, and we want it to be equal to zero, so we say $\frac{dA}{dr}=0$
12. MoreForJoe
• 3 years ago
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where did dA/dr come from?
13. Dalvoron
• 3 years ago
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Have you started calculus yet?
14. MoreForJoe
• 3 years ago
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no...
15. Dalvoron
• 3 years ago
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Good grief. I have no idea how to solve this problem without calculus. You've not seen anything like dy/dx yet?
16. MoreForJoe
• 3 years ago
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no... my class is intermediate algebra
17. Dalvoron
• 3 years ago
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Well my apologies, I don't see how it's possible to solve this with just algebra.
18. MoreForJoe
• 3 years ago
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hmm... can you type how to solve this your way and i can just write it down?
19. Dalvoron
• 3 years ago
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It's pretty advanced mathematics compared to what you're currently doing, there's just no way you'd be expected to do it this way.
20. MoreForJoe
• 3 years ago
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true...
21. alexxis15
• 3 years ago
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WELL THAT'S EASY
22. MoreForJoe
• 3 years ago
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huh?
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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": 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": 0.9988707304000854, "perplexity": 1740.437952398689}, "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-2015-32/segments/1438042981856.5/warc/CC-MAIN-20150728002301-00103-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://proxies-free.com/real-analysis-proof-of-iiimplies-iii-in-theorem-4-18-rudins-rca/ | # real analysis – Proof of (ii)\$implies\$ (iii) in Theorem \$4.18\$, Rudin’s RCA
Rudin says that (ii)$$implies$$(iii) of Theorem $$4.18$$ follows from Theorem $$4.17$$. I do not see how, although I have tried two different approaches. I have described my work below and attached the theorems for reference below at the end of the post.
Approach 1: Assume (ii), i.e. $$P$$ is dense in $$H$$. Then for every $$epsilon > 0$$ and $$xin H$$, there exists a finite $$Fsubset A$$ and $$c_alpha$$‘s such that $$left|x – sum_{alpha in F} c_alpha u_alpharight| < epsilon$$
We have $$sum_{alphain A} |hat x(alpha)|^2 le |x|^2$$ from Theorem $$4.17$$ already. If we can show $$sum_{alphain A} |hat x(alpha)|^2 ge |x|^2$$, we are done. I observed that
$$|x| le left|x – sum_{alpha in F} c_alpha u_alpharight| + left|sum_{alphain F}c_alpha u_alpha right| < epsilon + sum_{alphain F}|c_alpha|^2$$
which doesn’t really help.
Approach 2: The proof of Theorem $$4.17$$ says that $$f$$ is an isometry of $$P$$ onto the dense subspace of $$ell^2(A)$$ consisting of finitely supported functions. Since we assume $$P$$ is dense in $$H$$, there is a sequence $$(x_n)$$ of elements in $$P$$, such that $$x_nto xin H$$. Since $$0in P$$ and $$f$$ is an isometry of $$P$$, we see that
$$|x_n|^2 = |f(x_n)|^2$$
$$|x_n|^2 = |hat x_n|^2 = sum_{alphain A}|hat x_n(alpha)|^2$$
Taking limits,
$$lim_{ntoinfty}|x_n|^2 = lim_{ntoinfty} sum_{alphain A}|hat x_n(alpha)|^2$$
Rudin defines $$sum_{alphain A} phi(alpha)$$ as $$sup_{Fsubset A, |F| < infty} sum_{alphain F}phi(alpha)$$ where $$F$$ is a finite subset of $$A$$, i.e. we take supremum over all finite sums.
Then,
$$lim_{ntoinfty}|x_n|^2 = lim_{ntoinfty} sup_{Fsubset A} sum_{alphain F, |F| < infty}|hat x_n(alpha)|^2$$
If we can switch $$lim$$ and $$sup$$ in the expression above, we are essentially done – but I’m not sure if the swapping is allowed.
Attached for reference:
Theorem 4.18:
Theorem 4.17:
Thank 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": 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": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9970653057098389, "perplexity": 124.01633630083037}, "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-2021-25/segments/1623487608702.10/warc/CC-MAIN-20210613100830-20210613130830-00485.warc.gz"} |
https://shaoshuanglin.wordpress.com/ | Posted by: Shuanglin Shao | June 14, 2010
## On localization of the Schrodinger maximal function
I was interested in the almost everywhere pointwise convergence problem of the Schrödinger solution $e^{it\Delta}f$ as time goes to zero since I was a graduate student. The question was raised by Carleson, and remains open in high dimensions. It is closely related to the boundedness of the Schrodinger maximal operator (local or global); in turn it is closely related to the interesting oscillatory integrals in harmonic analysis and Strichartz estimates in PDE. Recently I understood some of the problem and wrote down a note to re-construct the two dimensional proof. These results contained in this note are not new; in the note I explore them from a slightly different perspective.
Posted by: Shuanglin Shao | December 6, 2009
## Kato-smoothing effect
“Kato smoothing” “is ” important in dispersive PDE.“Smoothing”, as it stands, sounds like a very good word. But why is it good? How will it be used remains as vague questions to me.
In the last week, I began to report a paper by Alazard-Burq-Zuily, “On the water waves questions with surface tension”, which contains a “local-smoothing” result: roughly speaking, solutions to 2-D water waves with surface tension in $C^0_tH^s_x$ will get $\frac 14$ regularity upgrade to $L^2_tH^{s+\frac 14}_x$, to the price of locally in space and being averaged in time. This result was first proven by Christianson-Hur-Staffilani.
A question came to me, why $1/4$? Where can I quickly see it? Soon I found out it is determined by the structure of the water-wave equation. To oversimplify the major result in the paper on the derivation of water-wave equations by means of paralinearization, it can be written as a type of dispersive equations,
$\partial_t u+iD^{3/2} u=0, \,D:=|\nabla|, \, (*)$
(I have dropped a lot of terms, for instance, only looking at a linear equation, and no flow-terms in (*); it is because I am only looking at main terms which I think reflect dispersion). (*) suggests a dispersive relation $\tau=|\xi|^{3/2}$. If one look for
$\int \int_{|x|\le 1} |D^\alpha u|^2 dxdt \lesssim \|u_0\|_{L^2}$, the maximum value of $\alpha$ turns out to be $1/4$. ($1/4$ was proved for 2D, does this suggest that it was the case in all dimensions?)
To understand this necessary condition on $\alpha$, I would like to draw an analogy with the “1/2-local smoothing” for the Schrodinger equations: for any $\epsilon >0$,
$\int\int \langle x \rangle^{-1-\epsilon} |D^{1/2} e^{it\Delta} f|^2 dx dt \le C \|f\|_2^2.$
Why $1/2$ there? and how it was proved? Then the clarifications of these matters provides a model (to me) that $1/4$ for water-wave sounds reasonable.
Let us first motivate “local-smoothing” for Schr\”odinger? It is well known that the solutions $e^{it\Delta} f$ to a free Schr\”odinger equation
$i\partial_t u+\Delta u=0, u(0,x)=f(x)$
obeys the conservation law of mass, i.e.,
$\| e^{it\Delta} f \|_{L^{\infty}_tL^2_x} =\|f\|_{L^2_x}. \,(1)$
Note that we can not add any $D^\alpha$ with $\alpha>0$ to the left hand side of (1) due to the Galilean transform (or simply by creating a bump at high frequency). One may argue that, is this no-gain-of-derivative due to we asking for too much in time by requiring a $L^\infty_t$? For instance, on a time interval, $[0,1]$, $L^\infty$-norm is stronger than any $L^q$ norm with $1\le q<\infty$? So is the following true,
$\|D^\alpha e^{it\Delta} f\|_{L^q_tL^2_x}\le C\|f\|_2,\,(2)$
for some $\alpha>0$ and $1\le q<\infty$. Unfortunately (2) does not hold for any $\alpha>0$ due to the same reason as above (one can take the same examples.).
Is there any hope that some variant of (2) holds true? By using the heuristic that solutions to a dispersive equations at high frequency travels much faster than low frequency, after waiting for a long time, only low-frequencies are left behind around spatial origin and they do not hurt positive derivatives. So if we were asking for an estimate locally in space, was it possible? The answer turns out to be Yes. We have the following estimate,
$\int \int_{|x|\le 1} |D^{1/2} e^{it\Delta}|^2 dxdt \lesssim \|f\|^2_2.\, (3)$
This is referred to as “Kato smoothing estimate" for Schr\"odinger in literature. Moreover $1/2$ is the most one can expect due to the obstruction (counterexample) we mentioned above. (3) will lead to a more general estimate, for any $\epsilon>0$
$\int \int_{\mathbb{R}^d} (1+|x|)^{-1-\epsilon} |D^{1/2} e^{it\Delta}|^2 dxdt \lesssim \|f\|^2_2.\, (4)$
The implication of (4) from (3) is easier by partitioning $\{|x|\ge 1\}$ and rescaling, and using the information that $\epsilon>0$.
We focus on proving (3). It will follow from a Plancherel argument.
Writing
$D^{1/2} e^{it\Delta} f(x)=\int_{\xi^\prime;, \tau} e^{ix^\prime\cdot \xi^\prime+it\tau}\int_{\xi_d} e^{ix_d\xi_d} \delta (\tau-|\xi|^2)|\xi|^{1/2} \widehat{f}(\xi)d\xi_d d\xi'd\tau,$
where $\delta$ is the Dirac mass, and $\xi=(\xi^\prime,\xi_d)$. We set
$F(\xi^\prime,\tau)=\int_{\xi_d} e^{ix_d\xi_d} \delta (\tau-|\xi|^2) |\xi|^{1/2}\widehat{f}(\xi)d\xi_d.$
To prove (3), it suffices to prove
$\int_{|x_d|\le 1} \int_{\mathbb{R}^d } |\widehat{F}(x^\prime,t)|^2 dx^\prime dt dx_d \le C\|f\|_2^2, \, (5)$
where $x=(x^\prime,x_d)$. Obviously by the Plancherel theorem, the left hand side of (5) is bounded by
$\int_{|x_d|\le 1} \int_{(\xi^\prime, \tau)} |F|^2 d\xi^\prime d\tau dx_d.$
Fixing $(\xi^\prime \tau)$, Cauchy-Schwarz yields
$|F|^2\le \int_{\xi_d} |\widehat{f}|^2 \delta(\tau-|\xi|^2 )d\xi_d \int_{\xi_d} |\xi|\delta(\tau-|\xi|^2) d\xi_d. \, (6)$
The second factor on the right hand side of (6) is bounded by an absolute if we restricting $\xi$ to to set $\{\xi: |\xi|\le \sqrt{d} |\xi_d|\}$; it is not hard to do so if at the beginning we aim to prove (3) under this restriction; then the general estimate follows from the triangle inequality and partition the frequency space into $laex d$ pieces.
So plugging (6) in (5) and interchanging the integration order, and using $|x_d|\le 1$, we find out the left hand side of (5) is bounded by $\|f\|_2^2$. This is exactly what we need. So we finish proving (3).
I like the previous type of argument very much; but I did not remember where it was the first place/time I saw it. So I recorded it for my own benefits.
Posted by: Shuanglin Shao | November 14, 2009
## Reading: GWP for critical 2D dissipative SQG
Last weekend I attended a wonderful conference SCAPDE at UC Irvine, where I learned an interesting theorem from Kiselev: solutions to critical 2D dissipative quasi-geostrophic equations (SQG) are globally wellposed
$\theta_t=u\cdot \nabla \theta-(-\Delta)^\alpha\theta, u=(u_1,u_2)=(-R_2\theta, R_1\theta),$
where $\theta:\mathbb{R}^2\to \mathbb{R}$ is a scalar function, $R_1, R_2$ are the usual Riesz transform in $\mathbb{R}^2$ defined via $\widehat{R_i(f)}(\xi)=\frac {i\xi_i}{|\xi|}\widehat{f}(\xi)$ and $\alpha\geq 0$.
This is his joint work with Nazarov and Volberg. I took a look at this paper in the past few days. The proof makes good use of classical tools in Fourier analysis such as singular integrals and modulus of continuity, which are familiar topics in Stein’s book, Singular integrals and differentiability properties of functions. It is a place where I do not see Strichartz estimates or Littlewood Paley decompositions and see the power of classical Fourier analysis.
The monotonicity formula they have is
$\|\nabla \theta\|_\infty\le C\|\nabla \theta_0\|_\infty \exp \exp\{C\|\theta_0\|_\infty\}, (1)$
for periodic smooth initial data $\theta_0$. I am new to this field of fluid dynamics but I would still like to say a few words on this estimate. The proof is to find an “upper bound “, modulus of continuity $\omega$, for solutions $\theta$; then they show that a family of modulus of continuity is preserved under evolution of the equation, which is strong enough to control $\|\nabla \theta\|_\infty$. Quoted from the paper, the idea is to show that the critical SQG possesses a stronger “nonlocal” maximum principle than $L^\infty$ control.
Recall a modulus of continuity $\omega: [0,\infty)\mapsto [0,\infty)$ is just an arbitrarily increasing continuous and concave function such that $\omega(0)=0$. A function $f: \mathbb{R}^n\mapsto \mathbb{R}^m$ has modulus of continuity $\omega$ if $|f(x)-f(y)|\le \omega (|x-y|)$ for all $x,y\in \mathbb{R}^n$.
We choose not to report the crucial/essential part of finding the continuity of modulus (maybe later). Instead we assume that $f$ has modolus of continuity $\omega$, which is unbounded and $\omega^\prime (0)<\infty$ and $\lim_{\xi\to 0+}\omega^{\prime\prime}=-\infty$. Then there holds
$\|\nabla f\|_\infty< \omega^\prime (0), (2)$.
The proof is actually very simple. The explicit form of $\omega$ will take care of the implication of (1) from (2).
Assume that $\|\nabla f\|_\infty=|\nabla f(x)|$ for some $x$. We consider the point $y=x+\xi e$ for $e=\frac {\nabla f}{|\nabla f|}$. On the one hand, we have
$f(y)-f(x)\le \omega (\xi), (3)$
for all $\xi\geq 0$. On the other hand, the left hand side of (3) is at least $|\nabla f(x)|\xi -C\xi^2$ where $C=\frac 12 \|\nabla^2 f\|_\infty$ while its right hand side can be represented as $\omega^\prime (0)-\rho(\xi)\xi^2$ with $\rho(\xi)\to\infty$ as $\xi\to 0+$.
Then
$|\nabla f(x)| \le \omega^\prime (0)-(\rho(\xi)-C)\xi$
for all sufficiently small $\xi>0$, and it remains to choose some $\xi>0$ satisfying $\rho(\xi)>C$.
Posted by: Shuanglin Shao | May 1, 2009
## The Cotlar-Stein lemma
The Coltar-Stein lemma is a powerful tool to deal with $L^2$ boundedness of some translation-invariant operators such as convolution operators, which can be expressed as
$Tf(x)=\int K(x-y) f(y)dy.$
I recently began to understand how powerful it might be via the $TT^*$ method. I would like to reproduce its proof following Fefferman’s presentation.
$\mathbf{Statement.}$
Suppose $T:=\sum_{k=1}^M T_k$ is a sum of operators on Hilbert spaces (say the usual $L^2$). Assume that
$\|T^*_jT_k\|\le a(j-k)$ and $\|T_jT^*_k\|\le a(j-k)$. Then $\|T\|\le \sum_{|j|\le M }\sqrt{a(j)}$.
Here $a(\cdot)$ is a non-negative even function.
$\mathbf{Proof.}$ The argument follows an idea of iteration.
Step 1. $\|Tf\|^2=\langle Tf, Tf \rangle =\sum_{j=1}^M\sum_{k=1}^M \langle T_j f, T_k f\rangle$. We estimate $\langle T_j f, T_k f\rangle$ into two ways. Firstly it is easy to see that
$\langle T_j f, T_k f\rangle \le (\max \|T_j\|)^2 \|f\|^2$; secondly it is also trival that
$\langle T_j f, T_k f\rangle = \langle T^*_kT_j f, f\rangle \le a(j-k) \|f\|^2$. Hence by taking the geometric means of these two bounds, we have
$\langle T_j f, T_k f\rangle \le \max \|T_j\| \sqrt {a(j-k)} \|f\|^2$.
Hence
$\|T\|^2\le \sum_{j=1}^M\sum_{k=1}^M \max \|T_j\| \sqrt {a(j-k)} \le M \max \|T_j\| \sum_{|j|\le M} \sqrt{a(j)}$.
This gives that
$\|T\|\le (M\max\|T_j\|)^{1/2} (\sum_{|j|\le M} \sqrt{a(j)})^{1/2}.$
Step 2. We investigate one more iteration of $TT^*$. We write
$\langle TT^* f, TT^* f\rangle= \sum_{1\le k_1,k_2,j_1,j_2\le M} \langle T_{k_1}T^*_{j_1}f, T_{k_2}T^*_{j_2}f\rangle$
$\qquad = \sum_{1\le k_1,k_2,j_1,j_2\le M} \langle T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}f, f\rangle.$
Also we estimate $\langle T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}f, f\rangle$ by organizing the operators in two ways.
Firstly since $T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}=\bigl(T_{j_2}T^*_{k_2}\bigr) \bigl(T_{k_1}T^*_{j_1}\bigr)$, we see that
$\langle T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}f, f\rangle \le a(j_2-k_2)a(j_1-k_1) \|f\|^2;$
secondly since $T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}=T_{j_2}\bigl(T^*_{k_2}T_{k_1}\bigr)T^*_{j_1}$,
$\langle T_{j_2}T^*_{k_2}T_{k_1}T^*_{j_1}f, f\rangle \le ( \max\|T_j\|)^2 a(k_2-k_1) \|f\|^2;$
By taking geometric means, we see that
$\|T\|^4 \le \sum_{1\le k_1,k_2,j_1,j_2\le M} \max\|T_j\| \sqrt{a(j_2-k_2)}\sqrt{a(k_2-k_1)}\sqrt{a(k_1-j_1)}$
$\qquad \le M\max(\sum_{|j|\le M}\sqrt{a(j)})^3 .$
Hence
$\|T\|\le (M\max\|T_j\|)^{1/4} (\sum_{|j|\le M}\sqrt{a(j)})^{3/4}$.
Step 3. by induction, we see that for any $n\ge 1$, we have
$\|T\|\le (M\max\|T_j\|)^{\frac 1{2n}} (\sum_{|j|\le M}\sqrt{a(j)})^{\frac {2n-1}{2n}}$.
Let $n\to\infty$, we see that
$\|T\|\le \sum_{|j|\le M} \sqrt{a(j)}$. The proof of this lemma is complete.
We remark that in the argument both bounds $\|T_jT^*_k\|$ and $\|T^*_jT_k\|$ are used.
$\mathbf{Example.}$
Let $P:=\sum_{k\in \mathcal{Z}}P_k$ where $P_k$ be the Littlerwood-Paley projection operator. It is easy to see that $P_k$ is a self-adjoint operator. Also $a(0)\le 1$ and $a(j)=0$ for $j\neq 0$.
Then Stein-Cotlar gives,
$\|P\|\le 1$,
which matches that $\|P\|=1$.
Posted by: Shuanglin Shao | February 28, 2009
## A misleading argument for the adjoint Fourier restriction for the sphere in 3 dimension
Today I thought that I had a new proof of the adjoint Fourier transform for the sphere in three dimensions; but it turns out it was wrong. I would like to record it here, which I hope is useful at some point.
First let us define the notion of Fourier transform. Let
$\widehat{f}(\xi) :=(2\pi)^{-3/2}\int_{\mathbf R^3} e^{-ix\xi} f(x)dx.$
$\|\widehat{f}\|_{L^2(\mathbf R^3)}=\|f\|_{L^2(\mathbf R^3)}.$
From this definition the convolution of two functions behaves under the Fourier transform like
$\widehat{ f\ast g}= (2\pi)^{3/2} \widehat{f}\widehat{g}.$
Now let us begin the argument. By Plancherel,
$\|\widehat{fd\sigma}\|^2_{L^2(\mathbf{R}^3)} =(2\pi)^{-3/2} \|fd\sigma \ast fd\sigma \|_{L^2(\mathbf{R}^3)}.$
In view of this, we may assume that $f\ge 0$.
We write the integrand out,
$fd\sigma \ast fd\sigma (x) =\int_{S^2} [f\sigma](x-y)[fd\sigma](y).$
(1) By Cauchy-Schwarz inequality, we have
$f(x-y)f(y)\le \dfrac {|f(x-y)|^2+|f(y)|^2}{2}.$
We observe that
$\int_{S^2}f^2(x-y)d\sigma(x-y)d\sigma(y)=\int_{S^2}d\sigma(x-y)f^2(y)d\sigma(y).$
Then
$\|fd\sigma \ast fd\sigma \|_{L^2(\mathbf{R}^3)}$
$\le\|\int_{S^2}f^2(y)d\sigma(x-y)d\sigma(y) \|_{L^2(\mathbf{R}^3)}.$
(2)Then by Minkowsik inequality,
$\|\int_{S^2}d\sigma(x-y)f^2(y)d\sigma(y) \|_{L^2(\mathbf{R}^3)}$
$\le \|d\sigma\|_{L^2} \int_{S^2} f^2(y)d\sigma=(4\pi)^{1/2} \|f\|^2_{L^2(S^2)}.$
Note that the best constant for Young’s inequality $L^1\times L^2 \to L^2$ is $1$.
(I orginially thought that $\|d\sigma\|_{L^2(\mathbf R^3)} =(\int_{S^2} d\sigma)^{1/2}=(4\pi)^{1/2}$. However, it is wrong since $d\sigma\neq d\sigma^2$ or $d\sigma\notin L^2(\mathbf R^3)$, which can be seen from the asympotics $\widehat{d\sigma}(\xi)\sim (1+|\xi|)^{-1}$ for large $\xi$ in $\mathbf R^3$. Incidentally, today I found that Terry remarked on his blog that the Dirac mass $\delta^2 \notin L^1$ by a trick of epsilon regularization (another way is to use Pancherel theorem and observe that $\widehat{\delta} =1$).)
At this point, there is no need to read on since the following is based on Step 2. I worte it down before I realized the mistake; so I backed it up as my notes.
{Then we can conclude that
$\|\widehat{fd\sigma}\|^2_{L^2 (\mathbf{R}^3)}\le (2\pi)^{-3/2}(4\pi)^{1/2}\|f\|^2_{L^2(S^2)}=2^{-1/2}\pi^{-1/2}\|f\|^2_{L^2(S^2)}.$
}
Posted by: Shuanglin Shao | February 18, 2009
## Pohazav argument: no moving-to-left “solitions” for the critical gKdV
I would like to show a short but beautiful Pohazav argument to exclude “moving-to-left” solitons in the form of $u(t,x):=u(x-ct)$ for the focusing generalized Korteweg de Vries equation
$u_t+u_{xxx}+ (u^5)_x=0.$
This is to show that $c>0$.
Firstly we assume that $u(x,t)=u(x-ct)$ is a solution for the equation above and also assume that $u$ decays to $0$ in the infinity (to justify the arguments). Then
$-cu_x+u_{xxx}+(u^5)_x=0.$
Then an integration yields $-cu+u_{xx}+u^5=0. \qquad (1)$.
We multiply (1) by $u$ and integrate,
$-c\int u^2-\int u_x^2 + \int u^6=0. \qquad (2)$
We multiply (1) by $xu$ and integrate,
$-c\int xuu_x +\int xu u_{xxx}+\lambda \int xu (u^5)_x=0,$
which is simiplified to
$\frac c2 \int u^2 +\frac 32 \int u_x^2-\int u^6=0. \qquad (3)$
Now we see that $(2)\times \frac 32+(3)$ yields that
$-c\int u^2 +\frac {4}3 \int \lambda^6=0$,
This forces $c>0$.
In general, for the defocusing critical gKdV, to exclude solutions in the “solitions” sense is really a hard topic.
A decay estimate on the right is desired.
Posted by: Shuanglin Shao | February 17, 2009
## A continuation in reading: heat flow monotonicity of Strichartz norms
It has been a long time that I haven’t updated this blog. I promised to post my readings on the Bennett-Bez-Carbery-Hundertmark’s short but beautiful paper, “heat flow monotonicity of Strichartz norm”. Here it goes!
In this paper, the authors applied the method of heat-flow in the setting of Strichartz inequality for the Schr\”odinger equation and amusingly obtained, among other things, the Strichartz norm
$\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)}$
is nondecreasing as the initial datum $f$ evolvs under a under quadratic flow when $d=1, 2$. This immediately yields that the Gaussians are extremisers to the classical Strichartz inequality,
$\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)} \le S_d\|f\|_{L^2(\bf{R}^d)}, \text{ with } S_d:=\frac {\|e^{is\Delta}f\|_{ L^{2+\frac 4d}_{s,x}(\mathbf{R}\times \mathbf{R}^d)}}{\|f\|_{ L^2(\bf{R}^d)}}.$
The paper also consider the monotonicity given by some other flows such as Mehler-flow; it also considers the higher dimensions’s analogues by embedding the usual Strichartz norm into one-parameter family of norms $\||\cdot\||_p$. In this post I will focus on the monotonicity induced by the heat-flow.
The method of heat-flow deformation is interesting in itself. It is expected to be applied to related problems such as the existence of extremisers to the adjoint Fourier restriction operators to the sphere; but it is a different story; so far I can see an immediate difficulty in the latter setting, i.e., an representation formula for $\|\widehat{fd\mu}\|_{L^{2+\frac 4d}_x(\bf{R}^{d+1})},$ where $d\mu$ is the standard surface measure for the unit sphere in $\bf R^{d+1}$.
Fortunately the representation formula of this kind is available in the paraboloid setting thanks to Hundertmark-Zharnitsky’s work; I already reported it in my previous post but I would like to record it here again basically because an “equation” is rare in analysis: for nonnegative $f\in L^2(\bf R)$,
$\|e^{is\Delta}f\|^6_{L^6_{s,x}(\bf R\times \bf R)}=\frac {1}{\sqrt{12}}\int_{\bf R^3} (f\bigotimes f\bigotimes f) (X)P_1( f\bigotimes f\bigotimes f)(X) dX,$
where $P_1: L^2(\bf{R}^3)\to L^2(\bf{R}^3)$ is the projection operator onto the subspace of functions on $\bf R^3$ which are invariant under the isometries which fix the direction $(1,1,1)$. ( We have a similiar representation formula when $d=2$. )
The paper innovatively combines this formula with monotonicity given by some heat-flow version of Cauchy-Schwartz inequality together to prove the following main theorem.
Let $f\in L^2(\bf{R}^d).$ If $(p,q,d)$ is Schr\”odinger admissible and $q$ is an even integer which divides $p$ then the one-parameter family of functions
$Q_{p,q}(t):= \|e^{is\Delta}(e^{t\Delta}|f|^2)^{1/2}\|_{ L^p_sL^q_x(\mathbf{R}\times \mathbf{R}^d)}$
is nondecreasing for all $t>0$; i.e., $Q_{p,q}(t)$ is nondecreasing in the case $(1,6,6), (1,8,4)$ and $(2,4,4)$. Next we show how to prove this theorem in the case $(1,6,6)$.
${\bf 1. \text{ Monotonicity of } \Lambda(t)}$
Given $n\in \mathcal{N}$ and nonnegative integrable functions $f_1, f_2$ on $\bf R^n$, we define $\Lambda(t):=\int_{\bf R^n} (e^{t\Delta}f_1)^{1/2}(e^{t\Delta}f_2)^{1/2}$. Then $\Lambda(t)$ is nondecreasing for all $t>0$. Instead of repeating the proof in the paper of using the convolution with the heat-kernel, we will try working out how it is deduced from the divergence theorem. Our goal is to prove the following explicit formula
$\Lambda'(t)=\frac 14 \int_{\bf R^n} |\nabla (\log e^{t\Delta}f_1)-\nabla (\log e^{t\Delta}f_2)|^2 (e^{t\Delta}f_1)^{1/2}(e^{t\Delta}f_2)^{1/2}.$
Let $U_j:=e^{t\Delta }f_j=\frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} e^{-\frac {|x-y|^2}{4t}}f_j(y)dy$ for $j=1,2$. Then
$\partial_t U_j=-\frac {n}{2t} U_j +\frac {1}{(4\pi t)^{n/2}}\int_{\bf R^n}e^{-\frac {|x-y|^2}{4t}}\frac {|x-y|^2}{4t^2}f_j(y)dy.$
We write $U_j= \frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} e^{-\frac {|y|^2}{4t}}f_j(x-y)dy$. Then
$\frac {d^2}{d^2x_i}U_j=\frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n} (\frac {y_i^2}{4t^2}-\frac {1}{2t})e^{-\frac {|y|^2}{4t}}f_j(x-y)dy$. Then
$\Delta U_j= \frac {1}{(4\pi t)^{n/2}} \int_{\bf R^n}(\frac {|y|^2}{4t^2} -\frac {n}{2t})e^{-\frac {|y|^2}{4t}}f_j(x-y)dy$.
Hence
$\partial_t U_j = \Delta U_j$. This is not surprising because $U_j$ formally solves the heat equation.
Let $V_j:= \nabla(\log U_j )$, then
$\partial_t (\log U_j) =\frac {(U_j)_t}{U_j}=\frac {\Delta U_j}{U_j}=div (V_j)+|V_j|^2$.
Hence we see
$\Lambda'(t) =\frac 12 \int_{\bf R^n} [\partial_t (\log V_1)+\partial_t \log V_2] \sqrt{U_1U_2}$
$\qquad =\frac 12 \int_{\bf R^n} [div (V_1)+div (V_2)+|V_1|^2+|V_2|^2] \sqrt{U_1U_2}$
$\qquad =\frac 14 \int_{\bf R^n } |V_1-V_2|^2 \sqrt{U_1U_2}.$
Here we have used,
$div(\sqrt(U_1U_2) V_1)=\sum_{i=1}^n (V_1)_i \sqrt{U_1U_2}+ (\frac {V_1}{2}+\frac {V_2}{2})\sqrt{U_1U_2}V_1$, which yields
$0=\int_{\bf R^n} div(\sqrt{U_1U_2} V_1)+div(\sqrt{U_1U_2} V_2)$
$\quad \int_{\bf R^n } \left(div (V_1)+div (V_2)+\frac {|V_1|^2}{2}+\frac {|V_2|^2}{2}+V_1V_2 \right) \sqrt{U_1U_2}$,
i.e.,
$\int_{\bf R^n } \left(div (V_1)+div (V_2)\right) \sqrt{U_1U_2}=-\int_{\bf R^n}\left(\frac {|V_1|^2}{2}+\frac {|V_2|^2}{2}+V_1V_2\right)\sqrt{U_1U_2}.$
${\bf 2. \text{ Monotonicity of } Q_{6,6}(t) }$
We start with a general representation formula for the projection map in $L^2$. For functions in $G\in L^2(\bf R^3)$ we write
$P_1 G(X) =\int_O G(\rho X) d\mathcal{H}(\rho),$
where $O$ is the group of isometries on $\bf R^3$ which fix the direction $(1,1,1)$, and $d\mathcal{H}(\rho)$ denotes the right-invariant Haar probability measure on $O$.
Let $F:=f\bigotimes f\bigotimes f$ and $F_\rho:=F(\rho \cdot)$. Then by using the fact the heat-flow operator commutes with tensor product and Hundertmark-Zharnistsky representation formula, we have
$Q_{6,6}^6=\frac {1}{\sqrt{12}} \int_O \int_{\bf R^3} (e^{t\Delta}|F|^2)^{1/2}(X) (e^{t\Delta}|F_\rho|^2)^{1/2}(X)dXd\mathcal{H}(\rho).$
This is in form of $\Lambda(t)$. Then the monotonicity of $Q_{6,6}$ follows from that of $\Lambda(t)$ and the non-negativity of the measure $d\mathcal{H}$.
${\bf 3. \text{ Gaussians are extremisers }}$
For $f\in L^2$, define $u(t,x)=H_t*|f|^2(x)$ with $H_t:=\frac {1}{(4\pi t)^{d/2}}e^{-\frac {|x|^2}{4t}}$, and the rescaled
$\tilde{u}(t,x) =t^{-d} u(t^{-2},t^{-1}x)=\frac {1}{(4\pi)^{d/2}}\int_{\bf R^d} e^{-\frac 14 |x-tv|^2}|f(v)|^2dv.$
Then
$Q_{6,6}(t^{-2}) =\|e^{is\Delta} (\tilde{u}(t,\cdot))^{1/2}\|_{L^6_{s,x}}.$
$\lim_{t\to \infty} Q_{6,6}(t)=\|e^{is\Delta}(H_1^{1/2})\|_{L^6_{s,x}} \|f\|_{L^2}.$
On the other hand,
$\lim_{t\to 0} Q_{6,6}(t)=\|e^{is\Delta}|f|\|_{L^6_{s,x}}.$
We observe that, by Plancherel theorem in both variables,
$\|e^{is\Delta}f\|^3_{L^6_{t,x}}=\|(fd\sigma)^\vee(fd\sigma)^\vee(fd\sigma)^\vee\|_{L^2_{s,x}}$
$\qquad = \|(fd\sigma)*(fd\sigma)*(fd\sigma)\|_{L^2_{\tau,\xi}}$
$\qquad \le \|(|f|d\sigma)*(|f|d\sigma)*(|f|d\sigma)\|_{L^2_{\tau,\xi}}=Q^3_{6,6},$
where $d\sigma$ denotes the surface measure of the paraboloid in $\bf R^2$. Hence we obtain
$\|e^{is\Delta}f\|_{L^6_{t,x}}\le \|e^{is\Delta}(H_1^{1/2})\|_{L^6_{s,x}} \|f\|_{L^2}.$
This shows that $H_1^{1/2}$ is an extremiser. So we are done.
${\bf 4. \text{ A final word}}$
You can see that the heat-flow deformation method is a very nice method; it has been applied by the authors in several settings such as the Hausdorff-Young inequality, and Young’s inequality (but I haven’t carefully read their previous works). It is also proved effective in treating $d$-linear analogues of the Strichartz estimate (Bennett-Carbery-Tao’s work) and in the setting of multilinear Brascamp_Lieb inequalities (Bennett-Carbery-Christ-Tao) (I admitted that I haven’t read them carefully either). The “disadvantage” is that this method doesn’t charaterize the set of extremisers. Of course, we can’t hope it is a tool of catch-all. The question of charaterization is a completely different problem.
Posted by: Shuanglin Shao | October 1, 2008
## Readings on sharp Strichartz inequalities for the Schrodinger equation
Since last weekend, I have read several papers due to Bennett-Bez-Carbery-Hundertmark, Carneiro, Foschi, Hundertmark-Zharnitsky and on the sharp Strichartz inequality
$\|e^{it\Delta}f\|_{L_{t,x}^{2+4/d}(\Bbb R\times \Bbb R^d)}\le C\|f\|_{L^2_x(\Bbb R^d)}$
where $e^{it\triangle}f$ is the solution to the following free Schrodinger equation
$iu_t+\triangle u=0$
with initial data $u(0,x)=f(x)$, where $u(t,x):\Bbb R\times {\Bbb R}^d\to \Bbb C$.
I am happy that the latest two papers cited mine on the existence of a maximiser for the nonendpoint Strichartz inequality. For my own benefits, I would like to record here the main idea of their proofs respectively.
I choose to start with Hundertmark-Zharnitsky’s paper, “on sharp Strichartz inequalities in low dimensions”(IMRN) since this paper introduces a beatiful argument which seems to be of general interest (at least essential to papers by Carneiro and Bennett-Bez-Carbery-Hundertmark). In this paper, Hundertmark-Zharnitsky obtained the sharp value for
$C_{p,q}:=\sup_{f\neq 0} \frac {\|e^{it\Delta}f\|_{L^q_sL^r_x(\Bbb R\times {\Bbb R}^d)}}{\|\|_{L^2_x({\Bbb R}^d)}}$
when $q=r=2+4/d$ in lower dimensions $d=1,2$. They first built an interesting representation formula for the Strichartz norm $\|e^{it\Delta}f\|^{2+4/d}_{L^{2+4/d}(\Bbb R\times {\Bbb R}^d)}$ as follows,
If $d=1$, then
$\int\int_{\Bbb R} |e^{it\Delta}f|^6dxdt=\frac 1{\sqrt{12}}\langle f\bigotimes f\bigotimes f, P_1(f\bigotimes f\bigotimes f)\rangle_{L^2_x({\Bbb R}^3)},$
where $f\bigotimes g$ denotes the usual tensor product and $P_1:L^2_x({\Bbb R}^3)\to L^2_x({\Bbb R}^3)$ denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of ${\Bbb R}^3$ which keep the $(1,1,1)$ direction fixed.
If $d=2$,
$\int\int_{{\Bbb R}^2} |e^{it\Delta}f|^4dxdt=\frac 14\langle f\bigotimes f, P_2(f\bigotimes f)\rangle_{L^2_x({\Bbb R}^4)},$
$P_2:L^2_x({\Bbb R}^4)\to L^2_x({\Bbb R}^4)$ denotes the orthogonal projection to the closed linear subspace which consists of functions invariant under rotations of ${\Bbb R}^4$ which keep the $(1,0,1,0)$ and $(0,1,0,1)$ directions fixed.
They proved them in a simple way. we only look at the $d=1$ case and its explicit maximizers. By using the definition of the delta function $\delta(\xi)=(2\pi)^{-1}\int e^{-ix\xi}dx,$ they expressed the left side $\int\int_{\Bbb R} |e^{it\Delta}f|^6dxdt$ as
$\frac 1{2\pi}\int \int_{{\Bbb R}^3}\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2)\overline{f\bigotimes f\bigotimes f(\eta)}f\bigotimes f\bigotimes f(\zeta)d\eta d\zeta.$
This leads to define the following symmetric linear operator: for $G\in \mathcal{C}_0^\infty({\Bbb R}^3)$,
$A_1(G)(\eta):=\frac 1{2\pi}\int_{{\Bbb R}^3}G(\zeta)\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2)d\zeta$
Then they showed that $A_1$ is a bounded operator with operator bound $\frac 1{\sqrt{12}}$ on $\mathcal{C}_0^\infty({\Bbb R}^3)$ by showing the measure $m_{\eta} (d\zeta):=\frac {\sqrt{3}}{\pi}\delta((1,1,1)\cdot(\eta-\zeta))\delta(|\eta|^2-|\zeta|^2)$ is a probability measure on ${\Bbb R}^3$ for almost every $\eta\in {\Bbb R}^3$. Finally they extended $A_1$ onto the whole $L^2_x({\Bbb R}^3)$ and showed that $A_1$ is a multiple of the orthogonal projection $P_1$. Now they obtain the representation formula for the Strichartz norm.
Once having representation formula, they obtained that
$\|e^{it\Delta}f\|^6_{L^6_{t,x}(\Bbb R\times {\Bbb R}^3)}\le \frac 1{\sqrt{12}}\langle f\bigotimes f\bigotimes f, P_1(f\bigotimes f\bigotimes f)\rangle_{L^2_x({\Bbb R}^3)},$
which gives $C_{6,6}\le \frac 1{\sqrt{12}}$. Also they saw that in order to have equality, the function $f\bigotimes f\bigotimes f$ must lie in the rangle of $P_1$, that is to say, it is invariant under rotations of ${\Bbb R}^3$ which fix $(1,1,1)$. Obviously, all functions of the form
$Ae^{(-\lambda+i\mu )x^2+cx}$
with $\lambda >0, \mu\in \Bbb R, A\in \Bbb C$, i.e., Gausssians, are maximisres.
In the second part of their paper, they managed to show that the Gausssians turn out to be the only maximizers in the following three steps,
“1”. Assume $f$ is a maximizer for the Strichartz inequality (then $f\bigotimes f\bigotimes f$ is in the range of $P_1$). Then $Q_t(f)$ never vanishes for all $t>0$. Here $Q_t(f)$ is the convolution of $f$ with the approximation to identity $Q_t(f):=\frac {1}{(2\pi t)^{1/2}}\int e^{-\frac{|x-y|^2}{2t} }f(y)dy$.
“2”. $Q_t(f)$ never vanishes and is differentiable, then $Q_t(f)$ is a Gausssian. hence as a limit, $f$ is a Gausssian too.
The fact “2” above is actually the following general theme: if $f$ is differentialable and never vanishes and $f\bigotimes f\bigotimes f$ is in the range of $P_1$, then $f$ is a Gausssian, which is proven by working out the explicit forms of the rotations $M(\theta)$ in ${\Bbb R}^3$ which keep $(1,1,1)$ invariant and use $h(\eta)=h(M(\theta)\eta)$ for all differentiable functions invariant under rotations. At this step, we note that $f\bigotimes f\bigotimes f$ is a product of one dimensional function, which we need in the one dimensional argument. In higher dimensions, for $f\in L^2_x({\Bbb R}^d)$ and $f\bigotimes \cdots\bigotimes f$ invariant under rotations which keep $(e_i,\cdots, e_i)_{1\le i\le d}$ with standard basis $e_i\in {\Bbb R}^d$, we will have to prove that $f$ is a product of one dimensional functions.
Recently, Carneiro generaized this argument to a prove a sharp form of “Strichartz inequalities” as follows, for $k\in \mathcal{Z}, k\ge 2$ and $(d,k)\neq (1,2)$,
$\|u(t,x)\|_{L^{2k}_tL^{2k}_x(\Bbb{R}\times \Bbb{R}^d)}\le \left(C_{d,k}\int_{\Bbb{R}^{dk}} |\hat{F}(\eta)|^2K(\eta)^{\frac {d(k-1)-2}{2}}\right)^{1/2k}d\eta$
with
$C_{d,k}=[2^{d(k-1)-1}k^{d/2}\pi^{(d(k-1)-2)/2}\Gamma(\frac {d(k-1)}{2})]^{-1}$
and $F(\eta)=f(\eta_1)\cdots f(\eta_d)$ with $\eta_i\in \Bbb{R}^d$ for $1\le i\le d$ and the kernel $K(\eta)=\frac{1}{k}\sum_{1\le i.
This inequality is sharp if and only if $f$ is a Gaussian.
How to find $K$ is mysterious for me here.
(I leave the discussion on Foschi’s and Bennett-Bez-Carbery-Hundertmark’s papers to a later time) | {"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": 331, "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": 0.9746270775794983, "perplexity": 359.1759335714007}, "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-2017-13/segments/1490218190295.4/warc/CC-MAIN-20170322212950-00637-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/175611-green-s-formula.html | # Math Help - Green's formula
1. ## Green's formula
Find by three methods a Green's formula for the operator $B=\frac{d^4}{dx^4}$.
$\displaystyle\int_a^bf^{(4)}g \ dx$
Need a lot of help here, thanks.
2. My hint was to use integration by parts 4 times.
$\displaystyle f'''g-fg'''+f'g''-f''g'+\int_a^bfg^{(4)}dx$
Now, it says notice that $B=A^2$ where $A=-\left(\frac{d^2}{dx^2}\right)$.
Show that $f'''g-fg'''$ is a derivative [analogous to formula (4.6.1)]
4.6.1:
$f''g-fg''=\frac{d}{dx}(f'g-fg')$
Not sure how to do that, but I assume it isn't too difficult. | {"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": 7, "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": 0.9079516530036926, "perplexity": 891.4300909487799}, "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/1413507443598.37/warc/CC-MAIN-20141017005723-00253-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=VectorCalculus/VectorPotential&L=F | VectorPotential - Maple Help
VectorCalculus
VectorPotential
compute a vector potential of a vector field in R^3
Calling Sequence VectorPotential(v)
Parameters
v - vector field or Vector valued procedure; specify the components of the vector field in R^3
Description
• The VectorPotential(v) command computes a vector potential of the vector field v. The result is a vector field F such that $\mathrm{Curl}\left(F\right)=v$. If a vector potential does not exist, NULL is returned.
• If v is a vector field, a vector field is returned. If v is a procedure, a procedure is returned.
Examples
> $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
> $\mathrm{SetCoordinates}\left('\mathrm{cartesian}'\left[x,y,z\right]\right)$
${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (1)
> $v≔\mathrm{VectorField}\left(⟨y,-x,0⟩\right)$
${v}{≔}\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (2)
> $\mathrm{VectorPotential}\left(v\right)$
$\left({-}{x}{}{z}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{y}{}{z}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (3)
>
$\left({y}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}{x}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (4)
> $\mathrm{SetCoordinates}\left('\mathrm{cylindrical}'\left[r,\mathrm{\theta },z\right]\right)$
${{\mathrm{cylindrical}}}_{{r}{,}{\mathrm{\theta }}{,}{z}}$ (5)
> $v≔\mathrm{VectorField}\left(⟨r,0,-2z⟩\right)$
${v}{≔}\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({-}{2}{}{z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (6)
> $\mathrm{VectorPotential}\left(v\right)$
$\left({0}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({-}{r}{}{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{z}{-}{r}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}{}{z}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{z}}$ (7)
>
$\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}{+}\left({0}\right){\stackrel{{_}}{{e}}}_{{\mathrm{θ}}}{+}\left({-}{2}{}{z}\right){\stackrel{{_}}{{e}}}_{{z}}$ (8) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 18, "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": 0.9967063665390015, "perplexity": 1674.6521874772832}, "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/1659882573197.34/warc/CC-MAIN-20220818124424-20220818154424-00747.warc.gz"} |
https://www.nature.com/articles/s41467-022-29808-1?error=cookies_not_supported&code=dbf54f16-ca86-4939-b227-f722a6b764d0 | ## Introduction
A crucial, but currently unresolved, question in climate change research is how the functioning of complex ecosystems will respond to long-term warming. This challenge has typically been approached using general relationships for the size- and temperature-dependence of species-level metabolic rates in ecosystem-scale mathematical models1,2. Most of these studies assume, based on the Metabolic Theory of Ecology (MTE), that whole-organism metabolic rate increases with size (typically measured as body mass) raised to the three-quarter power, and with temperature according to the Boltzmann–Arrhenius equation with an activation energy of ~0.65 eV3. The assumed universality of these specific values has been questioned more recently, however, given that they vary both within and across species4,5. Furthermore, species may have the capacity to alter their metabolic traits through acclimation, evolutionary adaptation or both6,7,8,9,10. We refer to this flexibility in species-level thermal responses henceforth as ‘metabolic plasticity’, which can be thought of simply as a group of similar organisms altering their metabolic rate in a similar way when the environment changes. Metabolic plasticity should ultimately have consequences for ecosystem functioning by altering energy flow through the food web11, but evidence for such changes across species and trophic levels in natural systems is still lacking.
To address this knowledge gap, we measured oxygen consumption rates (a standard measure of metabolic rate12) of freshwater invertebrates in a large-scale natural warming experiment. Our study site, the Hengill catchment in Iceland13,14,15,16,17,18, consists of multiple streams, each with a characteristic temperature regime (Supplementary Fig. 1), resulting from long-term geothermal heating of the underlying bedrock. The invertebrate populations in each stream have been exposed to a distinct thermal regime over many generations, and thus provide an ideal natural experiment for elucidating the effects of chronic warming on physiology across trophic levels in the food web. This space-for-time substitution allows us to glimpse the potential impacts of centennial-scale warming19, with previous research in the Hengill system revealing effects of temperature on biodiversity13,14, community structure17,18 and ecosystem functioning15,16.
Our central hypothesis was that metabolic plasticity consistently declines with body size, given that smaller organisms tend to have faster rates of acclimation20 and adaptation21. To test this, we collected thousands of invertebrates from nine different streams spanning mean annual temperatures of 5–20 °C (Supplementary Figs. 12). Invertebrates are the dominant primary and secondary consumers in the streams13, and therefore central to energy fluxes through the food web17,18. We measured the individual-level routine metabolic rate (i.e., organisms exhibited some activity12) under acute exposure (experiments lasting 10–60 min) to temperatures of 5, 10, 15, 20 or 25 °C in the laboratory (see Methods). This response of metabolic rate to short-term temperature change is a fundamental measure of organismal physiology and also governs the daily effects of temperature on energy fluxes through ecosystems1. Metabolic rate was only quantified for each individual at one acute temperature, after which we measured its body mass and confirmed its species identity via microscopy. After quality-control procedures (see Methods), this yielded data on 1359 individuals from 16 species representing 44 different populations (Supplementary Table 1). Note that these species were not found in every stream, but there were still multiple populations of each species, whereby a population is a unique species × stream combination.
## Results and discussion
We found that chronic exposure to warmer environments (i.e., stream temperatures) altered both the size- and (acute) temperature-dependence of metabolic rate in two fundamental ways (Table 1a). First, higher temperatures reduced the allometric scaling exponent of metabolic rate because smaller organisms had a more elevated metabolism after long-term warming than larger ones (Fig. 1a, b). This key finding suggests that metabolic plasticity can be modelled mechanistically, irrespective of species identity, through its relationship with body mass. Second, chronic warming raised the activation energy (thermal sensitivity) of metabolic rate, whereby organisms from warmer streams had a more elevated metabolism under acute warming (Fig. 1c, d). Higher metabolic rates can provide organisms with greater scope for faster growth or improved performance through increased activity, foraging, and competitive ability6,22,23. However, there is also a greater energy cost, and higher metabolism becomes disadvantageous if resource supply cannot keep pace23,24. These changes in two fundamental features of thermal physiology across multiple species exposed to chronic warming indicate a degree of metabolic plasticity that has never been documented before. This may have significant implications for ecosystem-level responses to climate change, such as altering total energy flux through the food web, or the amount of carbon emitted to the atmosphere through ecosystem respiration. Supplementary Analyses indicated no influence of spatial autocorrelation on these results, i.e., warm and cold streams are sufficiently mixed in the landscape (Supplementary Fig. 3). In addition, the optimal model describing our data (Table 1a) remained the same after accounting for phylogenetic information, suggesting that both evolutionary and acclimatory processes play important roles in shaping metabolic plasticity (Supplementary Tables 23; Supplementary Figs. 45; see Supplementary Analyses).
We used a general mathematical model of ecosystem energy fluxes to explore the potential implications of these empirical findings25. We parameterised the model with previously published empirical biomass and dietary data18 from the study system and physiological rates that incorporated either metabolic plasticity (our new finding; Table 1a) or fixed thermal responses (the classical MTE assumption; Table 1b; see Methods). Note that the size-dependent nature of the model makes it independent of species identity and enables it to span multiple trophic levels26, which should make it generally applicable to other ecosystems. Our estimates of energy flux reflected measurements of ecosystem respiration taken from the same streams15 (r2 > 0.70; Supplementary Fig. 6), showing that individual-level metabolic rate measurements can be used to predict ecosystem functioning.
Next, we used both models to estimate the change in energy flux under a global warming scenario of +2 °C, predicted for the end of the century under intermediate IPCC scenarios of greenhouse gas emissions19. We expected higher energy flux in the model with metabolic plasticity due to the greater scope for elevated metabolism following long-term warming by smaller organisms near the base of the food web (Fig. 1). We found that warming always increased energy flux from resources to consumers, and this was greater by 59 ± 9% (mean ± standard error) for the model that included metabolic plasticity (Wilcoxon test: V = 12, p = 0.004; Fig. 2a, b). This was largely driven by increased energy flux from primary producers to herbivores (Wilcoxon test: V = 11, p = 0.003; Fig. 2c), with no significant differences between the models for detritivorous (Wilcoxon test: V = 33, p = 0.121; Fig. 2d) or predatory fluxes (Wilcoxon test: V = 55, p = 0.572; Fig. 2e).
These results indicate that current predictive models that ignore metabolic plasticity may substantially underestimate the changes in ecosystem fluxes under future warming. The associated increase in respiratory losses could help to reconcile the apparent paradox of amplified ecosystem respiration after 7 years of warming in a pond mesocosm experiment, relative to the effects after just 1 year of warming27. If organisms were adapting to mitigate the effects of warming over longer timescales, then ecosystem respiration should converge with the controls in this experiment. The amplification of ecosystem respiration is thus surprising and should lead to greater carbon emissions to the atmosphere27. Our model predictions for amplified ecosystem fluxes following chronic exposure to warmer conditions thus highlight the value of using size-dependent metabolic plasticity to predict these surprising long-term warming effects on ecosystems. More generally, accounting for different rates of metabolic plasticity among species could provide a better understanding of how species-specific responses to global warming may sum up to alter ecosystem functioning.
To the best of our knowledge, this is the first study to identify consistent effects of chronic warming on the allometric scaling and activation energy of metabolic traits across trophic levels, and determining the mechanistic basis is an important area for future research. It is important to note that our study system is near the Arctic region, where organisms are more likely to be energetically constrained by the colder environment, and thus have greater scope for elevating their metabolism to increase foraging and growth rates as temperature increases (i.e., maximising their energy gain). In contrast, tropical organisms may find it more beneficial to downregulate their metabolism to minimise energy loss and avoid heat stress28, highlighting the potential for local climate to interact with metabolic plasticity. Our findings may thus be most relevant for high latitude ecosystems, with temperate and tropical comparisons a priority for further studies on the topic. Our modelling framework should also be tested against ecosystems with different distributions of body sizes and trophic interactions to test the generality of our findings beyond our focal study system.
The levels of metabolic plasticity we report here (Fig. 1) are likely due to a combination of acclimation to warmer conditions (non-heritable changes within an organism’s lifetime) and evolutionary adaptation (heritable changes over many generations). We did not empirically quantify the relative contribution of these two types of change, which is unfeasible to disentangle for multiple species in complex natural ecosystems. However, an exploration of phylogenetic structure in our data suggests that evolutionary processes explain approximately half of the variability in metabolic rate (Supplementary Fig. 4). Most of the organisms in our study (13 of 16 species; see Supplementary Table 1) are larval invertebrates with an aerial adult phase, so genetic mixing between populations of the same species across streams is likely given that all the streams lie within 2 km of each other14. This suggests that thermal adaptation plays a weaker role in driving metabolic plasticity in our space-for-time study than might occur in response to long-term warming. Our results may thus be a conservative estimate of future change if genetic adaptation further increases the scope for metabolic plasticity over time.
Activity levels can be an important source of variation in estimates of temperature effects on metabolic rate29. Organisms in our experiments were confined to small glass vials, and thus their activity was constrained during the experiments compared to normal activity. As such, our measurements are much closer to resting than maximum metabolic rate, and temperature effects are thus likely to indicate differences in energy allocated to growth, rather than activity. Nevertheless, quantification of activity levels and/or measurement of resting and maximum metabolic rates would be needed to disentangle the relative contributions of behavioural and physiological plasticity to the observed changes in rates of energy expenditure. Follow-up studies should prioritise this research gap.
Our findings have important implications for top predators, which are widely predicted to decline due to global warming11,30. Previous research in our study streams has shown that this is not necessarily the case and that larger organisms can thrive in warmer conditions if the production of resources is sufficient to meet their higher energy demands16,17. Thus, the scope for metabolic plasticity to increase energy flux from the base of the food web could help sustain large predators at higher trophic levels in the face of global warming. Understanding all these effects will require timescales of observation sufficient to disentangle adaptation from acclimation via decadal-scale warming experiments11,31 and meta-analyses of time series data that incorporate long-term temperature changes32. In the meantime, embedding metabolic plasticity in individual-to-ecosystem projections of climate change impacts may improve realism and help to account for some of the ecological surprises in response to warming that have been reported recently11,16,17,27.
## Methods
### Study system & organisms
The study was conducted in the Hengill valley, Iceland13,14,15,16,17,18 (N 64°03; W 21°18), which contains many streams of different temperature due to geothermal heating of the bedrock or soils surrounding the springs (Supplementary Fig. 1). The streams have been heated in this way for centuries33 and are otherwise similar in their physical and chemical properties13,18, providing an ideal space-for-time substitution in which to measure species responses after chronic exposure to different temperatures6,34. Fieldwork was performed in the summers of 2015–2018, between May and July. Stream temperatures were logged every 4 h using Maxim Integrated DS1921G Thermochron iButtons submerged in each stream (Supplementary Fig. 2). The average stream temperature over this study period was used as a measure of chronic temperature exposure, encompassing at least the lifetime of every invertebrate species under investigation (and potentially multiple generations6,35).
Invertebrates were collected from nine streams spanning a temperature gradient of 5–20 °C across the entire study system (Supplementary Figs. 12). The streams exhibit some differences in the annual variability of their thermal regimes, but there are examples of both cold and warm streams that have high (IS12 and IS2) and low (IS13 and IS8) variability throughout the year. Our main finding is also robust to the inclusion of stream temperature variability as a random effect in our modelling framework (Supplementary Table 4; Supplementary Fig. 7). Note that we present temperature data from 15 streams in Supplementary Fig. 2, but it was not logistically feasible to study acute thermal responses of invertebrates collected from all of them, thus we focused on a subset of nine streams that best spanned the temperature gradient. The remaining six streams were included in other studies from the system, quantifying the biomass of all the constituent species17, describing food web structure18, and measuring whole-stream respiration15 (described in detail below).
Individual organisms were stored in containers within their ‘home stream’ until the end of each collection day, when they were transported within 1 h to the University of Iceland and then transferred into 2 L aquaria filled with water from the main river in Hengill, the Hengladalsá. The water was passed through a 125 µm sieve to ensure no organisms or filamentous algae entered the aquaria, and thus limiting the potential food available to the study organisms. The aquaria were continuously aerated in temperature-controlled chambers set to the home stream temperature of the organisms during sampling, which were maintained without food for at least 24 h to standardise their digestive state prior to metabolic measurements36. While we did not observe any cannibalism or organisms feeding on dead bodies in the laboratory, we cannot rule out the possibility that organisms fed on fine algal or detrital particles in the water, thus increasing variability in our metabolic measurements due to differences in digestive state.
### Quantifying metabolic rates
Experiments were carried out to determine the effects of body mass, acute temperature exposure (5, 10, 15, 20 and 25 °C), and chronic temperature exposure (i.e., average stream temperature) on oxygen consumption rates as a measure of metabolic rate3,12. Before each experiment, individual organisms were confined in glass chambers in a temperature-controlled water bath and slowly adjusted to the (acute) experimental temperature over a 15 min period to avoid a shock response. Glass chambers ranged in volume from 0.8–5 ml and scaled with the size of the organism. The glass chambers were filled with water from the Hengladalsá, which was filtered through a 0.45 µm Whatman membrane after aeration to 100% oxygen saturation. A magnetic stir bar was placed at the bottom of each chamber and separated from the organism by a mesh screen. In each experiment, one individual organism was placed in each of seven chambers and the eighth chamber was used as an animal-free control to correct for potential sensor drift. The chambers were sealed with gas-tight stoppers after the 15 min acclimatisation period, ensuring there was no headspace or air bubbles.
Oxygen consumption by individual organisms was measured using an oxygen microelectrode (MicroRespiration, Unisense, Denmark), fitted through a capillary in the gas-tight stopper of each chamber37. A total of 330 s measurement periods were recorded for each individual, where dissolved oxygen was measured every second. Oxygen consumption rate was calculated as the slope of the linear regression through all the data points from a single chamber, corrected for differences in chamber volume and the background rate measured from the control chamber (which was never >5% of the measured metabolic rates). We converted the units of this rate (µmol O2 h−1) to energetic equivalents (J h−1) using atomic weight (1 mol O2 = 31.9988 g), density (1.429 g L−1), and a standard conversion38 (1 ml O2 = 20.1 J). Organisms generally exhibited some activity during experiments, thus these measurements can be classified as routine metabolic rates12, which are more reflective of energy expenditure in field conditions. Nevertheless, activity levels were minimal due to the space constraints of the chambers (volume equal to 5–100 times the mass of the measured organism), indicating that the measured rates were likely to be closer to resting metabolic rates. Oxygen concentrations were never allowed to decline below 70% to minimise stress and avoid oxygen limitation. The system was cleaned with bleach at the end of each measurement day to avoid accumulation of microbial organisms on the insides of glass chambers and the water bath. In total, oxygen consumption rates were measured for 1819 individuals, none of which were ever reused in another experiment, thus every data point in the analysis corresponds to a single new individual (see below for details of how this dataset was curated to the final analysed subset of 1359 individuals based on quality-control procedures).
Following each experiment, individuals were preserved in 70% ethanol and later identified to species level under a dissecting microscope, except for Chironomidae, which were identified by examining head capsules under a compound microscope39. A linear dimension was precisely measured for every individual using an eyepiece graticule and converted to dry body mass using established length-weight relationships (Supplementary Table 1).
### Statistical analysis
All statistical analyses were conducted in R 4.0.2 (see the Supplementary Note for full details of statistical R code). According to the Metabolic Theory of Ecology3 (MTE), metabolic rate, I, depends on body mass and temperature as:
$$I={I}_{0}{M}^{b}{e}^{{E}_{A}{T}_{A}},$$
(1)
where I0 is the intercept, M is dry body mass (mg), b is an allometric exponent, EA is the activation energy (eV), and TA is a standardised Arrhenius temperature:
$${T}_{A}=\frac{{T}_{{acute}}-{T}_{0}}{k{T}_{{acute}}{T}_{0}}.$$
(2)
Here, Tacute is an acute temperature exposure (K), T0 sets the intercept of the relationship at 283.15 K (i.e., 10 °C), and k is the Boltzmann constant (8.618 × 10−5 eV K−1). We performed a multiple linear regression (‘lm’ function in the ‘stats’ package) on the natural logarithm of Eq. (1) to explore the main effects of temperature and body mass on the metabolic rate of each population (i.e., species × stream combination) in our dataset3. Following these analyses, we excluded populations where n < 10 individuals, r2 < 0.5, and p > 0.05 for any term in the model (see Supplementary Table 5, Supplementary Figs. 89). This excluded any poor quality species-level data and resulted in 1359 individuals from 44 populations for further analysis. Note that we find the same overall conclusion if we analyse the entire dataset (Supplementary Table 6, Supplementary Fig. 10).
To determine whether chronic temperature exposure alters the size- and acute temperature-dependence of metabolic rate, we added a term for chronic temperature exposure to Eq. 1. We began our analysis by considering the natural logarithm of all possible combinations of the main and interactive effects in this model:
$${ln}I= {ln}{I}_{0}+b{ln}M+{E}_{A}{T}_{A}+{E}_{C}{T}_{C}+{b}_{A}{ln}M{T}_{A}+{b}_{C}{ln}M{T}_{C}+{E}_{{AC}}{T}_{A}{T}_{C}\\ +{b}_{{AC}}{ln}M{T}_{A}{T}_{C}.$$
(3)
Here, TC is a standardised Arrhenius temperature with Tchronic as a chronic temperature exposure (K) substituted for Tacute in Eq. (2). To determine the optimal random effects structure for this model, we compared a generalised least squares model of Eq. 3 with linear mixed-effects models (‘gls’ and ‘lme’ functions in the ‘nlme’ package) containing all possible subsets of the following random effects structure40:
$${random}={\sim} 1+{ln}\,M+{T}_{A}+{T}_{C}|{species}.$$
(4)
Here, we are accounting for the possibility that metabolic rate could be different for each species (i.e., a random intercept) and that the effect of body mass, acute temperature exposure, or chronic temperature exposure on metabolic rate could also be different for each species (i.e., random slopes).
The full random structure (Eq. 4) was identified as the best model using Akaike Information Criterion (ΔAIC > 31.2; see Supplementary Table 7). We used this random structure in subsequent analyses, set ‘method = ‘ML’’ in the ‘lme’ function, and performed AIC comparison on all possible combinations of the fixed-effect structure40 (i.e., Equation 3). The optimal model was identified as follows:
$${ln}I={ln}{I}_{0}+b{ln}\,M+{E}_{A}{T}_{A}+{E}_{C}{T}_{C}+{b}_{C}{ln}M{T}_{C}+{E}_{{AC}}{T}_{A}{T}_{C}.$$
(5)
Note that while the model with an additional interaction between ln(M) and TA performed similarly (ΔAIC = 0.2; see Supplementary Table 8), that term was not significant (t = −1.645; p = 0.1002). We set ‘method = ‘REML’’ before extracting model summaries and partial residuals from the best-fitting model40. Note that the models were always fitted to the raw metabolic rate data, with residuals only extracted for a visual representation of the best-fitting models, excluding the noise explained by the random effect of species identity (see R code in the Supplementary Note).
### Exploration of spatial autocorrelation
A Mantel test (‘mantel’ function in the ‘vegan’ R package) was used to test for spatial autocorrelation in the temperature gradient, by comparing pairwise temperature difference between streams to the pairwise distance between streams. Pairwise distances were calculated from GPS coordinates taken at the confluence of each stream with the main river and the ‘earth.dist’ function in the ‘fossil’ R package. This analysis revealed no significant relationship between pairwise temperature and pairwise distance between sites (Mantel r = −0.1293, p = 0.780).
In addition, we explored for spatial autocorrelation in the residuals of our optimal model (Table 1a) by generating an empirical semivariogram cloud, illustrating the squared difference between all pairwise residual data points as a function of the distance between the two points. We also calculated Moran’s I as a measure of spatial autocorrelation in the model residuals. The semivariogram indicated no clear patterns in the residuals as a function of the distance between data points (Supplementary Fig. 3) and there was no statistical evidence for spatial autocorrelation in the model residuals (Moran’s I = 0.1187, p = 0.453).
### Exploration of phylogenetic structure
To examine the influence of evolutionary relatedness on metabolic rate measurements, we reconstructed a time-calibrated phylogeny of the 16 species in our final dataset (Supplementary Table 1). To this end, we combined: (i) nucleotide sequences of the 5′ region of the cytochrome c oxidase subunit I gene (COI-5P) from the Barcode of Life Data System database41; (ii) tree topology information from the Open Tree of Life42 (OTL; v. 13.4); and (iii) previously reported divergence time estimates between pairs of genera from the TimeTree database43. More precisely, we were able to obtain COI-5P nucleotide sequences for 15 out of 16 species (Supplementary Table 2), which we aligned using the G-INS-i algorithm of MAFFT44 (v. 7.490). To constrain the topology of our phylogeny based on the results of previous studies, we queried the OTL via the ‘rotl’ R package45 (v. 3.0.11). This yielded topological information for all 16 species. Finally, we manually queried the TimeTree database to obtain node age estimates. We only used three such estimates that (a) were based on more than five previous studies and (b) did not force any tree branches to have a length of zero.
We next used MrBayes46 (v. 3.2.7a) to obtain a time-calibrated phylogeny based on the sequence alignment, the OTL topology, and the node ages from TimeTree. For this, we first determined the most appropriate nucleotide substitution model using ModelTest-NG47 (v. 0.1.7). This was the General Time-Reversible model with Gamma-distributed rate variation across sites and a proportion of invariant sites. To allow branches of the phylogeny to differ in their rate of sequence evolution, we specified the Independent Gamma Rates model48 and used a normal distribution with a mean of 0.00003 and a standard deviation of 0.00001 as the prior for the mean clock rate. Finally, we executed four MrBayes runs with two chains per run for 100 million generations, sampling from the posterior distribution every 500 generations. Samples from the first ten million generations were treated as burn-in and were discarded. We examined the remaining samples to ensure that the four MrBayes runs had converged on statistically indistinguishable posterior distributions (i.e., all potential scale reduction factor values were below 1.1) and the parameter space was sufficiently explored (i.e., all effective sample size values were higher than 200). We summarised the sampled trees into a single time-calibrated phylogeny by calculating the median age estimate for each node (Supplementary Fig. 4).
To investigate the influence of evolutionary and acclimatory processes on metabolic rate, we first estimated the phylogenetic heritability of metabolic rate, i.e., the extent to which closely related species have more similar trait values than species chosen at random49. This metric takes values from 0 (trait values are independent of the phylogeny) to 1 (trait values evolve similarly to a random walk in the parameter space), with intermediate values indicating deviations from a pure random walk. To estimate phylogenetic heritability, we fitted a generalised linear mixed-effects model using the ‘MCMCglmm’ R package50 (v. 2.32). We set the natural logarithm of metabolic rate as the response variable and only an intercept as a fixed effect. We also specified a phylogenetic species-level random effect on the intercept, using the phylogenetic variance-covariance matrix obtained from our time-calibrated phylogeny. We used the default (normal) prior for the fixed effect, an uninformative Cauchy prior for the random effect, and an uninformative inverse Gamma prior for the residual variance. We then executed four independent runs for 500,000 MCMC generations each, with parameter samples being obtained every 50 generations after the first 50,000. We verified that sufficient convergence was reached, based on potential scale reduction factor and effective sample size values, as described earlier. Phylogenetic heritability was calculated as the ratio of the variance captured by the species-level random effect to the sum of the random and residual variances. The mean posterior phylogenetic heritability estimate of the natural logarithm of metabolic rate was 0.48. This means that nearly half (48%) of the variation can be explained by the evolution of metabolic rate along the phylogeny (Supplementary Fig. 4), with the other half arising from other sources including (but not necessarily limited to) acclimation and measurement error.
To describe the remaining unexplained variation, we fitted a series of models using MCMCglmm in R with all possible combinations of log body mass, acute temperature exposure, and chronic temperature exposure (fixed effects, as in Eq. (3) of the main text) and species-level random effects on the intercept and slopes (as in Eq. (4) of the main text). Furthermore, we specified both phylogenetic and non-phylogenetic variants of each model to understand if such a correction is warranted when the fixed effects are included. We determined the most appropriate model based on the Deviance Information Criterion51 (DIC). The optimal model (ΔDIC > 19; Supplementary Table 3; Supplementary Fig. 5) was found to include the full random effects structure (Eq. 4), the main effects of log body mass, acute temperature exposure, and chronic temperature exposure, the interaction between log body mass and chronic temperature exposure, and the interaction between acute temperature exposure and chronic temperature exposure (as for Eq. 5 in the main text), i.e., the same optimal model as that containing only species-level, rather than phylogenetic, information (Table 1a; Fig. 1). We calculated the marginal and conditional coefficients of determination to report the amounts of variance explained by the fixed and random effects, or left unexplained52. We found that the unexplained variation dropped from 52% to 8%, indicating that metabolic rate is strongly influenced by acclimatory processes in addition to evolutionary processes (see above).
It should be noted, however, that a definitive empirical quantification of the relative strength of evolutionary and acclimatory processes would require population genetics (to determine evolutionary divergent populations among streams), transcriptomics (to identify the expression of genes associated with thermal adaptation), and exhaustive common garden experiments (to disentangle acclimation from adaptation in all populations). Such an undertaking was logistically unfeasible in this study, but should be a focus for follow-up research on this topic.
### Modelling ecosystem-level energy fluxes
We used a recently proposed approach for inferring energy fluxes through trophic links25 to predict the effects of climate warming on ecosystem-level energy fluxes. We began by assuming that each stream ecosystem is at energetic steady state, i.e., for all n consumer species in the system:
$${G}_{i}\,=\,{L}_{i},\,i\,=\,1,\,2,\,\ldots ,\,n,$$
(6)
where Gi and Li are the energy gain and loss rates [J h−1], respectively, of the ith species in that stream. All basal species are implicitly assumed to be at energy balance. The two terms in Eq. (6) can be specified in a general way as
$${G}_{i}=\mathop{\sum}\limits_{k\,\in \,{{{{{{\rm{R}}}}}}}_{i}}{e}_{{ki}}{w}_{{ki}}{F}_{{ki}},{{{{{\rm{and}}}}}}$$
(7)
$${L}_{i}={Z}_{i}+\mathop{\sum}\limits_{j\,\in \,{{{{{{\rm{C}}}}}}}_{i}}{w}_{{ij}}{F}_{{ij}}.$$
(8)
Here, for the ith species, Ri and Ci are the sets of its resource and consumer species respectively, and Zi is its population-level energy loss rate stemming from mortality and metabolic expenditure on various activities realised over the timescale of the system’s dynamics. For the jth species feeding on the ith species, Fij is the maximum population-level feeding rate, eij is the assimilation efficiency (expressed as a proportion), and wij is the consumer’s preference for that species (all preferences for a given consumer sum to 1). Thus, the effective flux through a trophic link is $${e}_{{ki}}{w}_{{ki}}{F}_{{ki}}$$. Next, assuming the energy balance condition in Eq. 6 holds for all species, there are n linear equations (corresponding to the n consumer species) of the form:
$${G}_{i}-{L}_{i}=\mathop{\sum}\limits_{k\in {{{{{{\rm{R}}}}}}}_{i}}{e}_{{ki}}{w}_{{ki}}{F}_{{ki}}-\left({Z}_{i}+\mathop{\sum}\limits_{j\in {{{{{{\rm{C}}}}}}}_{i}}{w}_{{ij}}{F}_{{ij}}\right)=0,$$
(9)
which can be solved iteratively to obtain the unknown fluxes $${F}_{{ij},{i}\ne j}$$ of all consumer species, provided all the Zi’s, eij’s, and wij’s are known.
For this, we used the ‘fluxing’ function in the ‘fluxweb’ R package, parameterised with: (1) binary predation matrices for 14 stream food webs, characterised by 49,324 directly observed feeding interactions18; (2) biomasses for every species in each food web, characterised by 13,185 individual body mass measurements17; (3) assimilation efficiencies (eij’s) based on an established temperature-dependence and resource type (i.e., plant, detritus, or invertebrate)53; (4) preferences (wij’s) depending on resource biomasses; and (5) metabolic rates estimated using Eqs. (1) and (5) (assuming that I approximates Z). We treated TA in Eqs. (1) and (5) as the short-term temperature of the streams during food web sampling17,18 and TC in Eq. (5) as the long-term average temperature of the streams measured over the current study (Supplementary Fig. 2). It is important to note that the energy balance assumption (Eq. 6) implies that Zi in Eq. (8) is a combination of basal, routine, and active metabolic rates, stemming from the combination of activities realised over the timescale of the system’s dynamics. Therefore, our use of routine metabolic rate I is an underestimate of Z, which in turn means that the fluxes (which must balance the losses) are an underestimate.
Biomass and food web data were sampled in August 2008, with extensive protocols described in previous publications17,18. Briefly, this involved three stone scrapes per stream for benthic diatoms, five Surber samples per stream for macroinvertebrates, and three-run depletion electrofishing for fish. All individuals in the samples were identified to species level where possible and counted. Linear dimensions were measured for at least ten individuals of each species in each stream, with body masses estimated from length-weight relationships17. The population biomass of each species in each stream was calculated as the total abundance [individuals m−2] multiplied by the mean body mass [mg dry weight]. Food web links were largely assembled from gut content analysis of individual organisms collected from the streams (>87% of all links in the database), but additional links were added from the literature when yield-effort curves indicated that the diet of a consumer species was incomplete18.
### Validation of the ecosystem flux model using field data
To test whether our model of energy fluxes through trophic links was empirically meaningful, we calculated the sum of all energy fluxes through each stream food web to get the total energy flux, F (i.e., the sum of all $${e}_{{ki}}{w}_{{ki}}{F}_{{ki}}$$’s in Eq. 7). This quantity is a measure of multitrophic functioning and is expected to be positively correlated with the total respiration of each stream25. To evaluate this, we compared F to whole-ecosystem respiration rates measured in the same study streams15. The ecosystem respiration estimates were based on a modified open-system oxygen change method using two stations corrected for lateral inflows54,55. Essentially, this was an in-stream mass balance of oxygen inflows and outflows along stream reaches (17–51 m long). Oxygen concentrations were measured during 24- to 48 h periods from 6th to 16th August 2008, i.e., the exact same time period during which biomass and food web data were sampled to parameterise the energy flux model15. Dissolved oxygen concentrations were measured every minute with optic oxygen sensors (TROLL9500 Professional, In-Situ Inc. and Universal Controller SC100, Hach Lange GMBF). Hourly ecosystem respiration was calculated from the net metabolism at night, i.e., when no primary production occurs due to lack of sunlight.
### Modelling the consequences of metabolic plasticity for global warming impacts on ecosystem-level energy flux
In addition to total energy flux, F, we also calculated a modified total energy flux, F*, for each food web after considering a global warming scenario, where we added 2 °C to TA in Eq. (1) and to both TA and TC in Eq. (5). We calculated the change in total energy flux as a result of the global warming scenario as ΔF = F*F. We tested whether the (statistically optimal) model with metabolic plasticity (Eq. 5) predicted a greater ΔF across the 14 empirical stream food webs from the Hengill system than the model without metabolic plasticity using paired Wilcoxon tests (since the data did not conform to homogeneity of variance). To determine whether our results were consistent for all major trophic groupings in the system, we repeated the analysis after calculating the change in energy flux to herbivores (ΔFH = FH*FH), detritivores (ΔFD = FD*FD), and predators (ΔFP = FP*FP) in each stream.
### Reporting summary
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https://lavelle.chem.ucla.edu/forum/search.php?author_id=14020&sr=posts | ## Search found 66 matches
Sat Mar 16, 2019 10:00 pm
Forum: Administrative Questions and Class Announcements
Topic: LYNDON'S PORK RAMEN REVIEW
Replies: 37
Views: 2902
### Re: LYNDON'S PORK RAMEN REVIEW
lyndon da goat fr fr
Tue Mar 12, 2019 9:43 pm
Forum: Reaction Mechanisms, Reaction Profiles
Topic: pre-equilibrium question
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Views: 80
### Re: pre-equilibrium question
Only everything before the intermediate being formed contributes to the rate law so if it comes after the slow step it is not really accounted for.
Tue Mar 12, 2019 9:41 pm
Forum: Reaction Mechanisms, Reaction Profiles
Replies: 4
Views: 80
the steady step is never used in this class according to Dr. Lavelle since it requires too much math.
Tue Mar 12, 2019 9:40 pm
Forum: Zero Order Reactions
Topic: Reaction Rate
Replies: 10
Views: 323
### Re: Reaction Rate
k is a specific for a certain temperature, so if the reaction takes place at the specified temperature then the k would be the same.
Thu Mar 07, 2019 3:51 pm
Forum: Method of Initial Rates (To Determine n and k)
Topic: Order of Reaction
Replies: 5
Views: 152
### Re: Order of Reaction
The graph posted above is a great tool to use in order to understand the order of reactions.
Thu Mar 07, 2019 3:43 pm
Forum: Method of Initial Rates (To Determine n and k)
Topic: Rate constant k
Replies: 3
Views: 93
### Re: Rate constant k
Temoerature and the addition of enzymes would affect the rate constant, as temperature plays a significant role when determining chemical reaction rates.
Sun Mar 03, 2019 7:32 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Cell diagrams and solids
Replies: 9
Views: 198
### Re: Cell diagrams and solids
The electrons can not be conducted by themselves therefore an inert solid like Platinum could be used to transfer the electrons over to the other side.
Sun Mar 03, 2019 7:31 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Cell Diagrams (Using Platinum)
Replies: 10
Views: 184
### Re: Cell Diagrams (Using Platinum)
We use platinum because It is highly inert and is great conductor therefore it is used when there are no present solids
Sun Mar 03, 2019 7:29 pm
Forum: Galvanic/Voltaic Cells, Calculating Standard Cell Potentials, Cell Diagrams
Topic: Does anyone know if test 2 will be curved?
Replies: 15
Views: 355
### Re: Does anyone know if test 2 will be curved?
the probability of this test being curved is very unlikely since the class itself is not curved.
Sun Mar 03, 2019 7:27 pm
Forum: Balancing Redox Reactions
Topic: Test #2
Replies: 10
Views: 273
### Re: Test #2
The higher the reduction potential the easier it is for a species to be reduced and the lower the reduction potential the easier it is foe a species to be oxidized.
Sun Mar 03, 2019 7:26 pm
Forum: Balancing Redox Reactions
Topic: Strength of reducing agent
Replies: 10
Views: 270
### Re: Strength of reducing agent
The more positive a reduction potential is the easier it is for it to be reduced, and vice versa for an element to be oxidized. Therefore a lower reduction potential would make the best reducing agent.
Sun Mar 03, 2019 7:25 pm
Forum: Balancing Redox Reactions
Topic: Acidic vs basic solutions
Replies: 10
Views: 194
### Re: Acidic vs basic solutions
Both are balanced by adding water molecules to a the reactants or products, but for acidic reactions, they are balanced with H^+ while basic reactions are balanced by adding hydroxides to a side.
Thu Feb 21, 2019 8:52 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Gibbs Free Energy Meaning
Replies: 6
Views: 154
### Re: Gibbs Free Energy Meaning
Gibbs free energy is the total amount o energy available to preform work as not all the energy of the system can be converted to work since it is impossible to fully convert energy to work as some energy would be lost through other means.
Thu Feb 21, 2019 8:50 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Delta H
Replies: 5
Views: 118
### Re: Delta H
A change In enthalpy alone cannot determine the spontaneous nature of a system as the entropy of the system and the temperature the reaction is taking place at are also needed to determine the spontaneity of a reaction.
Thu Feb 21, 2019 8:49 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Units of Delta G
Replies: 5
Views: 146
### Re: Units of Delta G
The units for delta G happen to be the same for enthalpy which are J, Kj, KJ/mol, J/mol.
Thu Feb 21, 2019 8:48 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Gibbs Free Energy vs Entropy
Replies: 4
Views: 86
### Re: Gibbs Free Energy vs Entropy
Gibbs free energy is the amount of energy available that can be used to preform work with entropy measures the disorder of the system. Gibbs free energy is so important because it allows us to determine if a reaction would proceed by itself to form products which is crucial under standard temp and p...
Thu Feb 21, 2019 8:45 pm
Forum: Gibbs Free Energy Concepts and Calculations
Topic: Negative ∆G means spontaneous reaction?
Replies: 5
Views: 151
### Re: Negative ∆G means spontaneous reaction?
Yes a negative value resulting from the Gibbs free energy equation means that the reactants will make products spontaneously.
Mon Feb 04, 2019 4:33 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Open vs Closed System
Replies: 13
Views: 242
### Re: Open vs Closed System
The simplest way to put it is that in an open system heat and material can be exchanged, in a closed system only heat can be exchanged with the surroundings, but in an isolated system nothing can be exchanged with the surroundings.
Mon Feb 04, 2019 4:30 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Constant P
Replies: 4
Views: 98
### Re: Constant P
At standard pressure, the pressure equals 1 atm, but unless otherwise specified it can be different.
Sun Feb 03, 2019 8:56 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: delta H vs q
Replies: 9
Views: 159
### Re: delta H vs q
Delta H represent the heat that changed in reaction, while q represents the heat.
Sun Feb 03, 2019 8:51 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: finding W
Replies: 6
Views: 114
### Re: finding W
Yes, the integral just implies that a very small volume is being changed at every instant therefore it all can be summed up with an integral, while if the change of volume is large and sudden ten delta v can just be used.
Tue Jan 29, 2019 8:07 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: What is the difference between delta H and q?
Replies: 3
Views: 353
### Re: What is the difference between delta H and q?
q just refers to heat while delta h refers to the change of heat within a specific reaction under standard and constant pressure
Tue Jan 29, 2019 8:01 pm
Forum: Reaction Enthalpies (e.g., Using Hess’s Law, Bond Enthalpies, Standard Enthalpies of Formation)
Topic: Best Method
Replies: 7
Views: 173
### Re: Best Method
It all depends on the initial information given in the problem, there is no true best way to solve this questions but some options are better for certain types of questions.
Tue Jan 29, 2019 7:57 pm
Forum: Phase Changes & Related Calculations
Topic: State Properites
Replies: 7
Views: 147
### Re: State Properites
State properties do not depend on the path that is taken to achieve the current state while properties such as work and heat do depend on which pathway is taken.
Tue Jan 29, 2019 7:55 pm
Forum: Phase Changes & Related Calculations
Topic: Standard entalpy of formation
Replies: 6
Views: 125
### Re: Standard entalpy of formation
When a molecule is at its basic state(elemental form including diatomic atoms) the standard enthalpy of formation is 0
Tue Jan 29, 2019 7:46 pm
Forum: Phase Changes & Related Calculations
Topic: Delta Hº versus delta H
Replies: 7
Views: 243
### Re: Delta Hº versus delta H
They mean the same thing, one is just referring to the initial delta H.
Tue Jan 29, 2019 7:45 pm
Forum: Phase Changes & Related Calculations
Topic: Enthalpy signs
Replies: 13
Views: 199
### Re: Enthalpy signs
The enthalpy change is positive when a reaction gains heat (endothermic reaction) and is negative when the reactants release energy to the universe (exothermic reactions)
Tue Jan 29, 2019 7:43 pm
Forum: Phase Changes & Related Calculations
Topic: Heat Capacity
Replies: 10
Views: 194
### Re: Heat Capacity
Heat Capacity is the energy required to raise the temp by one degree Celsius while the molar heat capacity is the energy required to raise the temperature of one mole of substance by one degree celsius.
Mon Jan 21, 2019 3:02 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: Inert Gases
Replies: 7
Views: 165
### Re: Inert Gases
Inert gases are excluded from the equilibrium expression because they do not change the equilibrium concentration nor presser of the aqueous or gaseous species.
Mon Jan 21, 2019 3:00 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: 5% rule
Replies: 12
Views: 215
### Re: 5% rule
When dealing with weak acids or bases, the 5% rule can be applied to neglect the change of the reactants towards the products in order to make calculating the change significantly easier without use of the quadratic formula.
Mon Jan 21, 2019 2:57 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: Change in temperature's effect on K
Replies: 4
Views: 82
### Re: Change in temperature's effect on K
If the reaction is endothermic then increasing the heat would favor an equilibrium sitting to towards the product. Decreasing the heat would favor an equilibrium sitting towards the reactants. For exothermic the process is reversed as expected.
Mon Jan 21, 2019 2:52 pm
Forum: Applying Le Chatelier's Principle to Changes in Chemical & Physical Conditions
Topic: The Conjugate Seesaw
Replies: 13
Views: 245
### Re: The Conjugate Seesaw
In very simple terms the conjugate seesaw basically states that the stronger the base or acid is, the weaker its conjugate acid or base is.
Mon Jan 21, 2019 2:51 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Weak acid and its salt
Replies: 3
Views: 59
### Re: Weak acid and its salt
As mentioned in 14A, Group 1 mentally and group 17 halogens do not affect the pH of a solution at all. On the other hand group two metals slightly change the pH but not by a drastic amount.
Mon Jan 21, 2019 2:49 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Pure substances
Replies: 3
Views: 74
### Re: Pure substances
While at equilibrium only aqueous and liquid species are represented in the equilibrium expression as they are the only species can change the concentrations.
Mon Jan 21, 2019 2:45 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Temperature for Equilibrium
Replies: 4
Views: 82
### Re: Temperature for Equilibrium
Most equilibrium expressions will be given at 25 C or 298K, but this is not mandatory as equilibrium can occur at any given temperature depending on the species.
Mon Jan 21, 2019 2:44 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Strong acids and bases
Replies: 4
Views: 81
### Re: Strong acids and bases
Strong acids and bases can also be recognized by their percent dissociation. Strong bases and acids fully dissociate while weak ones hardly do.
Mon Jan 21, 2019 2:42 pm
Forum: Equilibrium Constants & Calculating Concentrations
Topic: Water in ICE tables
Replies: 10
Views: 214
### Re: Water in ICE tables
Since water is a pour solvent in can be left out of the ice table when calculating equilibrium concentrations since it is in excess.
Wed Dec 05, 2018 8:55 pm
Forum: Bronsted Acids & Bases
Topic: acetic acid vs formic acid
Replies: 2
Views: 90
### Re: acetic acid vs formic acid
The strength of an acid is all in relation to how easily the Hydrogen proton can separate from the compound, in this case, it is easier for formic acid to lose a hydrogen proton.
Wed Dec 05, 2018 8:51 pm
Forum: Bronsted Acids & Bases
Topic: Water as an Acid or Base
Replies: 4
Views: 140
### Re: Water as an Acid or Base
Due to it's neutral pH, water can be considered as either a base or an acid. It could gain a proton and become hydronium or lose a proton and become hydroxide.
Wed Dec 05, 2018 8:45 pm
Forum: Properties & Structures of Inorganic & Organic Bases
Topic: “Soapy” feel of bases
Replies: 5
Views: 351
### Re: “Soapy” feel of bases
Bases have the unique ability of dissolving acids, therefore when they interact with our oily skin, a slippery feel is produced.
Wed Dec 05, 2018 8:43 pm
Forum: Properties & Structures of Inorganic & Organic Bases
Topic: HF ion
Replies: 3
Views: 201
### Re: HF ion
The bond length between the Hydrogen and Flourine atom is very small due to FLourine's high electronegativity, therefore it is harder for the Hydrogen ion to dissociate.
Sun Dec 02, 2018 4:17 pm
Forum: Hybridization
Topic: Hybridization
Replies: 2
Views: 99
### Re: Hybridization
They are completely different atomic orbitals are the electrons surrounding an atom, while hybridization refers to the mixing of different orbitals to form a mixture of bonds.
Tue Nov 27, 2018 8:20 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: Test 3
Replies: 38
Views: 850
### Re: Test 3
the best way of identifying molecular shapes is by memorizing the designated shape according to how many species and lone pairs there are
Tue Nov 27, 2018 8:14 pm
Forum: Hybridization
Topic: Molecular Shape
Replies: 2
Views: 52
### Re: Molecular Shape
Yes lone pairs play a very important role in hybridization as they determine what energy levels are hybridized, therefore lone pairs must be accounted for since they allow for the correct hybridized species to be identified.
Tue Nov 27, 2018 8:10 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: Sigma and Pi bonds
Replies: 4
Views: 132
### Re: Sigma and Pi bonds
The main difference between sigma and pi bonds is that sigma bonds consist of one bond, while pi bonds consist of 2 bonds. Another difference between these two bonds is that sigma bonds allow for rotation while sigma bonds do not allow for any movement.
Sun Nov 18, 2018 1:47 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: Dipole-Dipole forces
Replies: 4
Views: 127
### Re: Dipole-Dipole forces
Dipole-dipole forces occur when their a great difference between the electronegativity of two atoms. When determining the dipole forces it is important to consider if they cancel or not because then polarity can be inferred.
Sun Nov 18, 2018 1:43 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: Bond Angles
Replies: 16
Views: 253
### Re: Bond Angles
Each molecular shape has its own respective bond angles, the easiest way to figure out these angles is just by memorizing them.
Sun Nov 18, 2018 1:40 pm
Forum: Determining Molecular Shape (VSEPR)
Topic: lone pairs
Replies: 11
Views: 204
### Re: lone pairs
Thu Nov 08, 2018 5:00 pm
Forum: Interionic and Intermolecular Forces (Ion-Ion, Ion-Dipole, Dipole-Dipole, Dipole-Induced Dipole, Dispersion/Induced Dipole-Induced Dipole/London Forces, Hydrogen Bonding)
Topic: Potential Energy of London interactions
Replies: 2
Views: 119
### Re: Potential Energy of London interactions
London Forces are the weakest forms of bonds created by instances moments of ionization, therefore bond length does not necessarily matter when discussing these bonds as their potential energy Is more influenced on the type of bond rather than distance.
Thu Nov 08, 2018 4:43 pm
Forum: Interionic and Intermolecular Forces (Ion-Ion, Ion-Dipole, Dipole-Dipole, Dipole-Induced Dipole, Dispersion/Induced Dipole-Induced Dipole/London Forces, Hydrogen Bonding)
Topic: Effect of Size of Atom/Molecule on Distortion
Replies: 2
Views: 92
### Re: Effect of Size of Atom/Molecule on Distortion
The size of an atom is determined by its atomic number, as the higher the atomic number gets the more protons are located in the nucleus and more electrons in the electron sphere thus making the atom larger. They all have something to do with the size of an atom
Thu Nov 08, 2018 4:41 pm
Forum: Interionic and Intermolecular Forces (Ion-Ion, Ion-Dipole, Dipole-Dipole, Dipole-Induced Dipole, Dispersion/Induced Dipole-Induced Dipole/London Forces, Hydrogen Bonding)
Topic: Dipole Moment
Replies: 3
Views: 104
### Re: Dipole Moment
Dipole moments occur instanously as electrons are pulled or pushed away by the electronegativity of the neighboring atom. These interactions are instances and last for a short period of time as electrons are continuously being pulled or pushed away. These instaneous shifts of electrons creates posit...
Thu Nov 01, 2018 5:31 pm
Forum: Ionic & Covalent Bonds
Topic: Covalent Bonds
Replies: 16
Views: 423
### Re: Covalent Bonds
Non metals generally do not become cations because they have a very high electron affinity therefore their ionization energies would be very high making it unlikely for them to lose an electron.
Thu Nov 01, 2018 5:21 pm
Forum: Coordinate Covalent Bonds
Topic: Coordinate Covalent Bond Definition
Replies: 14
Views: 726
### Re: Coordinate Covalent Bond Definition
Yes! Instead of one electron coming from each atom, a coordinate covalent bond uses two electrons that come from the same atom.
Thu Nov 01, 2018 5:14 pm
Forum: Formal Charge and Oxidation Numbers
Topic: Formal charge purpose
Replies: 40
Views: 1665
### Re: Formal charge purpose
Formal charge has to be considered when drawing lewis structures as it will help you figure out if there is a double bond or not while trying to figure out the most stable form of the compound.
Fri Oct 26, 2018 12:26 am
Forum: Electron Configurations for Multi-Electron Atoms
Topic: Shielding effect
Replies: 6
Views: 248
### Re: Shielding effect
Shielding protects the outer electrons from the pull of the nucleus through the inner electrons absorbing the pull.
Thu Oct 25, 2018 11:53 pm
Forum: Quantum Numbers and The H-Atom
Topic: Degeneracy
Replies: 7
Views: 473
### Re: Degeneracy
As far as degeneracy is concerned it is important to know that electrons in the same hydrogen atom have the same energy level. However, multi electron systems are not degenerate as with different sub shells there are different energies.
Thu Oct 25, 2018 11:47 pm
Forum: DeBroglie Equation
Topic: De Broglie wavelengths
Replies: 3
Views: 137
### Re: De Broglie wavelengths
De Brogile equation is used for any object with moment and wavelength, so in reality it could be applied to almost all scenarios although it would be pointless if it used on an object with a large wavelength.
Tue Oct 16, 2018 6:04 pm
Forum: Quantum Numbers and The H-Atom
Topic: Balmer/Lyman... Series
Replies: 4
Views: 92
### Re: Balmer/Lyman... Series
In regards to energy, the Lyman and Balmer series allow us to infer where the electron returns back to ground state. In the Lyman series, n=1 so the electron would lose energy and return to the first energy level. On the other hand, when using the Balmer series, we can infer that the electron return...
Tue Oct 16, 2018 5:59 pm
Forum: Photoelectric Effect
Topic: 1B.9: Trouble with exponents and units
Replies: 3
Views: 100
### Re: 1B.9: Trouble with exponents and units
To solve this question, you must calculate the total amount of energy released (32W x 2s)=64J and then consider the energy released by one photon with a wavelength 420nm, then divide the total energy/the energy of one photon. This will result in the correct answer. Also consider the appropriate SI u...
Tue Oct 16, 2018 5:53 pm
Forum: Properties of Electrons
Topic: Energy emitted by electrons
Replies: 6
Views: 142
### Re: Energy emitted by electrons
When considering the release of energy, it is important to remember that energy can not be created nor destroyed, only transferred. With this in mind, the energy released from an excited electron does not disappear, but instead is released into the universe thus conserving the total energy from the ...
Thu Oct 11, 2018 2:22 pm
Forum: Photoelectric Effect
Topic: Threshold energy
Replies: 5
Views: 310
### Re: Threshold energy
The threshold value needed to remove an electron can be treated like any other requirement needed before an action can occur. A particle needs to have a minimum amount (varying on the type of substance) of energy in order for an electron to be ejected. Taking this into consideration, if we know how ...
Thu Oct 11, 2018 2:15 pm
Forum: Properties of Light
Topic: Balmer series vs Lyman series
Replies: 4
Views: 98
### Re: Balmer series vs Lyman series
The main difference between the two series is the quantum levels they encompass. The Lyman series includes quantum level one which is why ultraviolet radiation is observed within this series(the leap from n=1 to n=2 realeses more energy than any other quantum jump that is why the radiation is uv.) O...
Thu Oct 11, 2018 2:02 pm
Forum: Properties of Light
Topic: Why Short Wavelengths Can Eject e-
Replies: 6
Views: 142
### Re: Why Short Wavelengths Can Eject e-
Longer wavelengths have less energy than shorter wavelengths due the energy = hc/wavelength formula that denotes that if the wavelength increases, then the energy of the particle will decrease which establishes a negative correlation. In conclusion, shorter wavelengths shoot out electrons compared t...
Mon Oct 01, 2018 6:32 pm
Forum: Balancing Chemical Reactions
Topic: Law of Conservation of Mass [ENDORSED]
Replies: 7
Views: 210
### Re: Law of Conservation of Mass[ENDORSED]
This is as simple as it gets: mass can not be created and mass cannot be destroyed. There will always be the same amount mass before and after a chemical reaction since mass cannot be created nor destroyed.
Mon Oct 01, 2018 6:28 pm
Forum: Significant Figures
Topic: Significant Figures
Replies: 10
Views: 327
### Re: Significant Figures
The full value of molar mass can be used to calculate whatever needs to be calculated, but the least amount of sig figs must be used when determining the final answer, as it would be inaccurate to have an answer that is more than the initial significant figures there given.
Mon Oct 01, 2018 6:17 pm
Forum: SI Units, Unit Conversions
Topic: Grams/mole
Replies: 12
Views: 379
### Re: Grams/mole
Both g/mol and gmol^-1 represent grams divided by moles while the latter just uses textbook notation in order to represent that grams is being divided by moles in a more comprehensive way. Feel free to use either or as they both represent the same thing. | {"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": 0.8452269434928894, "perplexity": 4380.637369059298}, "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/1579250593937.27/warc/CC-MAIN-20200118193018-20200118221018-00205.warc.gz"} |
http://math.stackexchange.com/questions/1874/presentation-of-the-fundamental-group-of-a-manifold-minus-some-points | # Presentation of the fundamental group of a manifold minus some points
I recently noticed a few things in some recent questions on MO:
1) the fundamental group of $S^2$ minus, say, 4 points, is $\langle a,b,c,d\ |\ abcd=1\rangle$.
2) The fundamental group of a torus minus a point is $\langle a,b,c\ |\ [a,b]c=1\rangle$.
I was just wondering if you have a manifold $M$, and you know a presentation of its fundamental group, can you quickly get a presentation of the fundamental group of $M$ minus $n$ points?
Of course, this could merely be a coincidence. These are both surfaces, and both their fundamental groups only have one relation. But perhaps there is more to it than that?
Steve
-
Are you sure that $\pi_1(T^2- \{pt\})$ has that fundamental group? Because $T^2 - \{pt\}$ is homotpic to $S^1 \vee S^1$ and then its fundamental group need to be $\mathbb{Z}^{*2}$. – Gianluca Jun 27 '15 at 19:50
The basic tool here is Van Kampen's theorem. Let $M$ be the manifold, and let $M' = M\setminus {x}.$ Let $D$ be a small ball around $x$. Then $M = M' \cup D,$ and $M \cap D = D\setminus {x}$ is homotopic to $S^{n-1}$ (here $n$ is the dimension of $M$).
So Van Kampen's theorem says that $\pi_1(M) = \pi_1(M')*_{\pi_1(S^{n-1})} \pi_1(D)$. If $n > 2,$ then $S^{n-1}$ is simply connected, and of course $\pi_1(D)$ is simply connected, and so this reduces to $\pi_1(M) = \pi_1(M')$; in other words, deleting points doesn't change $\pi_1$ in dimensions $> 2$. (You can probably convince yourself of this directly: imagine you have some loop contracting in a 3-dimensional space; then if you remove a point, you can always just perturb the contraction slightly so that it misses the point. Of course, this is not the case in two dimensions.)
If $n = 2$, then $\pi_1(S^1) = \mathbb Z$ is infinite cyclic, while $\pi_1(D)$ is trivial again. So we see that $\pi_1(M)$ is obtained from $\pi_1(M')$ by killing a loop. One can be more precise, of course (and I will restrict myself to the orientable case, so as to make life easier): if you have a genus $g$ surface, with $r>0$ punctures, and also $g + r > 1$, then $\pi_1$ is a free group on $2 g + r - 1$ generators. (Every puncture after the first adds another independent loop, namely the loop around that puncture.) If $g \geq 1,$ and $r = 0$, then $\pi_1$ is obtained via Van Kampen as above: you begin with a free group on $2g$ generators for $M'$, and then you kill off the loop around the puncture when you fill it in (so you get the standard presentation for the $\pi_1$ of a compact orientable surface of genus $g \geq 1$). (Conversely, going from the compact surface $M$ to the once-punctured surface $M'$ does not add any generators to $\pi_1$, but gets rid of a relation.) If $g = 0$ and $r \leq 1,$ then you have either a disk ($r = 1$) or a sphere ($r = 0$) and so $\pi_1$ is trivial. | {"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": 0.9342769384384155, "perplexity": 87.58557297007627}, "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-2016-22/segments/1464049270513.22/warc/CC-MAIN-20160524002110-00177-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://dsp.stackexchange.com/questions/69444/how-to-prove-a-train-of-sinc-pulses-in-digital-communicaton-system-are-orthogona | # How to prove a train of sinc pulses in digital communicaton system are orthogonal to each other?
Consider a train of sinc pulses: $$\phi_n(t)= \frac{\sin(\omega_M(t-nT_s))}{\omega_M(t-nT_s)}\quad; n=0,\pm1,\pm2,\dots$$ $$\quad$$where,$$\quad T_s=\frac{\pi}{\omega_M}$$
Now ,in order to show sinc pulses are orthogonal we need to prove: $$\int_{-\infty}^{\infty}\phi_n(t)\phi_k(t)dt=T_s \delta_{nk} \quad \dots(1)$$ where, $$\delta_{nk}$$ is kronecker's delta.
So, i began doing it as follows: $$T_s=\frac{\pi}{\omega_M}=\frac{\pi}{2\pi f_M}=\frac{1}{2f_M}=\frac{1}{f_N} \quad \dots(2)$$ where, $$f_N$$ is the Nyquist frequency $$\phi_0(t)=\frac{\sin(\omega_Mt)}{\omega_Mt}=\frac{\sin(2\pi f_Mt)}{2\pi f_Mt}=sinc(2f_Mt)=sinc(f_Nt) \quad \dots(3)$$ Now, $$\mathscr{F}\{ sinc(f_Nt) \}=\frac{1}{f_N} rect(\frac{f}{f_N})$$ ,where $$rect$$ is a rectangular function centred at origin and having width= $$f_N$$ $$\implies \mathscr{F}\{ sinc(f_N(t-nT_s)) \}=\frac{1}{f_N} \exp(-i2\pi f n T_s) rect(\frac{f}{f_N}) \quad \dots(4)$$ Now we can write: $$\int_{-\infty}^{\infty}\phi_n(t)\phi_k(t)dt=\int_{-\infty}^{\infty} \{ \Phi_n(f) \circledast \Phi_k(f) \} df$$ $$=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\Phi_n(\tau) \Phi_k(f-\tau) d\tau df$$ $$=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp(-i2\pi \tau n T_s) \frac{1}{f_N} rect(\frac{\tau}{f_N}) \exp(-i2\pi (f-\tau) k T_s) \frac{1}{f_N} rect(\frac{f-\tau}{f_N}) d\tau df$$ $$=\int_{-\infty}^{\infty} \exp(-i2\pi \tau n T_s) \frac{1}{f_N} rect(\frac{\tau}{f_N}) \exp(i2\pi \tau k T_s) \{ \int_{-\infty}^{\infty} \exp(-i2\pi f k T_s) \frac{1}{f_N} rect(\frac{f-\tau}{f_N}) df \} d\tau \quad \dots(5)$$ The inner integral of $$(5)$$ can be simplified as: $$\int_{\tau -\frac{f_N}{2}}^{\tau +\frac{f_N}{2}} \frac{1}{f_N} \exp(-i2\pi f k T_s) df$$ $$=\frac{\exp(-i2\pi \tau k T_s) \sin(\pi k)}{\pi k} \quad \dots(6)$$ So, $$(5)$$ can be rewritten as: $$\int_{-\infty}^{\infty} \exp(-i2\pi \tau n T_s) \frac{1}{f_N} rect(\frac{\tau}{f_N}) \exp(i2\pi \tau k T_s) \frac{\exp(-i2\pi \tau k T_s) \sin(\pi k)}{\pi k} d\tau$$ $$=\frac{\sin(\pi k)}{\pi k} \int_{-\infty}^{\infty} \exp(-i2\pi \tau n T_s) \frac{1}{f_N} rect(\frac{\tau}{f_N})d\tau$$ $$=\frac{\sin(\pi k)}{\pi k} \frac{\sin(\pi n)}{\pi n} \quad \dots(7)$$ Now, $$(7)$$ is even equal to $$0$$ when $$k=2$$ and $$n=2$$
So,where i missed ? any help or suggestions please...
The idea of solving the integral in the frequency domain is good, but you made a mistake rewriting the integral. Note that
$$\int_{-\infty}^{\infty}\phi_n(t)\phi_k(t)dt\tag{1}$$
equals the Fourier transform of $$\phi_n(t)\phi_k(t)$$ evaluated at $$f=0$$. As you know, that Fourier transform is given by the convolution of the two individual Fourier transforms of $$\phi_n(t)$$ and $$\phi_k(t)$$, respectively:
$$\mathcal{F}\big\{\phi_n(t)\phi_k(t)\big\}=\int_{-\infty}^{\infty}\Phi_n(\xi)\Phi_k(f-\xi)d\xi\tag{2}$$
Evaluating $$(2)$$ at $$f=0$$ gives
$$\int_{-\infty}^{\infty}\phi_n(t)\phi_k(t)dt=\int_{-\infty}^{\infty}\Phi_n(\xi)\Phi_k(-\xi)d\xi=\int_{-\infty}^{\infty}\Phi_n(\xi)\Phi_k^*(\xi)d\xi\tag{3}$$
where the last equality in $$(3)$$ is true because $$\phi_k(t)$$ is real-valued. Eq. $$(3)$$ is just Parseval's theorem.
I'm sure you can continue from here and show that the right-hand side of $$(3)$$ equals zero for $$n\neq k$$.
Note that the integral you tried to compute equals the inverse Fourier transform of $$(\Phi_n\star\Phi_k)(f)$$ evaluated at $$t=0$$, i.e., it equals $$\phi_n(0)\phi_k(0)$$ which also satisfies
$$\phi_n(0)\phi_k(0)=\delta[n-k]\tag{4}$$
• Thank you so much sir – Suresh Jul 27 '20 at 13:11 | {"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": 45, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868747591972351, "perplexity": 194.4225591938831}, "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-2021-25/segments/1623487647232.60/warc/CC-MAIN-20210619081502-20210619111502-00377.warc.gz"} |
https://stacks.math.columbia.edu/tag/04B3 | Remark 7.7.2. If the covering $\{ U_ i \to U\} _{i \in I}$ is the empty family (this means that $I = \emptyset$), then the sheaf condition signifies that $\mathcal{F}(U) = \{ *\}$ is a singleton set. This is because in (7.7.1.1) the second and third sets are empty products in the category of sets, which are final objects in the category of sets, hence singletons.
There are also:
• 4 comment(s) on Section 7.7: Sheaves
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9437017440795898, "perplexity": 456.43701277043584}, "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-2023-14/segments/1679296943589.10/warc/CC-MAIN-20230321002050-20230321032050-00092.warc.gz"} |
http://mathhelpforum.com/trigonometry/81306-how-do-i-solve-sin-x-948-a.html | # Math Help - How do I solve "sin x = .948"?
1. ## How do I solve "sin x = .948"?
I know that it involves an arc(trig function) but I don't exactly know which. I'm having a (very) rough year in trigonometry but was hoping someone could teach me how to solve that basic problem.
2. Originally Posted by iggy1110
I know that it involves an arc(trig function) but I don't exactly know which.
To undo a sine, try using the arc-sine.
3. Originally Posted by stapel
To undo a sine, try using the arc-sine.
Thank you very much =)
So would the equation I use to solve look like this:
x = arcsin .9486
4. Originally Posted by iggy1110
Thank you very much =)
So would the equation I use to solve look like this:
x = arcsin .9486
Yep, since sin(x) repeats itself the answer would be $x = arcsin(.9486) \pm 2k\pi$ where k is an integer and radians are used | {"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": 1, "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": 0.9388941526412964, "perplexity": 1175.8202283061378}, "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-2016-30/segments/1469257824146.3/warc/CC-MAIN-20160723071024-00109-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=4162080 | ## Must choose between these two classes for NEng
Hey,
I am a math major looking to get my Ph.D. in Nuclear Engineering. Next semester I have to choose between two classes because of scheduling conflict (each class only offers one section; they are 30 minutes apart, but one is at another university which is 30min-1hr drive depending on traffic). I have to choose between:
Numerical Analysis 2
Topics included will be numerical methods for systems of equations, differentiation and integration, approximation theory, initial and boundary value problems and problems for ordinary differential equations. Extensive use of computing will be incorporated.
vs.
Characteristics of atomic and nuclear radiations, transition probabilities, radioactivity, classical and quantum-mechanical derivations of cross sections, interactions of photon, neutron, and charged particles with matter.
Which do I choose?
I would say it would depend upon what you plan to focus upon for your PhD. Numerical analysis is good for developing and refining codes for numerous applications such as reactor simulation, neutron transport, etc., assuming you then go on to numerical methods for PDEs. Radiation physics looks good for looking at radiation effects upon reactor materials.
I have pretty much decided on taking the Radiation physics class because it is a prerequisite for higher level nuclear engineering classes. I am very interested in numerical analysis and simulation but I should be able to pick it up next year. Also, my physics professor told me that my transcript is too purely-math oriented and I need to take some engineering and physics classes to be competitive for graduate applications. Thank you for the advice.
## Must choose between these two classes for NEng
Have you taken all the prereqs for the radiation physics class?
Yes the prerequisites are only Calculus 2 and Calc-based Physics 1. | {"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": 0.8126679062843323, "perplexity": 786.8182963251282}, "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/1368702414478/warc/CC-MAIN-20130516110654-00051-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/331103/intuitive-explanation-of-entropy | Intuitive explanation of entropy
I have bumped many times into entropy, but it has never been clear for me why we use this formula:
If $X$ is random variable then its entropy is:
$$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$
Why are we using this formula? Where did this formula come from? I'm looking for the intuition. Is it because this function just happens to have some good analytical and practical properties? Is it just because it works? Where did Shannon get this from? Did he sit under a tree and entropy fell to his head like the apple did for Newton? How do you interpret this quantity in the real physical world?
• That was well known in Statistical Mechanics. Max Planck derived it for a phonon gas. Take some visit to Jaynes stuff. – Felix Marin Apr 25 '14 at 5:18
• I just want to point out that there are simple pieces of software around on the Internet claiming to determine (and in fact they do correctly compute according to the formula of Shannon) the "entropy" of a user-given character sequence. However, entropy is a concept relevant to the source of randomness, not to a particular "given" sequence. Hence such calculations are problematical. See the thread crypto.stackexchange.com/questions/33231/entropy-calculation – Mok-Kong Shen Apr 10 '16 at 9:12
• plus.maths.org/content/information-surprise – Polisetty Sep 27 '19 at 20:23
We want to define a measure of the amount of information a discrete random variable produces. Our basic setup consists of an information source and a recipient. We can think of our recipient as being in some state. When the information source sends a message, the arrival of the message causes the recipient to go to a different state. This "change" is exactly what we want to measure.
Suppose we have a set of $$n$$ events with respectively the following probabilities
$$p_1,p_2,...,p_n.$$
We want a measure of how much choice we are to make, how uncertain are we?
Intuitively, it should satisfy the following four conditions.
Let $$H$$ be our "measure".
1. $$H$$ is continous at every $$p_i$$
2. If $$p_i = 1$$, then $$H$$ is minimum with a value of $$0$$, no uncertainty.
3. If $$p_1 = p_2= \dots = p_n$$, i.e. $$p_i=\frac{1}{n}$$, then $$H$$ is maximum. In other words, when every outcome is equally likely, the uncertainty is greatest, and hence so is the entropy.
4. If a choice is broken down into two successive choices, the value of the original $$H$$ should be the weighted sum of the value of the two new ones.
An example of this condition $$4$$ is that $$H\left(\frac1{2}, \frac1{3}, \frac{1}{6} \right) = H\left(\frac{1}{2}, \frac{1}{2} \right) + \frac{1}{2} H\left(1 \right) + \frac{1}{2} H\left(\frac{2}{3}, \frac{1}{3} \right)$$
Here we decided to either take the a) first element or b) one of the other two elements. Then in a) we had no further decision, but for b) we had to decide which of those two to take.
The only $$H$$ satisfying the conditions above is:
$$H = −K\sum^n_{i=1}p_i log(pi)$$
To see that this definition gives what we intuitively would expect from a "measure" of information, we state the following properties of $$H$$.
1. $$H = 0 \iff p_i = 1$$ and $$p_j= 0, \forall j \neq i$$
2. $$\forall n \in N$$, $$H$$ is maximum when $$p_1=,\cdots,= p_n$$
3. Suppose $$x$$ and $$y$$ are two events with $$x \in R^n$$, $$y \in R^m$$ and $$p(i,j)$$ is the probability that $$x$$ and $$y$$ jointly occur (i.e. occur at the same time).
• $$H(x, y) = −\sum_{i, j} p(i, j) \log(p(i, j))$$
• $$H(x, y) \leq H(x) + H(y)$$.
With equality only if the occurrences are independent.
• $$H_x(y) = −\sum_{i, j} p_i(j) \log(p_i(j))= H(x, y) − H(x).$$
The entropy of $$y$$ when $$x$$ is known.
• $$H(y) \geq H_x(y)$$.
The entropy of $$y$$ is never increased by knowing $$x$$.
4. Any change towards equalization of the probabilities increases $$H$$. Greater uncertainty $$\Rightarrow$$ greater entropy.
Here is a post with some illustrative R code
• IIRC this result is due to Faddeev. – Qiaochu Yuan Apr 28 '13 at 7:01
• What does it mean that H is continuous at every p(i)? – confused00 Nov 21 '15 at 14:39
• @confused00 it basically means that if you change p(i) a little bit then H will only change a little bit as well. – An.Ditlev Aug 31 '16 at 18:27
• "We want a measure of how much choice we are to make, how uncertain are we?" How much choice to do what? How uncertain are we about what? You should be more precise to avoid ambiguities. – nbro Mar 1 '18 at 10:17
• @nbro, I will leave it as is. That sentence is not supposed to convey any mathematilcal rigor but there is plente of that elsewhere in the answer. – An.Ditlev Mar 30 '18 at 20:19
Here's one mildly informal answer.
How surprising is an event? Informally, the lower probability you would've assigned to an event, the more surprising it is, so surprise seems to be some kind of decreasing function of probability. It's reasonable to ask that it be continuous in the probability. And if event $A$ has a certain amount of surprise, and event $B$ has a certain amount of surprise, and you observe them together, and they're independent, it's reasonable that the amount of surprise adds.
From here it follows that the surprise you feel at event $A$ happening must be a positive constant multiple of $- \log \mathbb{P}(A)$ (exercise; this is related to the Cauchy functional equation). Taking surprise to just be $- \log \mathbb{P}(A)$, it follows that the entropy of a random variable is its expected surprise, or in other words it measures how surprised you expect to be on average after sampling it.
Closely related is Shannon's source coding theorem, if you think of $- \log \mathbb{P}(A)$ as a measure of how many bits you need to tell someone that $A$ happened.
• @QiaochuYuan I'm guessing it's impossible to 'prove' formally why entropy in theory of information is defined this way. Rather, we should start with an intuitive concept and try to define a mathematical formula satisfying the properties we want it to satisfy in the informal sense. So 'informal' answers are the most formal. What do you think? – user4205580 May 5 '15 at 19:07
• How would you intuitively characterize temperature $T$ along these lines if we think of $1/T$ coming from $dS/dE = 1/T$ as giving us something like 'the change in our expected surprise as the random variable changes'? hyperphysics.phy-astr.gsu.edu/hbase/thermo/temper2.html – bolbteppa Sep 24 '15 at 18:39
• First paragraph was useful... the rest got too abstract – CodyBugstein Oct 27 '15 at 17:59
• What does this mean : "It's reasonable to ask that it be continuous in the probability"? – confused00 Nov 21 '15 at 14:40
• why is it that two disjoint events are the ones that their information is added...why is it a property of intersection of events and why does independence matter at all? (at least intuitively when one is coming up with these definitions) – Charlie Parker Apr 26 '18 at 20:38
Let me give you an intuitive (rather than mathematically rigorous) interpretation of entropy, denoted $H(X)$.
Let me start by giving you my interpretation first and then let me justify it.
Entropy can be viewed as the cost of encoding a specific distribution $X$.
Since I will describe it in terms of encoding messages, let me change the notation to make the description more intuitive. We want to transmit some message $(M=m)$ across some channel $C$. Intuitively, the cost of sending a messages across a channel is the length of the encoding of the message $m$. i.e. the longer the message, the more it will cost us to send the message since we have to send more (bits) of information. The frequency (and the probability) of getting each message is dictated by the language $\mathcal{L}$ which the message came from. For example, the language could be $\mathcal{L} = English$, were the word "the" is probably relatively common (i.e. high frequency and high probability) and thus, we should choose wisely how to encode this, since we will have to send it very often (or in the case of English, write it pretty pretty often!). So we want an efficient encoding for "the". By efficient, we want it to mean choosing a encoding that happens to choose less number of "stuff" (or information, bits etc) that we need to send through the channel. Since the messages we have to send are somewhat random, then its seems reasonable that we aim to send the least amount of bits that we can, at least on average. i.e intuitively, we want to minimize:
$$E[ |M|] = \sum_m Pr[M=m]|m|$$
where $|m|$ denotes the length of the encoding of message m.
For example, we might want to encode it this way: for common (high probability) messages lets use fewer bits of information to encode them since we have to send them very frequently. So we can encode them based of the relative frequency dictated by the distribution for $\mathcal{L}$. With a little more thought you can come up with Huffman coding or some other scheme similar to it, if you make sure that the messages can be decoded unambiguously, the main idea in my opinion is to encode frequent words with short code lengths and infrequent ones with longer code lengths.
It turns out that Shannon proved that the notion of entropy provides a precise lower bound for the expected number of bits required to encode instances/messages sampled from $P(M)$. i.e. if we consider any proper codebook for values of $M \in \mathcal{L}$, then the expected code length, relative to the distribution $P(M)$, cannot be less than the entropy $H(M)$:
$$H(M) \leq E[|M|]$$
Since there exists a scheme that makes this inequality tight, then we can expect to encode the messages $M$ as efficiently as possible (on average).
Thus, returning to the interpretation I suggested. Since, the cost of encoding something can be thought of as the number of bits we need to send through a channel, and the optimum value (entropy) can be achieved, then entropy becomes the expected cost of encoding a distribution of messages.
(or if you want to view it from the inequalities perspective, its the best/minimum expected cost you can have to encode any known distribution $P(M)$.)
• Well written! I am glad that it is clarified that entropy is property of the language and not the word that we are encoding. Also entropy is all about measure and not random variable. – user1700890 Sep 21 '15 at 15:02
• @user1700890 what do you mean its about measure and not r.v.? If u have some better view on entropy to provide, please do! We all would like to learn :) – Charlie Parker Feb 12 '16 at 1:26
• @CharlieParker well, values of r.v. do not matter for entropy, only distribution (measure) matters. – user1700890 Feb 12 '16 at 20:02
• @user1700890 oh I see what you mean. Yes sure. I guess I tried making an intuitive/conceptual explanation rather than a super rigorous one. If you want to add some supplementary remarks on my answer to point that out, I'd be happy to incorporate them. – Charlie Parker Feb 13 '16 at 1:30
• Is the scheme you refer to to make the inequality tight the Huffman encoding scheme? – vader999 Oct 13 '19 at 23:04
Here is a simple intuitive explanation of Shannon entropy.
The telegraph message "SOS" is encoded as "...---..." in Morse code. The thing to note is that the massage is made up of letters from the alphabet but what is transmitted down the communication line are only dots and dashes. The message is written in the alphabet but transmitted in dots and dashes. Morse code maps letters to dots and dashes.
Text messages, emails and instant messaging are all written in text (i.e. upper and lower case letters, space, tab, decimal digits, punctuation marks, etc) but transmitted as bits {0, 1}. For the mathematician the problem of communication is of finding the most efficient way of mapping text messages to streams of bits. By most efficient I mean the least number of bits. If I have a text message of 100 characters what is the smallest number of bits I need to transmit down the line?
From these examples we can see that message transmission involves 2 sets A and B. The message is a sequence of letters from set A but the communication line can only transmit characters from set B. Let $m_A$ be a message written in A and let $m_B$ be the same message written in B. Let E be an encoding function that maps messages from A to messages from B.
$$m_B = E(m_A)$$
We can measure the size of the message by counting the number of characters in the message. The length of $m_A$ is $L(m_A)$ and the length of $m_B$ is $L(m_B)$.
Clearly a short message will have a sort encoding and a long message will have a long encoding. If we double the length of the message we will double the length of the encoding. The length of the encoding will be proportional to the length of the message.
$$L(m_B) \varpropto L(m_A)$$
By introducing a constant of proportionality k we can turn this into an equation.
$$L(m_B) = kL(m_A)$$
The problem of finding the most efficient encoding reduces to find the minimum possible value of k. This minimum value is the entropy of the set A measured over the set B.
If
$$A = \{A_1, A_2, A_3, ..., A_n\}$$
and the probability of $A_i$ being in the message is $p_i$ and B is the set
$$B = \{B_1, B_2, B_3, ..., B_m\}$$
then the entropy is given by
$$Entropy = \sum_{i=1}^n p_i\log_m(\frac{1}{p_i})$$
Note that log is taken to the base m which is the size of the set B.
So far I have dealt with the most general case but now I will switch to the simple case when each of the n characters in A are equally likely so $p_i = \frac{1}{n} \forall i$.
$$Entropy = \sum_{i=1}^n \frac{1}{n} \log_m(n)$$
This simplifies to
$$Entropy = \log_m(n)$$
In this case the entropy only depends on the of the sizes of A and B.
To prove this is correct function for the entropy we consider an encoding $E: A^r \rightarrow B^s$ that encodes blocks of r letters in A as s characters in B.
$$L(m_A) = r, \space L(m_B) = s, \space m_B = E(m_A)$$
The range of E must be greater than or equal to the size of the domain or otherwise two different messages in the domain would have to map to the same encoding in the range. The size of the domain is $n^r$ and the size of the range is $m^s$. We chose s to satisfy the following inequalities.
$$m^{(s-1)} \lt n^r \le m^s$$
The right hand inequality ensures the range is greater than or equal to the domain. The left hand inequality ensures this is the smallest such s that has this property.
Taking the log to base m of both side of these inequalities gives us.
$$\log_m(m^{(s-1)}) \lt \log_m(n^r) \le \log_m(m^s)$$
$$(s-1)\log_m(m) \lt r\log_m(n) \le s\log_m(m)$$
$$\frac{(s-1)}{r} \lt \log_m(n) \le \frac{s}{r}$$
The constant of proportionality k we introduced earlier is the ratio $\frac{s}{r}$.
$$k = \frac{s}{r}$$
So the right hand inequality proves
$$\log_m(n) \le k$$
This proves that $\log_m(n)$ is a lower bound for k but how close can an encoding come to the lower bound? We note that
$$\frac{(s-1)}{r} \lt \log_m(n) \le \frac{s}{r}$$
implies
$$\frac{s}{r}-\log_m(n) \lt \frac{s}{r}-\frac{(s-1)}{r}$$
$$\frac{s}{r}-\log_m(n) \lt \frac{1}{r}$$
$$k-\log_m(n) \lt \frac{1}{r}$$
We can make k as close as we like to $\log_m(n)$ by increasing the block size r.
This finishes our treatment of the special case of equal probabilities. Hopefully having proved that in this case the entropy is $\log_m(n)$ the more general formula should not come as too much of a surprise.
I hope you find this simple explanation more intuitive than the usual approach. I searched the internet for an explanation of entropy but didn't like any of the results so I came up with my own. It took me 5 years but now I understand Shannon entropy.
The three postulates in this answer are the ones used in Shannon's original 1948 paper. If you skip over to Appendix II in that paper, you can find the remainder of the derivation.
1. Derive the expression for $H \left(\tfrac{1}{n}, \tfrac{1}{n}, \ldots, \tfrac{1}{n} \right)$ as $-K \log n$.
2. If all the $p_i$'s are rational, we can find an $m$ such that $m p_i \in \mathbb{N}, \forall i$. Now, use postulate $3$ to derive the entropy formula.
3. Using the continuity postulate (first postulate), you can directly extend the formula to the case where the $p_i$'s are not necessarily rational.
• Thanks for outlining Shannon's proof. It helps me to understand his proof. – Eli4ph Oct 27 '19 at 16:10
Consider transmitting long numbers, e.g. values between 0 and 999,999 (decimal).
Each value can take one of out of a million possible states, and yet we can transmit each number with only 6 digits.
Noting that:
$$\log_{10}(1,000,000) = 6$$
Note that I've set the log base to match the number of symbols (0 to 9), and that the result is the number of (decimal) digits needed to encode a number with one million possible states.
For binary we get:
$$\log_{2}(1,000,000) \approx 19.93 \text{ bits}$$
So, hopefully, you can see that log({number of possibilities}) inherently gives a measure of how much information (how many digits) we need to encode a variable with $N$ possible states.
It may also be useful to move the minus sign inside the log, recalling that:
$$-\log{x} = \log{\frac{1}{x}}$$
Thus:
$$H(X) = \sum{p(x)}\log{\frac{1}{p(x)}}$$
So, in the above discussion, it looks like we just equated these two terms:
$$\{\text{number of possibilities}\} = \frac{1}{p(x)}$$
Which sort of makes sense. For example, if $p(x) = 1$, then $x$ can be the only possible value; if $p(x) = 0.5$, then there may be lots of other values, but there can be only one other at most with that same share of the probability, i.e. $0.5$.
So, $\log \left(\frac{1}{p(x)} \right)$ is giving us an amount of information fitting for a value with probability $p(x)$. We then multiply that amount of information by the probability of that value actually occurring, effectively calculating a weighted sum over all of the possible values. Giving:
$$H(X) = \sum{p(x)}\log \left( \frac{1}{p(x)} \right)$$
The physical meaning of information entropy is:
the minimum number of storage "bits" needed to capture the information.
This can be less than implied by the number of different values a variable can take on. For example, a variable may take on $4$ different values, but if it takes on one of these values more often than the others, then one would need less than $\log(4)=2$ bits to store the information, if we choose an efficient way of storing the information.
We get entropy in terms of "bits" when the base of the log in the entropy equation is $2$. For some other technology, e.g., some esoteric memory based on tri-state devices, we would use log of base $3$ in the entropy equation. And so on.
For a verbose explanation of the intuition behind Shannon's entropy equation, you could check out this document: Understanding Shannon's Entropy metric for Information.
• Log p(x) represents bits that would be needed to store the information. Can you explain what is the intuition of multiplying it with p(x) in formula. @Sriram V – Akki Sep 5 '17 at 17:23
I see entropy as a number that gives you an idea of how random an outcome will be based on the probability values of each of the possible outcomes in a situation.
Let's start with a simple case. Suppose only a single outcome is possible, then there is only one value of $$i$$ ($$=1$$) and $$p_{1}=1$$. From the formula, the entropy is then zero:
$$-p_{1} \log(p_{1}) = p_{1} \log \left(\frac{1}{p_{1}} \right) = 1 * 0 = 0$$
This is cool! When the outcome will be the same every single time, the "randomness" is zero, and so the entropy does indeed correspond to a measure of randomness.
Now, before moving to more complicated cases, let's look at a plot of the factors involved in the entropy formula. Let me rewrite the formula first as follows:
$$- \sum_{i} p_{i} \log(p_{i}) = \sum_{i} p_{i} \log \left(\frac{1}{p_{i}} \right)$$
Looking at this plot you see that there is nothing really special about $$\log \left( \frac{1}{p} \right)$$, really any function of $$p$$ such that $$f(1) = 0$$ would have done the trick.
Now, you might wonder, what if I have two possible outcomes, one that is nearly certain and one that is very unlikely; for example $$p_{1} = 0.999$$ and $$p_{2} = 0.001$$. This case is tricky!
For the first outcome, we see that $$p_{1} \log\left(\frac{1}{p_{1}} \right)$$ is a number very close to zero. That first outcome is not too different from the single-outcome situation we looked at before.
For the second outcome, $$p_{2} = 0.001$$, let's think about the limit of the product $$p\log(\frac{1}{p})$$ as $$p \rightarrow 0$$. Intuitively, we know that if we add an extremely unlikely event, such as the one with $$p_{2} = 0.001$$, the "randomness" situation should not really be that different from our original single-outcome process.
Let's look at a graph to see what the definition of entropy does for us in this case:
Beautiful! This means that an extremely unlikely event contributes nearly zero to the entropy of the system. Extremely likely and extremely unlikely are similar in terms of their "randomness": they have pretty much none of it!
Why the logarithm?
At this point you might be wondering, what is so special about the logarithm? It does seem kind of an arbitrary choice. There certainly must be other functions of $$p$$ that have the same convergence properties as $$p$$ goes to $$0$$ and $$p$$ goes to $$1$$.
So, I'll give you a situation to think about. Suppose you have a system where there are two equally likely choices $$1$$ and $$2$$, with probabilities $$p_{1} = p_{2} = \frac{1}{2}$$. That situation will have some entropy, let's call it $$S_{2}$$. Consider also a second system with an entropy $$S_{3}$$ where there are three equally likely choices $$A$$, $$B$$ and $$C$$, with probabilities $$p_{A} = p_{B} = p_{C} = \frac{1}{3}$$.
It would be nice if the entropy were a function such that if I considered the union of the two independent systems, the resulting entropy of the global system would be additive, that is
$$S_{g} = S_{2} + S_{3}$$
In simpler words, it would be nice for our measure of "randomness" to be additive.
Let's be explicit here and write down the full expression for $$S_{g}$$, assuming that the events from one system are completely independent from events in the other system.
\begin{align} S_{g} = p_{1} p_{A} \log \left (\frac{1}{p_{1}p_{A}} \right) + p_{1}p_{B} \log \left(\frac{1}{p_{1}p_{B}} \right) + p_{1}p_{C} \log \left( \frac{1}{p_{1}p_{C}} \right) + \\ p_{2}p_{A} \log \left( \frac{1}{p_{2}p_{A}} \right) + p_{2}p_{B} \log \left( \frac{1}{p_{2}p_{B}} \right) + p_{2}p_{C} \log \left( \frac{1}{p_{2}p_{C}} \right) \end{align}
The property of the logarithm that makes it a good choice for defining entropy is then more clear:
$$\log \left( \frac{1}{p_{1}p_{A}} \right) = \log \left( \frac{1}{p_1} \right) + \log \left( \frac{1}{p_A} \right)$$
Given this property, we can simplify $$S_{g}$$ as
$$S_{g} = p_{1}\log\left( \frac{1}{p_{1}} \right) (p_{A} + p_{B} + p_{C}) + p_{1} S_{3} + p_{2} \log \left( \frac{1}{p_{2}} \right) (p_{A} + p_{B} + p_{C}) + p_{2} S_{3}$$
$$S_{g} = S_{2} (p_{A} + p_{B} + p_{C}) + S_{3} (p_{1} + p_{2})$$
Since probabilities add up to $$1$$, this gives us the desired property:
$$S_{g} = S_{2} + S_{3}$$
This culminates our motivation for why the formula for entropy is what it is!
Key takeaway
I will summarize by saying that the key point is that "randomness" is hard thing to quantify. We can choose a measure for "randomness" (such as Shannon's entropy formula), and that choice is only informed by the properties that we want the measure to have.
When you look at
$$S = - \sum_{i} p_{i} \log(p_{i})$$
for the first time in your life you might think: where on earth did they pull this out from? But it turns out that it was a definition only informed by the properties that it holds.
An informal enumeration of these properties is given below:
1. An extremely likely event should not contribute much to the randomness measure.
2. An extremely unlikely event should not contribute much to the randomness measure.
3. Randomness should be additive.
• That's the answer I was looking for!! Thank you so much for such a clear explanation! Greetings – Phatee P Feb 24 '19 at 21:26
Entropy is often too abstract to me! The following perspective from statistical physics (instead of informational entropy) is what I got so far.
Let $N = n_1 + ... + n_k$ and $p_i = \frac{n_i}{N}$. $$\log \left( \frac{N!}{n_1 ! \cdots n_k ! } \right) \approx - N \sum_i p_i \log p_i$$ by Stirling's formula.
I wonder if this was the first time ever in human history that such an expression $$\sum_i p_i \log p_i$$ appeared!
The approximation above is a link between counting combinations and entropy, and it seems to provide the most concrete grasp. This is the genius of Boltzmann, Maxwell and Gibbs which leads to the development of statistical mechanics.
If you understand expected value $\mathbb{E}$ of a random variable $X$, then the concept of entropy should be easier to understand. With no mathematical rigor, I'll express the entropy $H$ (of a discrete random variable $X$) in the following way:
$$H(X) = \mathbb{E}[\text{surprise from outcome encoded in 2-bits}] = \mathbb{E}[\log(\text{surprise from outcome})].$$
For an event $A$ with probability $p$, the surprise is inversely proportional to the probability of the event, $\frac{1}{p}$.
Although this question has been asked a long time ago and has received very relevant answers, I would like to add my personnal touch on the subject. Actually, since some time, I am wondering (like jjepsuomi) why the Shannon entropy is so ubiquitous and fascinating. Here is a summary of my current insight :
Information as reduction of uncertainty
An intuitive idea for understanding the information given by the unveiling of a random variable is to see this unveiling as a reduction of the space of what was possible before the variable was unveiled.
Suppose that you have a mysterious point $$U$$ that can be anywhere in a volume $$V$$ with uniform probability. A random variable X can be seen as a partition of $$V$$ into $$N$$ different smaller volumes $$p_iV$$. Due to uniformity, $$U$$ has the probability $$p_i$$ to be in the volume $$V_i$$.
Unveiling X will let us know in which domain $$U$$ stands. So, the possible area for $$U$$ will be reduced. The value of the new volume will be $$p_iV$$. We can define the reduction $$r$$ by $$p_i$$ itself. It can be seen as a function of $$X$$ and so, also as a random variable (that is why I prefer to call it r instead of $$p_i$$ that would be confusing). When $$X$$ has been unveiled, the intuitive notion of received information can safely be linked to the reduction $$r$$ (the smaller $$r$$, the more information I have received). Anyway, introducing a measure of information (like $$\log(p_i)$$) although tempting, is not useful right now. It will give some magic to the concept that will leads us too easily to the Shannon entropy that we want to discover without magic...
Shannon entropy as a metric to summarize reduction of uncertainty associated with a random variable
Now, suppose that $$X$$ has not yet been unveiled. We might find it useful to summarize the ability $$X$$ has to reduce $$V$$, through a single number.
Of course, a good candidate is the Shannon entropy, the opposite of the expected log of the reduction $$r$$ : $$H=-\sum_ip_i\log(p_i)$$.
A concurrent for Shannon Entropy ?
But, another metric is maybe more natural : the expected reduction itself : $$\sum_ip_i^2$$. I have spent some time looking at this metric and trying to figure out why it has not been as popular as the Shannon Entropy.
First, it is more elegant to introduce its logarithm : $$G=-\log(\sum_ip_i^2)$$ that have many intersting properties, similar to Shannon entropy's. (actually, only property 4 from the list given by @An.Ditlev is not valid).
In particular :
• We have $$G(X,Y)=G(X)+G(Y)$$ when $$X$$ and $$Y$$ are independent.
• We can devise a symetrical "mutual information" $$I_G(X;Y)=G(X)+G(Y)-G(X,Y) \ge 0$$
$$H$$ and $$G$$ are not measuring the same thing and we can find two random variables $$X$$ and $$Y$$ such that $$H(X) \gt H(Y)$$ while $$G(X) \lt G(Y)$$
Shannon Entropy is fitted for i.i.d sequences of random variables
Now, why Shanon Entropy has had a much better career that my $$G$$ metric ? Well, I guess it can be understood with the Asymptotic Expectation Property : In the case of a sequence of i.i.d random variables, we can prove (by the weak law of large numbers) that $$\log(P(X_1,X_2,...,X_n)) \approx 2^{-nH}$$. To put it simply, we are almost certain that the $$\log()$$ will be $$2^{-nH}$$ (that is a decreasing function of the Shannon Entropy). The higher the Shannon entropy, almost certainly the smaller the reduction!.
Strange examples
The success of the Shannon Entropy is due to the fact that we can decently forget the "almost" word in the last sentence.
That leads to a strange result. Let's go back to my previous example of two random variable $$X$$ and $$Y$$, $$X$$ having a higher $$H$$ and $$Y$$ a higher $$G$$. If we can chose between repeating a sequence $$X_i$$ or a sequence $$Y_i$$ :
If we chose $$X$$, we can be almost certain that the reduction $$r$$ will be better for $$X$$, but the average reduction will be beter with $$Y$$... That's because a set of sequences with a very small cumulated weight is sufficient to have a deep impact on the average in $$G$$.
It is analoguous to the problem of a gambler that can repeat an experiment where he has 50% chance to lose all his money and 50% chance to triple his bet. If he repeats the experiment a lot of times, he is almost certain to lose all its money, but on average, he will be very very rich... due to the very unlikely event that he will always win.
Things get stranger if we do not consider i.i.d variables but just a sequence of independent variables. Then, AEP don't stand and the Shanon Entropy can, in some cases, be quite a strange metric.
For instance, we can devise a sequence of independent random variables having each a Shannon Entropy of 1 bit while the probability $$Pr(r \ge 0.5) \ge 0.5$$. That is, the received Shanon Entropy increases to $$\infty$$, but the probability that we learn almost nothing remains quite important. (with random variables having the probability of the state 1 increasing rapidly to 1 with i but the remaining probability uniformly shared between a very rapidly number of states).
This is a work in progress... In particular, I would like to see the implication for the coding. Is there a coding that is best fitted for $$G$$? Is it true that, on average the $$G$$-coding will be shorter on average, while larger almost certainly ?
Conclusion
As a conclusion, I will say that the Shannon Entropy is just a metric that gives useful information about the ability a random variable has to reduce uncertainty. It does not describe it perfectly. (it is not a sufficient statistic of the probability distribution). While, there are some traps, in many case (i.i.d sequence) it is the most useful metric.
• Very nice @Arnaud, thank you for your contribution! – jjepsuomi Feb 28 '20 at 8:03
• You might want to look up Renyi entropy - your $G$ happens to be $H_2$! I know one operational interpretation of $H_2$ - if symbols $X_i$ are drawn i.i.d. from a distribution $P$, and you are asked to guess a $k$-length sequence, then for large $k$ the mean number of guesses you need to make are approximately $\exp(k H_2(P))$ - this has been significantly generalised, see some of Arikan's work on guessing. – stochasticboy321 Feb 29 '20 at 6:15
There is an easy to understand video that illustrates, in layman terms, how to arrive at the formula for entropy:
The context of the video is information theory.
In my view, the most intuitive way to look at entropy, which I also think is also the most correct one is simply as follows:
1. If kilo grams is the measure of mass, seconds is the measure of time, meters is the measure of distance, then what is the measure of information? Say, how much information is inside a closed letter? In other words, if you open the letter and read it, how much information will you obtain?
2. Shannon's entropy basically tries to answer that question. The answer is very simple and extremely intuitive: how about we measure the quantity of information based on the number of questions that must be asked in order to fully discover all unknowns that were hidden in that thing. So in the case of the letter, the amount of information in the letter is the quantity of questions that we need to get answered in order to fully know the content of the letter.
But obviously there is a problem with (2), which is: how can the number of questions be meaningful? Someone may ask a single broad question like "what is inside the letter?" and fully obtain the letter. Or someone may ask really stupid questions like "what is the 1st letter?" then "what is the 2nd letter?", …, ending up with thousands of silly questions for a tiny letter!
Shannon's entropy solves this problem by standardising the questions, such that their numbers are stable and meaningful, and it does so by defining the qualities the questions must have. Turns out this is very easy!
Here is how the questions are standardised so that their quantity is stable:
The average number of perfect n-nary questions that, if answered, we would end up fully resolving all unknowns about the subject.
An $$n$$-nary question is "perfect" if it splits the search space evenly into $$n$$ many equal sub-spaces. So if a question is binary ($$n=2$$), then every time you ask a perfect binary question, it must split the search space in half.
Wait, that looks like a balanced tree! Balanced $$n$$-nary trees split the search space in $$n$$-many equal parts every time we move down by a single node.
In balanced $$n$$-nary trees, we have a tree root on top, and branches, until we finally reach terminal leaf nodes. Also, in a balanced $$n$$-nary tree, the total number of nodes that you need to cross until you finally reach the terminal leaf nodes is $$\log_n \texttt{m}$$, where $$m$$ is total number items stored in the balanced $$n$$-nary tree that you are trying to lookup — this is where the $$\log$$ part in entropy comes from!
Now, let's have a closer look at entropy:
$$H(\mathcal{X}) = -\sum_{x\in\mathcal{X}} p_x \log_n(p_x)$$
Now let's unwrap it piece by piece. 1st let's get rid of that negative:
$$H(\mathcal{X}) = \sum_{x\in\mathcal{X}} p_x \log_n(\frac{1}{p_x})$$
That's it for the negative! It was added there only because $$\log_n(\frac{1}{p_x})$$ doesn't look neat enough. Basically $$\log_n(\frac{1}{p_x}) = -\log_n(p_x)$$.
HOMEWORK 1: The $$\sum_{x\in\mathcal{X}} p_x \ldots$$ is just a weighted sum. I leave you figure out why we need it on your own.
But what about $$\log_n(\frac{1}{p_x})$$? First let's remind ourselves what is $$p_x$$:
$$p_x = \frac{\text{number of times x happened}}{\text{number of times everything happened}}$$
And what does $$\frac{1}{p_x}$$ mean?
$$\begin{split} \frac{1}{p_x} &= \frac{1}{\frac{\text{number of times x happened}}{\text{number of times everything happened}}}\\ &= \frac{\text{number of times everything happened}}{\text{number of times x happened}}\\ \end{split}$$
Wow! You see! It's getting very similar to that $$m$$ in the balanced $$n$$-nary tree example (i.e. $$\log_n m$$ part above).
HOMEWORK 2: Now, I leave it to you to figure out how $$\frac{\text{number of times everything happened}}{\text{number of times x happened}}$$ corresponds to $$m$$ in my balanced $$n$$-nary tree example earlier.
Now, only if you solve HOMEWORK 1 and HOMEWORK 2, you'd fully understand Shannon's entropy! It's actually very simple and intuitive.
• Thank you for your contribution! – jjepsuomi Sep 17 '20 at 18:00
As a physicist, I can confess that not many people grasp what entropy is (regardless of the different attempts to nail it with mathematical definitions)!
There are also other definitions of entropy flying around in the physics community which, for certain situations, are more consistent than the standard definition.
• How does this answer the questions? This should be a comment. – nbro Mar 1 '18 at 11:08
Let me try to answer to part of your question where you ask how it's related to the "physical world" taking the same direction as MichaelNgelo's answer.
Let's say you have $N$ objects, out of them there are $n_1$ many $A$s and $n_2$ many $B$s.
How many ways can you order them? For example for $N = 4$, with one $A$ and three $B$s we have: $$ABBB$$ $$BABB$$ $$BBAB$$ $$BBBA$$
This is a combinatorics problem and we can find the formula: $$\frac{N!}{n_1!n_2!}$$
Note that there are more cases with $N=4$ and a arbitrarly number of $A$s and $B$s ($16$ cases) or with $N=4$ and with two $A$s and two $B$s ($6$ cases).
Given the proportions of $A$s and $B$s, it reduces the number of possibilities and we just need a way to differentiate between them.
This is where the logarithm comes in to tell the least amount of bits to differentiate between these $M$ cases. Explicitly we need $\log_2(M)$ bits.
Now divide by $N$ to find the number of bits needed per symbol in the sequence. So if put all together, the formula becomes:
$$\frac{\log_2(\frac{N!}{n_1!n_2!})}{N}$$
and for convenience instead of talking about $n_1$ and $n_2$ in absolute number, we talk about the fraction of $N$, respectively $f_1$ and $f_2$. Finally tending $N$ to infinity gives the shannon entropy.
Example with $f_1 = 0.25$ and $f_2 = 0.75$:
Someone experienced can probably derive the Shannon entropy formula from the formula above with $N$ tending to infinity.
So a possible interpretation of the Shannon formula would be: The number of bits needed per symbol given the proportions of each symbol in advance in an infinite sequence. | {"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": 162, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8973124623298645, "perplexity": 335.3051070033448}, "config": {"markdown_headings": false, "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-2021-04/segments/1610703564029.59/warc/CC-MAIN-20210125030118-20210125060118-00568.warc.gz"} |
https://openreview.net/forum?id=SygT21SFvB | ## Towards Understanding Generalization in Gradient-Based Meta-Learning
Sep 25, 2019 Withdrawn Submission readers: everyone
• TL;DR: We study generalization of neural networks in gradient-based meta- learning by analyzing various properties of the objective landscape.
• Abstract: In this work we study generalization of neural networks in gradient-based meta-learning by analyzing various properties of the objective landscapes. We experimentally demonstrate that as meta-training progresses, the meta-test solutions obtained by adapting the meta-train solution of the model to new tasks via few steps of gradient-based fine-tuning, become flatter, lower in loss, and further away from the meta-train solution. We also show that those meta-test solutions become flatter even as generalization starts to degrade, thus providing an experimental evidence against the correlation between generalization and flat minima in the paradigm of gradient-based meta-leaning. Furthermore, we provide empirical evidence that generalization to new tasks is correlated with the coherence between their adaptation trajectories in parameter space, measured by the average cosine similarity between task-specific trajectory directions, starting from a same meta-train solution. We also show that coherence of meta-test gradients, measured by the average inner product between the task-specific gradient vectors evaluated at meta-train solution, is also correlated with generalization.
• Code: https://github.com/anonymousauthor181/anonymous_repository
• Keywords: meta-learning, objective landscapes
• Original Pdf: pdf
0 Replies
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http://clay6.com/qa/9340/if-overrightarrow-overrightarrow-overrightarrow-are-a-right-handed-triad-of | Browse Questions
# If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are a right handed triad of mutually perpendicular vectors of magnitude $a, b , c$ than value of $[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]$ is
$\begin{array}{1 1}(1) a^{2} b^{2} c^{2} &(2)0\\(3)\frac{1}{2} abc&(4)abc \end{array}$
Given $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular vectors of magnitude a,b,c then
$[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}] =abc$
Hence 4 is the correct 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": 2, "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": 0.912916362285614, "perplexity": 733.9495334211219}, "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-2017-13/segments/1490218187193.40/warc/CC-MAIN-20170322212947-00564-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.cheenta.com/tag/dimensional-analysis/ | Select Page
## A Tricky Integral
Let $$I=\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$J= \int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$. Find the value of $$J-I$$. Solution: $$I=\int e^x/(e^{4x}+e^{2x}+1) dx$$ $$J= \int e^{-x}/(e^{-4x}+e^{-2x}+1)dx$$ Let $$e^x$$=$$z$$ ...
## Dimensional Analysis
Consider an expression F=Ax$$sin^{-1} (Bt)$$ where F represents force, x represents distance and t represents time. Dimensionally the quantity AB represents (A) energy (B) surface tension (C) intensity of light (D) pressure Solution: The quantity Ax on RHS must have...
## Dimensional Analysis
The distance travelled by an object is given by x=(at+bt2)/(c+a) where t is time and a,b,c are constants. The dimension of b and c respectively are: [L2T-3], [LT-1] [LT-2], [LT-1] [LT-1], [L2T1] [LT-1], [LT2] Solution: The dimension of x is L. Hence, the dimension of... | {"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": 2, "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": 0.9142362475395203, "perplexity": 4553.328776630445}, "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-2019-18/segments/1555578841544.98/warc/CC-MAIN-20190426153423-20190426175423-00453.warc.gz"} |
http://mathoverflow.net/questions/21775/spectrum-of-a-generic-integral-matrix/21906 | # Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle.
We obtain a result under two additional assumptions
1) Characteristic polynomial of the matrix A is irreducible
2) Every circle contains no more than two eigenvalues of A (i.e. no more than two eigenvalues have the same absolute values)
We feel that the second assumption holds for a "generic" matrix. Is it true?
To be more precise, consider the set X of integral hyperbolic matrices which have determinant 1 and irreducible characteristic polynomial. What are the possible ways to speak of a generic matrix from X? Does assumption 2) hold for generic matrices?
• Assumption 1) doesn't bother us as it is a necessary assumption.
• Probably it is easier to answer the question when X is the set off all integral matrices. In this case we need to know that hyperbolicity is generic, 2) is generic and how generic is irreducibility.
-
Unless all eigenvalues are collinear, there must be a circle containing 3 of them. Or does 2) mean that no more than two eigenvalues have the same modulus? – Gjergji Zaimi Apr 18 '10 at 22:46
Yes circle centered at origin, you are right, 2) just means that no more than two eigenvalues have the same absolute value. – Andrey Gogolev Apr 18 '10 at 23:07
Yes, a generic integer matrix has no more than two eigenvalues of the same norm. More precisely, I will show that matrices with more than two eigenvalues of the same norm lie on a algebraic hypersurface in $\mathrm{Mat}_{n \times n}(\mathbb{R})$. Hence, the number of such matrices with integer entries of size $\leq N$ is $O(N^{n^2-1})$.
Let $P$ be the vector space of monic, degree $n$ real polynomials. Since the map "characteristic polynomial", from $\mathrm{Mat}_{n \times n}(\mathbb{R})$ to $P$ is a surjective polynomial map, the preimage of any algebraic hypersurface is algebraic. Thus, it is enough to show that, in $P$, the polynomials with more than two roots of the same norm lie on a hypersurface. Here are two proofs, one conceptual and one constructive.
Conceptual: Map $\mathbb{R}^3 \times \mathbb{R}^{n-4} \to P$ by $$\phi: (a,b,r) \times (c_1, c_2, \ldots, c_{n-4}) \mapsto (t^2 + at +r)(t^2 + bt +r) (t^{n-4} + c_1 t^{n-5} + \cdots + c_{n-4}).$$
The polynomials of interest lie in the image of $\phi$. Since the domain of $\phi$ has dimension $n-1$, the Zariski closure of this image must have dimension $\leq n-1$, and thus must lie in a hyperplane.
Constructive: Let $r_1$, $r_2$, ..., $r_n$ be the roots of $f$. Let $$F := \prod_{i,j,k,l \ \mbox{distinct}} (r_i r_j - r_k r_l).$$ Note that $F$ is zero for any polynomial in $\mathbb{R}[t]$ with three roots of the same norm. Since $F$ is symmetric, it can be written as a polynomial in the coefficients of $f$. This gives a nontrivial polynomial condition which is obeyed by those $f$ which have roots of the sort which interest you.
-
Although this does not really affect the answer, there is one more component: one of the roots is real and the other two are complex conjugate. – damiano Apr 20 '10 at 7:22
Good point. So one needs a second component, parameterized by $(t-r)(t^2+at+r^2)(t^{n-3}+\ldots)$ or, from the second perspective, one needs to consider $\prod (r_i r_j - r_k^2)$. – David Speyer Apr 20 '10 at 11:51
Thank you very much! This is really nice, especially the "conceptual proof". I understand that the estimate $O(N^{n^2-1}$ should follow from the fact that "bad" matrices lie on an algebraic hypersurface. This is because all the "folding" occurs in a compact core, outside of which the hypersurface is sufficiently "straight". But is it really so obvious? – Andrey Gogolev Apr 20 '10 at 17:08 | {"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": 0.9656683206558228, "perplexity": 200.4788116416279}, "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/1466783396945.81/warc/CC-MAIN-20160624154956-00085-ip-10-164-35-72.ec2.internal.warc.gz"} |
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# Oligarchic planetesimal accretion and giant planet formation
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### Abstract
Aims. In the context of the core instability model, we present calculations of in situ giant planet formation. The oligarchic growth regime of solid protoplanets is the model adopted for the growth of the core. Methods. The full differential equations of giant planet formation were numerically solved with an adaptation of a Henyey-type code. The planetesimals accretion rate was coupled in a self-consistent way to the envelope's evolution. Results. We performed several simulations for the formation of a Jupiter-like object by assuming various surface densities for the protoplanetary disc and two different sizes for the accreted planetesimals. We find that the atmospheric gas drag gives rise to a major enhancement on the effective capture radius of the protoplanet, thus leading to an average timescale reduction of 30% -- 55% and ultimately to an increase by a factor of 2 of the final mass of solids accreted as compared to the situation in which drag effects are neglected. With regard to the size of accreted planetesimals, we find that for a swarm of planetesimals having a radius of 10 km, the formation time is a factor 2 to 3 shorter than that of planetesimals of 100 km, the factor depending on the surface density of the nebula. Moreover, planetesimal size does not seem to have a significant impact on the final mass of the core.
### Author and article information
###### Journal
0709.1454
10.1051/0004-6361:20066729
General astrophysics | {"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": 0.8648399114608765, "perplexity": 1468.4810114303273}, "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/1593655924908.55/warc/CC-MAIN-20200711064158-20200711094158-00191.warc.gz"} |
http://www.lmfdb.org/EllipticCurve/?field=5.5.14641.1&jinv=-91210781%2F43%2Aa%5E4+%2B+244652383%2F43%2Aa%5E3+-+46805862%2F43%2Aa%5E2+-+4530150%2Aa+%2B+54191702%2F43 | ## Results (unique match)
curve label base field conductor norm conductor label isogeny class label Weierstrass coefficients
43.3-a2 $$\Q(\zeta_{11})^+$$ 43 43.3 43.3-a $$\bigl[a^{3} + a^{2} - 2 a - 2$$ , $$a^{4} + a^{3} - 5 a^{2} - 4 a + 5$$ , $$a^{4} + a^{3} - 3 a^{2} - 2 a + 1$$ , $$-a^{4} + 8$$ , $$a^{4} + a^{3} - 5 a^{2} - 4 a + 4\bigr]$$ | {"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": 0.9994028806686401, "perplexity": 4783.799860749923}, "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-2019-35/segments/1566027331485.43/warc/CC-MAIN-20190826085356-20190826111356-00306.warc.gz"} |
http://clay6.com/qa/50957/a-solution-containing-one-mole-per-litre-each-of-cu-no-3-2-agno-3-hg-2-no-3 | A solution containing one mole per litre each of $Cu(NO_3)_2,AgNO_3,Hg_2(NO_3)_2$ is being electrolysed by using inert electrodes.The values of standard potentials are $E^{\large\circ}_{Ag^+|Ag}=0.80V,E^{\large\circ}_{Hg_2^{2+}|Hg}=0.79V,E^{\large\circ}_{Cu^{2+}|Cu}=0.34V,E^{\large\circ}_{Mg^{2+}|Mg}=-2.3V$ with increasing voltage,the sequence of deposition of metals on the cathode will be
$\begin{array}{1 1}Ag,Hg,Cu,Mg\\Mg,Cu,Hg,Ag\\Ag,Hg,Cu\\Cu,Hg,Ag\end{array}$
Electrolysis will occur if $E_{ext} > E_{galvanic}$.The value of $E_{galvanic}$ (in which metal is oxidized) is $E_{H_2O|O_2|Pt}-E_M^{n+}|M$.Hence,$E_{ext}$ will be reached in the order Ag,Hg and Cu.Hence,the sequence of deposition of metals on the cathode will be Ag,Hg and Cu.Before Mg is deposited,$H_2$ gas will be obtained by the reduction of $H_2O$
answered Jul 22, 2014 | {"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": 0.8663633465766907, "perplexity": 2336.274987884434}, "config": {"markdown_headings": false, "markdown_code": false, "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-2017-43/segments/1508187825174.90/warc/CC-MAIN-20171022094207-20171022114207-00113.warc.gz"} |
http://mathhelpforum.com/differential-geometry/123929-proof-question-what-s-happening.html | # Thread: Proof question: What's happening?
1. ## Proof question: What's happening?
Okay, so I did get the problem right the first try. However I am not sure what's going on.
I have the problem: Give an epsilon-delta proof of the limit(x^2-5x-3) = 3 as X approaches -1.
At a point in the proof I come to a point where I have ...|(x-6)(x+1)|<E
My notes and book have some really, really wild stuff going on. Basically "Agree that delta is less than or equal to 1", then it does something really wild. The problem ends with it being epsilon over 8, then the proof continues like normal.
I'm not sure what's happening mathematically here. The PDF on limits has only confused me further, as the notes I have ( which are a copy of the instructor's ) don't mention using a variable such as "M" in them.
Thanks!
2. Originally Posted by Wolvenmoon
Okay, so I did get the problem right the first try. However I am not sure what's going on.
I have the problem: Give an epsilon-delta proof of the limit(x^2-5x-3) = 3 as X approaches -1.
At a point in the proof I come to a point where I have ...|(x-6)(x+1)|<E
My notes and book have some really, really wild stuff going on. Basically "Agree that delta is less than or equal to 1", then it does something really wild. The problem ends with it being epsilon over 8, then the proof continues like normal.
I'm not sure what's happening mathematically here. The PDF on limits has only confused me further, as the notes I have ( which are a copy of the instructor's ) don't mention using a variable such as "M" in them.
Thanks!
You have $\lim_{x \rightarrow -1} x^2-5x-3 =3$
$|(x^2-5x-3)-3| < \epsilon$ If $0<|x+1|<\delta$
That's just using the definition to prove the limit. The first part means that value of your function $f(x)=x^2-5x-3$ will fall between $3- \epsilon$ and $3+ \epsilon$. The second bit just means that x lies in the interval $(-1-\delta , -1+\delta)$. To complete your proof you only need to discover a value of $\delta$ for which this statement holds, and then you must prove that the statement holds for $\delta$.
By the way, $|(x^2-5x-3)-3|= |x^2-5x-6|$ $= |(x-6)(x+1)| < \epsilon$
It's just been factorized, if you were wondering where it came from. | {"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": 11, "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": 0.8584966659545898, "perplexity": 336.66255683735926}, "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-2017-13/segments/1490218191353.5/warc/CC-MAIN-20170322212951-00437-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://proceedings.neurips.cc/paper/2019/hash/ea9268cb43f55d1d12380fb6ea5bf572-Abstract.html | #### Authors
Nicolas Keriven, Gabriel Peyré
#### Abstract
Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or \emph{equivariant} (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we prove new universality theorems. More precisely, we consider networks with a single hidden layer, obtained by summing channels formed by applying an equivariant linear operator, a pointwise non-linearity, and either an invariant or equivariant linear output layer. Recently, Maron et al. (2019) showed that by allowing higher-order tensorization inside the network, universal invariant GNNs can be obtained. As a first contribution, we propose an alternative proof of this result, which relies on the Stone-Weierstrass theorem for algebra of real-valued functions. Our main contribution is then an extension of this result to the \emph{equivariant} case, which appears in many practical applications but has been less studied from a theoretical point of view. The proof relies on a new generalized Stone-Weierstrass theorem for algebra of equivariant functions, which is of independent interest. Additionally, unlike many previous works that consider a fixed number of nodes, our results show that a GNN defined by a single set of parameters can approximate uniformly well a function defined on graphs of varying size. | {"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": 0.9277783036231995, "perplexity": 369.0454171242562}, "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-2021-31/segments/1627046155529.97/warc/CC-MAIN-20210805095314-20210805125314-00520.warc.gz"} |
https://en.wikipedia.org/wiki/Empty_product | # Empty product
In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity 1 (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.[1][2][3]
The term "empty product" is most often used in the above sense when discussing arithmetic operations. However, the term is sometimes employed when discussing set-theoretic intersections, categorical products, and products in computer programming; these are discussed below.
## Nullary arithmetic product
### Justification
Let a1, a2, a3,... be a sequence of numbers, and let
$P_m = \prod_{i=1}^m a_i = a_1 \cdots a_m$
be the product of the first m elements of the sequence. Then
$P_m = a_m \cdot P_{m-1}$
for all m = 1,2,... provided that we use the following conventions: $P_1 = a_1$ and $P_0=1$. In other words, a "product" $P_1$ with only one factor evaluates to that factor, while a "product" $P_0$ with no factors at all evaluates to 1. Allowing a "product" with only one or zero factors reduces the number of cases to be considered in many mathematical formulas. Such "products" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty product is one convention" is common practice in mathematics and computer programming.
### Relevance of defining empty products
The notion of an empty product is useful for the same reason that the number zero and the empty set are useful: while they seem to represent quite uninteresting notions, their existence allows for a much shorter mathematical presentation of many subjects.
For example, the empty products 0! = 1 and x0 = 1 shorten Taylor series notation (see zero to the power of zero for a discussion when x=0). Likewise, if M is an n × n matrix then M0 is the n × n identity matrix.
As another example, the fundamental theorem of arithmetic says that every positive integer can be written uniquely as a product of primes. However, if we do not allow products with only 0 or 1 factors, then the theorem (and its proof!) become longer.[4][5]
More examples of the use of the empty product in mathematics may be found in the binomial theorem (which assumes and implies that x0=1 for all x), Stirling number, König's theorem, binomial type, binomial series, difference operator and Pochhammer symbol.
### Logarithms
Since logarithms turn products into sums, they should map an empty product to an empty sum. So if we define the empty product to be 1, then the empty sum should be $\ln(1)=0$. Conversely, the exponential function turns sums into products, so if we define the empty sum to be 0, then the empty product should be $e^0 = 1$.
$\prod_i x_i = e^{\sum_i \ln x_i}$
## Nullary Cartesian product
Consider the general definition of the Cartesian product:
$\prod_{i \in I} X_i = \{ g : I \to \bigcup_{i \in I} X_i\ |\ \forall i\ g(i) \in X_i \}.$
If I is empty, the only such g is the empty function $f_\varnothing$, which is the unique subset of $\varnothing\times\varnothing$ that is a function $\varnothing \to \varnothing$, namely the empty subset $\varnothing$ (the only subset that $\varnothing\times\varnothing = \varnothing$ has):
$\prod_\varnothing{} = \{ f_\varnothing: \varnothing \to \varnothing \} = \{ \varnothing\}.$
Thus, the cardinality of the Cartesian product of no sets is 1.
Under the perhaps more familiar n-tuple interpretation,
$\prod_\varnothing{} = \{ ( ) \},$
that is, the singleton set containing the empty tuple. Note that in both representations the empty product has cardinality 1.
### Nullary Cartesian product of functions
The empty Cartesian product of functions is again the empty function.
## Nullary categorical product
In any category, the product of an empty family is a terminal object of that category. This can be demonstrated by using the limit definition of the product. An n-fold categorical product can be defined as the limit with respect to a diagram given by the discrete category with n objects. An empty product is then given by the limit with respect to the empty category, which is the terminal object of the category if it exists. This definition specializes to give results as above. For example, in the category of sets the categorical product is the usual Cartesian product, and the terminal object is a singleton set. In the category of groups the categorical product is the Cartesian product of groups, and the terminal object is a trivial group with one element. To obtain the usual arithmetic definition of the empty product we must take the decategorification of the empty product in the category of finite sets.
Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may not exist in a given category; e.g. in the category of fields, neither exists.
## In computer programming
Many programming languages, such as Python, allow the direct expression of lists of numbers, and even functions that allow an arbitrary number of parameters. If such a language has a function that returns the product of all the numbers in a list, it usually works like this:
listprod( [2,3,5] ) --> 30
listprod( [2,3] ) --> 6
listprod( [2] ) --> 2
listprod( [] ) --> 1
This convention sometimes helps avoid having to code special cases like "if length of list is 1" or "if length of list is zero" as special cases.
Many programming languages do not permit the direct expression of the empty product, because they do not allow expressing lists. Multiplication is taken to be an infix operator and therefore a binary operator. Languages implementing variadic functions are the exception. For example, the fully parenthesized prefix notation of Lisp languages gives rise to a natural notation for nullary functions:
(* 2 2 2) ; evaluates to 8
(* 2 2) ; evaluates to 4
(* 2) ; evaluates to 2
(*) ; evaluates to 1 | {"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": 17, "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": 0.9058471322059631, "perplexity": 381.2535721069904}, "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-2014-15/segments/1398223206672.15/warc/CC-MAIN-20140423032006-00581-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://gradestack.com/CBSE-Class-11th-Science/Trigonometric-Functions/Trigonometric-Ratios-and/17567-3565-29689-study-wtw | # Trigonometric Ratios and Identities
This topic of Trigonometry is already introduced in Standard X.
Recollect b. given any one Trigonometric ratio, you know how to find the other ratios.
Or
sin and cos are complements of each other
tan and cot are complements of each other
sec and cosec are complements of each other
e. Simple Identities
f. Values of Trigonometric ratios of angles 0, 30°, 45°, 60° and 90°.
sin cos tan cosec sec cot 0° 0 1 0 not defined 1 not defined 30° 2 45° 1 1 60° 2 90° 1 0 not defined 1 not defined 0
You have used all these properties in heights and distance problems involving right angled triangles. | {"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": 0.9738025069236755, "perplexity": 3150.8351733824843}, "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/1500549448095.6/warc/CC-MAIN-20170728062501-20170728082501-00691.warc.gz"} |
https://math.stackexchange.com/questions/2152348/does-the-matrix-exponential-have-this-convexity-property | # Does the matrix exponential have this convexity property?
Let $A$ be a real matrix (not necessarily symmetric) whose off-diagonal elements are all non-negative, so that the elements of the matrix exponential $\exp(A)$ are all non-negative. Let $\mathbf x$ be a vector whose elements are all non-negative. Let $\sum(\cdot)$ denote the sum of all the terms in a vector.
I hypothesise that the function $f(t) = \log \,\sum\big(\exp(At)\, \mathbf x\big)$ is a convex function of $t$.
Note that '$\log$' in this formula is an ordinary logarithm while '$\exp$' is a matrix exponential. From playing around with this I haven't found an easy way to show it, but I haven't found an obvious counterexample either. So my question is, is $f(t) = \log\, \sum\big(\exp(At)\, \mathbf x\big)$ a convex function of $t$, if the entries of $A$ and $\mathbf x$ are non-negative?
For the avoidance of doubt, and for ease of searching for counterexamples, here is some Python code that calculates $f(t)$. (A and x should be numpy arrays of the appropriate dimensions.)
import numpy as np
import scipy.linalg as spl
def f(t, A, x):
return np.log(np.sum(spl.expm(A*t).dot(x)))
Notes
1. Originally I was using the convention that "convex" means a downward curve, i.e. the second derivative is never positive. However, Jonas Meyer points out in a comment that there are examples where the second derivative is consistently positive. So now I guess my question is, can the second derivative of $f(t)$ change sign as a function of $t$?
2. I'm mostly only interested in part of the function where $t>0$, though in examples it seems to work for $t<0$ as well, until the sum of the elements becomes negative.
3. It might be necessary to make additional assumptions about $A$ and $\mathbf x$, e.g. that $\exp(A)$ is irreducible, or that $\mathbf x$ has all positive elements. If that's the case I'd like to know.
4. The motivation has to do with growing populations in biology. It's equivalent to asking whether the per capita rate of population growth is always non-increasing in a simple model of a growing (or shrinking) population.
5. This question is obviously closely related but not the same, and my hypothesis doesn't seem to immediately follow from that result.
6. The result will be true if the elements of $\exp(At)$ are log-convex, which would be a generally useful result to know about if it's true. It turns out not to be the case that all the elements of $\exp(At)$ are either consistently all convex upward or convex downward; they can change from one to the other as a function of $t$. See my related MathOverflow question. This makes me skeptical that the hypothesis in this (Math.SE) question is true.
Here is an example of what it tends to look like. For this example, $$A = \begin{pmatrix}1,1\\4,1\end{pmatrix}, \qquad \mathbf{x} = \begin{pmatrix}1\\0\end{pmatrix}.$$
• So my first reflex when you gave that context was to make up an example with a negative eigenvalue. This is easy to do, and if $x$ is such an eigenvector then $|e^{At} x|$ is decreasing...but then it is still log-convex (exponentials are both log-convex and log-concave), and at any rate such an $x$ must have negative entries. So I think the only way to get a counterexample will be with complex eigenvalues. This is impossible in 2D (check that $(a_{11}+a_{22})^2-4(a_{11}a_{22}-a_{12}a_{21}) = (a_{11}-a_{22})^2 + 4a_{12} a_{21} \geq (a_{11}-a_{22})^2 \geq 0$) but it is already possible in 3D. – Ian Feb 20 '17 at 3:02
• Thanks for the helpful comment. The fact that a log of a sum of exponentials is convex is one motivation for the hypothesis, but I agree the possibility of complex eigenvalues makes it not immediately clear. – Nathaniel Feb 20 '17 at 3:04
• When I've played around with randomly generated examples, whenever there were complex eigenvalues in 3D they had negative real part. That meant that the real or imaginary parts of these eigenvectors provided an "irrelevant counterexample", i.e. they failed to satisfy the conclusion but they also failed to have nonnegative entries. So I think you might be good... – Ian Feb 20 '17 at 6:43
• With diagonal $2$-by-$2$ examples you can have $f''>0$ instead of $f''<0$. So is that a couterexample for you? E.g., take $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$, $x=(1,1)^T$. – Jonas Meyer Feb 28 '17 at 2:45
• @JonasMeyer that's odd - thank you very much. That example makes sense now that I see it. I suppose now I have to change the question to ask whether the second derivative can change sign. – Nathaniel Feb 28 '17 at 3:23
Ok, so I can finally put this question to bed with a numerical example. Here is a plot of $\log Z$ against $t$ for some particular non-negative matrix $A$:
The second derivative clearly changes sign twice, once at around 0.3 and again just after 0.5.
It's probably not worth printing the whole matrix here since it's $100\times 100$ and there must be smaller, more instructive examples, but it was generated with the following rather arbitrary Python code:
n = 100
A = np.random.random((n,n)) # should be a Metzler matrix
A = np.exp(W*2)
A = W - np.eye(n)
A[np.random.random((n,n))>0.02] = 1e-3
(The point was to put the leading eigenvalue close to the cloud of other eigenvalues that random matrices tend to exhibit.)
Here is a random counter example:
A =
0.3384042 0.1355143 0.2244029 0.0094600 0.3658720
0.7335746 0.5823969 0.9182229 0.8483730 0.9743839
0.3992142 0.6501968 0.8946143 0.8120155 0.5938882
0.8255626 0.0570781 0.2135843 0.8498636 0.1053567
0.4915833 0.7814256 0.1870035 0.4286809 0.6484581
x =
0.384444
0.441505
0.623156
0.016760
0.791119
This produces this function like this:
Looks pretty convex (although I am not sure if you are probably after concavity), but its not. Around zero there is slightly decreasing derivative…
• It doesn't look remotely like that when I plot it. Are you sure you're calculating it correctly? – Nathaniel Feb 28 '17 at 1:48
• (By the way, in my experience there is no consistent convention for the meaning of 'convex' vs 'concave', except that they're the opposite of each other. Perhaps it's a physics versus mathematics thing. For me, in this context, 'convex' means the second derivative is less than or equal to zero, as per the plot in the question. I'll edit to make that clear.) – Nathaniel Feb 28 '17 at 1:49
• Oh, I just realized that by $|\cdot|$ you meant the sum of the elements (and not the sum of the absolute values) - quite unfortunate notation I think. Since $\exp(At)$ has some negative entries, this makes the difference. – Dirk Feb 28 '17 at 6:55
• Also I have never seen any ambiguity about what is a convex or concave function, also in physics. – Dirk Feb 28 '17 at 7:01
• I changed the notation in the question. In physics the entropy is referred to as a convex function, but it's concave according to your convention. Not sure there's much value in discussing that issue further. – Nathaniel Feb 28 '17 at 7:26 | {"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": 0.8604424595832825, "perplexity": 278.30802155207414}, "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-2019-39/segments/1568514574039.24/warc/CC-MAIN-20190920134548-20190920160548-00262.warc.gz"} |
http://physics.aps.org/articles/v6/15 | # Focus: Landmarks—Computer Simulations Led to Discovery of Solitons
Published February 8, 2013 | Physics 6, 15 (2013) | DOI: 10.1103/Physics.6.15
#### Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States
N. J. Zabusky and M. D. Kruskal
Published August 9, 1965
Landmarks articles feature important papers from the archives of the Physical Review journals.
The emergence of computers as a tool for doing science didn’t merely help researchers solve difficult problems. It also led to the discovery of entirely unexpected phenomena. An example appeared in Physical Review Letters in 1965, in a report describing solitary waves—dubbed “solitons” by the authors—that moved somewhat like individual particles. Solitons are now recognized as a widespread phenomenon, occurring in fluids, optics, and even the ultracold atomic clouds known as Bose-Einstein condensates.
In 1955 Enrico Fermi, John Pasta, and Stanislaw Ulam (FPU) came across a puzzling result when using the MANIAC I computer at what was then called the Los Alamos Scientific Laboratory in New Mexico. They wrote a program to follow the motion of up to 64 masses connected by springs in a horizontal line. Each mass could move only in the direction of the line, stretching or compressing the two springs connected to it.
The team started the simulation by displacing each mass from its initial position in a pattern that formed one half of a sine wave, with the end masses having zero displacement and the middle masses having the greatest displacement. The masses would then oscillate, and if the springs were strictly linear—that is, if their force were proportional to the amount of stretch or compression—then a snapshot of the motion at any time in the future would show the masses still in a sine wave pattern. But Fermi and his colleagues added a small degree of nonlinearity to the springs’ force, expecting it to break up the sine wave and cause the oscillation energy to become, in time, equally distributed among all the masses.
That’s not what happened. Although the sine wave indeed evolved into a more complex oscillation, the motion of the masses never became completely disorderly, and in fact it periodically returned to the initial state [1].
A decade later, Norman Zabusky, then at Bell Labs in Whippany, New Jersey, collaborated with Martin Kruskal of Princeton University to re-examine the FPU work. They transformed the discrete masses-and-springs equation into one for a continuous system similar to water waves. The team then programmed a computer to calculate the wave motion over a fixed horizontal distance but in such a way that a disturbance passing out of one side of the range reappeared at the other side.
Like FPU, Zabusky and Kruskal set the system in motion with an initial sine wave pattern. As the wave rolled along, its leading edge became steadily steeper and then developed smaller-wavelength ripples. These ripples eventually grew into individual waves that moved independently, with a velocity that depended on their height. Remarkably, when these new waves occasionally collided, they passed through each other, emerging almost wholly unscathed from their encounters. In addition, the waves would regularly align to reproduce the initial sine wave, momentarily, before separating again and repeating—not quite perfectly—the same cycle. This phenomenon was similar to the periodic return to the initial state that FPU had observed.
Zabusky and Kruskal soon learned that their equation had a name: It was the Korteweg-de Vries equation, devised in 1895 by two Dutch physicists to explain solitary waves—single, isolated peaks—that had been observed occasionally in canals as early as the 1830s. Any interaction between such waves was expected to be complex and hard to predict because of the nonlinearity. However, Zabusky, now at the Weizmann Institute of Science in Rehovoth, Israel, says that he and Kruskal were amazed to find that their solitary waves could run through each other and remain intact. He says this behavior was so striking that these waves deserved a name, and he came up with “solitons.”
Zabusky says that the discovery of solitons met with some skepticism, which he was able to overcome in part by using Bell Labs’ facilities to make movies of the waves. Some years later, physicists began to find soliton solutions in other wave equations, making their credibility unarguable.
Gennady El of Loughborough University in England calls the Zabusky-Kruskal paper a classic example of how numerical simulations can “provide insight into deep and fundamental properties of a mathematical model and lead to the discovery of completely new phenomena.” An important modern offshoot, he adds, is the theory of “dispersive shock waves”—coherent, nonlinear structures that can be regarded as systems of interacting solitons, and which show up in Bose-Einstein condensates and nonlinear optics.
–David Lindley
David Lindley is a freelance writer in Alexandria, Virginia, and author of Uncertainty: Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science (Doubleday, 2007).
### References
1. E. Fermi, J. R. Pasta, and S. Ulam, “Studies of nonlinear problems. I.,” Report LA-1940, Los Alamos Scientific Laboratory (1955). | {"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": 0.818981409072876, "perplexity": 1391.7515483160362}, "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/1414637900397.29/warc/CC-MAIN-20141030025820-00066-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.ideals.illinois.edu/handle/2142/16340/browse?type=contributor&value=Katz%2C+Sheldon | Browse Dissertations and Theses - Mathematics by Contributor "Katz, Sheldon"
• (2014-05-30)
The Hilbert scheme of $n$ points in a smooth del Pezzo surface $S$ parameterizes zero-dimensional subschemes with length $n$ on $S$. We construct a flat family of deformations of Hilb$^n S$ which can be conceptually ...
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• (2012-09-18)
This thesis consists of three parts. In the first part, we compute the topological Euler characteristics of the moduli spaces of stable sheaves of dimension one on the total space of rank 2 bundle on P1 whose determinant ...
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• (2006)
We consider the problem of constructing K3-fibered and elliptically fibered Calabi-Yau threefolds over P1 and P2 respectively. We first show how to write weighted projective space bundles as toric varieties. We then ...
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• (2017-07-10)
In this thesis we mainly consider supermanifolds and super Hilbert schemes. In the first part of this dissertation, we construct the Hilbert scheme of $0$-dimensional subspaces on dimension $1 | 1$ supermanifolds. By ...
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• (2010-08-20)
In this thesis, we first use the ${\mathbb C^*}^2$-action on the Hilbert scheme of two points on a Hirzebruch surface to compute all one-pointed and some two-pointed Gromov-Witten invariants via virtual localization, then ...
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• (2017-07-05)
Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \cc$. The {\it $K$-rank of $f$} is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear ...
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• (2011-08-25)
This thesis is composed of two parts. In the first part we introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold $X$. More precisely, we develop a moduli theory for ...
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http://math453spring2009.wikidot.com/pop-culture | Lecture 28 - Math in Pop Culture
## Summary
We started class by introducing the play Proof. Proof, written by David Auburn and published in 2001, deals with three mathematicians: a mathematics Professor, his daughter, and his former student. The class read two scenes aloud that gave way to our first two topics. The first scene was an argument between the Professor and his daughter about how many days she has wasted. After some back and forth, they realize that she has wasted a little over 33 days, which, if treated as years, and then converted to weeks, equals 1729 weeks. This number is special because it is expressible as the sum of two cubes in two different ways. Wael showed us a theorem showing how to characterize numbers that are the sums of two cubes. The second scene from the play that the class read aloud was about the daughter and the former student discussing mathematics conferences. The daughter tells Sophie Germain's story, which leads into our second theorem below presented by Jen, regarding when Germain primes occur.[1]
Afterwards, Tom discussed a conversation from the television show Bones in which the topic of perfect numbers came up. He followed with some brief history of perfect numbers, by proving and disproving some claims by Nicomachus, and following up with a neat characteristic of perfect numbers.
## Characterizing the sum of two cubes
Theorem: Let $n \in \mathbb{Z^+}$. Then $\exists \;\; x,y \in \mathbb{Z^+}$ satisfying the equation $n=x^3+ y^3$ if and only if conditions (a), (b) and (c) are satisfied:
• (a) $\exists \;\; m \in \mathbb{Z^+}$ such that $m \mid n$ and $n^{1/3} \le m \le 2^{2/3} \cdot n^{1/3}$ such that,
• (b) $\exists \;\; k \in \mathbb{Z^+}$ such that $m^2 - \frac{n}{m} = 3k$ such that,
• (c) $m^2 - 4k$ is a perfect square.
Proof: $( \; \Longrightarrow \;)$ Let $n = u^3 + v^3$ with $n,u,v \in \mathbb{Z^+}$. We need to show that the three conditions (a), (b) and (c) are satisfied.
We have that $n = u^3 + v^3 = (u+v)(u^2 - uv+v^2)$. Letting $m = u+v$, then $m \in \mathbb{Z^+}$ and $m \mid n$ so we need to verify that $n^{1/3} \le m \le 2^{2/3} \cdot n^{1/3}$ in order to prove that (a) is satisfied. Notice that
(1)
\begin{align} u^2 - uv + v^2 = \frac{n}{m} \end{align}
(2)
$$u + v = m$$
so it can be easily seen that the point $(u,v)$ satisfy the following two equations
(3)
\begin{align} x^2 - xy + y^2 = \frac{n}{m} \end{align}
(4)
$$x+y = m$$
Now let's treat both equations geometrically. The second equation is an equation of a line with a slope of $-1$. For the first equation, we know that the general equation for a conic section is
(5)
$$ax^2 + bxy + cy^2 + dx + ey = f$$
and that equation represents an Ellipse if and only if $f \neq 0$ and $f\cdot (b^2 - 4ac) < 0$. In Equation (3), we have that $a=1, b=-1, c=1, d=0, e=0, f=\frac{n}{m}$ so we can deduce that $f = \frac{n}{m} \neq 0$ because $n,m \in \mathbb{Z^+}$ so dividing them will result always in a positive real number. Moreover, $f\cdot (b^2-4ac) = \frac{n}{m} \cdot(1 - 4(1)(1)) = - \frac{3n}{m} < 0$ so we conclude that Equation (3) represents an Ellipse. It can be shown that this Ellipse has a major axis lying on the line $y=x$ which has a slope of $+1$ (showing that requires cumbersome computations, and it can be easily checked with graphing programs so we will avoid it here).
Now, we know that the point $(u,v)$ satisfies both Equations (3) and (4), and that $u$ and $v$ are both positive, so the point $(u,v)$ lies on the first quadrant, and so the line in Equation (4) intersects with the ellipse in Equation (3) in the first quadrant only. Observe also that the lines $x+y=m$ and $y=x$ are perpendicular since their slopes' product is $(+1) \times (-1) = -1$. Using those two observations we can sketch the following graph
In the graph above, it should be clear that the line $x+y=m$ (represented by the dashed line) has no fixed place, but we are sure that it lies somewhere on or between the parallel lines $L_1$ and $L_2$ (and parallel to them) since it must intersect with the Ellipse in the first quadrant. Now, we know that the line $x+y=m$ will cut the $x$-axis when $y=0$ and that implies $x=m$. We also know that the ellipse will cut the $x$-axis when $y=0$ and that implies $x=\sqrt{\frac{n}{m}}$. Since the line will cut the $x$-axis on the point $A$ or beyond, we deduce
(6)
\begin{align} \sqrt{\frac{n}{m}} \le m \Longrightarrow \frac{n}{m} \le m^2 \Longrightarrow n^{1/3} \le m \end{align}
Now, since the line $x+y=m$ never exceeds the line $L_2$, so the distance from the origin $(0,0)$ to the line $x+y=m$ is less than or equal to the semi-major axis of the Ellipse (which is the line segment $\overline{OB}$). To calculate the distance from the origin to the line $x+y=m$ we use the general equation of calculating the distance from a point $(x_1,y_1)$ to the line $ax+bx+c=0$ which is
(7)
\begin{align} D = \frac{|a\cdot x_1 + b \cdot y_1 + c|}{\sqrt{a^2 + b^2}} = \frac{|1\cdot (0) + 1 \cdot (0) + (-m)|}{\sqrt{1^2+1^2}} = \frac{m}{\sqrt2} \end{align}
we also can deduce (from the properties of ellipses and the Pythagorean Theorem) that the semi-major axis has a length of
(8)
\begin{align} |OB|^2 = \left( \sqrt{\frac{n}{m}} \right)^2 + \left( \sqrt{\frac{n}{m}} \right)^2 \Longrightarrow |OB| = \sqrt{2} \cdot \sqrt{\frac{n}{m}} \end{align}
we conclude that
(9)
\begin{align} D \le |OB| \Longrightarrow \frac{m}{\sqrt2} \le \sqrt{2} \cdot \sqrt{\frac{n}{m}} \Longrightarrow \frac{m^2}{2} \le 2 \cdot \frac{n}{m} \Longrightarrow m^3 \le 2^2 \cdot n \Longrightarrow m \le 2^{2/3} \cdot n^{1/3} \end{align}
combining (6) and (9), we get
(10)
\begin{align} n^{1/3} \le m \le 2^{2/3} \cdot n^{1/3} \end{align}
so condition (a) is satisfied.
For condition (b), note that $m^2 - \frac{n}{m} = (u+v)^2 - (u^2 -uv +v^2) = 3uv$, letting $k=uv \in \mathbb{Z^+}$ we see that condition (b) is satisfied.
For condition (c), note that $m^2 - 4k = (u+v)^2 -4uv = (u-v)^2 \in \mathbb{Z^+}$ so $m^2 -4k$ is a perfect square, so condition (c) is satisfied.
$( \Longleftarrow )$ Suppose that conditions (a), (b) and (c) are satisfied and let's prove that $n = x^3 + y^3$ for some positive integers $x$ and $y$. Define $m$ as in (a), let $k$ be a positive integer satisfying $m^2 - \frac{n}{m} = 3k \Longrightarrow \frac{n}{m} = m^2 - 3k$ as in (b), and let $m^2 - 4k$ be a perfect square as in (c).
Now, let's examine the equation $x^2 -mx + k=0$. This equation has a determinant $\Delta = m^2 - 4k$, and that's a perfect square, thus $\exists \; s \in \mathbb{Z}$ such that $m^2-4k=s^2$. Now consider the two solutions (let's call them $u$ and $v$) of the equation $x^2 -mx+k=0$. Using the general method of solving quadratic equations, we get
(11)
\begin{align} u = \frac{m + s}{2} , \; \; v = \frac{m-s}{2} \end{align}
(where $u$ and $v$ can be switched) Now, consider the parity of $m$ and $s$. We have that $m$ is either $\text{even}$ or $\text{odd}$. If $m = \text{odd}$ then by $s = \sqrt{m^2 - 4k}$ we get that the parity of $s$ is
(12)
\begin{align} \sqrt{ (\text{odd})^2 - \text{even} } = \sqrt{ \text{odd} - \text{even} } = \sqrt{\text{odd}} = \text{odd} \end{align}
from the above argument we conclude that if $m = \text{odd}$ then $s = \text{odd}$ and thus $m - s$ and $m + s$ are both $\text{even}$, hence $\frac{m+s}{2}, \frac{m-s}{2} \in \mathbb{Z} \Longrightarrow u,v \in \mathbb{Z}$. By a similar argument, when $m = \text{even}$ we get that the parity of $s$ is
(13)
\begin{align} \sqrt{ (\text{even})^2 - \text{even} } = \sqrt{ \text{even} - \text{even} } = \sqrt{\text{even}} = \text{even} \end{align}
and thus $\frac{m+s}{2}, \frac{m-s}{2} \in \mathbb{Z} \Longrightarrow u,v \in \mathbb{Z}$. So we conclude that in both cases, $u$ and $v$ will always be integers. It can be readily seen that $m=u+v$ and $k=uv$. Now let's look at the signs (positive or negative) of $u$ and $v$. Since $k$ is positive, we get that $u$ and $v$ have the same sign. But since their sum (which is $m$) is also positive, then they can't be both negative, and thus $u$ and $v$ are both positive, or $u,v \in \mathbb{Z^+}$. Now we deduce that
(14)
\begin{align} n = m \cdot \frac{n}{m} = (u+v)(m^2 -3k) = (u+v)((u+v)^2 - 3uv) = (u+v)(u^2 - uv + v^2) = u^3 + v^3 \end{align}
and thus $n$ can be represented as a sum of two positive cubes. $\Box$
## Germain Primes
Define: A prime $p$ is said to be a Sophie Germain prime if both $p$ and $2p+1$ are prime.
Examples: The first few Sophie Germain primes are $2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ...$ It is not known if there are an infinite number of Sophie Germain primes.
The largest known Sophie Germain prime as of Feb. 2009 has $p= 48047305725 \cdot 2 ^{172403} - 1$, which has $51910$ decimal digits and was found by Chris Caldwell on Jan. 25, 2007.
Theorem: Let $p \equiv 3 \mod{4}$ be prime. $2p+1$ is also prime if and only if $2p+1$ divides $M_p$, a Mersenne number of the form $2^{p} - 1$.
Proof: $( \Longrightarrow )$ Suppose $q = 2p+1$ is prime. Then $q = 2p + 1 = 2 (4k+3) + 1 \;\; \text{[by how we defined p]} \; = 8k+7 \equiv 7 \mod{8}$ so $q \equiv 7 \mod{8}$ so $2$ is a quadratic residue $\text{mod} \; q$ and $\left( \frac{2}{q} \right) = 1$. This shows,
(15)
\begin{align} 2^{p} \equiv 2^{\frac{q-1}{2}} \equiv \left( \frac{2}{q} \right) \; \; \text{[by Euler's Criterion]} \; \equiv 1 \mod{q} \Longrightarrow q \mid M_p \end{align}
$( \Longleftarrow )$ Conversely, let $2p+1$ be a factor of $Mp = 2^{p} -1$. So $2p+1 \mid 2^{p} -1$. We want $2p+1$ to be prime. For contradiction, suppose $2p+1$ is composite and let $m$ be its least prime divisor, which implies $2p+1 \ge m^2$. This is because the prime factorization of $2p+1$ must include at least one other prime divisor, call it $d$. So we get that $2p+1 \ge m\cdot d$ but $d \ge m$ because $m$ is the least prime divisor of $2p+1$, so we get that $2p+1 \ge m \cdot d \ge m \cdot m = m^2$. Now we get,
(16)
\begin{align} m \mid 2p+1 \;\; \text{and} \; \; 2p+1 \mid 2^{p} -1 \Longrightarrow m \mid 2^{p} -1 \Longrightarrow 2^{p} \equiv 1 \mod{m} \Longrightarrow \text{ord}_m(2) \mid p \end{align}
Now, $p$ is prime, so its only divisors are $1$ and $p$, which means $\text{ord}_m(2) = 1 \; \text{or} \; p$. If $\text{ord}_m(2)= 1$, then $2^{1} \equiv 1 \mod{m} \Leftrightarrow m \mid 2-1 \Leftrightarrow m \mid 1$, which can't happen. So $\text{ord}_m(2) = p$.
Since $m$ is prime, Fermat's little theorem gives $2^{m-1} \equiv 1 \mod{m} \Longrightarrow \text{ord}_m(2) \mid m-1$, but $\text{ord}_m(2) = p$ so $p \mid m-1 \Longrightarrow m > p \Longrightarrow m^{2} > p^{2}$. We know $2p+1 \ge m^{2}$, so $2p+1 \ge m^2 > p^2 \Longrightarrow 2p+1 > p^2$. However, since $p>2$, this is a contradiction, so $2p+1$ is prime. $\Box$
## Perfect Numbers
Euclid gave in Book IX of the Elements a very wordy description of perfect numbers as seen below:
"If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect."
Which roughly translates to: $\text{if } \;\; 1+2+2^{2}+...+2^{k} \;\; \text{ is prime, then that sum times } \;\; 2^{k} \;\; \text{ must be perfect}$.
Examples:
(17)
\begin{align} 1+2=3 \;\; \Longrightarrow 3 \cdot 2=6 \;\; \text{ is perfect and } \;\; 1+2+2^{2}=7 \;\; \Longrightarrow7 \cdot 2^{2}=28 \;\; \text{is perfect} \end{align}
We then listed the first four perfect numbers: 6, 28, 496, and 8128 which led to Nicomachus five unproven claims of perfect numbers:
1. The $n^{th}$ perfect number has $n$ digits.
2. All perfect numbers are even.
3. All perfect numbers end in either $6$ or $8$, alternately.
4. Euclid's Algorithm to generate perfect numbers will give all perfect numbers. $2^{p-1} \cdot (2^{p} - 1) \;\; \text{for} \;\; p\ge 1 \;\; \text{ and } \;\; (2^{p} - 1) \;\; \text{ prime}$. Euler later proved this was only the case when $p$ was prime.
5. There are infinitely many perfect numbers.
Claim 4 if we remember, was proven by Andy in lecture 15. Click here for a refresher Lecture 15 - Perfect Numbers. Thus we were able to check this claim as being true.
Claim 2 is believed to be true as no odd perfect numbers have yet to be found. Many are trying to prove the existence of odd perfect numbers and have a lower bound of $10^{300}$. However many are also trying to prove the non existence of odd perfect numbers. Check out the links below if interested to learn more about odd perfect numbers.
Claim 5 is also accepted to be true but still unproven since the number of Mersenne primes is still unknown and since whenever you find a new Mersenne prime, you also find an even perfect number, and vice versa.
Disproving Claim 1
Having Euclid's Algorithm, Haldrichus Regins in 1536 was able to show $(2^{11} - 1) = 2047 =23 \cdot 89$ which is not a Mersenne prime and thus not leading to a perfect number. By then testing $(2^{13} - 1) =8191$ he was able to show it was prime and thus resulting in the fifth perfect number: $2^{12} \cdot (2^{13} -1) = 33,550,336$. He was able to show that $8191$ is prime by taking the square root of $8191$ and then testing whether any of the primes below that value were able to divide $8191$.
As we can see, the fifth perfect number here has $n=8$ digits and thus disproves claim 1 listed above.
Claim 3
In a similar fashion as Regins, Cataldi in 1603 constructed two tables. The first was a table of the numbers $1$ through $800$ and all of the factors for each number. He was then able to construct a table with all the prime numbers between $1$ and $750$. Since $750^{2} = 562,500$ Cataldi knew that he had all prime divisors possible for $(2^{17} - 1) = 131,071 < 562,500 = 750^2$.
Through tedius calculations he was able to show that $131,071$ is prime, thus resulting in the sixth perfect number: $2^{16} \cdot (2^{17}-1) = 8,589,869,056$. However because the fifth and sixth perfect numbers both end in $6$, claim 3 is partially false. We know they don't alternate between $6$ and $8$, but do all perfect numbers end in $6$ or $8$? The proof proving this follows:
Theorem: Every perfect number ends in either $6$ or $8$.
Proof: Every prime number greater than $2$ is of the form $4m+1 \;\; \text{or} \;\; 4m+3$.
Case 1: $p=4m+1$. We will use this property: $6^m \equiv{6} \pmod{10} ; \forall \; m \in \mathbb{Z^+}$ which can be proven easily by induction: for $m=1$ we have $6^1 \equiv{6} \pmod{10}$, now suppose that $6^m \equiv{6} \pmod{10}$ for some integer $m \ge 2$, now notice that $6^{m+1} \equiv{6 \cdot 6^m} \equiv{ 6 \cdot 6} \equiv{36} \equiv{6} \pmod{10}$ so the statemnet holds for all integers $m \ge 1$. Now let $N$ be our perfect number
(18)
\begin{align} N=2^{4m} \cdot (2^{4m+1}-1) = 16^{m} \cdot (2 \cdot 16^{m}-1) \equiv 6^{m} \cdot (2 \cdot 6^{m}-1) \equiv 6 \cdot (2 \cdot 6 -1) \equiv{ 6 \cdot 11} \equiv{6} \pmod{10} \end{align}
Thus showing when a prime is of the form $4m+1$ and results in a Mersenne prime, the resulting perfect number will end in $6$.
Case 2: $p=4m+3$. Let $N$ be our perfect number
(19)
\begin{align} N=2^{4m+2}\cdot (2^{4m+3}-1) = 4 \cdot 16^{m} \cdot (8 \cdot 16^{m}-1) \equiv 4 \cdot 6 \cdot (8 \cdot 6-1) \equiv{24 \cdot 47} \equiv{ 4 \cdot 7 } \equiv{8} \pmod{10} \end{align}
Thus showing when a prime is of the form $4m+3$ and results in a Mersenne prime, the resulting perfect number will end in $8$.
Case 3: $p=2$. Let $N$ be our perfect number
(20)
\begin{align} N=2^{1} \cdot (2^{2}-1) = 2 \cdot 3 = 6 \end{align}
Since when our prime is $2$ our perfect number is $6$, we can conclude that all even perfect numbers end in $6$ or $8$ and thus claim 3 was partially true.$\Box$
To finish the presentation Tom presented a couple other facts about perfect numbers. First off at the moment there are 46 known even perfect numbers with the largest having 26 million digits! And secondly, the reciprocals of the divisors of a perfect number add up to 2.
Proof:
Let $N$ be our perfect number. We have $\sigma (N) = 2N$ . Let $\lbrace d_1 , d_2 , ... , d_k \rbrace$ be the set of positive divisors of $N$.
(21)
\begin{align} \frac{1}{d_1} + \frac{1}{d_2} + ...+ \frac{1}{d_{k-1}} + \frac{1}{d_k} \text{ has common denominator N, since all divisors divide evenly into N.} \end{align}
In addition if you divide $N$ by one of its divisors, you result in another divisor of $N$. Without loss of generality, let $1 = d_1 < d_2 < .... < d_{k-1} < d_k = N$. Now, since $N$ has an even number of divisors (because $N$ is of the form $2^{p-1} \cdot (2^p - 1)$ and this form has only one copy of the prime $2^p -1$ and thus $N$ can't be a perfect square, and only perfect squares have an odd number of divisors) then we have "divisor pairs" where $(d_1 \cdot d_k) = N , (d_2 \cdot d_{k-1}) = N, ....$ and so on for all divisors of $N$. Thus if we adjust for a common denominator in the summation of the reciprocals above we simply get
(22)
\begin{align} \frac{d_1 + d_2 +...+ d_{k-1} + d_k}{N} \Longrightarrow \frac{\sigma (N)}{N} = \frac{2N}{N} =2 \end{align}
$\Box$
Bibliography
1. Auburn, David. Proof. New York: Faber and Faber, Inc., 2001. | {"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": 19, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.99144446849823, "perplexity": 284.16902083968836}, "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/1534221217901.91/warc/CC-MAIN-20180820234831-20180821014831-00078.warc.gz"} |
https://www.physicsforums.com/threads/many-worlds-and-many-minds.224789/ | # Many Worlds and Many Minds
1. Mar 27, 2008
### Artificial
Anyone know anything about creating artificial consciousness through the blueprints of people's consciousness, thus creating an artificial society. Through this you can predict how people will interact in an environment through social simulations and manipulate them.
Through quantum computers, I believe that the theory of many worlds would be computed by evolutionary algorithms to predict the most likely reality and by changing variables you can manipulate reality itself.
2. Mar 28, 2008
### Schrodinger's Dog
I don't think Many Minds/MWI, has much to do with practical theory atm outside of the hypothetical. So to give an answer would mean assuming that MWI is correct. Which is a bit of a stretch for your average (Copenhagen Interpretation) CI advocate. That said, I suppose in "theory", given that the quantum computer is modelled on a MWI then it ought to be perfectly possible to create an AI type environment in the computer, based on the same idea as MWI by modelling just such a reality.
That said though I doubt you'd get much interest in funding such a project as it's all a bit arm wavey methinks atm. I have read some interesting stuff about reality being a computer simulation, which was quite plausible in a self contained sort of way if not slightly bizarre. So for all we know we're already living in just such a Universe, modelled by just such a race of people. Anyone seen The Matrix.
3. Mar 28, 2008
### vanesch
Staff Emeritus
This thread is way over-speculative. This has nothing to do anymore with quantum theory. | {"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": 0.8045833706855774, "perplexity": 1315.2184623399805}, "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-2018-47/segments/1542039742685.33/warc/CC-MAIN-20181115120507-20181115142507-00484.warc.gz"} |
https://ai.stackexchange.com/tags/policy-gradients/hot | Tag Info
36
However, both approaches appear identical to me i.e. predicting the maximum reward for an action (Q-learning) is equivalent to predicting the probability of taking the action directly (PG). Both methods are theoretically driven by the Markov Decision Process construct, and as a result use similar notation and concepts. In addition, in simple solvable ...
14
Just ignore the invalid moves. For exploration it is likely that you won't just execute the move with the highest probability, but instead choose moves randomly based on the outputted probability. If you only punish illegal moves they will still retain some probability (however small) and therefore will be executed from time to time (however seldom). So you ...
11
Usually softmax methods in policy gradient methods using linear function approximation use the following formula to calculate the probability of choosing action $a$. Here, weights are $\theta$, and the features $\phi$ is a function of the current state $s$ and an action from the set of actions $A$. $$\pi(\theta, a) = \frac{e^{\theta \phi(s, a)}}{\sum_{b \... 9 An important thing we're going to need is what is called the "Expected Grad-Log-Prob Lemma here" (proof included on that page), which says that (for any t):$$\mathbb{E}_{\tau \sim \pi_{\theta}(\tau)} \left[ \nabla_{\theta} \log \pi_{\theta}(a_t \mid s_t) \right] = 0.Taking the analytical expression of the gradient (from, for example, slide 9) ... 8 This Tutorial by OpenAI offers a great comparison of different RL methods. I'll try to summarize the differences between Q-Learning and Policy Gradient methods: Objective Function In Q-Learning we learn a Q-function that satisfies the Bellman (Optimality) Equation. This is most often achieved by minimizing the Mean Squared Bellman Error (MSBE) as the loss ... 8 Using the law of iterated expectations one has: \triangledown _\theta \sum_{t=1}^T \mathbb{E}_{(s_t,a_t) \sim p(s_t,a_t)} [b(s_t)] = \nabla_\theta \sum_{t=1}^T \mathbb{E}_{s_t \sim p(s_t)} \left[ \mathbb{E}_{a_t \sim \pi_\theta(a_t | s_t)} \left[ b(s_t) \right]\right] = written with integrals and moving the gradient inside (linearity) you get = \sum_{... 7 The "trick" of subtracting a (state-dependent) baseline from the Q(s, a) term in policy gradients to reduce variants (which is what is described in your "baseline reduction" link) is a different trick from the modifications to the rewards that you are asking about. The baseline subtraction trick for variance reduction does not appear to be present in the ... 7 As you has said, actions chosen by Actor-Critic typically come from a normal distribution and it is the agent's job to find the appropriate mean and standard deviation based on the the current state. In many cases this one distribution is enough because only 1 continuous action is required. However, as domains such as robotics become more integrated with AI, ... 7 I faced a similar issue recently with Minesweeper. The way I solved it was by ignoring the illegal/invalid moves entirely. Use the Q-network to predict the Q-values for all of your actions (valid and invalid) Pre-process the Q-values by setting all of the invalid moves to a Q-value of zero/negative number (depends on your scenario) Use a policy of your ... 7 IMHO the idea of invalid moves is itself invalid. Imagine placing an "X" at coordinates (9, 9). You could consider it to be an invalid move and give it a negative reward. Absurd? Sure! But in fact your invalid moves are just a relic of the representation (which itself is straightforward and fine). The best treatment of them is to exclude them completely ... 6 It depends on your loss function, but you probably need to tweak it. If you are using an update rule like loss = -log(probabilities) * reward, then your loss is high when you unexpectedly got a large reward—the policy will update to make that action more likely to realize that gain. Conversely, if you get a negative reward with high probability, this will ... 6 In order for the algorithm to have stable behavior, the replay buffer should be large enough to contain a wide range of experiences, but it may not always be good to keep everything. The larger the experience replay, the less likely you will sample correlated elements, hence the more stable the training of the NN will be. However, a large experience replay ... 5 The first part of this answer is a little background that might bolster your intuition for what's going on. The second part is the more practical and direct answer to your question. The gradient is just the generalization of the derivative to multivariable functions. The gradient of a function at a certain point is a vector that points in the direction of ... 5 You need to read this 2020 paper by Deepmind: "Revisiting Fundamentals of Experience Replay" Also, to add to the answer by @nbro Assume you implement experience replay as a buffer where the newest memory is stored instead of the oldest. Then, if your buffer contains 100k entries, any memory will remain there for exactly 100k iterations. Such a ... 5 Absolutely, it’s a really interesting problem. Here is a paper detailing off policy actor critic. This is important because this method can also support continuous actions. The general idea of off-policy algorithms is to compare the actions performed by a behaviour policy (which is actually acting in the world) with the actions the target policy (the ... 5 Calculation of gradient \begin{align} \nabla_{\theta} \log(\pi_{\theta}(a|s)) &= \phi(s,a) - \mathbb E[\phi (s, \cdot)]\\ &= \phi(s,a) - \sum_{a'} \pi(a'|s) \phi(s,a') \end{align} is only valid for linear function approximation with action preferences of form $$h(s, a, \theta) = \theta^T \phi(s,a)$$ and softmax policy \... 4 We subtract mean from values and divide it with standard deviation to get data with mean of zero and variance of one. The range of values per episode does not matter, it will always make it to have zero mean and variance of one in all cases. If the range is bigger ([100, 200]) then deviation will be bigger as well than for smaller range ([0, 1]) so we will ... 4 When using the loss function for the critic described in your question, the Actor-Critic is an on-policy approach (as are most Actor-Critic methods). Your intuition as to what it is learning seems to be quite close, but the notation/terminology is not quite on point. First it's important to realize that the Q(s, a) critic is an estimator, we're training ... 4 The loss function is estimated in every batch training cycle. Gradients of the loss are computed and propagation back to the network happens in every cycle. This means that you use a small batch (e.g. 100 instances) from the replay memory, and by having the states you can input them to the respective network and have the Q(s) for every state in your batch. ... 4 I think what you mean to ask is how can differentiation occur when there's no obvious neural network function to differentiate? Don't worry - lots of people get confused about this, because it seems like an obvious hole in the puzzle. As mentioned by @AtillaOzgur, neural networks use partial differentiation through backpropagation. First, take the output ... 4 First, the derivative is usually taken with respect to a variable (input) of the function. Hence the notation \frac{df}{dx} for some function f(x). If you look at your equation more carefully\nabla log P(\tau^{i};\theta) = \Sigma_{t=0}\nabla_{\theta}log\pi(a_{t}|s_t, \theta).You will see that the gradient is taken with respect to \theta, which ... 4 DQN on the other hand, explores using epsilon greedy exploration. Either selecting the best action or a random action. This is a very common choice, because it is simple to implement and quite robust. However, it is not a requirement of DQN. You can use other action choice mechanisms, provided all choices are covered with a non-zero probability of being ... 4 In the policy gradient theorem, we don't need to write r as a function of a because the only time we explicitly 'see' r is when we are taking the expectation with respect to the policy. For the first couple lines of the PG theorem we have \begin{align} \nabla v_\pi(s) &= \nabla \left[ \sum_a \pi(a|s) q_\pi (s,a) \right] \;, \\ &= \sum_a \left[ \... 4 The key to REINFORCE working is the way the parameters are shifted towards G \nabla \log \pi(a|s, \theta). Note that \nabla \log \pi(a|s, \theta) = \frac{ \nabla \pi(a|s, \theta)}{\pi(a|s, \theta)}. This makes the update quite intuitive - the numerator shifts the parameters in the direction that gives the highest increase in probability that the action ... 3 This question is discussed in detail, in the following NeurIPS 2016 paper by David Silver: Learning values across many orders of magnitude. They also give experimental results over the Atari domain. 3 A stateless RL problem can be reduced to a Multiarmed Bandit (MAB) problem. In such a scenario, taking an action will not change the state of the agent. So, this is the setting of a conventional MAB problem: at each time step, the agent selects an action to either perform an exploration or exploitation move. It then records the reward of the taken action ... 3 Usually it is assumed that there is no correlation between different actions, so the covariance matrix will be zero everywhere except on the main diagonal. Diagonal will represent variances of actions. Diagonal covariance matrix will be positive semidefinite if all values on diagonal are \geq 0 so you need to insure that output of final layer is \geq 0, ... 3 Let's say your old policy is \pi_b and your current one is \pi_a. If you collected trajectory by using policy \pi_b you would get return G whose expected value is \begin{align} E_{\pi_b}[G_t|S_t = s] &= E_{\pi_b}[R_{t+1} + G_{t+1}]\\ &= \sum_a \pi_b(a|s) \sum_{s', r} p(s', r|s, a) [r + E_{\pi_b}[G_{t+1}|S_{t+1} = s']]\\ \end{align} You can ... 3 Softmax policy \pi_\theta(s,a) is defined as \frac{\exp{(\phi(s,a)^T \theta})}{\Sigma \exp{(\phi(s,a) ^T \theta) }}, where the summation is over the action space. Taking log, this becomes \log \pi_\theta(s,a) = log(e^{\phi(s,a) ^T \theta}) - log({\Sigma e^{\phi(s,a) ^T \theta }}) \\ = \phi(s,a) ^T \theta - log({\Sigma e^{\phi(s,a)^T \theta }}) $$... 3 The unrolling step is due to the fact you end up with an equation that you can keep expanding indefinitely. Note that we start with calculating \nabla v_\pi(s) and arrive at$$\nabla v_\pi(s) = \sum_a\left[ \nabla \pi(a|s) q_\pi(s,a) + \pi(a|s) \sum_{s'}p(s'|s,a) \nabla v_\pi (s') \right]\;, which contains a term for $\nabla v_\pi(s')$. This is a ...
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https://www.ias.ac.in/listing/bibliography/jcsc/Hemant_K._Kashyap | • Hemant K Kashyap
Articles written in Journal of Chemical Sciences
• Non-ideality in Born-free energy of solvation in alcohol-water and dimethylsulfoxide-acetonitrile mixtures: Solvent size ratio and ion size dependence
Recent extension of mean spherical approximation (MSA) for electrolyte solution has been employed to investigate the non-ideality in Born-free energy of solvation of a rigid, mono-positive ion in binary dipolar mixtures of associating (ethanol-water) and non-associating (dimethylsulfoxide-acetonitrile) solvents. In addition to the dipole moments, the solvent size ratio and ion size have been treated in a consistent manner in this extended MSA theory for the first time. The solvent-solvent size ratio is found to play an important role in determining the non-ideality in these binary mixtures. Smaller ions such as Li+ and Na+ show stronger non-ideality in such mixtures compared to bigger ions (for example, Cs+ and Bu4N+). The partial solvent polarization densities around smaller ions in tertiary butanol (TBA)-water mixture is found to be very different from that in other alcohol-water mixtures as well as to that for larger ions in aqueous solutions of TBA. Non-ideality is weaker in mixtures consisting of solvent species possessing nearly equal diameters and dipole moments and is reflected in the mole fraction dependent partial solvent polarization densities.
• Solvent sorting in (mixed solvent + electrolyte) systems: Time-resolved fluorescence measurements and theory
In this manuscriptwe explore electrolyte-induced modification of preferential solvation of a dipolar solute dissolved in a binary mixture of polar solvents. Composition dependence of solvation characteristics at a fixed electrolyte concentration has been followed. Binary mixtures of two different polarities have been employed to understand the competition between solute-ion and solute-solvent interactions. Time-resolved fluorescence Stokes shift and anisotropy have been measured for coumarin 153 (C153) in moderately polar (ethyl acetate + 1-propanol) and strongly polar (acetonitrile + propylene carbonate) binary mixtures at various mixture compositions, and in the corresponding 1.0M solutions of LiClO4. Both the mixtures show red shifts in C153 absorption and fluorescence emission upon increase of mole fraction of the less polar solvent component in presence of the electrolyte. In addition, measured average solvation times become slower and rotation times faster for the above change in the mixture composition. A semi-molecular theory based on solution density fluctuations has been developed and found to successfully capture the essential features of the measured Stokes shift dynamics of these complex multi-component mixtures. Dynamic anisotropy results have been analyzed by using both Stokes-Einstein-Debye (SED) and Dote-Kivelson-Schwartz (DKS) theories. The importance of local solvent structure around the dissolved solute has been stressed.
• Three-dimensional Morphology and X-ray Scattering Structure of Aqueous tert-Butanol Mixtures: A Molecular Dynamics Study
It is well established that water-alcohol mixtures exhibit anomalous properties at very low as well as at very high alcohol concentrations. Almost all the studies in this regard intend to link these anomalies to the microscopic structural changes as water (or alcohol) concentration increases in the mixture. However, it isimportant to note that the nature of these structural changes could be different at the water- and TBA-rich concentrations. In this article, our goal is to address such structural change overs, if really present, in the mixtures of water and tert-butanol (TBA) by using simulated X-ray scattering structure function, S(q), real space radial and spatial distribution functions and heterogeneity order parameter. By using a judicial partitioning scheme, we show that structural characteristic of pure water is qualitatively retained for XTBA < 0.1. The simulated S(q) peaks at around q=2 and q=2.8 Å⁻¹, which correspond to water oxygen correlations, begin to fade away only after XTBA ≥ 0.1. This is a clear indication of microscopic structural transition at XTBA ≈ 0.1. Beyond XTBA = 0.1, the TBA structural features begin to take over to that of water. The peak at q=1.3 Å⁻¹ which primarily corresponds to nonpolar-nonpolar correlations in pure TBA begin to rise at XTBA ≈ 0.1. However, the pre-peak at around q=0.75 Å⁻¹, which is due to polar-polar and nonpolar-polar correlations in pure TBA, seems to appear at lower q value only at the equi-molar concentration of the mixture. From the solvent cage surrounding the TBA molecules, we observe that while the aggregation of TBA alkyl groups, due to hydrophobic interaction, is maximum at 10% TBA, the intervening hydrogen bonding interactions between water and TBA molecules tend to lower the hydrophobic interactions between the alkyl groups of alcohol with increasing concentration of TBA. In addition to this, we also observe dimers and small clusters of water molecules in the TBA-rich regime. The computed heterogeneity order parameters for the individual components of the mixture reveal enhanced non-uniform distribution of the TBA molecules near XTBA ≈ 0.1 to 0.3. These results are also supported by the radial distribution functions and nearest neighbour coordination numbers of water and TBA oxygen atoms around TBA oxygen.
• # Journal of Chemical Sciences
Volume 135, 2023
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
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http://math.stackexchange.com/questions/868997/principal-bundle-definition-cum-exercise-from-geometry-and-topology-by-bredon | # Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon
The definition of fiber bundle can be found from here: Definition of Fiber Bundle
Then Bredon defines Principal bundle in the exercise as follows:
I am not able to show how K acts naturally on X and the rest of the exercise.
My try was-
We need to get a map $X \times K \rightarrow X$. Let $(k,x) \in K\times X$.
Then $p(x)\in B$ and by the trivial fibration condition there exists $U \ni p(x)$ such that there is homeomorphism $\varphi :U\times F \rightarrow p^{-1}(U)$ .
Send $(k,x)$ to $\varphi(p(x),k)$.
Note: By definition of Principal Bundle F and K are same.
Am I correct?
How to prove the next part of the exercise?
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## 1 Answer
I dont think you have defined an action yet. If we take $g \in K$ then we use the map $\varphi$ to define the action where if $(u,h) \in U\times K$ then $$g(u,h)=(u,gh)$$ now you need to use the fact that $K$ is the structural group to show this this is well defined.
The orbit space is $B$ since $K$ acts transitively.
If you have a section, set the values equal to the group identity element and use this to define a homeomorphism with the trivial bundle.
Both these last statement need to have the details supplied.
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Given an element $(x,k)\in X\times K$ we have to define an element of $X$. How does another element $g\in K$ come into picture? – Babai Jul 16 '14 at 15:59
I didnt understand your notation, you are still not defining an action. How should $K$ act n a fibre ? It should act by group multiplication. In your definition $k$ send the entire fibre to the element $k$, and this is not invariant under change of coordinates. – Rene Schipperus Jul 16 '14 at 16:13
K acts on F by right translation, yes that is correct. But then what is the action of K on X ? And K and F are same by definition. – Babai Jul 16 '14 at 16:20
I changed my notation to avoid confusion. – Rene Schipperus Jul 16 '14 at 16:20
I guess you want to say this- for $(x,g)\in X\times K$ consider the fibration $\varphi$ over $U$ such that $p(x) \in U$ So $x\in p^{-1}(U)$. Let $(u,h)=\varphi ^{-1}(x)$ So,send $(x,g)$ to $\varphi (x,gh)$. Am I correct? – Babai Jul 16 '14 at 16:31 | {"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": 0.9797492623329163, "perplexity": 328.5229621829912}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2015-18/segments/1429246633512.41/warc/CC-MAIN-20150417045713-00185-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/11537/calculating-fourier-transform-at-any-frequency | Calculating fourier transform at any frequency
I know that if we have some data representing some wave, for example image line values, we can use fourier transform to get frequency function of that wave. But we have N values at points x=0...N-1 And we get only N frequencies at the output. So I want to analyze the wave everywhere in the range [0, N-1] For example at point u = 1.5. How can I do it?
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You need to have some a priori knowledge about your wave between and beyond the sampling points to get a meaningful guess about the full Fourier transform. The $N$ values that you get doing the discrete Fourier transform have anything to do with the continuous Fourier transform only for indices much less than $N$. Note that different assumptions will lead to different answers for large frequences. If you need only relatively low frequences, your signal is compactly supported in time, and your sampling points cover the entire support, you can safely interpolate the values of the discrete FT to guess the continuous one but there is no way to get reliable high precision values for the continuous Fourier transform at frequences comparable to $N$. The $N$-point resolution on $[0,1]$ is just not high enough to catch those frequences without an error.
Also note that you want the frequences not on "on the interval $[0,N-1]$" as you wrote, but rather on the interval $[-M,M]$ where $M\ll N$. For decent signals, the frequences with indices close to $N$ in the discrete $FT$ actually catch the negative low frequences in the continuous FT, not the high ones.
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Strictly speaking, you can't. That is, you have to interpolate to do it, and to do that in a meaningful way requires some knowledge of the problem domain. However, the easiest way to do so is to note that the given function is naturally written in terms of its discrete Fourier transform as a trigonometric polynomial, so you can just go ahead and evaluate that trigonometric polynomial at any intermediate point. The same thing goes with the function and its Fourier transform interchanged, of course. Though it does not really make good mathematical sense to do this interpolation in both domains simultaneously.
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Here is one answer from the electrical engineering(or, image processing) point of view. However this answer much non-mathematical and more for practical illustration. Since this is a non-mathematical answer, I have made it community wiki so that I don't get reputation.
In digital signal processing, you sample an analog signal into digital encoding, and let your microprocessor work with it. When you do sampling, you clearly lose some information and there is no way you can recover it. When you become too greedy and actually try to get more, the problem of "aliasing" will happen.
Here is the wikipedia link for aliasing.
For image processing, situation is even better for an illustration, since you can actually see wavy patterns. These are called Moiré patterns, and you must have seen it sometime in the real world. Here is a wikipedia link.
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Pretty silly argument to make the answer community wiki! :D – Mariano Suárez-Alvarez Jan 12 '10 at 20:55
If what you need is a simple practical method to do the interpolation, then just multiply the "time domain" samples by the linear-phase signal $\{\exp\left(\alpha\cdot i 2\pi k /N\right)\}_{k=0}^{N-1}$, where $\alpha\in(0,1)$ is the sub-sample shift in the "frequency domain." (I'm not sure if an additional constant of absolute value 1 is required here). As already noted in previous answers, the motivation for this is the assumption that your samples vector $\{x_k\}$ comes from sampling some continuous-time signal $\{X(t)\}_{t\in \mathbb{R}}$, as in $x_k=X(kT_s)$ for $k=0,\ldots, N-1$, where $T_s\in \mathbb{R}_{++}$ is the sampling period. If the continuous-time signal has a compactly
supported Fourier transform, than by Shannon's sampling theorem $X$ is determined by the infinite sequence $\{X(kT_s)\}_{k\in \mathbb{Z}}$ for small enough $T_s$. Since we only have $N$ entries of the infinite sequence, we loose some information on $X$. But if $X$ decays rapidly (rather than having a compact support), then at least intuitively we don't loose much (I know that this is a dangerous and imprecise statement :) ).
For example, try the following Matlab lines:
ttt=-32:31;
x_time=exp(-ttt.^2/10);
figure;plot(x_time);
x_freq = fft(x_time);
figure;plot(fftshift(abs(x_freq)));
figure
for alpha=-4:0.5:4
x_time_2 = x_time .* exp(2 * pi * i * (0:63)/64 * alpha);
x_freq=fft(x_time_2);
plot(fftshift(abs(x_freq)));
axis([26,37,0,10]);
grid on;
pause(1);
end
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https://stats.stackexchange.com/questions/245166/least-squares-coefficient-estimates-in-calculus-and-matrix-calculus?noredirect=1 | # least squares coefficient estimates in calculus and matrix calculus
I am a beginner stat learner and recently I've been reading Introduction to Statistical Learning with Applications in R by Hastie and Tibshirani.
On linear regression, it says:
$$RSS=(y_1 -\hat{B_0} - \hat{B_1x_1})^2 + (y_2 -\hat{B_0} - \hat{B_1x_2})^2+...+(y_n -\hat{B_0} - \hat{B_1x_n})^2.$$
The least squares approach chooses $\hat{B_0}$ and $\hat{B_1}$ to minimize the RSS. Using some calculus, one can show that the minimizers are:
$$\hat{B_1} =\frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i-\bar{x})^2},$$
$$\hat{B_0} = \bar{y}-\hat{B_1}\bar{x}.$$
In a different book, The Elements of Statistical Learning by the same authors, $\hat{\beta}$ is defined as:
$$\hat{\beta} = (X^T X)^{-1} X^Ty.$$
Are the $\hat{\beta}$ the same as $\hat{B_1}$? I assumed they were the same until I realized that they don't seem to produce the same result.
$X^T y$ seems to match with $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})$ and $(X^T X)^{-1}$ seems to match with $\sum_{i=1}^n (x_i-\bar{x})^2$ but I don't know why the algebraic solution subtracts the mean while the matrix computation doesn't.
• It is likely helpfull to note that $X$ (the design matrix) is an $n \times (k+1)$ matrix. Nov 10 '16 at 7:49
They are talking about the same thing. They simply used different notations and one is a particular case of the other one.
I'll start with The Elements of Statistical Learning which is the general case. We have:
$$\hat{\beta} = (X^TX)^{-1}X^Ty$$
Here $\hat{\beta}$ is a vector of the form $(\hat{\beta_1},\hat{\beta_2},..\hat{\beta_p})$ and is the vector of fitted coefficients for a linear regression with $p$ variables, including intercept. We also have $X$ the design matrix having each $x_i$ as columns and $y$ the vector with the independent variable. Those equations are well known and sometimes are named normal equations.
Let's move to the ITSL book. The exposition from there discuss a particular case of a multivariate linear regression. Specifically, it describes the linear regression with a single dependent variable and an intercept. That means in out case the design matrix $X$ has two columns: the intercept (all ones) and the single dependent variable $x$. So, $X = \begin{bmatrix}1 &x\end{bmatrix}$. Also we have $\hat{\beta}$ is the vector of the two fitted model parameters, so $\hat{\beta}=\begin{bmatrix}\hat{\beta_0} & \hat{\beta_1}\end{bmatrix}^T$ or in your notation $\begin{bmatrix}\hat{B_0} & \hat{B_1}\end{bmatrix}^T$. I will use beta instead of B, since I am more comfortable with it.
A preliminary calculus shows us that:
$$\begin{bmatrix}1 &x\end{bmatrix}^T \begin{bmatrix}1 &x\end{bmatrix} = \begin{bmatrix}n & \sum x \\ \sum x & n\end{bmatrix} = n\begin{bmatrix}1 & \bar{x}\\ \bar{x} & \frac{x^Tx}{n}\end{bmatrix}$$
Here we used the fact that:
$$\begin{bmatrix}1 &x\end{bmatrix}^T \begin{bmatrix}1 &x\end{bmatrix}=\begin{bmatrix}1 & 1 &.. & 1 \\ x_1 & x_2 & .. & x_n\end{bmatrix}\begin{bmatrix}1 & x_1 \\ 1 & x_2 \\ .. & .. \\1 & x_n\end{bmatrix} = \begin{bmatrix}n & \sum x\\\sum x & x^Tx\end{bmatrix}= n\begin{bmatrix}1 & \bar{x} \\ \bar{x} & \frac{x^Tx}{n}\end{bmatrix}$$
Considering that we now have the normal equations for your particular case as
$$\begin{bmatrix}\hat{\beta_0} \\ \hat{\beta_1} \end{bmatrix} = (n\begin{bmatrix}1 & \bar{x}\\ \bar{x} & \frac{x^Tx}{n}\end{bmatrix})^{-1} \begin{bmatrix}1 & x\end{bmatrix}^T y$$
Notice is not easy to invert in formula the covariance matrix, so we will multiply with that matrix on the right to get rid of the inverse. Thus we will obtain:
$$n\begin{bmatrix}1 & \bar{x}\\ \bar{x} & \frac{x^Tx}{n}\end{bmatrix} \begin{bmatrix}\hat{\beta_0} \\ \hat{\beta_1} \end{bmatrix} = \begin{bmatrix}1 & x\end{bmatrix}^T y$$
Moving $n$ to the right we have
$$\begin{bmatrix}1 & \bar{x}\\ \bar{x} & \frac{x^Tx}{n}\end{bmatrix} \begin{bmatrix}\hat{\beta_0} \\ \hat{\beta_1} \end{bmatrix} = \begin{bmatrix}\bar{y} \\ \frac{x^Ty}{n}\end{bmatrix}$$
What we obtained right now is a system of two equations, so both can be used. The first equation is what you already have:
$$\hat{\beta_0}-\bar{x}\hat{\beta_1} = \bar{y}$$
I am convinced that the second equations after some substitutions looks the way you saw it in the book.
As a conclusion the ISTL talks about a particular case and all the beta coefficients are scalars, and the other description works for generic case and beta from there is a vector of coefficients. Hope that helped.
• ah i see. That makes a lot more sense. This may be a very elementary question but how do you get $\sum x$ from multiplying $[1 \space x]^T$ and $[1 \space x]$? Nov 14 '16 at 5:19
• I had a mistake in calculus (sorry, it's always easier on paper). I updated the answer to help you with that also. Basically $\sum x$ is obtained by multiplying a row of $1$ with the column $x$ Nov 14 '16 at 8:41 | {"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": 0.9103161096572876, "perplexity": 220.9569547654707}, "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/1642320306301.52/warc/CC-MAIN-20220128152530-20220128182530-00399.warc.gz"} |
https://www.arxiv-vanity.com/papers/hep-th/9604128/ | OUT–4102–62
21 April 1996
hep-th/9604128
On the enumeration of irreducible -fold Euler sums
[3pt] and their roles in knot theory and field theory
D.B
Physics Department, Open University, Milton Keynes MK7 6AA, UK
http://yan.open.ac.uk/
Abstract A generating function is given for the number, , of irreducible -fold Euler sums, with all possible alternations of sign, and exponents summing to . Its form is remarkably simple: , where is the Möbius function. Equivalently, the size of the search space in which -fold Euler sums of level are reducible to rational linear combinations of irreducible basis terms is . Analytical methods, using Tony Hearn’s REDUCE, achieve this reduction for the 3698 convergent double Euler sums with ; numerical methods, using David Bailey’s MPPSLQ, achieve it for the 1457 convergent -fold sums with ; combined methods yield bases for all remaining search spaces with . These findings confirm expectations based on Dirk Kreimer’s connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron’s magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings.
## 1 Introduction
Recent progress in number theory [1, 2] interacts strongly with the connection between knot theory and quantum field theory, discovered by Dirk Kreimer [3, 4], and intensively investigated to 7 loops [5] by analytical and numerical techniques. The sequence of irreducible non-alternating double Euler sums studied in [1] starts with a level-8 sum that occurs in the 6-loop renormalization of quantum field theory [5], where its appearance is related [4] to the uniquely positive 8-crossing knot ; the sequence of irreducible non-alternating triple Euler sums in [2] starts with a level-11 sum that occurs at 7 loops, where its appearance is associated with the uniquely positive hyperbolic 11-crossing knot [5].
These exciting connections, between number theory, knot theory, and quantum field theory, led to work with Bob Delbourgo and Dirk Kreimer [6], on patterns [7] of transcendentals in perturbative expansions of field theories with local gauge invariance, and with John Gracey and Dirk Kreimer [8], on transcendentals generated by all-order [9] results in field theory, obtainable in the limit of a large number, , of interacting fields [10, 11].
In the course of the large- analysis [8], the number theory in [1] appeared to constitute a severe obstacle to the development of the connection between knot theory and field theory. From the skeining of link diagrams that encode the flow of momenta in Feynman diagrams, we repeatedly obtained a family of knots, associated with the occurrence of irreducible Euler sums in counterterms. The obstacle was created by the (indubitably correct) ‘rule of 3’ discovered in [1], for non-alternating sums of level . The analysis of [1] shows that non-alternating sums of odd levels are reducible, while at even level there are irreducibles, where is the integer part. At levels 8 and 10 this made us very happy, since we had the 8-crossing knot to associate with , and the 10-crossing knot to associate with . Thereafter the knots increase in number by a ‘rule of two’, giving knots with crossings. So there are two 12-crossing knots, while [1] has only one level-12 irreducible, and two 14-crossing knots, which is [1] the number of level-14 irreducibles.
Faced with a 12-crossing knot in search of a number, we saw two ways to turn: to study 4-fold non-alternating sums, or 2-fold sums with alternating signs. The first route is numerically intensive: it soon emerges that well over 100 significant figures are needed to find integer relations between 4-fold sums at level 12. The second route is analytically challenging; it soon emerges that at all even levels there are relations between alternating double sums that cannot be derived from any of the identities given in [1].
Remarkably, these two routes lead, eventually, to the same answer. The extra 12-crossing knot is indeed associated with the existence of a 4-fold non-alternating sum, , which cannot be reduced to non-alternating sums of lower levels. It is, equivalently, associated with the existence of an irreducible alternating double sum, . The equivalence stems from the unsuspected circumstance that the combination , and only this combination, is reducible to non-alternating double sums. Moreover, is the lowest level at which the reduction of non-alternating 4-fold sums necessarily entails an alternating double sum. The ‘problem pair’ of knots are a problem no more. Their entries in the knot-to-number dictionary [12] record that they led to a new discovery in number theory: the reduction of non-alternating sums necessarily entails alternating sums.
This discovery led me to study the whole universe of -fold Euler sums, with all possible alternations of sign, at all levels . As will be shown in this paper, it is governed by beautifully simple rules, which might have remained hidden, were it not for Dirk Kreimer’s persistent transformation of the Feynman diagrams of field theory to produce a pair of 12-crossing knots.
The remainder of the paper is organized as follows. Section 2 states111It would be very difficult to prove. One cannot even prove that is irrational. the formula for the number, , of irreducible -fold sums, with all possible alternations of sign, at level . Section 3 outlines the process by which it was discovered. The anterior numerics of Section 4 describe the high-precision evaluation methods and integer-relation searches that helped to produce the formula; the posterior analytics of Section 5 describe computer-algebra proofs of rigorous upper bounds on that are respected (and often saturated) by it. Section 6 summarizes numerical and analytical findings by listing an instructive choice of concrete bases. Section 7 shows that all 12 of the irreducible sums with appear in perturbative quantum electrodynamics. Section 8 considers the import, for quantum field theory, for knot theory, and for number theory, of results obtained by calculations up to level 23, corresponding to knots with up to 23 crossings, and to Feynman diagrams with up to 13 loops.
## 2 Result
To specify an alternating -fold Euler sum, one may give a string of signs and a string of positive integers. It is very222Without such a convention, results such as (22,30) become almost unreadable. convenient to combine these strings, by defining
ζ(a1,…,ak)=∑ni>ni+1k∏i=1(signai)nin|ai|i, (1)
for , on the strict understanding that the arguments are non-zero integers, and that , to prevent a divergence. Hence one avoids a proliferation of disparate symbols for the types of -fold sum. The correspondence with the double-sum notations of [1] is
ζ(s,t) = σh(t,s), (2) ζ(−s,t) = αh(t,s), (3) ζ(s,−t) = −σa(t,s), (4) Us,t≡ζ(−s,−t) = −αa(t,s), (5)
for positive integers and , with emphatically no implication of analytic continuation.
In perturbative quantum field theory, three-loop radiative corrections [13, 14] involve
U3,1=∑n>m>0(−1)n+mn3m=12ζ(4)−2{Li4(12)+124ln22(ln22−π2)}. (6)
At six [15, 16, 17] and seven [5] loops, counterterms associated with the and torus knots, [4] and , involve and , whose irreducibility is equivalent to that of and , respectively. In higher counterterms, the independent irreducibility of and is associated [8] with a pair of 12-crossing knots.
In what follows, the number of summations, , is referred to as the depth of the sum (1), and is referred to as its level. The set of convergent sums of level and depth has
N(l,k)=2k(l−1k−1)−2k−1(l−2k−2) (7)
elements, as is easily proven by induction. The number of convergent sums at level is
N(l)=l∑k=1N(l,k)=4×3l−2, (8)
provided that .
Table 1: Euler’s triangle333It seems appropriate to call it Euler’s triangle; see Section 8.3 for the connection with Pascal’s. of irreducibles, at level and depth , for .
l∖k123456789101112131415…11213141511611712182219133110253111157411238941131714125114314201551151925301861164204240226117112426655267118430759970307…19115661321439135…20540132212200112…2111899245333273…22555212429497…23122143429715…24670333800…25126200715…26691497…27130273…287112…29135…307…311…
The main purpose of this work is to determine, as in Table 1, the number, , of irreducible sums in , i.e. the minimum number of sums which, together with sums of lesser depth and products of sums of lower level, furnish a basis for expressing the elements of as linear combinations of terms, with rational coefficients. The conclusion is that
E(l,k)=δl,2δk,1+2l+k∑2d|l±kμ(d)(l+k2dl−k2d), (9)
where the summation is over the positive integers such that are integers, and is weighted by the Möbius function, , which vanishes if is divisible by the square of a prime and otherwise is , according as whether has an even or odd number of prime divisors. With the exception of , from , and , from , the irreducibles come from , with . Moreover, contains the same number of irreducibles as , as illustrated in Table 1.
From (9) one obtains the number of irreducibles at level :
E(l)=l∑k=1E(l,k)=1l∑d|lμ(l/d){Fd+1+Fd−1}, (10)
in terms of the Fibonacci numbers. The integer sequence (10) is tabulated as M0317 by Sloane and Plouffe [18], who record its origin in the study [19] of congruence identities. It shows that the irreducibles are an exponentially decreasing fraction of the number of convergent sums at level :
E(l)N(l)∼94lexp(−βl);β=ln6√5+1≈0.6174. (11)
This relative sparsity is illustrated in Table 2.
Table 2: The numbers, and , of irreducibles and sums, at level , for .
l12345678910111213E(l)11112245811182540N(l)141236108324972291687482624478732236196708588
From (9) one may obtain the size, , of the search space for sums of level and depth , i.e. the minimum number of terms that allow one to express every element of as a linear combination, with rational coefficients. This basis consists of the irreducibles with level and depths no greater than , together with all the independent terms that are formed from products of sums whose levels sum to and whose depths sum to no more than .
One may generate from , by expanding
lmax∑n=1⎡⎣lmax∑l=1⎛⎝(X2,1,1x2)ly+l∑k=1E(l,k)∑i=1Xl,k,ixlyk⎞⎠⎤⎦n (12)
to order in both and , where is greatest level required, and serves as a symbol for the th irreducible in , so that stands for any rational multiple of , and hence for . Then is obtained by selecting the terms of order , dropping powers of higher than , setting , and counting the length of the resulting expression. This procedure is easily implemented in REDUCE, Maple, Mathematica, etc.
Table 3: The size, , of the search space for sums in , for .
l∖k123456789101112131415112123123413455136786147111213714101419202181511212632333491515254046535455101616365172798788891116214176971251331421431441217225792148176212221231232233131728631331892733093543643753763771418298515528136548553058559660860961015183692218344554674839894960972985986987
Inspection of Table 3 reveals that the sizes satisfy the recurrence relation
S(l+1,k)=S(l,k−1)+S(l−1,k),for l>k>1. (13)
Moreover, the Fibonacci numbers appear at maximum depth:
S(l,l)=S(l,l−1)+1=Fl+1. (14)
From the Lucas relation between Fibonacci and binomial numbers, one obtains
S(l,k)=k−1∑n=0(⌊l+n−12⌋n), (15)
as the solution to (13,14).
Since the process of generating Table 3 from Table 1 is reversible, formula (9) may be replaced by the simple statement that at level the size of the search space increases by the binomial coefficient
S(l,k+1)−S(l,k)=(⌊l+k−12⌋k), (16)
when the depth is increased from to . This pragmatic formulation444Computation leaves no doubt as to the equivalence of (9) and (16), though it is not yet proven. is rather helpful, when using an integer-relation search algorithm, such as PSLQ [20]. An even simpler, though informal, restatement appears in Section 8.3.
## 3 Discovery
Some brief historical remarks seem in order at this stage, since my route to (9) in fact began with the observation of 7 members of the Fibonacci sequence (14), during the course of 800-significant-figure integer-relation searches entailed by the relation between knot theory and field theory.
Using David Bailey’s magnificent MPPSLQ [21] implementation of PSLQ [20], I succeeded in reducing all 1457 sums in to just 12 numbers and their products. These irreducible numbers may conveniently be taken as
ln2,π2,{ζ(l)∣l=3,5,7},{α(l)∣l=4,5,6,7},U5,1,ζ(5,1,−1),ζ(3,3,−1), (17)
with the polylogarithms
α(l)=(−ln2)ll!{1−l(l−1)12(πln2)2}+∞∑n=112nnl (18)
populating the deepest diagonal of Table 1. As in (6), the definition (18) postpones the appearance of and to depths and , respectively, which is required by (16). Moreover , and , which is not a new irreducible.
After noting the Fibonacci sequence for the maximum sizes , with , I sought a combinatoric form for . Formula (15) suggested itself on the empirical basis of the 28 cases with , and was then submitted to intense numerical and analytical tests at higher levels, as indicated in Sections 4 and 5. I implemented the generator (12), using the weight and length commands of REDUCE [22], and constructed Table 1, working backwards from (16). Next came the observation that the generating functions for the and columns of Table 1 have comparable forms:
G2(x)=12{1/(1−x)2−1/(1−x2)},G3(x)=13{1/(1−x)3−1/(1−x3)}. (19)
The symmetry of Table 1 was vital to the discovery of the simple formula
Gk(x)=∞∑n=0E(k+2n,k)xn=1k∑d|kμ(d)(1−xd)−k/d, (20)
which produces formula (9). Computation of , for , revealed that it produced the integers nearest to , where is the golden section. The equivalent form (10) was obtained by submitting the integer sequence to Neil Sloane’s helpful on-line [23] version of [18]. It was noteworthy that this lookup returned the values , from the Fibonacci form (10), agreeing with the appearance of , at , and , at . Their inclusion in the column of Table 1, above Euler’s triangle, thus became appropriate, as they appear to usher in the higher transcendentals, in much the same way that seed the exponential growth of MPPSLQ’s CPUtime (or Fibonacci’s rabbits) along the deepest diagonal of Table 3.
It remains to describe yet more probing numerical and analytical tests, in further support of the claim in (16), and its equivalent version in (20). Nonetheless, it is hoped that the reader already shares some of my feeling that these two formulæ are simply too beautiful to be wrong.
## 4 Anterior numerics
### 4.1 Numerical evaluation
Suppose that one wishes to obtain a high-precision approximation to , where the truncated sum has a summand with an asymptotic series in , starting at , with . From , one may form a table , by the procedure
T(n,m+1)=(C+m+n)T(n,m)−nT(n−1,m)C+m, (21)
with . The method exploits the vanishing of as . It takes applications of (21) to produce . Provided that , and that rounding errors have been controlled, one obtains the approximation , where the factorial becomes significant for .
This elementary and economical method of accelerated convergence is applicable to every Euler sum of the form (1) that has no argument equal to unity, in which case repeated appeal to the Euler-Maclaurin formula [24] underwrites the absence of logarithms in the expansion of the truncation error, and the outermost summation is of the form , with , for an argument . Thus one should store truncations after even increments of , and set for , or for , in procedure (21). To obtain the starting values, one has merely to set up a single loop that updates and stores each layer of the nest as its particular summation variable, , assumes the even and odd values within the loop. Thus the evaluation time for a truncation at of a -fold sum is roughly proportional to .
For sums with no unit arguments, one needs therefore only a few lines of conventional FORTRAN, which may be handed over to David Bailey’s TRANSMP [25] utility, to produce code that calls his MPFUN [26] multiple-precision subroutines. As a rule of thumb, the working accuracy should be somewhat better than the square of the desired output accuracy, when using (21). When, and only when, rounding errors are so controlled, an output accuracy of very roughly is achieved by iterations of (21), with input data obtained from looping over pairs of successive even and odd integers. For a 4-fold sum, the accumulation of data takes roughly calls of MPFUN subroutines, and the acceleration of the convergence takes roughly calls. Thus, to achieve significant figures for a 4-fold sum, one should choose a value of that keeps the time factor, , close to its absolute minimum. I find that between 780 and 800 significant figures are reliably and efficiently achieved with iterations, and truncation at , which entails calls to MPFUN subroutines, operating at multiple precision 1700. This takes less than half an hour on a DEC Alpha 3000-600 machine, corresponding to a call rate that is faster than 1 kHz. The memory requirement is less than 1 MB: an array of 4-byte cells holds the truncations in multiple precision and is updated iteratively by (21).
Of course, the above method fails as soon as one sets any of the arguments to unity. However, all is not lost. By iteratively applying the Euler-Maclaurin formula, one arrives at the conclusion that the maximum power of in the truncation error is the largest number, , of successive units in the string of arguments, since only for does the integration of increase the power of . Consider, for example, , with and . By the time the logarithm generated by the summation is felt by the summation, it has acquired an inverse power of , thanks to the benignity of the Euler-Maclaurin formula for the summation. Thus this sum has demon-number , even though it contains two unit arguments. On the other hand, and have , as does , for example. At level the most demonic convergent sum has unity for all its arguments, except for the first, which must therefore be . Fortunately, it is possible to give exact expressions for this beast, with , and another, with . With a string of unit arguments denoted by , the all-level results
ζ(−1,{1}l−1)=(−ln2)ll!,−ζ(−1,−1,{1}l−2)=Lil(12)=∞∑n=112nnl, (22)
were inferred numerically, and then obtained analytically, by Jon Borwein and David Bradley, as the coefficients of in and , generated by the trivially summable hypergeometric series . From one generates the corollary, , of a theorem [27]
∑ai>δi,1, l=Σiaiζ(a1,a2,…,ak)=ζ(l), (23)
from Andrew Granville, which was used to check evaluations of non-alternating sums.
To mitigate the computational difficulties caused by unit arguments, truncation errors may be obtained analytically for combinations of Euler sums of the form
S(a;b1,…,bk−1)=∞∑n=1(signa)nn|a|k−1∏i=1n−1∑mi=1(signbi)mim|bi|i. (24)
It seems appropriate to call (24) a boxed sum, since the symmetrical inner summations span a lattice that is confined to a -dimensional hypercube by the outer summation variable. It is built out of symmetrical combinations of Euler sums with depths no greater than , and is, so to speak, a ‘cheap boxed set’, available at a reduced [28] computational price, since it requires only the multiplication of polygamma Euler-Maclaurin series, followed by a single further application of the Euler-Maclaurin formula, to determine the truncation error to any order for which ones favorite computer-algebra engine has the power to multiply, differentiate, and integrate double series in and . Only depth-1 data is needed, since the inner summations can be rewritten as , with the first term giving a constant and the second an asymptotic series in , except for , which gives Euler’s constant, a log, and an asymptotic series.
This Euler-Maclaurin method is the obvious generalization of that used in [28], in the restricted cases with , or . In the general case, much computer-algebra time may be consumed by multiplying asymptotic series for the distinct values of and then integrating a long expression involving many powers of and , though the algorithm is straightforward to implement. To obtain a few hundred significant figures in a short time one may stay within REDUCE, without feeding thousands of lines of FORTRAN statements to TRANSMP; at higher precision one benefits from this transfer of function, at the expense of non-trivial file management, when handling many boxed sums.
From an analytical point of view, it is highly significant that every 3-fold Euler sum can be transformed into a boxed sum: , where will henceforward stand for ‘equality modulo terms of lesser depth, their products, and consequent questions of convergence’. The argument is simple [2]: to the summation with , for the truncation of , add that with , for the truncation of . Adding the case , which has lesser depth, one has covered the values , and hence has a product of truncated sums of lesser depth. Thus . The questions of convergence obviously concern the case with , in the limit . Such issues are handled with great dexterity in [2], in the case of non-alternating triple sums. Dealing, as now, with truncated sums, no problem of convergence arises. The snag is that one must multiply truncations to obtain the product term, so the process becomes both messy and ad hoc, from the perspective of a programmer seeking a systematic algorithm for sums of any depth. From a numerical point of view, little is gained from boxability at depth .
Truly nested Euler sums (1) have depth , where only combinations of them can be boxed. In the next subsection it will be shown that at there first occurs a significant phenomenon, of central relevance to the claim of Section 2, and to knot-theoretical studies [3, 4, 5, 8]. But first, there is an outstanding computational dilemma to confront. Should one attempt to automate the process of chaining applications of the Euler-Maclaurin formula, feeding in the required values for multiple Euler sums from runs of lesser depth, and carefully separating the odd and even summations, at each link of a chain whose length one would like to vary? Or should one, rather, use the empirical truncation data to accelerate the convergence? In the absence of logs, the one-line procedure (21) settles the issue, to my mind. Why use masses of computer algebra to feed into TRANSMP information that is already sitting in the numerical data, in such immediately usable form? When there are logs, from unit arguments, the matter is moot. Having taken the empirical approach when the going was easy, I opted to stick with it, when the going got tough, at depths , with demons.
It is clear that high-precision knowledge of truncated values, for a sum with demon-number , should suffice to accelerate the convergence by a factor of roughly , as was achieved with (21), for . How best to achieve this, for , is another matter. Unable to devise an easily programmable iterative method like (21), I returned to the brute-force method of using truncations to solve directly for unknowns, as was done a decade ago [15] in the investigations which first suggested the appearance of irreducibles with depth at the six-loop level of renormalization of quantum field theory, a prediction amply confirmed by recent six- and seven-loop analysis [5], and illuminated by knot theory [3, 4].
After translating the Gauss-Jordan [29] method into MPFUN calls, one may systematically obtain the coefficients, in the approximation
S(n)≈S∞+d∑i=0(lnn)iC+M−1∑j=CAi,jnj, (25)
from truncations, and in particular find an accurate value for . I chose the truncations , taken distances apart, starting at some value . The choices of , , , and working accuracy, to achieve a desired output accuracy for a sum with demon-number , are an art learnt by experience, not yet a science that is fit to be explained here. Suffice it to say that it is wise to work at the cube of the desired accuracy, and that solving sets of equations, at a working accuracy of one part in , in order to perform Euler-sum searches with the MPPSLQ [21] integer-relation finder, is demanding of core memory, CPUtime, vigilance, and patience. The rewards, in increased analytical understanding, are considerable.
### 4.2 Exact numerical results
By an exact numerical result, I mean an equation whose exactness is beyond intelligent doubt, yet is validated, to date, only by very high precision numerical evaluation. It is, strictly speaking, possible that an equation presented here is a divine hoax, just as the rationality of is still a possibility. The improbability beggars all description.
The exact numerical result that sparked the genesis of Tables 1 and 3, and revealed their utter simplicity, was found at level and depth . It reads
25⋅33ζ(4,4,2,2)=25⋅32ζ4(3)+26⋅33⋅5⋅13ζ(9)ζ(3)+26⋅33⋅7⋅13ζ(7)ζ(5) +27⋅35ζ(7)ζ(3)ζ(2)+26⋅35ζ2(5)ζ(2)−26⋅33⋅5⋅7ζ(5)ζ(4)ζ(3) −28⋅32ζ(6)ζ2(3)−13177⋅15991691ζ(12) +24⋅33⋅5⋅7ζ(6,2)ζ(4)−27⋅33ζ(8,2)ζ(2)−26⋅32⋅112ζ(10,2) +214U9,3 (26)
and the sting is in its tail.
Exceptionally, factorizations of rationals were written in (26), with a central dot (not to be confused with a decimal point) denoting multiplication. If one dislikes the number 691, one may remove it, using . Factorizations were given above lest the reader found the unfactorized rationals implausible, on first encountering an exact numerical result obtained by MPPSLQ. The practice will not be continued.
It is apparent that one needs at least 100 significant figures to discover (26), because of the product of two 5-digit primes in the term. MPPSLQ (almost always) finds the integer relation, , with smallest euclidean norm, , consistent with the requested accuracy of fit. If one knew to only 100 significant figures, the routine would be at perfect liberty to return 13 ‘random’ 8-digit integers that happened to fit at 100 significant figures. It would not care that the true form has attractive factorizations of all integers save one. At 800 significant figures, which is the accuracy to which (26) has been validated, the probability of it being spurious is of order .
The import of (26) is dramatic: non-alternating sums, with exclusively positive arguments in (1), do not inhabit a cosy little world of their own, uncontaminated by contact with their alternating cousins, as the presence of clearly demonstrates.
As explained in the introduction, this was wonderful news for the connection between knot theory and quantum field theory [3, 4, 5, 8]. It was also what sparked the present systematic inquiry into the whole universe of Euler sums of the form (1), setting firmly aside the notion that there is something special about non-alternating sums. The liberating effect is apparent in the discovery of the simple formulæ (16,20), which give order to the larger universe so embraced.
Inspection of Table 3 reveals some practical limits to exploration. Even with 800 significant figures one might expect to encounter problems finding relations in , or , each of which has 36 basis terms. If just one of the 37 integers in a desired integer relation exceeds , then MPPSLQ may (quite properly) fit 800-significant-figure data with 37 ‘random’ 22-digit integers. Experience shows that integers of order are produced by successful searches in , with 28 basis terms. So investigation of was judged to be imprudent, with ‘only’ 800 significant figures at hand. Because of the importance of level 10 in [5], investigation of is reported in Section 6.
The successful fit at all levels up to and including , using the 12 numbers (17), has been reported, as have the all-level results (22). There remains the apparently trifling matter of an exact numerical relation in :
U4,2=9796ζ(6)−34ζ2(3), (27)
with coefficients that Euler could, no doubt, have found by mental arithmetic. Surprisingly, nothing in the most recent work on double sums [1] suggested the existence of such a relation. It turned out to be relatively easy to devise an analytical proof, when thus apprised of its need, so the subject is postponed to the next section, where the armory of analytical tools is augmented. Yet the relation belongs here, since it was David Bailey’s engine that disclosed it. The analytical techniques, developed to derive (27), feed back useful information for further numerical analysis.
First, they clear up the whole of the double-sum sector, for good and all, confirming the knot-theoretic expectation of a ‘rule of 2’, with , and thus allowing numerical exploration to progress to . Secondly, they provide rigorous (though non-optimal) upper bounds:
E(7,3)≤3,E(8,4)≤4,E(9,3)≤6,E(11,3)≤11,E(13,3)≤17. (28)
Thirdly, for each bound it is possible to find an overcomplete basis, whose size is determined by the bound. Thus one needs to evaluate only a small fraction of the sums in these sectors, and then use MPPSLQ to reduce an overcomplete basis to a minimal basis of size . If the formula for were false, MPPSLQ might sometimes fail to reduce the basis down to the claimed size, or might reduce it down to a size smaller than that claimed by (15). Of course, it never did the latter, else the claim would not have been made. The fact that it never did the former, with 800-significant-figure data, in search spaces of sizes up to 36, is a testament to its author [21] as well as to the formula. Finally, and most fortunately, it is possible to construct overcomplete bases, in the spaces bounded by (28), that are demon-free. Hence 800 significant figures are available, in less than half an hour per sum, at the touch of button (21).
Thanks to these circumstances, all Euler sums in and have been shown to be reducible to bases of the sizes given in Table 3, by the operation of MPPSLQ on overcomplete bases. Relatively simple examples of such reductions, in , and , are provided by:
ζ(3,−3,−3) = 6ζ(5,−1,−3)+6ζ(3,−1,−5)−31532ln2ζ(3)ζ(5)+6U5,1ζ(3) + 40005128ζ(2)ζ(7)−3964ζ3(3)+1993256ζ(3)ζ(6)+8295128ζ(4)ζ(5)−226369384ζ(9), ζ(3,−5,−3) = 105980ζ(5,3,3)+15ζ(7,−1,−3)+15ζ(3,−1,−7)+70169U5,3ζ(3) + 15U7,1ζ(3)−6615256ln2ζ(3)ζ(7)−118529672560ζ(11)+301599128ζ(2)ζ(9) − 1249435888ζ2(3)ζ(5)+175357735328ζ(3)ζ(8)+29601035120ζ(4)ζ(7)+340532ζ(5)ζ(6), ζ(3,−1,3,−1) = 6127ζ(−3,−3,−1,−1)−143ζ(−5,−1,−1,−1)−18527U5,1ζ(2) (29) − 16349922356U5,3+205154U7,1+289ln22U5,1+3596ln22ζ2(3) − 58164ln22ζ(6)−8735576ln2ζ(2)ζ(5)−90364ln2ζ(3)ζ(4) − 1441288ζ(2)ζ2(3)+10365875476928ζ(3)ζ(5)+369164351907712ζ(8).
In , the relations are more complex, with the prime factor 102149068537421 appearing in one case. Nonetheless, the probability of a spurious fit is less than , in all cases, and is often much less than this. The existence of further relations, forbidden by (20), cannot be disproved by numerical methods. The euclidean norms of such unwanted relations would, however, exceed those of the discovered relations by factors ranging between and , which makes it extremely implausible that the formula is in error in any of the spaces with .
## 5 Posterior analytics
### 5.1 Analytical tools
The analysis of [1, 2] makes use of two very simple types of relation between Euler sums. In the general case of -fold sums, with all possible alternations of sign, it is somewhat difficult to notate these relations, in all generality. To avoid cumbersome formulæ, terms that involve sums of lower depth, and their products, will be omitted, as in the case of .
The first type of relation involves permutations of arguments:
0 ≃ ζ(a1,a2,a3,…,ak)+ζ(a2,a1,a3,…,ak)+ζ(a2,a3,a1,…,ak)+… (30) + ζ(a2,a3,a4,…,a1).
The proof is trivial: by including all insertions of in the string , one obtains a combination of sums that differs from the product only by terms in which summation variables are equal, corresponding to sums of depth . For 4-fold sums, such relations reduce a set of 24 possible permutations to a set of 9, when the arguments are distinct.
The second type of relation follows from use of the partial-fraction identity [1, 2]
1AaBb=∑s>01(A+B)a+b−s{(a+b−s−1a−s)1As+(a+b−s−1b−s)1Bs} (31)
for positive integer and . To see how this is used, consider the product , with positive arguments. It may be written as , where each summation variable runs over all the positive integers. Now apply (31), setting and . The second type of resulting partial fraction is of the form , which is an Euler sum. To the first type, apply (31) with and . Its second terms are also Euler sums. To its first, apply (31) with and . Each term so produced is an Euler sum. Thus one has obtained a relation for non-alternating sums. By including signs, 16 such relations can be generated. In general, one gets relations by applications of (31) for every set of exponents that one chooses for the initial product of sums.
It can seen that there is no scarcity of trivially derivable relations between Euler sums. The notable achievement of [1] was to organize the relations between double sums in such a way as to prove the reducibility of all double sums of odd level. In [2] non-alternating triple sums of even level were proven to be reducible. It was conjectured that non-alternating sums of level and depth are reducible whenever is odd. The stronger claim made by (9) is that this applies to alternating sums as well. In the course of the present work, reducibility has been demonstrated, by a combination of analytical and numerical methods, for all odd values of such that .
As remarked previously, the identities of [1] are insufficient to derive the simple relation (27). No tally was given in [1] of the numbers of alternating double sums left unreduced at even levels, though the tally was made for non-alternating double sums at level . Using REDUCE, one easily discovers that the relations given in [1] allow reduction of double sums to the set and that no further reduction is possible without additional input. Using MPPSLQ, one easily discovers that a truly irreducible set is furnished by . Thus the relations derived in [1] are insufficient in a way that is very easy to state: they fail to relate the even cases to the odd cases .
To remedy this failure, it was sufficient to derive the further555Jon Borwein later told me that (32,33,34) were known to, though not used by, the authors of [1]. relation
ζ(a,b)+ζ(−a,−b) = ∑s>0( | {"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": 0.899201512336731, "perplexity": 1066.2467741772955}, "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-2021-43/segments/1634323588282.80/warc/CC-MAIN-20211028065732-20211028095732-00627.warc.gz"} |
https://www.dlubal.com/en-US/support-and-learning/support/faq/003290 | # I design surfaces in RF‑CONCRETE Surfaces and get a high minimum reinforcement in some areas as a result of the SLS design. In other areas, however, the minimum reinforcement is significantly smaller. What is the reason?
By default, the determination of the minimum reinforcement is based on the stress distribution in the tension zone before initial cracking (coefficient kc according to EN 1992‑1‑1, 7.3.2 (7.2.)), which depends on the external loading (Figure 01). It means that kc is variable by deafult.
Depending on the state of stress (for example, pure bending, bending and tension, bending and compression, and pure tension), the minimum reinforcement can vary along the surface.
However, it is also possible to adjust the presetting. It is possible to select a bending constraint (kc = 0.4) or a centric constraint (kc = 1.0) (Figure 02).
As soon as the constant value for kc is assumed, the minimum reinforcement is independent of the external load and is applied uniformly over the structural component.
In this context, you can also select the direction in which the force should be considered (Figure 03). | {"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": 0.8425002694129944, "perplexity": 1063.7183852856301}, "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/1596439737225.57/warc/CC-MAIN-20200807202502-20200807232502-00450.warc.gz"} |
http://mathhelpforum.com/geometry/42028-finding-area-volume-helix.html | # Math Help - finding the area and volume of a helix
1. ## finding the area and volume of a helix
hi,
how do i find the area and volume helix.
thanks
2. Originally Posted by qwertypunk
hi,
how do i find the area and volume helix.
thanks
This is, what, the fourth or fifth time you have posted this question since March? I think the problem is that we are confused about what you are asking about. The double helix is made from two "bands" the spiral around each other. There really is no enclosed volume. The area, I presume, is the cross-section, which is no more than a circle, so $A = \pi r^2$.
That's probably the best answer you are going to get for this unless you can tell us why this is not a sufficient answer.
-Dan
3. ## volume of a helix
i came here looking for the volume of a helical pipe.
4. Originally Posted by robc
i came here looking for the volume of a helical pipe.
The only thing I can say is, for a volume can you model this as a cylinder? Then $V = \pi r^2 L$ where L is the distance along the helix's axis.
Or is the pipe wrapped around in a helical shape?
-Dan | {"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": 2, "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": 0.8242236971855164, "perplexity": 375.5489499174335}, "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-2014-52/segments/1419447562872.37/warc/CC-MAIN-20141224185922-00011-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/269942/liouvilles-proof-of-the-existence-of-transcendental-numbers | # Liouville's proof of the existence of transcendental numbers
The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers.
It is written here that http://en.wikipedia.org/wiki/Transcendental_number Liouville first proved the existence of transcendental numbers in 1844. I doubt that he proved this result by a cardinality argument, as I think that these ideas were first brought by Cantor and this was well after 1844. How did Liouville prove the existence of transcendental numbers.
-
Cantor's argument would only appear 15-20 years later. Read about Liouville numbers. – Asaf Karagila Jan 3 '13 at 21:08
See en.wikipedia.org/wiki/Liouville_number. Basically, the irrational algebraic numbers (non-transcendental numbers) "poorly approximate" rational numbers. – Andrew Salmon Jan 3 '13 at 21:09
Roughly speaking, irrational algebraic numbers are poorly approximable by rationals. Liouville showed that numbers whose decimal expansion involves only $0$'s and $1$'s, where the number of $0$'s between consecutive $1$'s grows rapidly, are well approximable by rationals. There is a hint of diagonalization in the construction. – André Nicolas Jan 3 '13 at 21:15 | {"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": 0.9325495362281799, "perplexity": 543.1219064135402}, "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/1429246641393.36/warc/CC-MAIN-20150417045721-00051-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://napitupulu-jon.appspot.com/posts/frequentist-vs-bayesian-inference-coursera-statistics.html | # Frequentist vs. Bayesian Approach
| Source
In this blog we're going to discuss about frequentist approach that use p-value, vs bayesian approach that use posterior.
Screenshot taken from Coursera 01:04
The study will help us make a comparison of frequentist vs bayesian approach. We have a population, and your task is to test whether the yellow is whether 10% or not.
Screenshot taken from Coursera 01:52
Then from the study, you make a decision table. If your decision is right, you're going to get a bonus, otherwise you lose a job.
Screenshot taken from Coursera 02:26
Then you're presented the money and the cost to gather the data. Remember, often it's pretty costly to get more data. So this example representing that condition.
Using frequentist approach, you're going to use hypothesis testing. To set the hypothesis:
• H0 : 10% yellow M&Ms
• HA : 20% yellow M&Ms
Using test statistic, because it's talking about the proportion, is the number of yellow observed in the sample. The p-value is calculating the probability of this many or more yellow M&M in the sample that you have buy, given the null hypothesis test. You might want to ask this, "If I have bought 3 times and observe all the p-value, can I predict what is the p-value for the fourth time?"
So how many sample that you think you would buy? 5,10,15,20? Recall that if you fail to predict the p-value, you would lose your job. But if you buy large sample size, it will be very costly. The decision is how to get the right sample size, that's enough to make it practically significant.
Let's choose 5 for this state. If some of you have known bootstrapping, this is one of the most important technique to engineer a new sample. In hypothesis testing, you're collecting the data, build test statistic, p-value, and compare it to significance level. The next question then become, what is significance level for this problem? Recall that significance level is all about type 1 error, rejecting the null hypothesis when the null hypothesis is true.
So which level should you choose? Using higher significance level, you can have a type 2 error rate. But using smaller alpha, can get you miss any true significant p-value. For this case, we stick with 5% significance level.
With sample size this small, we can use binomial distribution.Because we set 10% as our null hypothesis, we set the probability of success 10%. Recall that null hypothesis is a null value for true population,hence the proportion is equal the probability of success. Suppose we have yellow (among 4 other colors) once.
p-value = P(1 or more yellows | n=5, p = 10%)
Since we're observing the probability of at least 1, and there are only two chances, whether it's yellow or not, we can simplify this calculation as,
P(k>=1) = 1 - P(none yellows)
The complement probability is 0.9, then
P(k>=1) = 1 - (0.9)^5 = 0.41
The result is 0.41, our p-value is greater than the significance level, we failed to reject the null hypothesis.
Since the sample size is indeed small, and because of that, we want to increase our sample size to 10. Then for the 10 draws, we get two yellows. The p-value is then
p-value = P(2 or more yellows | n=10,p=10%)
Since this is getting too complicated(we can calculate the probability of 2,3,4..10, or 1-(k=0+k=1)), we can use R.
In [2]:
sum(dbinom(2:10,10,0.1))
Out[2]:
[1] 0.2639011
Again, based on this p-value, we fail to reject the null hypothesis.
How about the 15 sample size? Again when doing 15 draws, we get 3 yellow. So,
p-value = P(3 or more yellows | n=15, p = 10%)
Using R,
In [3]:
sum(dbinom(3:15,15,0.1))
Out[3]:
[1] 0.1840611
Again we failed to reject the null hypothesis.
For the sake of the argument, we use 20 sample size, and have 2 yellows in 20 draw. Again we set our binomial,
p-value = P(4 or more yellows | n=20,p=10%)
In [4]:
sum(dbinom(4:20,20,0.1))
Out[4]:
[1] 0.1329533
And then, once again, we failed to reject the null hypothesis.
Screenshot taken from Coursera 14:33
If we're looking at the possibilities earlier(1 out of 5, 2 out of 10, 3 out of 15, and 4 out of 20), we know that the proportion is actually 20%. Since we failed to reject our null hypothesis, we would lose our job.So you see, it's important to see from looking at the problem.
Screenshot taken from Coursera 15:30
Now let's use bayessian approach. Again, only two conditional probabilities.You can either have 10% or 20%. Since we don't how is the true population, we make a fair judgement 50:50. In Bayes this is our prior probability. As you recall in Bayes, we can be presented with the data, calculate the posterior, make that as an input of next data, calculate the posterior and keep doing that.So the p-value in bayesian is the probality that given the observed data, what are th posterior probability.
Screenshot taken from Coursera 17:16
So we can calculate the probability of 10% given data. This is what Bayesian can solve. We can use Bayes to calculate like the one in the examples. Recall that in Bayes, we have
P(A and B) = P(B and A), if A and B are dependent.
P(A|B) * P(B) = P(B|A) * P(A)
We subtitute the equation like the one in the example. Since there either 10% or 20%, the probability of 20% is the complement of 10% yellow.
In [3]:
dbinom(1,5,0.2)
Out[3]:
[1] 0.4096
Screenshot taken from Coursera 20:18
In [4]:
dbinom(1,5,0.1)
Out[4]:
[1] 0.32805
In [5]:
dbinom(1,5,0.2)
Out[5]:
[1] 0.4096
Once again, since we only have two conditions in our probability space, and we're observing the exact successes for n trial, we can use dbinom function. Recall that we have 1 yellow in our first trial. We calculate the Bayesian as observing the probability of 10% given the data. And by calculating the probability of data, what is the probability that we have 10% yellow or 20% yellow given the data. Since we have an or in probability, we join by addition. In dbinom we can calcute the probability of k success, given n trial, knowing the probability of success. We can use P(data | 10%) with dbinom in R. So we're calculating P(data|10%) is 0.33 and P(data|20%) is 0.41. We incorporate the formula, and have 0.44.
Screenshot taken from Coursera 21:40
You may want to mark your results in the table. Since 20% yellow is the complement probability of 10%, we take it 0.56 for the 20%. And we repeat our process for 10,15,20.
Screenshot taken from Coursera 23:05
In [8]:
dbinom(2,10,0.1)
Out[8]:
[1] 0.1937102
In [7]:
dbinom(2,10,0.2)
Out[7]:
[1] 0.3019899
for 10 value, we can get 0.39 for P(10%|data). And the complement for 20% is 0.61
In [9]:
p_data_10 = dbinom(3,15,0.1) * 0.5
p_data_20 = dbinom(3,15,0.2) * 0.5
p_data_10/(p_data_10+p_data_20)
Out[9]:
[1] 0.339383
So we have 0.34 for P(10%|data) and for the complement of 20%, we have 0.66. That is our posterior probability of step 3.
In [10]:
p_data_10 = dbinom(4,20,0.1) * 0.5
p_data_20 = dbinom(4,20,0.2) * 0.5
p_data_10/(p_data_10+p_data_20)
Out[10]:
[1] 0.2915103
Finally, for 20 M&M we have 0.29 for 10% and the complement 0.71 for 20%. Let's take a look at the overall table, all 4 steps of frequentist vs bayesian approach.
Screenshot taken from Coursera 27:31
The frequentist approach, p-value makes HT failed to reject, and keep siding with 10%. On the other hand, Bayesian consistently prefer 20%. So there's two contradicting results on two approach. Which is right? Since you know that 20% is the true population, Bayesian is the winning side. Indeed sometimes two approach could yield slightly different result.
Recall that in Bayesian, you'll always update prior according to your posterior. Earlier, we don't update our prior. It keeps at constant 0.5.How about we keep updating prior based on resulted posterior? You could also using Bayesian approach like this
In [13]:
p_data_10 = dbinom(1,5,0.1) * 0.5
p_data_20 = dbinom(1,5,0.2) * 0.5
p_10 = p_data_10/(p_data_10+p_data_20)
c(p_10,1-p_10)
Out[13]:
[1] 0.4447231 0.5552769
In [14]:
p_data_10 = dbinom(2,10,0.1) * 0.44
p_data_20 = dbinom(2,10,0.2) * 0.56
p_10 = p_data_10/(p_data_10+p_data_20)
c(p_10,1-p_10)
Out[14]:
[1] 0.3351035 0.6648965
In [18]:
p_data_10 = dbinom(3,15,0.1) * 0.34
p_data_20 = dbinom(3,15,0.2) * 0.66
p_10 = p_data_10/(p_data_10+p_data_20)
c(p_10,1-p_10)
Out[18]:
[1] 0.2092687 0.7907313
In [19]:
p_data_10 = dbinom(4,20,0.1) * 0.21
p_data_20 = dbinom(4,20,0.2) * 0.79
p_10 = p_data_10/(p_data_10+p_data_20)
c(p_10,1-p_10)
Out[19]:
[1] 0.09859043 0.90140957
So there you go, using Bayesian approach, you get 0.1, which is near to 0.13 frequentist.
REFERENCES:
Dr. Mine Çetinkaya-Rundel, Cousera | {"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": 0.8202666640281677, "perplexity": 1271.6551804531246}, "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/1500549423812.87/warc/CC-MAIN-20170721222447-20170722002447-00066.warc.gz"} |
https://email.esm.psu.edu/pipermail/macosx-tex/2010-June/044289.html | # [OS X TeX] hebrew?
David Derbes loki at uchicago.edu
Fri Jun 11 08:27:42 EDT 2010
On Jun 11, 2010, at 7:20 AM, Peter Dyballa wrote:
>
> Am 11.06.2010 um 12:37 schrieb David Derbes:
>
>> Imagine an astronomical object so massive and so compact that even \emph{light} is too slow to leave the object's surface; for this object, $v_{\text{esc}} \ge c$. Presumably it would be \emph{invisible}.
>
> Is this really correct? Most artificial objects that left our Earth started with zero velocity (relative to the surface where they stood before ignition of the rocket engines). Isn't the curvature of space and time responsible for the effect that a black hole's supposed surface would look kind of dark? (Actually no-one would see it, because light from back or the sides is redirected into the beholder's eyes. And at the event horizon there's the effect of spontaneous pair production of particles which could involve some delusive gleam of light.)
This is always the danger of excerpting. Black holes were dreamt up in the eighteenth century assuming that light was a Newtonian object. It isn't, nor is gravity as Newton supposed it. Curiously, treating light as a Newtonian object and gravity according to Newton leads to the right expression for the Schwarzschild radius (the event horizon.) I was describing light according to Newtonian physics and then go on to do it more correctly, but not yet going into the full curved spacetime stuff.
It is well known that black holes radiate, according to the discoveries of Hawking and Bekenstein as you describe.
David Derbes
>
> --
> Greetings
>
> Pete
>
> A blizzard is when it snows sideways.
>
> ----------- Please Consult the Following Before Posting -----------
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> | {"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": 0.8486570119857788, "perplexity": 3126.16431650226}, "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/1600402131986.91/warc/CC-MAIN-20201001174918-20201001204918-00387.warc.gz"} |
https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_20&diff=151159&oldid=145076 | # Difference between revisions of "2021 AMC 12B Problems/Problem 20"
## Problem
Let and be the unique polynomials such thatand the degree of is less than What is
## Solution 1
### Solution 1.1
Note that so if is the remainder when dividing by , Now, So , and The answer is
### Solution 1.2 (More Thorough Version of Solution 1.1)
Instead of dealing with a nasty , we can instead deal with the nice , as is a factor of . Then, we try to see what is. Of course, we will need a , getting . Then, we've gotta get rid of the term, so we add a , to get . This pattern continues, until we add a to get rid of , and end up with . We can't add anything more to get rid of the , so our factor is . Then, to get rid of the , we must have a remainder of , and to get the we have to also have a in the remainder. So, our product is Then, our remainder is . The remainder when dividing by must be the same when dividing by , modulo . So, we have that , or . This corresponds to answer choice . ~rocketsri
## Solution 2 (Complex numbers)
One thing to note is that takes the form of for some constants A and B. Note that the roots of are part of the solutions of They can be easily solved with roots of unity: Obviously the right two solutions are the roots of We substitute into the original equation, and becomes 0. Using De Moivre's theorem, we get: Expanding into rectangular complex number form: Comparing the real and imaginary parts, we get: The answer is . ~Jamess2022(burntTacos;-;)
## Solution 3 (Directly finding the quotient by using patterns)(Probably one for MSM peeps)
Note that the equation above is in the form of polynomial division, with being the dividend, being the divisor, and and being the quotient and remainder respectively. Since the degree of the dividend is and the degree of the divisor is , that means the degree of the quotient is . Note that R(x) can't influence the degree of the right hand side of this equation since its degree is either or . Since the coefficients of the leading term in the dividend and the divisor are both , that means the coefficient of the leading term of the quotient is also . Thus, the leading term of the quotient is . Multiplying by the divisor gives . We have our term but we have these unnecessary terms like . We can get rid of these terms by adding to the quotient to cancel out these terms, but this then gives us . Our first instinct will probably be to add , but we can't do this as although this will eliminate the term, it will produce a term. Since no other term of the form where is an integer less than will produce a term when multiplied by the divisor, we can't add to the quotient. Instead, we can add to the coefficient to get rid of the term. Continuing this pattern, we get the quotient as The last term when multiplied with the divisor gives . This will get rid of the term but will produce the expression , giving us the dividend as . Note that the dividend we want is of the form . Therefore, our remainder will have to be in order to get rid of the term in the expression and give us , which is what we want. Therefore, the remainder is
~ rohan.sp
## Solution 4 (Division Analysis Without Finding Q(z))
By the difference of cubes or the short geometric series, we get We rewrite by the polynomial division algorithm: where and are unique polynomials such that Taking in modulo (in which ), we have Substituting back into gives which almost resembles to the original equation Since we require that the divisor goes into the remaining for one more time. Rewriting produces from which
~MRENTHUSIASM
## Video Solution using long division(not brutal)
https://youtu.be/kxPDeQRGLEg ~hippopotamus1
~ pi_is_3.14 | {"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": 0.9556529521942139, "perplexity": 307.0196587531068}, "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-2021-21/segments/1620243991801.49/warc/CC-MAIN-20210515100825-20210515130825-00161.warc.gz"} |
https://math.stackexchange.com/questions/2895271/leibniz-notation-for-derivation/2895514 | # Leibniz notation for derivation
I searched through a bunch of sites and articles but I could not find the answer to my question anywhere. Hopefully not because of my complete search incompetence.
Is it okay to write a function in the "denominator" of Leibniz notation like this:
$$\frac{d}{d(x+2)}[(x+2)^5]$$
To mean differentiation with respect to x+2, treating x+2 as the variable. This seems extremely useful to me, especially when writing things like the chain rule; This notation makes it clear and understandable in my opinion.
One problem is when the expression that is differentiated can be given by the "denominator" in more than one way. For example, if you write $\frac{d}{d\sin x}\cos x,$ should it be interpreted as $\frac{d}{d\sin x}\sqrt{1-\sin^2 x}$ or as $\frac{d}{d\sin x}\left(-\sqrt{1-\sin^2 x}\right)$? It needs to be specified here whether $\cos x>0$ or $\cos x<0.$ | {"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": 0.8834273815155029, "perplexity": 204.03897575220319}, "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-2020-16/segments/1585370510846.12/warc/CC-MAIN-20200403092656-20200403122656-00335.warc.gz"} |
https://www.physicsforums.com/threads/precise-definition-of-a-limit.52980/ | # Precise Definition of a Limit
1. Nov 16, 2004
### JasonRox
How do I find g?
It's so confusing.
I'm trying to learn this on my own, so bare with me.
I'm going with an example I know the answer to, and maybe someone can work with me here. I'll ask questions through the solution.
We will do x^3 since that is complicated enough, but I understand the steps, just not the logic to moving on to the next step.
$$|x^3-a^3|<e$$
$$0<|x-a|<g$$
Find a value for g that satisfies the above.
$$|x^3-a^3|=|x-a||x^2+ax+a^2|$$
$$|x^2+ax+a^2|<=|x|^2+|a||x|+|a|^2$$
Note: $$|x|-|a|<=|x-a|<1$$
If you don't understand why one is chosen maybe this isn't for you. In case you forgot, we take 1 because that guarantees that the difference won't be too big.
$$|x|<1+|a|$$
Take the above and you get...
[tex](1+|a|)^2+|a|(1+|a|)+|a|^2
WARNING: This is in the works. I will be back to complete it.
2. Nov 17, 2004
### matt grime
You'd be better using that |a|< |x| +g
or better yet, assuming that g is chosen such that |a|<2|x|, since if some g works, a smaller g has to work too, so there's no harm in placing a maximal size on g that helps eliminate a (assuming x is not zero. if x is zero it's quite easy)
Last edited: Nov 17, 2004
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https://www.physicsforums.com/threads/anti-photons.23122/ | # Anti photons
1. Apr 28, 2004
### cangus
Since an anti photon is a photon travelling backwards in time, is the max speed of an anti photon 186,000miles/sec, or is it -186,000?
2. Apr 28, 2004
### chroot
Staff Emeritus
Actually, the photon is its own antiparticle. A photon and an antiphoton are exactly the same thing. Further, the concept of time really has no meaning for a photon, since it always travels the speed of light.
Also, a negative velocity has the same meaning as a positive velocity -- it just means the object is moving in the opposite direction.
- Warren
3. Apr 29, 2004
### kurious
What about a negative energy photon - does that have momentum in the same direction as it is moving?
4. Apr 29, 2004
### chroot
Staff Emeritus
There is no such thing as a negative energy photon.
- Warren
5. Apr 29, 2004
### LURCH
You know, it's never occurred to me before that I've never heard of a negative energy photon. Is negative energy not considered to be quantized?
6. Apr 29, 2004
### chroot
Staff Emeritus
I'm not really sure I understand lurch; I've never really heard of any "negative energy" particle.
- Warren
7. Apr 29, 2004
### kurious
Negative energy could account for the attraction of masses by carrying momentum in the opposite direction to which it is moving and also it could be causing the acceleration of the universe because negative energy would repel positive masses whereas normal positive energy would deccelerate masses.
For more on this issue go to theory development - the mechanism of gravity .
8. Apr 29, 2004
### arivero
Actually the attractive/repulsive character of a carrier has a lot of its spin. 0 and 2 are attractive, 1 is repulsive (for equal charges, this is).
9. Apr 29, 2004
### LURCH
I haven't either, that's what I find so strange. Negative energy is known to exist, and I've done a little bit of research into it (though not very much, I'm afraid). It is only in the context of this discussion that I came to realize that I have never heard of a negative energy particle. In fact, I have never heard of negative energy being discussed in any quantized form.
10. Apr 29, 2004
### Tom Mattson
Staff Emeritus
There are no negative energy photons because photon energies are differences in atomic energy states. The absolute values of atomic energy states have no significance in and of themselves, but their differences do. Since photons are only emitted in transitions from higher states to lower states, you are bound to get a positive photon energy. Of course, you can always adopt the point of view that absorbed photons (those that promote electrons from lower states to higher states) are actually negative energy photons that move backwards in time.
No, the energy levels of the hydrogen atom are En=(-13.6eV)/n2, where n=1,2,3,...
As I said, that's just a matter of convention, and we can just as easily define the states to have positive energies, but there's something appealing about having the zero of energy at n-->infinity. That way, we associate negative energies with bound states, and nonnegative energies with free states.
11. Apr 29, 2004
### ZapperZ
Staff Emeritus
Maybe this is way too obvious to mention, but in case anyone forgets, the relativistic Schrodinger equation, as solved by Dirac, contains both positive and negative energy eigenvalues. When Dirac solve this, he still had fermions, but with these negative energies. This was the first indication of antiparticles that was verified later.
http://www.phys.ufl.edu/~korytov/phz5354/lecture_D06.pdf
So yes, there can be particles with "negative" energies. It is interesting to note that in condensed matter, these antiparticles correspond to the "holes" in the fermi sea below the vacuum state. So these holes also have negative energies.
Zz.
12. Jun 22, 2005
### Solasis
Okay, this is probably going to sound idiotic, but it's 3:30 in the morning, I'm tired, and my mind is mixed up in Nemesis Theory, Evolutionism, Anti-Particles, and a story, but... Photons are particles, correct? And it doesn't make much sense to me to be your own anti-particle; if you are, you can't be annihalated. So, isn't it possible, by some weird anomaly, that anti-photons are not oppositely charged to Photons, but, rather, have the opposite gravitational pull?
13. Jun 22, 2005
### dextercioby
Nope.In particle physics whe have a clear distinction of "antiparticle" and we have a clear rule of what might actually happens to the quantum field operators in order that that particle coincides with its own antiparticle.
The photon is its own antiparticle,just like the $\pi^{0}$ meson.
Daniel.
14. Jun 22, 2005
### dextercioby
According to the famous analysis of the irreducible unitary representations of the Poincaré group by E.P.Wigner who published an article [1] in 1939,QFT can accomodate,basically for any value of spin and mass,negative energy particles.
$$\hat{P}^{0}|\vec{p},s_{3}\rangle =\mbox{sign}\left(p^{0}\right) \sqrt{\vec{p}^{2}+m^{2}} |\vec{p},s_{3}\rangle$$
for arbitrary spin massive particles and
$$\hat{P}^{0}|\vec{p},\lambda\rangle =\mbox{sign}\left(p^{0}\right) \sqrt{\vec{p}^{2}} |\vec{p},\lambda\rangle$$
for arbitary spin massless particles.
Upon reading more from [2],i realize that the third axiom of QFT (axiom III.a from [2]) states
"The spectrum of the energy momentum operator $\hat{P}^{\mu}$ belongs to the closed future light cone $\bar{V}^{+}$."
,therefore all negative energy unitary irreds of the restricted Poincaré group do not lead to physical states.
Daniel.
[1]E.P.Wigner (1939),"On Unitary Representations of the Inhomogeneous Lorentz Group",Ann.Math.,40,149.
[2]N.N.Bogolubov,A.A.Logunov,I.T.Todorov,"Introdiction to Axiomatic Quantum Field Theory",Benjamin/Cummings,NY,1975.
Last edited: Jun 22, 2005
15. Jun 22, 2005
### Kruger
I have the following definition problem (maybe it's just my teacher's problem):
He said in radioactivity decay (beta+ decay) the positron that gets free isn't an antiparticle. Is that true? In my mind not.
16. Jun 22, 2005
### El Hombre Invisible
I've heard of negative energy in two contexts:
1) the curvature of space - positive energy curves space one way, negative the other... one is well-like, one is saddle-like but I can't remember which way round it is;
2) Hawking radiation, in which a pair of positive and negative energy particles are created on the event horizon of a black hole and separated by its gravity - the negative energy particle is pulled into the hole and reduces its mass and the positive energy particle is expelled away from it. I believe this might be what the OP is referring to, since as I recall it should appear as if the positive particle has come from inside the black hole.
If this 'negative energy' particle is indeed travelling backwards in time, then if it were to be absorbed it would look, forward in time, exactly like a positive energy particle being emitted. This, I guess, demonstrates why the photon is its own antiparticle: a 'negative' photon being absorbed is indistinguishible from a 'positive' photon being emitted and vice versa. So as Tom said, you could talk in terms of emission events being absorption of negative energies and absorption events being the emission of negative energies - convention and everyday experience dictate otherwise.
17. Jun 22, 2005
### dextercioby
The positron is a particle,the quanta of the positronic quantum field.It's the electron's antiparticle.Under any possible circumstances.
Daniel.
18. Jun 22, 2005
The positron IS an antiparticle - the antiparticle of the electron. Are you getting positrons and photons mixed up?
19. Jun 23, 2005
### Kruger
My teacher is doctor in Physics. So, might be he is wrong.
20. Jun 30, 2005
### El Hombre Invisible
Don't ask me why I looked at this thread again, but what is the positronic quantum field? I just googled it and got exactly zero matches.
Have something to add?
Similar Discussions: Anti photons | {"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": 0.8854215145111084, "perplexity": 1279.3180054382}, "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-2016-44/segments/1476988721405.66/warc/CC-MAIN-20161020183841-00125-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://clay6.com/qa/40289/find-the-value-of-p-so-that-the-three-lines-3x-y-2-0-px-2-y-3-0-and-2x-y-3- | # Find the value of $p$ so that the three lines $3x + y – 2 = 0, px + 2 y – 3 = 0$ and $2x – y – 3 = 0$ may intersect at one point.
$\begin {array} {1 1} (A)\;p=-5 & \quad (B)\;p=+5 \\ (C)\;p = \pm 5 & \quad (D)\;\text{ none of the above} \end {array}$
Toolbox:
• If three lines may intersect at one point, then the point of intersection of lines (1) and (3) will also satisfy line (2).
The equation of the given lines are
$\qquad 3x+y-2=0$---------(1)
$\qquad px+2y-3=0$---------(2)
$\qquad 2x-y-3=0$-----------(3)
Let us solve equation (1) and (3) we get,
$\qquad 3x+y-2=0$
$\qquad 2x-y-3=0$
$\qquad 5x-5=0 \Rightarrow x = 1$
$\qquad \therefore y = -1$
Hence the point of intersection of line (1) and line (3) is (1, -1)
Now substituting the value of $x$ and $y$ in equation (2), we get,
$\qquad p(1)+2(-1)-3=0$
$\qquad \Rightarrow p-5=0$
$\qquad \therefore p = 5$
Hence the required value is 5. | {"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": 0.8415870666503906, "perplexity": 155.53946227832714}, "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-20170817170613-20170817190613-00404.warc.gz"} |
http://repository.uwyo.edu/ela/vol31/iss1/8/ | •
•
#### Article Title
Reverse Jensen-Mercer Type Operator Inequalities
#### Keywords
Jensen-Mercer operator inequality, convex function, superquadratic function, operator power mean, operator quasi-arithmetic mean
#### Abstract
Let $A$ be a selfadjoint operator on a Hilbert space $\mathcal{H}$ with spectrum in an interval $[a,b]$ and $\phi:B(\mathcal{H})\rightarrow B(\mathcal{K})$ be a unital positive linear map, where $\mathcal{K}$ is also a Hilbert space. Let $m,M\in J$ with \$m
abs_vol31_pp87-99.pdf (28 kB)
Abstract
COinS | {"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": 0.981095552444458, "perplexity": 1511.6197938218068}, "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/1500549426161.99/warc/CC-MAIN-20170726122153-20170726142153-00367.warc.gz"} |
http://physics.stackexchange.com/questions/62009/question-on-hadamard-gate-and-cnot-gate-circuit-tables | # Question on hadamard gate and cnot gate circuit tables
I'm trying to solve this problem for homework:
Now show that if the CNOT gate is applied in the Hadamard basis - i.e. apply the Hadamard gate to the inputs and outputs of the CNOT gate - then the result is a CNOT gate with the control and target qubit swapped.
So I've computed the truth table for $a/b$ with just a CNOT gate, but I'm not sure how to "apply the Hadamard gate". I know that applying Hadamard twice will leave the bit unchanged (verified this via matrix multiplication), but I'm stuck at this part:
Let's say $a$ and $b$ are qubits and both go through the Hadamard gate. Then they go into a CNOT, where $b$ is the control. If $a$ and $b$ started off at $|0 \rangle$, then their matrix version is $\frac{1}{\sqrt 2}$ $[ 1 1 ]$, but how do you apply that to a CNOT? Alternatively, if you leave them as kets, then how do you represent $H |0 \rangle$? It was straight forward just applying a CNOT since the states remain as either $|0 \rangle$ or $|1 \rangle$, but I don't understand what happens if you apply a Hadamard gate to a $0$ or $1$ qubit.
-
I am guessing you know the action of a Hadamard gate on the computational basis: H(|0>)=1/sqrt(2)(|0>+|1>) and H(|1>)=1/sqrt(2)(|0>-|1>). What state do you get when you apply a Hadamard to each qubit when the initial state is |00>? To compute the action of the CNOT gate, just use linearity the of quantum mechanics. For example CNOT(|00>+|11>)=CNOT|00>+CNOT|11>. – Juan Miguel Arrazola Apr 24 '13 at 4:06
Here's a calculation to get you started.$\def\ket#1{\lvert#1\rangle}$
• Define $\ket{h_j} = H\ket{j}$: \begin{align*} \ket{h_0} &= \tfrac1{\sqrt 2}\Bigl(\ket0 + \ket1\Bigr) \\ \ket{h_1} &= \tfrac1{\sqrt 2}\Bigl(\ket0 - \ket1\Bigr) \end{align*}
• Compute (for example) $\ket{h_0}\ket{h_1}$ by distributing the tensor product over the addition and subtraction: $$\ket{h_0}\ket{h_1} = \tfrac12 \Big( \ket0\ket0 - \ket0\ket1 + \ket1\ket0 - \ket1\ket1 \Bigr)$$
• Compute the effect of CNOT on this state, distributing it over the additions and subtractions, applying it to each standard basis state in the expression: \begin{align*} \mathbf{CNOT} \ket{h_0}\ket{h_1} &= \tfrac12\Bigl( \mathbf{CNOT} \ket0\ket0 - \mathbf{CNOT} \ket0\ket1 + \mathbf{CNOT} \ket1\ket0 - \mathbf{CNOT} \ket1\ket1 \Bigr) \\ &= \tfrac12\Bigl( \ket0\ket0 - \ket0\ket1 + \ket1\ket1 - \ket1\ket0 \Bigr) \\ &= \tfrac12\Bigl( \ket0\ket0 - \ket0\ket1 - \ket1\ket0 + \ket1\ket1 \Bigr) \end{align*}
• Identify which state $\ket{h_j}\ket{h_k}$ (if any) this output represents. It may be useful to compute them ahead of time, for the purposes of comparison.
Alternatively, there is another way you can do it: compute $(H \otimes H) \mathbf{CNOT} (H \otimes H)^\dagger$ and determine what operation it performs on the standard basis — this is a more direct way of solving the 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": 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": 0.9985771179199219, "perplexity": 803.3491160005013}, "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/1427131302318.44/warc/CC-MAIN-20150323172142-00113-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/estimate-of-the-probability.342449/ | # Estimate of the probability
1. Oct 3, 2009
### kristymassi
i need this information..please someone help me!!
What useful tool allows us to refine?
Last edited: Oct 4, 2009
2. Oct 3, 2009
### SW VandeCarr
If you are sampling from the same population and your estimate of p(x) is (x)/n (n=sample size) then the result of repeated random samples from the population will converge to the population value according to the central limit theorem. The underlying population doesn't need to be normally distributed.
Last edited: Oct 3, 2009
3. Oct 3, 2009
### kristymassi
you are right..i must study on this..
thank you very much | {"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": 0.9864950180053711, "perplexity": 3718.9106168477615}, "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-2018-22/segments/1526794867841.63/warc/CC-MAIN-20180526170654-20180526190654-00113.warc.gz"} |
https://www.researchwithnj.com/en/publications/nonequilibrium-mean-field-theory-of-resistive-phase-transitions-2 | Nonequilibrium mean-field theory of resistive phase transitions
Jong E. Han, Jiajun Li, Camille Aron, Gabriel Kotliar
Research output: Contribution to journalArticlepeer-review
8 Scopus citations
Abstract
We investigate the quantum mechanical origin of resistive phase transitions in solids driven by a constant electric field in the vicinity of a metal-insulator transition. We perform a nonequilibrium mean-field analysis of a driven-dissipative symmetry-broken insulator, which we solve analytically for the most part. We find that the insulator-to-metal transition (IMT) and the metal-to-insulator transition (MIT) proceed by two distinct electronic mechanisms: Landau-Zener processes and the destabilization of the metallic state by Joule heating, respectively. However, we show that both regimes can be unified in a common effective thermal description, where the effective temperature Teff depends on the state of the system. This explains recent experimental measurements in which the hot-electron temperature at the IMT was found to match the equilibrium transition temperature. Our analytic approach enables us to formulate testable predictions on the nonanalytic behavior of I-V relation near the insulator-to-metal transition. Building on these successes, we propose an effective Ginzburg-Landau theory which paves the way to incorporating spatial fluctuations and to bringing the theory closer to a realistic description of the resistive switchings in correlated materials.
Original language English (US) 035145 Physical Review B 98 3 https://doi.org/10.1103/PhysRevB.98.035145 Published - Jul 27 2018
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
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Dive into the research topics of 'Nonequilibrium mean-field theory of resistive phase transitions'. Together they form a unique fingerprint. | {"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": 0.8002647757530212, "perplexity": 2650.0895409143477}, "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/1637964358560.75/warc/CC-MAIN-20211128134516-20211128164516-00373.warc.gz"} |
http://xbeams.chem.yale.edu/%7Ebatista/vaa/node39.html | # Mean Field Theory
The goal of this section is to introduce the so-called mean field theory (also known as self consistent field theory) and to illustrate the theory by applying it to the description of the Ising model.
The main idea of the mean field theory is to focus on one particle and assume that the most important contribution to the interactions of such particle with its neighboring particles is determined by the mean field due to the neighboring particles. In the 1-dimensional Ising model, for instance, the average force exerted on spin is
(300)
where the index includes all the nearest neighbors of spin . Therefore, the average magnetic field acting on spin is
(301)
where
(302)
is the contribution to the mean field due to the nearest neighbors. Note that when all spins are identical. Eq. (301) defines the self consistent aspect of the theory, since according to such equation the mean field acting on spin is determined by its own mean value . The assumption that the interactions of a spin with its neighboring spins can be approximately described by the mean field, introduced by Eq. (302), introduces an enormous simplification. Such mean field approximation simplifies the many body statistical mechanics problem to a one-body problem (i.e., Eq. (301) transforms the problem of N interacting spins influenced by an external magnetic field to a problem of N non-interacting spins influenced by the mean field ).
The partition function, under the mean field approximation, is
cosh (303)
and the average value of is
(304)
where is the probability of state . The average value of spin is
(305)
Note that Eq. (305) involves a transcendental equation. Its solution corresponds to the value of for which the function on the left hand side of Eq.(305) (i.e., ) equals the function on the right hand side of Eq. (305) (i.e., tanh). In the absence of an external magnetic field (i.e., when ), Eq. (305) always has the trivial solution and a non-trivial solution only when . Such solution is represented by the following diagram:
The diagram shows that the mean field theory predicts spontaneous magnetization (i.e., magnetization in the absence of an external magnetic field) for the 1-dimensional Ising model at any temperature , since there is a non-trivial solution for which Eq. (305) is satisfied. Unfortunately, however, this result is erroneous! The 1-dimensional Ising model does not undergo spontaneous magnetization at any finite temperature, since each spin has only two nearest neighbors and the stabilization energy due to two nearest neighbors is not enough to overcome the randomization process due to thermal fluctuations. This simple example, however, illustrates the theory including the fact that it is sometimes inaccurate near critical points. The theory works better in higher dimensionality, e.g., in the 2-dimensional Ising model where the theory predicts spontaneous magnetization at a critical temperature that is close to the experimental value 2.3 J/K.
Exercise:
Show that there is no spontaneous magnetization in the 1-dimensional Ising model at finite temperature by computing the average magnetization from the exact canonical partition function. Hint: Compute the average magnetization in the presence of an external magnetic field and show that in the limit when such magnetization becomes negligible.
Subsections | {"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": 0.9777206778526306, "perplexity": 381.75319736006344}, "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/1471982950827.61/warc/CC-MAIN-20160823200910-00066-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://es.wikidoc.org/index.php/Ionic_bond | # Ionic bond
File:Ionic bonding animation.gif
Sodium and chlorine bonding ionically to form sodium chloride. Sodium loses its outer electron to give it a noble gas electron configuration, and this electron enters the chlorine atom exothermically. The oppositely charged ions are then attracted to each other, and their bonding releases energy. The net transfer of energy is that energy leaves the atoms, so the reaction is able to take place.
An ionic bond (or electrovalent bond) is a type of chemical bond that can often form between metal and non-metal ions (or polyatomic ions such as ammonium) through electrostatic attraction. In short, it is a bond formed by the attraction between two oppositely charged ions.
The metal donates one or more electrons, forming a positively charged ion or cation with a stable electron configuration. These electrons then enter the non metal, causing it to form a negatively charged ion or anion which also has a stable electron configuration. The electrostatic attraction between the oppositely charged ions causes them to come together and form a bond.
For example, common table salt is sodium chloride. When sodium (Na) and chlorine (Cl2) are combined, the sodium atoms each lose an electron, forming a cation (Na+), and the chlorine atoms each gain an electron to form an anion (Cl-). These ions are then attracted to each other in a 1:1 ratio to form sodium chloride (NaCl).
2Na + Cl2 → 2Na+ + 2Cl- → 2NaCl
File:Ionic bonding.png
Electron configurations of lithium and fluorine. Lithium has one electron in its outer shell, held rather loosely because the ionization energy is low. Fluorine carries 7 electrons in its outer shell. When one electron moves from lithium to fluorine, each ion acquires the noble gas configuration. The bonding energy from the electrostatic attraction of the two oppositely-charged ions has a large enough negative value that the overall bonded state energy is lower than the unbonded state
The removal of electrons from the atoms is endothermic and causes the ions to have a higher energy. There may also be energy changes associated with breaking of existing bonds or the addition of more than one electron to form anions. However, the attraction of the ions to each other lowers their energy.
Ionic bonding will occur only if the overall energy change for the reaction is favourable – when the bonded atoms have a lower energy than the free ones. The larger the resulting energy change the stronger the bond. The low electronegativity of metals and high electronegativity of non-metals means that the energy change of the reaction is most favorable when metals lose electrons and non-metals gain electrons.
Pure ionic bonding is not known to exist. All ionic bonds have a degree of covalent bonding or metallic bonding. The larger the difference in electronegativity between two atoms, the more ionic the bond. Ionic compounds conduct electricity when molten or in solution. They generally have a high melting point and tend to be soluble in water.
## Polarization effects
Ions in crystal lattices of purely ionic compounds are spherical; however, if the positive ion is small and/or highly charged, it will distort the electron cloud of the negative ion. This polarization of the negative ion leads to a build-up of extra charge density between the two nuclei, i.e., to partial covalency. Larger negative ions are more easily polarized, but the effect is usually only important when positive ions with charges of 3+ (e.g., Al3+) are involved (e.g., pure AlCl3 is a covalent molecule). However, 2+ ions (Be2+) or even 1+ (Li+) show some polarizing power because their sizes are so small (e.g., LiI is ionic but has some covalent bonding present).
## Ionic structure
Ionic compounds in the solid state form a continuous ionic lattice structure in an ionic crystal. The simplest form of ionic crystal is a simple cubic. This is as if all the atoms were placed at the corners of a cube. This unit cell has a weight that is the same as 1 of the atoms involved. When all the ions are approximately the same size, they can form a different structure called a face-centered cubic (where the weight is 4${\displaystyle *}$atomic weight), but, when the ions are different sizes, the structure is often body-centered cubic (2 times the weight). In ionic lattices the coordination number refers to the number of connected ions.
## Ionic versus covalent bonds
In an ionic bond, the atoms are bound by attraction of opposite ions, whereas, in a covalent bond, atoms are bound by sharing electrons. In covalent bonding, the molecular geometry around each atom is determined by VSEPR rules, whereas, in ionic materials, the geometry follows maximum packing rules.
## Electrical conductivity
Ionic substances in solution conduct electricity because the ions are free to move and carry the electrical charge from the anode to the cathode. Ionic substances conduct electricity when molten because atoms (and thus the electrons) are mobilised. Electrons can flow directly through the ionic substance in a molten state.
## Substances in ionic form
Common Cations
Stock System Name Formula Historic Name
Simple Cations
Aluminium Al3+
Barium Ba2+
Beryllium Be2+
Caesium Cs+
Calcium Ca2+
Chromium(II) Cr2+ Chromous
Chromium(III) Cr3+ Chromic
Chromium(VI) Cr6+ Chromyl
Cobalt(II) Co2+ Cobaltous
Cobalt(III) Co3+ Cobaltic
Copper(I) Cu+ Cuprous
Copper(II) Cu2+ Cupric
Copper(III) Cu3+
Gallium Ga3+
Gold(I) Au+
Gold(III) Au3+
Helium He2+ (Alpha particle)
Hydrogen H+ (Proton)
Iron(II) Fe2+ Ferrous
Iron(III) Fe3+ Ferric
Lithium Li+
Magnesium Mg2+
Manganese(II) Mn2+ Manganous
Manganese(III) Mn3+ Manganic
Manganese(IV) Mn4+ Manganyl
Manganese(VII) Mn7+
Mercury(II) Hg2+ Mercuric
Nickel(II) Ni2+ Nickelous
Nickel(III) Ni3+ Nickelic
Potassium K+
Silver Ag+
Sodium Na+
Strontium Sr2+
Tin(II) Sn2+ Stannous
Tin(IV) Sn4+ Stannic
Zinc Zn2+
Polyatomic Cations
Ammonium NH4+
Hydronium H3O+
Nitronium NO2+
Mercury(I) Hg22+ Mercurous
Common Anions
Formal Name Formula Alt. Name
Simple Anions
Arsenide As3−
Azide N3
Bromide Br
Chloride Cl
Fluoride F
Hydride H
Iodide I
Nitride N3−
Oxide O2−
Phosphide P3−
Sulfide S2−
Peroxide O22−
Oxoanions
Arsenate AsO43−
Arsenite AsO33−
Borate BO33−
Bromate BrO3
Hypobromite BrO
Carbonate CO32−
Hydrogen carbonate HCO3 Bicarbonate
Chlorate ClO3
Perchlorate ClO4
Chlorite ClO2
Hypochlorite ClO
Chromate CrO42−
Dichromate Cr2O72−
Iodate IO3
Nitrate NO3
Nitrite NO2
Phosphate PO43−
Hydrogen phosphate HPO42−
Dihydrogen phosphate H2PO4
Permanganate MnO4
Phosphite PO33−
Sulfate SO42−
Thiosulfate S2O32−
Hydrogen sulfate HSO4 Bisulfate
Sulfite SO32−
Hydrogen sulfite HSO3 Bisulfite
Anions from Organic Acids
Acetate C2H3O2
Formate HCO2
Oxalate C2O42−
Hydrogen oxalate HC2O4 Bioxalate
Other Anions
Hydrogen sulfide HS Bisulfide
Telluride Te2−
Amide NH2
Cyanate OCN
Thiocyanate SCN
Cyanide CN | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.8144387602806091, "perplexity": 4900.02810616132}, "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-2020-10/segments/1581875145654.0/warc/CC-MAIN-20200222054424-20200222084424-00515.warc.gz"} |
https://math.stackexchange.com/questions/3367300/probability-of-picking-a-fair-dice-given-certain-rolls | # Probability of picking a fair dice given certain rolls
Say I have 8 fair dice and one trick die that always rolls "1."
I pick one die out of the nine and roll it three times, getting three 1's in a row.
What is the probability the dice I picked was fair?
My thought process:
I want to use conditional probability in the following way:
$$P(F | 3x1)$$ = $$P(F \cap3x1)/P(3x1)$$.
I know that if I roll a fair die 3 times, there are $$6^3$$ possibilities, but if I roll the trick die three times, there is only one possibility. How do I put these ideas together?
• Is this true? For example, (fair, 1, 1, 2) lives in F, but does not live in $F \cap 3x1$ – theta Sep 23 at 21:52
The probability you pick a fair die is $$7/8$$, and the probability (given that) you roll three $$1$$s is $$\left( \frac{1}{6}\right)^3$$.
The probability you pick the trick die is $$1/8$$ and the probability (given that) you roll three $$1$$s is 1.0.
$$\frac{\frac{1}{8}}{\frac{7}{8} \left( \frac{1}{6} \right)^3 + \frac{1}{8}} = \frac{216}{223} = 0.96861.$$ | {"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": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9167695045471191, "perplexity": 130.16926065272523}, "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-2019-51/segments/1575540527205.81/warc/CC-MAIN-20191210095118-20191210123118-00094.warc.gz"} |
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1858705/?tool=pubmed | • We are sorry, but NCBI web applications do not support your browser and may not function properly. More information
BMC Bioinformatics. 2007; 8: 115.
Published online Apr 3, 2007.
PMCID: PMC1858705
# Dependence of paracentric inversion rate on tract length
## Abstract
### Background
We develop a Bayesian method based on MCMC for estimating the relative rates of pericentric and paracentric inversions from marker data from two species. The method also allows estimation of the distribution of inversion tract lengths.
### Results
We apply the method to data from Drosophila melanogaster and D. yakuba. We find that pericentric inversions occur at a much lower rate compared to paracentric inversions. The average paracentric inversion tract length is approx. 4.8 Mb with small inversions being more frequent than large inversions.
If the two breakpoints defining a paracentric inversion tract are uniformly and independently distributed over chromosome arms there will be more short tract-length inversions than long; we find an even greater preponderance of short tract lengths than this would predict. Thus there appears to be a correlation between the positions of breakpoints which favors shorter tract lengths.
### Conclusion
The method developed in this paper provides the first statistical estimator for estimating the distribution of inversion tract lengths from marker data. Application of this method for a number of data sets may help elucidate the relationship between the length of an inversion and the chance that it will get accepted.
## Background
Reconstructing the history of inversions and/or translocations separating two chromosomes or genomes is a classical problem in computational biology dating back as far as early work by the pioneers of genetic research from the 1930's (eg. [1]). In many applications, this problem has been treated as a problem of finding the minimum number of events required in the evolutionary history of the two genomes. The computational problem involved is known as sorting by reversal (e.g. [2-4]). An alternative approach is to estimate the number of events using statistical estimators that take into account that more inversions (and translocations) may have occurred than the minimum possible number. Larget et al. [5] and York et al. [6] have developed Bayesian methods based on Markov Chain Monte Carlo (MCMC) for estimating the history of inversions separating two chromosomes. The following description is based on the method of York et al. [6]. In brief, a Markov chain is established that has, as its stationary distribution, the posterior distribution of inversion paths (possible histories of inversions).
The likelihood function is calculated assuming inversions occur according to a Poisson process and assuming a uniform prior over all possible inversions paths of a fixed length. The inversion path is then represented explicitly in the computer memory and updates are proposed according to a proposal kernel, allowing exploration of the posterior distribution. The update kernel is guided by the parsimony distance computed from the breakpoint graphs developed for solving the sorting by reversal problem [4,7]. Using the parsimony distance to guide updates greatly increases convergence rates of the Markov chain. Point estimates of the number of inversions, with associated measures of statistical confidence are then obtained from the posterior distribution. The method of [6] was extended in [8] to the case of multiple chromosomes differing by an unknown number of translocations and inversions. Similarly, [9] extends this type of approach to rearrangements due to transpositions and inverted transpositions in addition to inversions. The advantage of these Bayesian approaches is that they use all of the information in the marker data to obtain a statistical estimate of the number of inversions and translocations. However, so far these approaches have assumed that a long chromosomal segment is as likely to be inverted as a short one, and have lumped together pericentric and paracentric inversions rather than distinguishing between them. Pericentric inversions appear to be rarer than paracentric ones, and there is evidence for a length-dependent effect also [10,11], with selection related to recombination in the inverted region of inversion heterozygotes as a possible cause. Another simplification made hitherto is that only the order of markers in a set (and their orientations, in the case of signed data) has been used, so no account is taken of the uneven spacing of the markers. The objective of this paper is to modify the previous methods to take into account these factors. This will allow us to estimate the relative frequency of pericentric and paracentric inversions and to estimate the distribution of inversion tract lengths. We apply the new method to genomic data from D. melanogaster and D. yakuba.
The assumption of tract-length independence is relaxed also in [12], which considers the problem of finding the optimal inversion path when the cost of an inversion depends on its tract-length; they do not address determining that dependence from data.
## Results and Discussion
### Results
We have analyzed a set of 388 markers on the three major chromosomes, 2, 3 and X, using distance information from D. yakuba but only marker order information from D. melanogaster. The positions of the centromeres are also used. For chromosome 3 we found 163 markers in 23 conserved blocks as shown in figure figure1.1. For the purposes of finding the inversion distance between these two marker arrangements this may be represented as the following signed permutation:
Position in D. melanogaster vs. position in D. yakuba for chains after filtering, chromosome 3. Lines show blocks.
1, -4, 3, -2, 5, -7, 6, -8, 9, -10, 11, 20, -13, 14, -18, 16, -17, -15, 19, 12, 21, -22, 23.
The inversion distance is 13. The centromere is between blocks 10 and 11. For chromosome X we found 84 markers in 14 blocks as shown in figure figure2.2. The signed permutation is:
Position in D. melanogaster vs. position in D. yakuba for chains after filtering, chromosome X. Lines show blocks.
1, -2, 3, -10, -4, 11, -9, -13, -7, 5, 12, 8, 14.
The inversion distance is 7. The centromere is between block 14 and the chromosome end. Similarly for chromosome 2 we found 141 markers in 19 blocks as shown in figure figure3.3. The signed permutation is:
Position in D. melanogaster vs. position in D. yakuba for chains after filtering, chromosome 2. Lines show blocks.
1, -4, 6, -2, 5, -3, 7, 12, -8, -11, -13, -9, 10, -14, 15, -18, -16, -17, 19.
The inversion distance is 11 including one pericentric inversion. The centromere is in the middle of block 11. Thus, it takes at least 31 inversions, including at least one pericentric inversion, to turn the D. yakuba marker arrangement on chromosomes 2, 3 and X into the D. melanogaster arrangement.
A run of 1.1 × 106 updates was performed, taking 90 cpu hours on a 1.8 GHz Athlon processor. The first 1.1 × 105 updates were discarded as burn-in. Figures Figures44 through through66 show histograms of quantities of interest using the remainder of the MCMC output; agreement among the four replicate chains (shown with dotted and dashed lines) is very good, indicating good MCMC convergence.
Posterior distribution of the number of inversions LI. Vertical lines indicate the 95% credible interval.
Posterior distribution of exponential tract-length dependence parameter β. Vertical lines indicate the 95% credible interval.
Posterior distribution of λpa. Vertical lines indicate the 95% credible interval.
Table Table11 lists 95% credible intervals and maximum a posteriori (MAP) estimates of the number of parametric inversions, Lpa, the rate parameters for paracentric, and pericentric, λpa and λpe, and the parameter β which describes the strength of the tract-length dependent effect in our model. These parameters are more fully defined in the methods section.
From figure figure44 it is clear that the number of inversions is compatible with the parsimony estimate of 31 inversions. However, the most likely number of inversions is 33. The credible interval (Table (Table1)1) excludes more than 37 inversions at the 95% level. The 95% credible interval for the rate parameter λpa is [0.028, 0.094] Mb-2.
A minimum of one pericentric inversion (on chromosome 2) is needed to rearrange the markers, and the posterior probability of > 1 pericentric inversions is < 1 × 10-4. The pericentric rate parameter, λpe, is correspondingly small, with a MAP estimate of 7.5 × 10-4 Mb-2.
The MAP estimate of the tract length dependence parameter (β) is 0.130 Mb-1 with a 95% credible interval of [0.044, 0.22] Mb-1 (Table (Table11 and Figure Figure6).6). Using (6), and the chromosome arm lengths (20.7, 22.3, 21.2, 24.2 and 28.7 Mb), we find that βMAP = 0.130 Mb-1 corresponds to a mean tract length of 4.8 Mb, compared with 8.1 Mb assuming β = 0.
The fact that β is positive, and that values of β close to zero receive very little support shows that small tract lengths are favored over large tract lengths.
Figure Figure77 shows the posterior joint distribution of β and λpa, together with the corresponding MAP estimate (β, λpa)MAP = (0.122 Mb-1, 0.053 Mb-2). The observed positive correlation between the two parameters is not surprising. The rate of inversions of tract length τ is proportional to λpae-βτ( - τ), which for λpa > 0 and 0 <τ is a decreasing function of β. Unless increasing β (favoring short tract lengths more) allows the observed rearrangement to be accomplished with fewer inversions, then λpa must increase as β does.
Posterior joint distribution of λpa and β. The triangle marks the mode.
## Discussion
One drawback of the current method is that, as is the case for many other MCMC methods, it is computationally slow. Nonetheless, the speed of the program is not so slow that it is prohibitive, as illustrated in the analysis of the Drosophila data. The existing program should be able to handle somewhat larger data sets (up to perhaps 100 blocks and 800 markers) by some combination of running longer, running replicate chains on separate processors, and, in some cases breaking a multi-chromosome data set down into individual chromosomes or arms and analyzing each one separately. To go beyond that would require substantial work to improve the algorithm. A related issue is the resolution at which we can analyze genome rearrangements. In addition to the increased computational burden of analyzing more and shorter blocks, the assumption implicit in our method that the observed rearrangement is due solely to inversions (and translocations) becomes more problematic for smaller scale rearrangements. The method is, therefore, more suitable for making statements regarding inversions occurring at the scale of hundreds of kilobases or megabases than at the scale of a few kilobases. Nonetheless, with several blocks less than 200 kilobases long in our data, and 5 chromosome arms totaling 117 megabases, we are apparently sensitive to inversion tract-lengths down to about 1% of the chromosome arm length.
## Conclusion
The method developed in this paper provides the first statistical estimator for estimating the distribution of inversion tract lengths from marker data. Application of this method to a number of data sets may help elucidate the relationship between the length of an inversion and the chance that it will get accepted.
## Methods
### The model
#### Using marker order information only
Previously [6,8] we have considered models of rearrangements of M markers on C chromosomes, in which only the order of the markers is used. Two arrangements of a set of markers are considered to be the same if and only if every pair of markers adjacent in one arrangement is also adjacent in the other, and every marker adjacent to a chromosome end in one arrangement is adjacent to a chromosome end in the other. In the case of a single chromosome the markers divide it into M + 1 segments and we can distinguish NI = M(M + 1)/2 inversions corresponding to unordered pairs of distinct segments. Assuming a Poisson process with rate Λ, and assuming the NI inversions to be equiprobable, the probability of a particular path X consisting of L inversions is:
$P(X|Λ)=e−ΛΛLL!NI-L=e−ΛL!λL MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaGaemiuaaLaeiikaGIaemiwaGLaeiiFaWNaeu4MdWKaeiykaKIaeyypa0ZaaSaaaeaacqWGLbqzdaahaaWcbeqaaiabgkHiTiabfU5ambaakiabfU5amnaaCaaaleqabaGaemitaWeaaaGcbaGaemitaWKaeiyiaecaaiabd6eaonaaDaaaleaacqqGjbqsaeaacqqGGaaicqqGTaqlcqqGmbataaGccqGH9aqpdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4MdWeaaaGcbaGaemitaWKaeiyiaecaaGGaciab=T7aSnaaCaaaleqabaGaemitaWeaaaaa@5CE0@$
where λ = Λ/NI. The posterior probability density is then
$p(X,λ|D)∝P(D|X)e−ΛL!λLp(λ). MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWGybawcqGGSaaliiGacqWF7oaBcqGG8baFcqWGebarcqGGPaqkcqGHDisTcqWGqbaucqGGOaakcqWGebarcqGG8baFcqWGybawcqGGPaqkdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4MdWeaaaGcbaGaemitaWKaeiyiaecaaiab=T7aSnaaCaaaleqabaGaemitaWeaaOGaemiCaaNaeiikaGIae83UdWMaeiykaKIaeiOla4caaa@4CED@$
The prior p(λ) is taken to be uniform between 0 and λmax, and zero elsewhere. The data in this case are the marker orders D1 and D2 observed in two taxa. A path X starting at D1 either ends up at D2, in which case P(D|X) = 1, or it ends up at some order other than D2, in which case P(D|X) = 0.
We construct an initial path by starting with D1, and performing inversions and translocations until D2 is obtained. Using the Hannenhalli-Pevzner breakpoint graph theory of sorting by reversal (i.e. by inversions) [4,7], which has been used in all the MCMC sampling approaches to the inversion problem [5,6,9,8], we preferentially choose rearrangements that lead to short paths. Proposed updates are constructed by choosing two points along the existing path and constructing a path between them in the same way, thus guaranteeing P(D|X) = 1.
Starting from a particular marker order there are NI distinct inversions, each occurring with rate λ, i.e., the probability of a particular inversion occurring in a short time t is λt where time is scaled such that the whole rearrangement process takes unit time.
In order to handle multiple chromosomes and translocations [8], we require distinct parameters λI and λT for the rates of inversions and translocations respectively. For each arrangement of markers there will be some number of inversions NI and some number of translocations NT; both of these depend on how many markers are on the various chromosomes, and therefore can change along a path. For this reason we now uniformize the process by defining a total event rate Λ(λI, λT) which is guaranteed to be at least as great as the sum of the total inversion and total translocation rates,
Λ(λI, λT) > Λreal ΛI + ΛT NIλI + NTλT, with "dummy" rearrangments (which have no effect on the genome) occurring with rate λd = Λ - Λreal. Now Λ(λI, λT) is fixed along the path and we may write
$P(X|λI,λT)=e−ΛL!λILIλTLT∏kλdk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqGGOaakcqWGybawcqGG8baFiiGacqWF7oaBdaWgaaWcbaGaemysaKeabeaakiabcYcaSiab=T7aSnaaBaaaleaacqWGubavaeqaaOGaeiykaKIaeyypa0ZaaSaaaeaacqWGLbqzdaahaaWcbeqaaiabgkHiTiabfU5ambaaaOqaaiabdYeamjabcgcaHaaacqWF7oaBdaqhaaWcbaGaemysaKeabaGaemitaW0aaSbaaWqaaiabdMeajbqabaaaaOGae83UdW2aa0baaSqaaiabdsfaubqaaiabdYeamnaaBaaameaacqWGubavaeqaaaaakmaarafabaGae83UdW2aaSbaaSqaaiabdsgaKnaaBaaameaacqWGRbWAaeqaaaWcbeaaaeaacqWGRbWAaeqaniabg+Givdaaaa@5358@$
where the product is over the dummy events on the path, indexed by k. Note that the path X here is a sequence of inversions, translocations and dummies.
#### Using distance information
Often, in addition to knowing the order of markers, we have some form of distance information, such as recombination distance or number of nucleotides between markers. We use this information by generating proposed paths which start from one of the genomes (call it genome 1) specified in the data, with not only the marker order being as specified, but also the distances. The distance information for genome 2 is ignored. A path is then constructed which has distance information at every step, but only the marker order at the end of the path is required to agree with genome 2. If distance information is available for both genomes, we can choose to use the distance information from either genome but not from both. We would like to be able to use the distance information from both genomes, where available, but we don't know how to construct a path which ends not only with a specified marker order, but also with (or close to) a specified set of inter-marker distances. This is particularly difficult because in reality the sum of these distances is not conserved.
Consider again for the moment a single chromosome, with M markers and length . When using only marker order information, we distinguished NI = M(M + 1)/2 inversions and assumed equiprobability. Now, using distance information, an inversion is specified by the distances x1 and x2 of the breakpoints from one end of the chromosome, with (x1, x2) lying in the triangle 0 <x1 <x2 < , and we assume (for now) a uniform distribution over this region. The total rate of inversions is then λI2/2 including inversions of segments containing zero markers. If we exclude these the rate is $ΛI=λI(ℓ2−∑i=0Msi2)/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemysaKeabeaakiabg2da9GGaciab=T7aSnaaBaaaleaacqWGjbqsaeqaaOGaeiikaGIaeS4eHW2aaWbaaSqabeaacqaIYaGmaaGccqGHsisldaaeWaqaaiabdohaZnaaDaaaleaacqWGPbqAaeaacqaIYaGmaaaabaGaemyAaKMaeyypa0JaeGimaadabaGaemyta0eaniabggHiLdGccqGGPaqkcqGGVaWlcqaIYaGmaaa@44B1@$ where the si are distances separating adjacent markers. In the multiple chromosome case this becomes:
$ΛI=λI2∑j=1C(ℓj2−∑i=0mjsij2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemysaKeabeaakiabg2da9maalaaabaacciGae83UdW2aaSbaaSqaaiabdMeajbqabaaakeaacqaIYaGmaaWaaabCaeaadaqadaqaaiabloriSnaaDaaaleaacqWGQbGAaeaacqaIYaGmaaGccqGHsisldaaeWbqaaiabdohaZnaaDaaaleaacqWGPbqAcqWGQbGAaeaacqaIYaGmaaaabaGaemyAaKMaeyypa0JaeGimaadabaGaemyBa02aaSbaaWqaaiabdQgaQbqabaaaniabggHiLdaakiaawIcacaGLPaaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdoeadbqdcqGHris5aaaa@4F1A@$
where j now indexes chromosomes, and mj is the number of markers on chromosome j. In the corresponding expression for translocations:
$ΛT=λTF∑j=1C∑k=j+1Cℓjℓk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemivaqfabeaakiabg2da9GGaciab=T7aSnaaBaaaleaacqWGubavaeqaaOGaemOray0aaabCaeaadaaeWbqaaiabloriSnaaBaaaleaacqWGQbGAaeqaaOGaeS4eHW2aaSbaaSqaaiabdUgaRbqabaaabaGaem4AaSMaeyypa0JaemOAaOMaey4kaSIaeGymaedabaGaem4qameaniabggHiLdaaleaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGdbWqa0GaeyyeIuoaaaa@49C3@$
the factor F is the number of allowed translocations for each choice of breakpoints. After breaking two chromosomes into four pieces, there are 2 ways to put them back together (in addition to the initial configuration); if both of these are allowed then F = 2, but if we require every chromosome to always have exactly one centromere (as we will do later) then one of these is disallowed, and F = 1. Now instead of (3) we have
$p(X|λI,λT)=e−ΛL!λILIλTLT∏kλdk, MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWGybawcqGG8baFiiGacqWF7oaBdaWgaaWcbaGaemysaKeabeaakiabcYcaSiab=T7aSnaaBaaaleaacqWGubavaeqaaOGaeiykaKIaeyypa0ZaaSaaaeaacqWGLbqzdaahaaWcbeqaaiabgkHiTiabfU5ambaaaOqaaiabdYeamjabcgcaHaaacqWF7oaBdaqhaaWcbaGaemysaKeabaGaemitaW0aaSbaaWqaaiabdMeajbqabaaaaOGae83UdW2aa0baaSqaaiabdsfaubqaaiabdYeamnaaBaaameaacqWGubavaeqaaaaakmaarafabaGae83UdW2aaSbaaSqaaiabdsgaKnaaBaaameaacqWGRbWAaeqaaaWcbeaaaeaacqWGRbWAaeqaniabg+GivdGccqGGSaalaaa@5482@$
which differs from (3) in that it is a density and because the dummy event rates, $λdk MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF7oaBdaWgaaWcbaGaemizaq2aaSbaaWqaaiabdUgaRbqabaaaleqaaaaa@317B@$, now depend on the continuous breakpoint positions.
An earlier version of our software, implementing this method of using distance information, but ignoring tract-lengths, was used in a comparative analysis of Arabidopsis thaliana, Arabidopsis lyrata and Capsella [13].
#### Inversion tract lengths
We are interested in investigating how the rate at which inversions occur depends on the inversion tract length, i.e., the distance between inversion breakpoints. To make this question more precise, we note that if the two breakpoints defining an inversion are distributed uniformly and independently along a chromosome, then the tract length, τ |x2 - x1|, is distributed as p(τ) ( - τ), 0 <τ < , and the mean tract length is /3. Now let us consider a joint distribution of the breakpoints which falls exponentially with tract length, i.e., of the form $p(x1,x2)∝e−β|x2−x1| MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWG4baEdaWgaaWcbaGaeGymaedabeaakiabcYcaSiabdIha4naaBaaaleaacqaIYaGmaeqaaOGaeiykaKIaeyyhIuRaemyzau2aaWbaaSqabeaacqGHsisliiGacqWFYoGycqGG8baFcqWG4baEdaWgaaadbaGaeGOmaidabeaaliabgkHiTiabdIha4naaBaaameaacqaIXaqmaeqaaSGaeiiFaWhaaaaa@44AD@$. With this distribution of breakpoints, the tract-length distribution is p(τ) ( - τ)e-βτ, and the mean tract length is
$τ¯=e−βℓ(2+βℓ)+βℓ−2β(e−βℓ+βℓ−1). MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFepaDgaqeaiabg2da9maalaaabaGaemyzau2aaWbaaSqabeaacqGHsislcqWFYoGycqWItecBaaGccqGGOaakcqaIYaGmcqGHRaWkcqWFYoGycqWItecBcqGGPaqkcqGHRaWkcqWFYoGycqWItecBcqGHsislcqaIYaGmaeaacqWFYoGycqGGOaakcqWGLbqzdaahaaWcbeqaaiabgkHiTiab=j7aIjabloriSbaakiabgUcaRiab=j7aIjabloriSjabgkHiTiabigdaXiabcMcaPaaacqGGUaGlaaa@4FCD@$
Now, defining
$A(ℓ,β)=∫0ℓ∫0x2e−β|x2−x1|dx1dx2=ℓ2(e−βℓ+βℓ−1)/(βℓ)2, MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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j7aIjabloriSbaakiabgUcaRiab=j7aIjabloriSjabgkHiTiabigdaXiabcMcaPiabc+caViabcIcaOiab=j7aIjabloriSjabcMcaPmaaCaaaleqabaGaeGOmaidaaOGaeiilaWcaaa@6A8A@$
the total rate of inversions is λIA(, β) including inversions of segments containing no markers. Excluding these and summing over chromosomes:
$ΛI=λI∑j=1C(A(ℓj,β)−∑i=0mjA(sij,β)). MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemysaKeabeaakiabg2da9GGaciab=T7aSnaaBaaaleaacqWGjbqsaeqaaOWaaabCaeaadaqadaqaaiabdgeabjabcIcaOiabloriSnaaBaaaleaacqWGQbGAaeqaaOGaeiilaWIae8NSdiMaeiykaKIaeyOeI0YaaabCaeaacqWGbbqqcqGGOaakcqWGZbWCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcYcaSiab=j7aIjabcMcaPaWcbaGaemyAaKMaeyypa0JaeGimaadabaGaemyBa02aaSbaaWqaaiabdQgaQbqabaaaniabggHiLdaakiaawIcacaGLPaaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdoeadbqdcqGHris5aOGaeiOla4caaa@57A7@$
We will analyze unsigned data, i.e., we use the positions of the markers but not the orientations of individual markers. In this case inversions containing one marker are undetectable. However, since finding the shortest inversion path is hard for the unsigned case, we study the unsigned problem by working with signed arrangements of markers, and sampling from the set of all (signed) paths consistent with the unsigned data, as described in [6]. This means our paths may include one or more 1-marker inversions.
#### Paracentric and pericentric inversions
We want to allow pericentric and paracentric inversions to occur at different rates. Under the uniform independent breakpoint distribution assumption the mean tract length of pericentric inversions is $ℓ¯pe MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWItecBgaqeamaaBaaaleaacqWGWbaCcqWGLbqzaeqaaaaa@30DD@$ = /2 independent of the centromere position. For paracentric inversions on a chromosome arm of length q the mean tract length is q/3 and the inversion rate is proportional to q2. For a chromosome with arm lengths ξ and (1 - ξ), this leads to a mean paracentric tract length of
$ℓ¯pa=(ξ2+(1−ξ)3ξ2+(1−ξ)2)ℓ/3. MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWItecBgaqeamaaBaaaleaacqWGWbaCcqWGHbqyaeqaaOGaeyypa0ZaaeWaaeaadaWcaaqaaGGaciab=57a4naaCaaaleqabaGaeGOmaidaaOGaey4kaSIaeiikaGIaeGymaeJaeyOeI0Iae8NVdGNaeiykaKYaaWbaaSqabeaacqaIZaWmaaaakeaacqWF+oaEdaahaaWcbeqaaiabikdaYaaakiabgUcaRiabcIcaOiabigdaXiabgkHiTiab=57a4jabcMcaPmaaCaaaleqabaGaeGOmaidaaaaaaOGaayjkaiaawMcaaiabloriSjabc+caViabiodaZiabc6caUaaa@4BF9@$
Depending on ξ, $ℓ¯pa MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWItecBgaqeamaaBaaaleaacqWGWbaCcqWGHbqyaeqaaaaa@30D5@$ lies between /3 and /6, so for any centromere position $ℓ¯pa MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWItecBgaqeamaaBaaaleaacqWGWbaCcqWGHbqyaeqaaaaa@30D5@$ <$ℓ¯pe MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWItecBgaqeamaaBaaaleaacqWGWbaCcqWGLbqzaeqaaaaa@30DD@$. This means that either a tract-length dependent effect (β > 0), or an effect which distinguishes only between paracentric and pericentric inversions, can have the effect of suppressing longer tract-length inversions. In order to know whether there is a tract-length effect independent of a possible paracentric/pericentric effect, we keep track of the two kinds of inversions separately, and assume a tract-length dependent paracentric rate
$Λpa=λpa∑j=12C(A(ℓj,β)−∑i=0mjA(sij,β)), MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemiCaaNaemyyaegabeaakiabg2da9GGaciab=T7aSnaaBaaaleaacqWGWbaCcqWGHbqyaeqaaOWaaabCaeaadaqadaqaaiabdgeabjabcIcaOiabloriSnaaBaaaleaacqWGQbGAaeqaaOGaeiilaWIae8NSdiMaeiykaKIaeyOeI0YaaabCaeaacqWGbbqqcqGGOaakcqWGZbWCdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcYcaSiab=j7aIjabcMcaPaWcbaGaemyAaKMaeyypa0JaeGimaadabaGaemyBa02aaSbaaWqaaiabdQgaQbqabaaaniabggHiLdaakiaawIcacaGLPaaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabikdaYiabdoeadbqdcqGHris5aOGaeiilaWcaaa@5BC7@$
where the sums are now over chromosome arms and j and mj are the length and number of markers for the jth arm. We assume a tract-length independent pericentric rate
$Λpe=λpe∑j=1C(ℓ2j−1ℓ2j), MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHBoatdaWgaaWcbaGaemiCaaNaemyzaugabeaakiabg2da9GGaciab=T7aSnaaBaaaleaacqWGWbaCcqWGLbqzaeqaaOWaaabCaeaacqGGOaakcqWItecBdaWgaaWcbaGaeGOmaiJaemOAaOMaeyOeI0IaeGymaedabeaakiabloriSnaaBaaaleaacqaIYaGmcqWGQbGAaeqaaOGaeiykaKcaleaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGdbWqa0GaeyyeIuoakiabcYcaSaaa@494F@$
where here 2j - 1 and 2j are the lengths of the two arms of chromosome j.
Now that we distinguish between paracentric and pericentric inversions and allow for a tract-length dependent rate, (5) becomes
$p(X|λpa,λpe,λT,β)=e−ΛL!λpaLpaλpeLpeλTLT∏kλdke−β∑lτl, MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWGybawcqGG8baFiiGacqWF7oaBdaWgaaWcbaGaemiCaaNaemyyaegabeaakiabcYcaSiab=T7aSnaaBaaaleaacqWGWbaCcqWGLbqzaeqaaOGaeiilaWIae83UdW2aaSbaaSqaaiabdsfaubqabaGccqGGSaalcqWFYoGycqGGPaqkcqGH9aqpdaWcaaqaaiabdwgaLnaaCaaaleqabaGaeyOeI0Iaeu4MdWeaaaGcbaGaemitaWKaeiyiaecaaiab=T7aSnaaDaaaleaacqWGWbaCcqWGHbqyaeaacqWGmbatdaWgaaadbaGaemiCaaNaemyyaegabeaaaaGccqWF7oaBdaqhaaWcbaGaemiCaaNaemyzaugabaGaemitaW0aaSbaaWqaaiabdchaWjabdwgaLbqabaaaaOGae83UdW2aa0baaSqaaiabdsfaubqaaiabdYeamnaaBaaameaacqWGubavaeqaaaaakmaarafabaGae83UdW2aaSbaaSqaaiabdsgaKnaaBaaameaacqWGRbWAaeqaaaWcbeaakiabdwgaLnaaCaaaleqabaGaeyOeI0Iae8NSdi2aaabeaeaacqWFepaDdaWgaaadbaGaemiBaWgabeaaaeaacqWGSbaBaeqaoiabggHiLdaaaaWcbaGaem4AaSgabeqdcqGHpis1aOGaeiilaWcaaa@749C@$
where now Λ(λpa, λpe, λT) > Λreal = Λpa + Λpe + ΛT, and λd = Λ - Λreal as before, and τl is the tract length of the lth paracentric inversion.
Now we can write down the posterior probability:
$p(X,λpa,λpe,λT,β|D)∝P(D|X)e−ΛL!λpaLpaλpeLpeλTLT×∏kλdke−β∑lτlp(λpa)p(λpe)p(λT)p(β). MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeGabaaabaGaemiCaaNaeiikaGIaemiwaGLaeiilaWccciGae83UdW2aaSbaaSqaaiabdchaWjabdggaHbqabaGccqGGSaalcqWF7oaBdaWgaaWcbaGaemiCaaNaemyzaugabeaakiabcYcaSiab=T7aSnaaBaaaleaacqWGubavaeqaaOGaeiilaWIae8NSdiMaeiiFaWNaemiraqKaeiykaKIaeyyhIuRaemiuaaLaeiikaGIaemiraqKaeiiFaWNaemiwaGLaeiykaKYaaSaaaeaacqWGLbqzdaahaaWcbeqaaiabgkHiTiabfU5ambaaaOqaaiabdYeamjabcgcaHaaacqWF7oaBdaqhaaWcbaGaemiCaaNaemyyaegabaGaemitaW0aaSbaaWqaaiabdchaWjabdggaHbqabaaaaOGae83UdW2aa0baaSqaaiabdchaWjabdwgaLbqaaiabdYeamnaaBaaameaacqWGWbaCcqWGLbqzaeqaaaaakiab=T7aSnaaDaaaleaacqWGubavaeaacqWGmbatdaWgaaadbaGaemivaqfabeaaaaaakeaacqGHxdaTdaqeqbqaaiab=T7aSnaaBaaaleaacqWGKbazdaWgaaadbaGaem4AaSgabeaaaSqabaGccqWGLbqzdaahaaWcbeqaaiabgkHiTiab=j7aInaaqababaGae8hXdq3aaSbaaWqaaiabdYgaSbqabaaabaGaemiBaWgabeGdcqGHris5aaaakiabdchaWjabcIcaOiab=T7aSnaaBaaaleaacqWGWbaCcqWGHbqyaeqaaOGaeiykaKIaemiCaaNaeiikaGIae83UdW2aaSbaaSqaaiabdchaWjabdwgaLbqabaGccqGGPaqkcqWGWbaCcqGGOaakcqWF7oaBdaWgaaWcbaGaemivaqfabeaakiabcMcaPiabdchaWjabcIcaOiab=j7aIjabcMcaPiabc6caUaWcbaGaem4AaSgabeqdcqGHpis1aaaaaaa@9A2C@$
The λ's priors are all uniform between 0 and λmax and zero elsewhere. We assume β ≥ 0 with a uniform prior. We assume that chromosomes always have exactly one centromere. In the computer code the breakpoint graph only considers marker-marker adjacencies, not marker-centromere adjacencies, and this means the way proposed rearrangement paths are constructed does not guarantee that centromeres end up in the right place. The centromeres are just passively carried along by the inversions and rearrangements dictated by the breakpoint graph. If the centromeres do not end up in the right place, the proposed path is rejected in the MCMC updating step, leading to loss of efficiency, but not loss of correctness. If the centromere lies within a region of conserved marker order its probability of ending up in the right place will typically be high, but if it lies between conserved regions this probability may be quite low, contributing to a low MCMC acceptance probability.
### Data processing
We chose D. yakuba to compare with D. melanogaster. This choice was dictated by the need to have sufficiently many inversions that the biological problem is interesting, but not so many inversions that computational complexity becomes too large. We started with chained and netted alignments as described in [14]. We used the "net" file droYak2.dm2.net.gz, downloaded from the UC Santa Cruz website. This file contains information on chained alignments ('chains"), organized into hierarchies called "nets". These alignments are based on the Nov. 2005 WUSTL version 2.0 D. yakuba assembly and the Apr. 2004, BDGP v. 4/DHGP v. 3.2 D. melanogaster assembly.
Figure Figure88 shows the position in D. melanogaster of each chain, plotted versus it's position in D. yakuba. The figure shows the 982 chains located on chromosome arm 3L in both species and the 1,322 chains located on 3R in both species. Many points lie along lines with slopes close to ± 1, as expected for markers rearranged by inversions and translocations. There are, however, many other points scattered about, requiring further processing. First, chains not labeled in the net file as of type "syn" (i.e., syntenic) are eliminated. The chains left after some additional processing will be the markers used by the analysis program; from here on we refer to markers rather than chains.
Position in D. melanogaster vs. position in D. yakuba for chains identified as on the same arm of chromosome 3 in both species.
Remaining markers are further processed by defining blocks within which adjacency is conserved. Two markers which are adjacent in both species are in the same block; if adjacent in just one species they are in different blocks. Blocks containing only a single marker are discarded, and blocks shorter than a minimum length are replaced by a single marker at the block's average position. This procedure is then repeated and the number of blocks may decrease, both directly because of discarding one-marker blocks, and also because when a block is discarded, or when a block is shortened to one marker, neighboring blocks will often join into one block. This procedure is repeated several times while the minimum block length is gradually increased from 100 bases to some final value Lmin. Thus, a long block can emerge from a set of short blocks as some are eliminated and others join together. In some cases a block which ideally would be retained and incorporated into a long block may be lost during this process, if it is shorter than Lmin and doesn't join another block soon enough. This can cause gaps in the spacing of markers on the resulting long block or the shortening of the block at an end. Neither of these is a big problem, although shortening at the ends of blocks means breakpoints are less well localized. The set of blocks generated is insensitive to Lmin over a broad range: for our data, any value of Lmim between 25 kilobases and 115 kilobases gives the set of blocks that we analyzed.
Finally, markers are thinned from blocks containing many markers, until no block has more than 8 markers. Markers at the ends of blocks are kept, and the thinning of the others is done so as get a fairly even spacing. This reduces the time and memory requirements of the program, while having little effect on posterior distributions, according to our studies.
Applied to chromosomes X, 2, and 3, this procedure gives the 388 markers in 56 blocks shown in figures figures1,1, ,2,2, and and33.
## Authors' contributions
RN had the idea of using MCMC methods to study chromosomal rearrangements and of using distance information to study tract lengths. RD contributed key mathematical techniques. TY wrote the MCMC code, analyzed the data, and wrote most of the manuscript. RN wrote parts of the manuscript and all authors participated in revising it. All authors have read and approved the final manuscript.
## Acknowledgements
This work was supported by joint NSF/NIGMS grant DMS-02-01037.
## References
• Sturtevant AH, Dobzhansky T. Inversions in the third chromosome of wild races of Drosophila pseudoobscura, and their use in the study of the history of the species. Proceedings of the National Academy of Sciences USA. 1936;22:448–450. doi: 10.1073/pnas.22.7.448. [PubMed]
• Watterson GA, Ewens WJ, Hall TE, Morgan A. The chromosome inversion problem. Journal of Theoretical Biology. 1982;99:1–7. doi: 10.1016/0022-5193(82)90384-8.
• Kececioglu J, Sankoff D. Exact and approximation algorithms for the inversion distance between two permutations. Algortihmica. 1995;13:180–210. doi: 10.1007/BF01188586.
• Hannenhalli S, Pevzner PA. Transforming cabbage into turnip (polynomial algorithm for sorting signed permutations by reversals) Proceedings of the 27th Annual ACM Symposium on the Theory of Computing. 1995. pp. 1–27.
• Larget B, Simon DL, Kadane JB. Bayesian phylogenetic inference from animal mitochondrial genome arrangements (with discussion) J R Stat Soc Ser B. 2002;64:681–693. doi: 10.1111/1467-9868.00356.
• York TL, Durrett R, Nielsen R. Bayesian Estimation of the Number of Inversions in the History of Two Chromosomes. Journal of Computational Biology. 2002;9:805–818. doi: 10.1089/10665270260518281. [PubMed]
• Bafna V, Pevzner PA. Genome rearrangements and sorting by reversals. SIAM Journal of Computing. 1996;25:272–289. doi: 10.1137/S0097539793250627.
• Durrett R, Nielsen R, York TL. Bayesian Estimation of Genomic Distance. Genetics. 2004;166:621–629. doi: 10.1534/genetics.166.1.621. [PubMed]
• Miklos I. MCMC genome rearrangement. Bioinformatics. 2003;19:130–137. [PubMed]
• Caceres M, Barbadilla A, Ruiz A. Inversion Length and Breakpoint Distribution in the Drosophila buzzatii Species Complex: Is Inversion Length a Selected Trait? Evolution. 1997;51:1149–1155. doi: 10.2307/2411044.
• Brehm A, Krimbas C. Inversion polymorphism in Drosophila obscura. Journal of Heredity. 1991;82:110–117. [PubMed]
• Pinter R, Skiena S. Sorting with length-weighted reversals. Proceedings of the 13th international conference on genome informatics. 2002. pp. 103–111.
• Yogeeswaran K, Frary A, York TL, Amenta A, Lesser AH, Nasrallah JB, Tanksley SD, Nasrallah ME. Comparative genome analyses of Arabidopsis spp.: inferring chromosomal rearrangement events in the evolutionary history of A. thaliana. Genome Res. 2005;15:505–515. doi: 10.1101/gr.3436305. [PubMed]
• Kent WJ, Baertsch R, Hinrichs A, Miller W, Haussler D. Evolution's cauldron: duplication, deletion, and rearrangement in the mouse and human genomes. Proc Natl Acad Sci USA. 2003;100:11484–11489. doi: 10.1073/pnas.1932072100. [PubMed]
Articles from BMC Bioinformatics are provided here courtesy of BioMed Central
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See all... | {"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": 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": 0.8110785484313965, "perplexity": 1782.8551973985864}, "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/1406510274967.3/warc/CC-MAIN-20140728011754-00050-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/physics-major-help.139170/ | Physics major help =(
1. Oct 19, 2006
almost__overnow
A railroad diesel engine weights 4 times as much as a flatear. If a diesel coasts a 5km/h into a flatear that is initally at rest. How fast do the two coast after they couple together?
2. Oct 19, 2006
Staff: Mentor
Hint: What's conserved in any collision?
3. Oct 19, 2006
almost__overnow
Momentum is conserved in any collison.
4. Oct 19, 2006
Staff: Mentor
Right! That's all you need to solve this one.
5. Oct 19, 2006
almost__overnow
But dont I have to do some kind of math since it says how fast do the two coast after they couple together? =\
6. Oct 19, 2006
Staff: Mentor
The fact that they couple together tells you that they have the same final speed. Now express the conservation of momentum mathematically and solve for that speed.
7. Oct 19, 2006
almost__overnow
okay wait one more question how do you find the kg since i already have the speed? thanks
8. Oct 19, 2006
OlderDan
The actual masses do not matter. All that matters is that the masses have a known ratio. If you do the problem using m for the mass of the flatear and 4m for the engine the m will divide out of the problem. There is no way for you to determine the value of 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": 0.8783378601074219, "perplexity": 1280.868224718073}, "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-2016-50/segments/1480698542323.80/warc/CC-MAIN-20161202170902-00249-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.math.kyoto-u.ac.jp/en/event/seminar/3917 | # Two-dimensional Brownian random interlacement
Date:
2019/05/24 Fri 15:00 - 16:30
Room:
Room 552, Building No.3
Speaker:
Francis Comets
Affiliation:
Université de Paris
Abstract:
Abstract: The cover time is the time needed for the Wiener sausage of radius 1 to cover the torus of linear size n. In dimension $d \geq 3$, A. Sznitman introduced random interlacements to describe the local covering picture at a fixed intensity; They still give a good account at large densities, bridging up to cover time. In dimension 2, with S. Popov and M. Vachkovskaia, we construct random interlacements to describe the neighborhood of an unvisited site at times proportional to the cover time. In this talk, I will explain the Brownian case. (Joint works with Serguei Popov and Marina Vachkovskaia.) | {"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": 0.890648365020752, "perplexity": 2178.43263766051}, "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-2019-30/segments/1563195525187.9/warc/CC-MAIN-20190717121559-20190717143559-00175.warc.gz"} |
http://blog.fox.geek.nz/2012/10/how-to-create-files-with-leading-period.html | ## 2012-10-11
### How to create files with a leading period in the filename with Microsoft Explorer
I have to post this, because this seems a very frequently asked question on the internet, and while there is a straight-forward solution, most people propose bizarre solutions that circumvent the problem by using some other tool.
If you dig deeper, you'll find working solutions in comments, but they're incomplete and its not obvious at first that it even works.
### The Problem
The problem is simple: You wish to create a unix-style hidden file, such as "`.htaccess`" , "`.gitignore`", or "`.netrc`", or a unix-style hidden folder, for whatever reason.
While this is not a problem for literally any tool other than Windows Explorer, attempting to do this in Explorer yields the following error:
You must type a file name.
### The Solution
Most proponents suggest strange solutions such as using `cmd.exe` or `notepad` to do your dirty work, and even Microsoft Developers seem to think that letting Explorer do this is crazy and suggest using some other tool
However, all that is unnecessary.
All that is required is writing an additional dot ( period ) at the tailing end.
If you wanted '`.htaccess`', instead, write '`.htaccess.`'
If you wanted '`.gitignore`' , instead, write '`.gitignore.`'
Explorer will silently strip the last dot and give you the file name you wanted, with no fuss.
#### 1 comment:
1. That's a nice trick. | {"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": 0.8403974771499634, "perplexity": 2524.7947508682482}, "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/1414119646944.14/warc/CC-MAIN-20141024030046-00059-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://smhasan.com/tag/neyman-pearson-criterion/ | # Build the most powerful hypothesis test: Part 2
In the post Build the most power hypothesis test: Part 1 we introduced Neyman-Pearson criterion and Sufficient Statistic. In this post, using both of these ideas we will develop an optimum decision rule. To interactively generate likelihood functions and see the impact of choosing $latex \mathrm{P_{FA}}$, do not forget to try the Shiny app linked … Continue reading Build the most powerful hypothesis test: Part 2 | {"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": 0.840357780456543, "perplexity": 1164.8611109097849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "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-2021-43/segments/1634323585183.47/warc/CC-MAIN-20211017210244-20211018000244-00656.warc.gz"} |
https://www.physicsforums.com/threads/potential-vs-potential-energy.641935/ | # Potential vs. Potential energy
1. Oct 7, 2012
### Shan K
I was reading electrostatic and found out that the electric field is just the negetive divergence of potential . My questions are why are we taking the negetive divergence ? Electric field is a conservative field and so it can be written as divergence of a scaler function but why we ar taking the negetive one instead of positive . And is this potential and potential energy are same ?
2. Oct 7, 2012
### Staff: Mentor
If you go down in potential energy, you lose energy. Therefore, the force should point towards points of lower potential. The direction of electric field is the force for positive charges (convention), therefore you have to take the negative derivative of the potential to get the electric field.
The potential energy depends on the charge of the object.
3. Oct 7, 2012
### Shan K
what do u mean by 'go down in the potential energy ' is it going nearer to the object who is responsible for the electric field ?
4. Oct 8, 2012
### Staff: Mentor
If that object has a negative charge, yes, otherwise no.
"down" -> lower potential (=lower potential energy with positive charges)
5. Oct 8, 2012
### sophiecentaur
If you 'get work out' when you move from A to B, then B is at a lower potential than A. If you have to put work in, B is a higher potential. That's the convention that's used and always applies. It avoids any confusion for situations of attraction and repulsion.
6. Oct 8, 2012
### Shan K
So are u saying that potential and potential energy are same ?
7. Oct 8, 2012
### Staff: Mentor
No, potential energy is the potential multiplied by the charge of your object.
8. Oct 8, 2012
### Shan K
O. Thats great . Thanks
Similar Discussions: Potential vs. Potential energy | {"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": 0.8967975378036499, "perplexity": 1630.9094955253395}, "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-2017-43/segments/1508187823260.52/warc/CC-MAIN-20171019084246-20171019104246-00325.warc.gz"} |
https://www.physicsforums.com/threads/binomial-theorm.126062/ | # Binomial theorm
1. Jul 14, 2006
### fork
Hello, everybody,
I know how to get the first step, but I don't know how can the first step be changed to the format of the final answer, can anyone help?
Thanks.:rofl:
#### Attached Files:
• ###### mstat.jpg
File size:
26.5 KB
Views:
67
2. Jul 15, 2006
How many terms are in the product "(-4)(-5)...(-4-r+1)"? How can this product be combined with (-1)^r?
Also, writing out r! as a product of consecutive integers should help.
3. Jul 15, 2006
### HallsofIvy
Staff Emeritus
Interesting! I was wondering how that r! in the denominator magically became 6.
My suggestion (in fact what I did) is calculate both formulas for r= 1, 2, 3, 4, 5,... and see what happens. | {"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": 0.8505750894546509, "perplexity": 2022.9006682353318}, "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/1487501171176.3/warc/CC-MAIN-20170219104611-00201-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://learn.careers360.com/ncert/ncert-solutions-class-12-physics-chapter-5-magnetism-and-matter/ | # NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter
NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter: You have studied in the first chapter that isolated charge can be obtained. Can you obtain magnetic monopole? If we cut a magnet into several parts each part will act as a magnet with south and north pole or magnetic dipole. This shows that magnetic monopole cannot exist. And you will study such properties of the magnet in NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter. While studying this chapter you can compare it with electrostatics. The comparison of electric dipole in chapter 1 and magnetic dipole in NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter is given below.
Quantity Electrostatics Magnetism Dipole Moment p m The field at the equatorial point of a dipole $\frac{-p}{4\pi\epsilon_0r^3}$ $\frac{\mu m}{4 \pi r^3}$ The field at the axial point of a dipole $\frac{2p}{4\pi\epsilon_0r^3}$ $\frac{-2\mu m}{4\pi r^3}$ The torque due to a dipole in the external field $p*E$ $m*B$ Energy due to a dipole in the external field $-p.E$ $-m.B$
## The main subheadings of NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter is given below.
5.1 Introduction
5.2 The Bar Magnet
5.2.1 The Magnetic field lines
5.2.2 Bar magnet as an equivalent solenoid
5.2.3 The dipole in a uniform magnetic field
5.2.4 The electrostatic analog
5.3 Magnetism and Gauss's Law
5.4 The Earth's Magnetism
5.4.1 Magnetic declination and dip
5.5 Magnetisation and Magnetic Intensity
5.6 Magnetic Properties of Materials
5.6.1 Diamagnetism
5.6.2 Paramagnetism
5.6.3 Ferromagnetism
5.7 Permanent Magnets and Electromagnets
The NCERT class 12 chapter 5 Magnetism and Matter chapter is theoretically very important for CBSE board exam. There are also a few derivations mentioned in the chapter.
## We provide with NCERT solutions for class 12 Physics chapter 5 Magnetism and Matter in detail.
View all solved exercise questions of Chapter 5 Magnetism and Matter here.
## NCERT Solutions for Class 12 Physics- Chapter wise
Chapter 1 Electric Charges and Fields Chapter 2 Electrostatic Potential and Capacitance Chapter 3 Current Electricity Chapter 4 Moving charges and magnetism Chapter 6 Electromagnetic Induction Chapter 7 Alternating Current Chapter 8 Electromagnetic Waves Chapter 9 Ray Optics and Optical Instruments Chapter 10 Wave Optics Chapter 11 Dual nature of radiation and matter Chapter 12 Atoms Chapter 13 Nuclei Chapter 14 Semiconductor Electronics: Materials, Devices and Simple Circuits | {"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": 11, "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": 0.8929005265235901, "perplexity": 1308.5379404274006}, "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/1573496665573.50/warc/CC-MAIN-20191112124615-20191112152615-00084.warc.gz"} |
https://www.physicsforums.com/threads/crazy-e-m-problem.35915/ | # Crazy e/m problem
1. Jul 20, 2004
### imationrouter03
Thank you for viewing this problem, hope u can help me.
In one type of computer keyboard, each key holds a small metal plate that serves as one plate of a parallel-plate, air-filled capacitor. When the key is depressed, the plate separation decreases and the capacitance increases. Electronic circuitry detects the change in capacitance and thus detects that the key has been pressed. In one particular keyboard, the area of each metal plate is A , and the separation between the plates is "s" before the key is depressed.
If the circuitry can detect a change in capacitance of DeltaC, how far must the key be depressed before the circuitry detects its depression? Use epsilon_0 for the permittivity of free space.
The correct answer involves the variable "s" adn it does not depend on the variable DeltaS
Thanks again for your time and concern
)
2. Jul 22, 2004
### Phymath
Well it really depends on how sensitive the circuitry is, but u'd need to know the minium change (delta C) for the keyboard to detect a key pressed then you can just...
DeltaC = (Epislion_0 * A)(1/(deltaS))
(Epislion_0 *A)/ (DeltaC) = deltaS, idk if i helped hopefully..
3. Jul 22, 2004
### Staff: Mentor
parallel plate capacitor
Model the key as a parallel plate capacitor, the formula for which is: $C = \frac{\epsilon_0 A}{d}$. Now find the change in capacitance when d changes from s to s - $\Delta x$. Solve for $\Delta x$.
4. Jul 22, 2004
### imationrouter03
I've tried the following of deltax=s-(epsilon_0*A)/(C) but the correct answer didn't involve the variable C =/
but y would u be solving for deltax?
I've also tried (epsilon_0*A)/DeltaC but the correct answer involves the variable "s"
I've also tried (epsilon_0*A)/(s-DeltaS) but the correct answer doesn't depend on DeltaS
And i've still haven't been able to solve this problem... any feedback will be appreciated..thanks
5. Jul 23, 2004
### maverick280857
Doc Al:
I posted a half solution to this problem a while back right here...it is missing strangely :-(
Cheers
Vivek
EDIT: Not quite...I posted it here: https://www.physicsforums.com/showthread.php?t=36180 (hope that helps)
6. Jul 23, 2004
### Staff: Mentor
I'm not sure what that equation is supposed to be. We need to solve for $\Delta x$ because that's what the problems asks us to find: "how far must the key be depressed". (I just happen to call the distance the key is depressed $\Delta x$.)
In any case, maverick280857 gave you some good advice on solving this problem (in the other thread! please don't post the same question twice), but here's a bit more. Starting with the equation for capacitance:
$C = \frac{\epsilon_0 A}{d}$
now find $\Delta C$:
$\Delta C = \frac{\epsilon_0 A}{s - \Delta x} - \frac{\epsilon_0 A}{s}$
Now rearrange and solve this equation for $\Delta x$ in terms of s and $\Delta C$. (It's easy.)
(Note: in maverick280857's other post, he calls the distance x instead of $\Delta x$; it doesn't matter--take your pick!)
7. Jul 23, 2004
### maverick280857
Retrospectively this is quite an interesting situation since the circuit--which you're not concerned with if you're solving this problem per se--has been made to detect a minimum threshold of DeltaC (or exactly DeltaC, as the case may be). Another twist of the problem would be to consider the capacitor to be filled with a dielectric and to have the student/problem-solver find the minimum distance that the key would have to be pressed so that the dilectric would break down. (Of course, you are less likely to have dielectrics for keyboards in computers still...) But this "twist" as I have simply put in words here, cannot be a reasonable question for starters though it does look like an interesting situation. Ah...one of the beauties of physics...somewhat easy to visualize, difficult to model mathematically :-D
Coming back to this problem and DocAl's situation, it is clear now that DeltaC is a definite quantity and so after rearranging the terms so as to get an expression for x (or $$\Delta x$$) you will discover that it is a function of DeltaC and s only (and more specifically, a function of s since DeltaC is constant preconfigured for your detection circuitry).
Cheers (and sorry for this rather verbose post...)
Vivek
8. Jul 23, 2004
### imationrouter03
thank u all for ur help.. the problem is finally resolved it can out to be the following:
deltax=(s^2*DeltaC)/(epsilon_0*A+s*DeltaC)
thanks =)
9. Jul 24, 2004
### Staff: Mentor
Exactly right.
Similar Discussions: Crazy e/m 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": 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": 0.8695176243782043, "perplexity": 1576.3241142142647}, "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-2017-43/segments/1508187826114.69/warc/CC-MAIN-20171023145244-20171023165244-00463.warc.gz"} |
http://dlmf.nist.gov/36.13 | # §36.13 Kelvin’s Ship-Wave Pattern
A ship moving with constant speed $V$ on deep water generates a surface gravity wave. In a reference frame where the ship is at rest we use polar coordinates $r$ and $\phi$ with $\phi=0$ in the direction of the velocity of the water relative to the ship. Then with $g$ denoting the acceleration due to gravity, the wave height is approximately given by
36.13.1 $z(\phi,\rho)=\int_{-\pi/2}^{\pi/2}\mathop{\cos\/}\nolimits\!\left(\rho\frac{% \mathop{\cos\/}\nolimits\!\left(\theta+\phi\right)}{{\mathop{\cos\/}\nolimits^% {2}}\theta}\right)d\theta,$
where
36.13.2 $\rho=\ifrac{gr}{V^{2}}.$ Symbols: $V$: speed and $g$: gravity Permalink: http://dlmf.nist.gov/36.13.E2 Encodings: TeX, pMML, png
The integral is of the form of the real part of (36.12.1) with $y=\phi$, $u=\theta$, $g=1$, $k=\rho$, and
36.13.3 $f(\theta,\phi)=-\frac{\mathop{\cos\/}\nolimits\!\left(\theta+\phi\right)}{{% \mathop{\cos\/}\nolimits^{2}}\theta}.$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function Permalink: http://dlmf.nist.gov/36.13.E3 Encodings: TeX, pMML, png
When $\rho>1$, that is, everywhere except close to the ship, the integrand oscillates rapidly. There are two stationary points, given by
36.13.4 $\displaystyle\theta_{+}(\phi)$ $\displaystyle=\tfrac{1}{2}(\mathop{\mathrm{arcsin}\/}\nolimits\!\left(3\mathop% {\sin\/}\nolimits\phi\right)-\phi),$ $\displaystyle\theta_{-}(\phi)$ $\displaystyle=\tfrac{1}{2}(\pi-\phi-\mathop{\mathrm{arcsin}\/}\nolimits\!\left% (3\mathop{\sin\/}\nolimits\phi\right)).$
These coalesce when
36.13.5 $|\phi|=\phi_{c}=\mathop{\mathrm{arcsin}\/}\nolimits\!\left(\tfrac{1}{3}\right)% =19^{\circ}.47122.$ Symbols: $\mathop{\mathrm{arcsin}\/}\nolimits z$: arcsine function Permalink: http://dlmf.nist.gov/36.13.E5 Encodings: TeX, pMML, png
This is the angle of the familiar V-shaped wake. The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency $\omega$ as a function of wavevector $\mathbf{k}$:
36.13.6 $\omega(\mathbf{k})=\sqrt{gk}+\mathbf{V}\cdot\mathbf{k}.$ Symbols: $k$: variable and $g$: gravity Permalink: http://dlmf.nist.gov/36.13.E6 Encodings: TeX, pMML, png
Here $k=|\mathbf{k}|$, and $\mathbf{V}$ is the ship velocity (so that $\mathrm{V}=|\mathbf{V}|$).
The disturbance $z(\rho,\phi)$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that $\theta_{\pm}(\phi)$ are real for $|\phi|<\phi_{c}$ and complex for $|\phi|>\phi_{c}$. (See also §2.4(v).) Then with the definitions (36.12.12), and the real functions
36.13.7 $\displaystyle u(\phi)$ $\displaystyle=\sqrt{\dfrac{\Delta^{1/2}(\phi)}{2}}\left(\dfrac{1}{\sqrt{f_{+}^% {\prime\prime}(\phi)}}+\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$ $\displaystyle v(\phi)$ $\displaystyle=\sqrt{\dfrac{1}{2\Delta^{1/2}(\phi)}}\left(\dfrac{1}{\sqrt{f_{+}% ^{\prime\prime}(\phi)}}-\dfrac{1}{\sqrt{-f_{-}^{\prime\prime}(\phi)}}\right),$ Symbols: $u(\phi)$: function and $v(\phi)$: function Permalink: http://dlmf.nist.gov/36.13.E7 Encodings: TeX, TeX, pMML, pMML, png, png
the disturbance is
36.13.8 $z(\rho,\phi)=2\pi\left(\rho^{-1/3}u(\phi)\mathop{\cos\/}\nolimits\!\left(\rho% \widetilde{f}(\phi)\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(-\rho^{2/3}% \Delta(\phi)\right)\*(1+\mathop{O\/}\nolimits\!\left(1/\rho\right))+\rho^{-2/3% }v(\phi)\mathop{\sin\/}\nolimits\!\left(\rho\widetilde{f}(\phi)\right){\mathop% {\mathrm{Ai}\/}\nolimits^{\prime}}\!\left(-\rho^{2/3}\Delta(\phi)\right)\*(1+% \mathop{O\/}\nolimits\!\left(1/\rho\right))\right),$ $\rho\to\infty$.
See Figure 36.13.1.
For further information see Lord Kelvin (1891, 1905) and Ursell (1960, 1994). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 64, "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": 0.9963006973266602, "perplexity": 1958.67605364262}, "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-2015-22/segments/1432207928780.77/warc/CC-MAIN-20150521113208-00276-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://gilkalai.wordpress.com/tag/graph-coloring/ | # When Do a Few Colors Suffice?
When can we properly color the vertices of a graph with a few colors? This is a notoriously difficult problem. Things get a little better if we consider simultaneously a graph together with all its induced subgraphs. Recall that an induced subgraph of a graph G is a subgraph formed by a subset of the vertices of G together with all edges of G spanned on these vertices. An induced cycle of length larger than three is called a hole, and an induced subgraph which is a complement of a cycle of length larger than 3 is called an anti-hole. As usual, $\chi (G)$ is the chromatic number of G and $\omega (G)$ is the clique number of G (the maximum number of vertices that form a complete subgraph. Clearly, for every graph G
$\chi(G) \ge \omega (G)$.
## Perfect graphs
Question 1: Describe the class $\cal G$ of graphs closed under induced subgraphs, with the property that $\chi(G)=\omega (G)$ for every $G\in{\cal G}$.
A graph G is called perfect if $\chi(H)=\omega (H)$ for every induced subgraph H of G. So Question 1 asks for a description of perfect graphs. The study of perfect graphs is among the most important areas of graph theory, and much progress was made along the years.
Interval graphs, chordal graphs, comparability graphs of POSETS , … are perfect.
Two major theorems about perfect graphs, both conjectured by Claude Berge are:
The perfect graph theorem (Lovasz, 1972): The complement of a perfect graph is perfect
The strong perfect graph theorem (Chudnovsky, Robertson, Seymour and Thomas, 2002): A graph is perfect if and only if it does not contain an odd hole and an odd anti-hole.
## Mycielski Graphs
There are triangle-free graphs with arbitrary large chromatic numbers. An important construction by Mycielski goes as follows: Given a triangle graph G with n vertices $v_1,v_2, \dots, v_n$ create a new graph G’ as follows: add n new vertices $u_1, u_2\dots u_n$ and a vertex w. Now add w to each $u_i$ and for every i and j for which $v_i$ and $v_j$ are adjacent add also an edge between $v_i$ and $u_j$ (and thus also between $u_i$ and $v_j$.)
## Classes of Graphs with bounded chromatic numbers
Question 2: Describe classes of graphs closed under induced subgraphs with bounded chromatic numbers.
Here are three theorems in this direction. The first answers affirmatively a conjecture by Kalai and Meshulam. The second and third prove conjectures by Gyarfas.
## Trinity Graphs
The Trinity graph theorem (Bonamy, Charbit and Thomasse, 2013): Graphs without induced cycles of length divisible by three have bounded chromatic numbers.
(The paper: Graphs with large chromatic number induce 3k-cycles.)
### Steps toward Gyarfas conjecture
Theorem (Scott and Seymour, 2014): Triangle-free graphs without odd induced cycles have bounded chromatic number.
(The paper: Coloring graphs with no odd holes.)
# Around Borsuk’s Conjecture 3: How to Save Borsuk’s conjecture
Borsuk asked in 1933 if every bounded set K of diameter 1 in $R^d$ can be covered by d+1 sets of smaller diameter. A positive answer was referred to as the “Borsuk Conjecture,” and it was disproved by Jeff Kahn and me in 1993. Many interesting open problems remain. The first two posts in the series “Around Borsuk’s Conjecture” are here and here. See also these posts (I,II,III, IV), and the post “Surprises in mathematics and theory” on Lipton and Reagan’s blog GLL.
Can we save the conjecture? We can certainly try, and in this post I would like to examine the possibility that Borsuk’s conjecture is correct except from some “coincidental” sets. The question is how to properly define “coincidental.”
Let K be a set of points in $R^d$ and let A be a set of pairs of points in K. We say that the pair (K, A) is general if for every continuous deformation of the distances on A there is a deformation K’ of K which realizes the deformed distances.
(This condition is related to the “strong Arnold property” (aka “transversality”) in the theory of Colin de Verdière invariants of graphs; see also this paper by van der Holst, Lovasz and Schrijver.)
Conjecture 1: If D is the set of diameters in K and (K,D) is general then K can be partitioned into d+1 sets of smaller diameter.
We propose also (somewhat stronger) that this conjecture holds even when “continuous deformation” is replaced with “infinitesimal deformation”.
The finite case is of special interest:
A graph embedded in $R^d$ is stress-free if we cannot assign non-trivial weights to the edges so that the weighted sum of the edges containing any vertex v (regarded as vectors from v) is zero for every vertex v. (Here we embed the vertices and regard the edges as straight line segments. (Edges may intersect.) Such a graph is called a “geometric graph”.) When we restrict Conjecture 1 to finite configurations of points we get.
Conjecture 2: If G is a stress free geometric graph of diameters in $R^d$ then G is (d+1)-colorable.
A geometric graph of diameters is a geometric graph with all edges having the same length and all non edged having smaller lengths. The attempt for “saving” the Borsuk Conjecture presented here and Conjectures 1 and 2 first appeared in a 2002 collection of open problems dedicated to Daniel J. Kleitman, edited by Douglas West.
When we consider finite configurations of points we can make a similar conjecture for the minimal distances:
Conjecture 3: If the geometric graph of pairs of vertices realizing the minimal distances of a point-configuration in $R^d$ is stress-free, then it is (d+1)-colorable.
We can speculate that even the following stronger conjectures are true:
Conjecture 4: If G is a stress-free geometric graph in $R^d$ so that all edges in G are longer than all non-edges of G, then G is (d+1)-colorable.
Conjecture 5: If G is a stress-free geometric graph in $R^d$ so that all edges in G are shorter than all non-edges of G, then G is (d+1)-colorable.
We can even try to extend the condition further so edges in the geometric graph will be larger (or smaller) than non-edges only just “locally” for neighbors of each given vertex.
1) It is not true that every stress-free geometric graph in $R^d$ is (d+1)-colorable, and not even that every stress-free unit-distance graph is (d+1)-colorable. Here is the (well-known) example referred to as the Moser Spindle. Finding conditions under which stress-free graphs in $R^d$ are (d+1)-colorable is an interesting challenge.
2) Since a stress-free graph with n vertices has at most $dn - {{d+1} \choose {2}}$ edges it must have a vertex of degree 2d-1 or less and hence it is 2d colorable. I expect this to be best possible but I am not sure about it. This shows that our “saved” version of Borsuk’s conjecture is of very different nature from the original one. For graphs of diameters in $R^d$ the chromatic number can, by the work of Jeff and me be exponential in $\sqrt d$.
3) It would be interesting to show that conjecture 1 holds in the non-discrete case when d+1 is replaced by 2d.
4) Coloring vertices of geometric graphs where the edged correspond to the minimal distance is related also the the well known Erdos-Faber-Lovasz conjecture..
See also this 1994 article by Jeff Kahn on Hypergraphs matching, covering and coloring problems.
5) The most famous conjecture regarding coloring of graphs is, of course, the four-color conjecture asserting that every planar graph is 4-colorable that was proved by Appel and Haken in 1976. Thinking about the four-color conjecture is always both fascinating and frustrating. An embedding for maximal planar graphs as vertices of a convex 3-dimensional polytope is stress-free (and so is, therefore, also a generic embedding), but we know that this property alone does not suffice for 4-colorability. Finding further conditions for stress-free graphs in $R^d$ that guarantee (d+1)-colorability can be relevant to the 4CT.
An old conjecture of mine asserts that
Conjecture 6: Let G be a graph obtained from the graph of a d-polytope P by triangulating each (non-triangular) face with non-intersecting diagonals. If G is stress-free (in which case the polytope P is called “elementary”) then G is (d+1)-colorable.
Closer to the conjectures of this post we can ask:
Conjecture 7: If G is a stress-free geometric graph in $R^d$ so that for every edge e of G is tangent to the unit ball and every non edge of G intersect the interior of the unit ball, then G is (d+1)-colorable.
### A question that I forgot to include in part I.
What is the minimum diameter $d_n$ such that the unit ball in $R^n$ can be covered by n+1 sets of smaller diameter? It is known that $2-C'\log n/n \le d_n\le 2-C/n$ for some constants C and C’. | {"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": 33, "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": 0.8610537052154541, "perplexity": 614.4203556668893}, "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-2015-22/segments/1432207928864.16/warc/CC-MAIN-20150521113208-00183-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/153769-test-series-convergence.html | # Thread: Test the series for convergence
1. ## Test the series for convergence
I am struggling through this new material.
Here is the problem:
I know from the limit comparison test that both b_n and a_n either converge or diverge, but I cant seem to find out which of the two it is...
How can I find whether b_n is convergent or divergent??
Thank you for the help!
2. first, note that ...
$\displaystyle \frac{1 + 2^n}{1 + 3^n} = \frac{1}{1+3^n} + \frac{2^n}{1+3^n}$
individually ...
$\displaystyle \frac{1}{1+3^n} < \frac{1}{3^n} = \left(\frac{1}{3}\right)^n$
$\displaystyle \frac{2^n}{1+3^n} < \frac{2^n}{3^n} = \left(\frac{2}{3}\right)^n$
... you should be able to figure it out from this point. | {"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": 3, "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": 0.9318633675575256, "perplexity": 552.9032295029692}, "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/1518891812293.35/warc/CC-MAIN-20180218232618-20180219012618-00117.warc.gz"} |
http://wikiwaves.org/Eigenfunction_Matching_Method_for_Floating_Elastic_Plates | # Eigenfunction Matching Method for Floating Elastic Plates
## Introduction
We show here a solution to the problem of wave propagation under many floating elastic plates of variable properties This work is based on Kohout et. al. 2006. This is a generalisation of the Eigenfunction Matching Method for a Semi-Infinite Floating Elastic Plate. We assume that the first and last plate are semi-infinite. The presentation here does not allow open water (it could be included but makes the formulation more complicated). In any case open water can be considered by taking the limit as the plate thickness tends to zero. The solution is derived using an extended eigenfunction matching method, in which the plate boundary conditions are satisfied as auxiliary equations.
## Equations
We consider the problem of small-amplitude waves which are incident on a set of floating elastic plates occupying the entire water surface. The submergence of the plates is considered negligible. We assume that the problem is invariant in the $y$ direction, although we allow the waves to be incident from an angle. The set of plates consists of two semi-infinite plates, separated by a region which consists of a finite number of plates with variable properties. We also assume that the plate edges are free to move at each boundary, although other boundary conditions could easily be considered using the methods of solution presented here. We begin with the Frequency Domain Problem for multiple Floating Elastic Plates in the non-dimensional form of Tayler 1986 (Dispersion Relation for a Floating Elastic Plate)
$\begin{matrix} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial z^2} - k_y^2\right) \phi = 0, \;\;\;\; \mbox{ for } -h \lt z \leq 0, \end{matrix}$
$\begin{matrix} \frac{\partial \phi}{\partial z} = 0, \;\;\;\; \mbox{ at } z = - h, \end{matrix}$
$\begin{matrix} \left( \beta_\mu \left(\frac{\partial^2}{\partial x^2} - k^2_y\right)^2 - \gamma_\mu\alpha + 1\right)\frac{\partial \phi}{\partial z} - \alpha\phi = 0, \;\;\;\; \mbox{ at } z = 0, \;\;\; l_\mu \leq x \leq r_\mu, \end{matrix}$
where $\alpha = \omega^2$, $\beta_\mu$ and $\gamma_\mu$ and the stiffness and mass constant for the $\mu$th plate. The conditions at the edges of the plates are
$\begin{matrix} \left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k^2_y\frac{\partial}{\partial x}\right) \frac{\partial\phi}{\partial z}= 0, \;\;\;\; \mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu, \end{matrix}$
$\begin{matrix} \left(\frac{\partial^2}{\partial x^2} - \nu k^2_y\right)\frac{\partial\phi}{\partial z} = 0, \;\;\;\;\mbox{ at } z = 0, \;\;\; \mbox{ for } x = l_\mu,r_\mu. \end{matrix}$
where $l_\mu$ and $r_\mu$ represent the left and right edge of the $\mu$th plate as shown in Figure~35.
## Method of solution
### Eigenfunction expansion
We will solve the system of equations using an Eigenfunction Matching Method. The method was developed by Fox and Squire 1994 for the case of a single plate as the research is described in Two-Dimensional Floating Elastic Plate. We show here how this method can be extended to the case of an arbitrary number of plates. One of the key features in the eigenfunction expansion method for elastic plates is that extra modes are required in order to solve the higher order boundary conditions at the plate edges.
The potential velocity of the first plate can be expressed as the summation of an incident wave and of reflected waves, one of which is propagating but the rest of which are evanescent and they decay as $x$ tends to $-\infty$. Similarly the potential under the final plate can be expressed as a sum of transmitting waves, one of which is propagating and the rest of which are evanescent and decay towards $+\infty$. The potential under the middle plates can be expressed as the sum of transmitting waves and reflected waves, each of which consists of a propagating wave plus evanescent waves which decay as $x$ decreases or increases respectively. We could combine these waves in the formulation, but because of the exponential growth (or decay) in the $x$ direction the solution becomes numerically unstable in some cases if the transmission and reflection are not expanded at opposite ends of the plate.
#### Separation of variables
The potential velocity can be written in terms of an infinite series of separated eigenfunctions under each elastic plate, of the form $\phi = e^{\kappa_\mu x} \cos(k_\mu(z+h)).$ If we apply the boundary conditions given we obtain the Dispersion Relation for a Floating Elastic Plate
$\begin{matrix} k_\mu\tan{(k_\mu h)}= & -\frac{\alpha}{\beta_\mu k_\mu^{4} + 1 - \alpha\gamma_\mu} \end{matrix}$
Solving for $k_\mu$ gives a pure imaginary root with positive imaginary part, two complex roots (two complex conjugate paired roots with positive imaginary part in all physical situations), an infinite number of positive real roots which approach ${n\pi}/{h}$ as $n$ approaches infinity, and also the negative of all these roots (Dispersion Relation for a Floating Elastic Plate) . We denote the two complex roots with positive imaginary part by $k_\mu(-2)$ and $k_\mu(-1)$, the purely imaginary root with positive imaginary part by $k_\mu(0)$ and the real roots with positive imaginary part by $k_\mu(n)$ for $n$ a positive integer. The imaginary root with positive imaginary part corresponds to a reflected travelling mode propagating along the $x$ axis. The complex roots with positive imaginary parts correspond to damped reflected travelling modes and the real roots correspond to reflected evanescent modes. In a similar manner, the negative of these correspond to the transmitted travelling, damped and evanescent modes respectively. The coefficient $\kappa_\mu$ is
$\kappa_\mu(n) = \sqrt{k_\mu(n)^2 + k_y^2},$
where the root with positive real part is chosen or if the real part is negative with negative imaginary part. Note that the solutions of the dispersion equation will be different under plates of different properties, and that the expansion is only valid under a single plate. We will solve for the coefficients in the expansion by matching the potential and its $x$ derivative at each boundary and by applying the boundary conditions at the edge of each plate.
### Expressions for the potential velocity
We now expand the potential under each plate using the separation of variables solution. We always include the two complex and one imaginary root, and truncate the expansion at $M$ real roots of the dispersion equation. The potential $\phi$ can now be expressed as the following sum of eigenfunctions:
$\phi \approx \left\{ \begin{matrix} { Ie^{\kappa_{1}(0)(x-r_1)}\frac{\cos(k_1(0)(z+h))}{\cos(k_1(0)h)} }+\\ { \qquad \qquad \sum_{n=-2}^{M}R_1(n) e^{\kappa_{1}(n)(x-r_1)} \frac{\cos(k_1(n)(z+h))}{\cos(k_1(n)h)} },& \mbox{ for } x \lt r_1,\\ { \sum_{n=-2}^{M}T_{\mu}(n) e^{-\kappa_\mu(n)(x-l_\mu)} \frac{\cos(k_\mu(n)(z+h))}{\cos(k_\mu(n)h)} } + \\ { \qquad \qquad \sum_{n=-2}^{M}R_{\mu}(n) e^{\kappa_\mu(n)(x-r_\mu)} \frac{\cos(k_\mu(n)(z+h))}{\cos(k_\mu(n)h)} }, &\mbox{ for } l_\mu\lt x \lt r_\mu,\\ { \sum_{n=-2}^{M}T_{\Lambda}(n)e^{-\kappa_{\Lambda}(n)(x-l_\Lambda)} \frac{\cos(k_\Lambda(n)(z+h))}{\cos(k_\Lambda(n)h)} }, &\mbox{ for } l_\Lambda\ltx, \end{matrix} \right.$
where $I$ is the non-dimensional incident wave amplitude in potential, $\mu$ is the $\mu^{th}$ plate, $\Lambda$ is the last plate, $r_\mu$ represents the $x$-coordinate of the right edge of the $\mu^{th}$ plate, $l_\mu$ ($=r_{\mu-1}$) represents the $x$-coordinate of the left edge of the $\mu^{th}$ plate, $R_\mu(n)$ represents the reflected potential coefficient of the $n^{th}$ mode under the $\mu^{th}$ plate, and $T_\mu(n)$ represents the transmitted potential coefficient of the $n^{th}$ mode under the $\mu^{th}$ plate. Note that we have divided by $\cos{(kh)}$, so that the coefficients are normalised by the potential at the free surface rather than at the bottom surface.
## Expressions for displacement
The displacement is given by
$\eta \approx \frac{i}{\omega}\left\{ \begin{matrix} { Ik_1(0)e^{\kappa_{1}(0)(x-r_1)}\tan{(k_1(0)h)} - } \\ { \qquad \sum_{n=-2}^{M}R_{1}(n)k_1(n)e^{\kappa_{1}(n)(x-r_1)} \tan{(k_1(n)h)} }, & \mbox{ for } x \lt r_1, \\ { -\sum_{n=-2}^{M}T_{\mu}(n)k_\mu(n)e^{-\kappa_\mu(n)(x-l_\mu)}\tan{(k_\mu(n)h)} - }\\ { \qquad \sum_{n=-2}^{M}R_{\mu}(n)k_\mu(n)e^{\kappa_\mu(n)(x-r_\mu)} \tan{(k_\mu(n)h)} },& \mbox{ for } l_\mu\lt x \lt r_\mu,\\ { -\sum_{n=-2}^{M}T_{\Lambda}(n)k_\mu(n)e^{-\kappa_\mu(n)(x-l_\Lambda)}\tan{(k_\mu(n)h)} }, &\mbox{ for } l_\Lambda\ltx. \end{matrix}\right.$
## Solving via eigenfunction matching
To solve for the coefficients, we require as many equations as we have unknowns. We derive the equations from the free edge conditions and from imposing conditions of continuity of the potential and its derivative in the $x$-direction at each plate boundary. We impose the latter condition by taking inner products with respect to the orthogonal functions $\cos \frac{m\pi}{h}(z+h)$, where $m$ is a natural number. These functions are chosen for the following reasons. The vertical eigenfunctions $\cos k_\mu(n)(z+h)$ are not orthogonal (they are not even a basis) and could therefore lead to an ill-conditioned system of equations. Furthermore, by choosing $\cos \frac{m\pi}{h}(z+h)$ we can use the same functions to take the inner products under every plate. Finally, and most importantly, the plate eigenfunctions approach $\cos{(m\pi/h)(z + h)}$ for large $m$, so that as we increase the number of modes the matrices become almost diagonal, leading to a very well-conditioned system of equations.
Taking inner products leads to the following equations
$\begin{matrix} { \int_{-h}^0 \phi_\mu(r_\mu,z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=& { \int_{-h}^0 \phi_{\mu+1}(l_{\mu+1},z)\cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z }\\ { \int_{-h}^0 \frac{\partial\phi_\mu}{\partial x}(r_\mu,z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } &=& { \int_{-h}^0 \frac{\partial\phi_{\mu+1}}{\partial x}(l_{\mu+1},z) \cos \frac{m\pi}{h}(z+h) \, \mathrm{d}z } \end{matrix}$
where $m\in[0,M]$ and $\phi_\mu$ denotes the potential under the $\mu$th plate, i.e. the expression for $\phi$ valid for $l_\mu \ltx\ltr_\mu$. The remaining equations to be solved are given by the two edge conditions satisfied at both edges of each plate
$\begin{matrix} { \left(\frac{\partial^3}{\partial x^3} - (2 - \nu)k_y^2\frac{\partial}{\partial x}\right)\frac{\partial\phi_\mu}{\partial z} } &=&0, & \mbox{ for } z = 0 \mbox{ and } x = l_\mu,r_\mu,\\ { \left(\frac{\partial^2}{\partial x^2} - \nu k_y^2\right)\frac{\partial\phi_\mu}{\partial z} } &=&0, & \mbox{ for } z = 0 \mbox{ and } x = l_\mu,r_\mu. \end{matrix}$
We will show the explicit form of the linear system of equations which arise when we solve these equations. Let ${\mathbf T}_\mu$ be a column vector given by $\left[T_{\mu}(-2), . . ., T_{\mu}(M)\right]^{{\mathbf T}}$ and ${\mathbf R}_\mu$ be a column vector given by $\left[R_{\mu}(-2) . . . R_{\mu}(M)\right]^{{\mathbf T}}$.
The equations which arise from matching at the boundary between the first and second plate are
$\begin{matrix} I{\mathbf C} + {\mathbf M}^{+}_{R_1} {\mathbf R}_1 ={\mathbf M}^{-}_{T_2} {\mathbf T}_2 + {\mathbf M}^{-}_{R_2} {\mathbf R}_2,\\ -\kappa_1(0)I\mathbf{C} + {\mathbf N}^{+}_{R_1} {\mathbf R}_1 = {\mathbf N}^{-}_{T_2} {\mathbf T}_2 + {\mathbf N}^{-}_{R_2} {\mathbf R}_2. \end{matrix}$
The equations which arise from matching at the boundary of the $\mu$th and ($\mu+1$)th plate boundary ($\mu\gt1$) are
$\begin{matrix} {\mathbf M}^{+}_{T_\mu} {\mathbf T}_\mu +{\mathbf M}^{+}_{R_\mu} {\mathbf R}_\mu ={\mathbf M}^{-}_{T_{\mu+1}} {\mathbf T}_{\mu+1} + {\mathbf M}^{-}_{ R_{\mu+1}} {\mathbf R}_{\mu+1}, \\ {\mathbf N}^{+}_{T_\mu} {\mathbf T}_\mu + {\mathbf N}^{+}_{R_\mu} {\mathbf R}_\mu ={\mathbf N}^{-}_{T_{\mu+1}} {\mathbf T}_{\mu +1} +{\mathbf N}^{-}_{ R_{\mu +1}} {\mathbf R}_{\mu +1}. \end{matrix}$
The equations which arise from matching at the ($\Lambda-1$)th and $\Lambda$th boundary are
$\begin{matrix} {\mathbf M}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1} + {\mathbf M}^{+}_{R_{\Lambda-1}} {\mathbf R}_{\Lambda-1} = {\mathbf M}^{-}_{T_\Lambda } {\mathbf T}_{\Lambda}, \\ {\mathbf N}^{+}_{T_{\Lambda-1}} {\mathbf T}_{\Lambda-1} + {\mathbf N}^{+}_{R_{\Lambda-1}} {\mathbf R}_{\Lambda-1} = {\mathbf N}^{-}_{T_\Lambda } {\mathbf T}_{\Lambda}, \end{matrix}$
where ${\mathbf M}^{+}_{T_\mu}$, ${\mathbf M}^{+}_{R_\mu}$, ${\mathbf M}^{-}_{T_\mu}$, and ${\mathbf M}^{-}_{R_\mu}$are $(M+1)$ by $(M+3)$ matrices given by
$\begin{matrix} { {\mathbf M}^{+}_{T_\mu}(m,n) = \int_{-h}^0 e^{-\kappa_\mu(n) (r_\mu-l_\mu )} \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z}, \\ { {\mathbf M}^{+}_{R_\mu}(m,n) = \int_{-h}^0 \frac{\cos \left(k_{\mu}(n) (z+h)\right)}{\cos \left(k_{\mu}(n) h\right)} \cos \left(\frac{m\pi}{h}(z+h)\right) \, \mathrm{d}z },\\ { {\mathbf M}^{-}_{T_\mu}(m,n) = {\mathbf M}^{+}_{R_\mu}(m,n) }\\ { {\mathbf M}^{-}_{R_\mu}(m,n) = {\mathbf M}^{+}_{T_\mu}(m,n). } \end{matrix}$
${\mathbf N}^{+}_{T_\mu}$, ${\mathbf N}^{+}_{R_\mu}$, ${\mathbf N}^{-}_{T_\mu}$, and ${\mathbf N}^{-}_{R_\mu}$ are given by
$\begin{matrix} {\mathbf N}^{\pm}_{T_\mu}(m,n)= -\kappa_\mu(n){\mathbf M}^{\pm}_{T_\mu}(m,n),\\ {\mathbf N}^{\pm}_{R_\mu}(m,n)= \kappa_\mu(n){\mathbf M}^{\pm}_{R_\mu}(m,n). \end{matrix}$
$\mathbf{C}$ is a $(M+1)$ vector which is given by
${\mathbf C}(m)=\int_{-h}^0 \frac{\cos (k_1(0)(z+h))}{\cos (k_1(0)h)} \cos \left(\frac{m\pi}{h}(z+h)\right)\, \mathrm{d}z.$
The integrals in the above equation are each solved analytically. Now, for all but the first and $\Lambda$th plate, the edge equation becomes
$\begin{matrix} {\mathbf E}^{+}_{T_\mu} {\mathbf T}_\mu + {\mathbf E}^{+}_{R_\mu} {\mathbf R}_\mu = 0,\\ {\mathbf E}^{-}_{T_\mu} {\mathbf T}_\mu + {\mathbf E}^{-}_{R_\mu} {\mathbf R}_\mu = 0. \end{matrix}$
The first and last plates only require two equations, because each has only one plate edge. The equation for the first plate must be modified to include the effect of the incident wave. This gives us
$\begin{matrix} I \left( \begin{matrix} {\mathbf E}^{+}_{T_1}(1,0)\\ {\mathbf E}^{+}_{T_1}(2,0) \end{matrix} \right) + {\mathbf E}^{+}_{R_1} {\mathbf R}_1 = 0,\\ \end{matrix}$
and for the $\Lambda$th plate we have no reflection so
$\begin{matrix} {\mathbf E}^{-}_{T_\mu} {\mathbf T}_\mu = 0.\\ \end{matrix}$
${\mathbf E}^{+}_{T_\mu}$, ${\mathbf E}^{+}_{R_\mu}$, ${\mathbf E}^{-}_{T_\mu}$ and ${\mathbf E}^{-}_{R_\mu}$ are 2 by M+3 matrices given by
$\begin{matrix} {\mathbf E}^{-}_{T_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(k_{\mu}(n)\kappa_\mu(n)\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{+}_{T_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(k_{\mu}(n)\kappa_\mu(n)e^{-\kappa_\mu(n)(r_\mu - l_\mu)}\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{-}_{R_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(-k_{\mu}(n)\kappa_\mu(n)e^{\kappa_\mu(n)(l_\mu - r_\mu)}\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{+}_{R_\mu}(1,n) = (\kappa_\mu(n)^2 - (2 - \nu)k_y^2)(-k_{\mu}(n)\kappa_\mu(n)\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{-}_{T_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{+}_{T_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)e^{-\kappa_\mu(n)(r_\mu - l_\mu)}\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{-}_{R_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)e^{\kappa_\mu(n)(l_\mu - r_\mu)}\tan{(k_{\mu}(n)h)}),\\ {\mathbf E}^{+}_{R_\mu}(2,n) = (\kappa_\mu(n)^2 - \nu k_y^2)(-k_{\mu}(n)\tan{(k_{\mu}(n)h)}).\\ \end{matrix}$
Now, the matching matrix is a $(2M+6)\times(\Lambda-1)$ by $(2M+1)\times(\Lambda -1)$ matrix given by
${\mathbf M} = \left( \begin{matrix} {\mathbf M}^{+}_{R_1} & -{\mathbf M}^{-}_{T_2} & -{\mathbf M}^{-}_{R_2} & 0 & 0 & & 0 & 0 & 0 \\ {\mathbf N}^{+}_{R_1} & -{\mathbf N}^{-}_{T_2} & -{\mathbf N}^{-}_{R_2} & 0 & 0 & & 0 & 0 & 0 \\ 0 & {\mathbf M}^{+}_{T_2} & {\mathbf M}^{+}_{R_2} & -{\mathbf M}^{-}_{T_3} & -{\mathbf M}^{-}_{R_3} & & 0 & 0 & 0 \\ 0 & {\mathbf N}^{+}_{T_2} & {\mathbf N}^{+}_{R_2} & -{\mathbf N}^{-}_{T_3} & -{\mathbf N}^{-}_{R_3} & & 0 & 0 & 0 \\ & & \vdots & & & \ddots & \\ 0 & 0 & 0 & 0 & 0 & & {\mathbf M}^{+}_{T_{\Lambda - 1}} & {\mathbf M}^{+}_{R_{\Lambda - 1}} & -{\mathbf M}^{-}_{ T_{\Lambda}} \\ 0 & 0 & 0 & 0 & 0 & & {\mathbf N}^{+}_{T_{\Lambda - 1}} & {\mathbf N}^{+}_{R_{\Lambda - 1}} & -{\mathbf N}^{-}_{T_{\Lambda }} \\ \end{matrix} \right),$
the edge matrix is a $(2M+6)\times(\Lambda-1)$ by $4(\Lambda-1)$ matrix given by
${\mathbf E} = \left( \begin{matrix} {\mathbf E}^{+}_{R_1} & 0 & 0 & 0 & 0 & & 0 & 0 & 0 \\ 0 & {\mathbf E}^{+}_{T_2} & {\mathbf E}^{+}_{R_2} & 0 & 0 & & 0 & 0 & 0 \\ 0 & {\mathbf E}^{-}_{T_2} & {\mathbf E}^{-}_{R_2} & 0 & 0 & & 0 & 0 & 0 \\ 0 & 0 & 0 & {\mathbf E}^{+}_{T_3} & {\mathbf E}^{+}_{R_3} & & 0 & 0 & 0 \\ 0 & 0 & 0 & {\mathbf E}^{-}_{T_3} & {\mathbf E}^{-}_{R_3} & & 0 & 0 & 0 \\ & & \vdots & & & \ddots & \\ 0 & 0 & 0 & 0 & 0 & & {\mathbf E}^{+}_{T_{\Lambda-1}} & {\mathbf E}^{+}_{R_{\Lambda-1}} & 0 \\ 0 & 0 & 0 & 0 & 0 & & {\mathbf E}^{-}_{T_{\Lambda-1}} & {\mathbf E}^{-}_{R_{\Lambda-1}} & 0 \\ 0 & 0 & 0 & 0 & 0 & & 0 & 0 & {\mathbf E}^{-}_{ T_\Lambda} \end{matrix} \right),$
and finally the complete system to be solved is given by
$\left( \begin{matrix} {\mathbf M}\\ {\mathbf E}\\ \end{matrix} \right) \times \left( \begin{matrix} {\mathbf R}_1\\ {\mathbf T}_2\\ {\mathbf R}_2\\ {\mathbf T}_3\\ {\mathbf R}_3\\ \vdots\\ {\mathbf T}_{\Lambda-1}\\ {\mathbf R}_{\Lambda-1}\\ {\mathbf T}_{\Lambda} \end{matrix} \right) = \left( \begin{matrix} -I{\mathbf C}\\ \kappa_{1}(0)I{\mathbf C}\\ 0\\ \vdots\\ -IE^{+}_{T_1}(1,0)\\ -IE^{+}_{T_1}(2,0)\\ 0\\ \vdots \end{matrix} \right).$
The final system of equations has size $(2M+6)\times (\Lambda - 1)$ by $(2M+6)\times (\Lambda - 1)$. | {"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": 0.9720136523246765, "perplexity": 217.50364779902972}, "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/1502886102393.60/warc/CC-MAIN-20170816191044-20170816211044-00579.warc.gz"} |
https://lucatrevisan.wordpress.com/2011/03/18/cs261-lecture-18-using-expert-advice/ | # CS261 Lecture 18: Using Expert Advice
In which we show how to use expert advice, and introduce the powerful “multiplicative weight” algorithm.
We study the following online problem. We have ${n}$ “experts” that, at each time step ${t=1,\ldots,T}$, suggest a strategy about what to do at that time (for example, they might be advising on what technology to use, on what investments to make, they might make predictions on whether something is going to happen, thus requiring certain actions, and so on). Based on the quality of the advice that the experts offered in the past, we decide which advice to follow, or with what fraction of our investment to follow which strategy. Subsequently, we find out which loss or gain was associated to each strategy, and, in particular, what loss or gain we personally incurred with the strategy or mix of strategies that we picked, and we move to step ${t+1}$.
We want to come up with an algorithm to use the expert advice such that, at the end, that is, at time ${T}$, we are about as well off as if we had known in advance which expert was the one that gave the best advice, and we had always followed the strategy suggested by that expert at each step. Note that we make no probabilistic assumption, and our analysis will be a worst-case analysis over all possible sequences of events.
The “multiplicative update” algorithm provides a very good solution to this problem, and the analysis of this algorithm is a model for the several other applications of this algorithm, in rather different contexts.
1. A Simplified Setting
We begin with the following simplified setting: at each time step, we have to make a prediction about an event that has two possible outcomes, and we can use the advice of ${n}$ “experts,” which make predictions about the outcome at each step. Without knowing anything about the reliability of the experts, and without making any probabilistic assumption on the outcomes, we want to come up with a strategy that will lead us to make not much more mistakes than the “offline optimal” strategy of picking the expert which makes the fewest mistakes, and then always following the prediction of that optimal expert.
The algorithm works as follows: at each step ${t}$, it assigns a weight ${w_i^t}$ to each expert ${i}$, which measures the confidence that the algorithm has in the validity of the prediction of the expert. Initially, ${w^0_i = 1}$ for all experts ${i}$. Then the algorithm makes the prediction that is backed by the set of experts with largest total weight. For example, if the experts, and us, are trying to predict whether the following day it will rain or not, we will look at the sum of the weights of the experts that say it will rain, and the sum of the weights of the experts that say it will not, and then we agree with whichever prediction has the largest sum of weights. After the outcome is revealed, we divide by 2 the weight of the experts that were wrong, and leave the weight of the experts that were correct unchanged.
We now formalize the above algorithm in pseudocode. We use ${\{a,b\}}$ to denote the two possible outcomes of the event that we are required to predict at each step.
• for each ${i\in \{1,\ldots, n\}}$ do ${w^1_i:= 1}$
• for each time ${t \in \{ 1 ,\ldots, T \}}$
• let ${w^t := \sum_i w_i^t}$
• if the sum of ${w^t_i}$ over all the experts ${i}$ that predict ${a}$ is ${\geq w^t/2}$, then predict ${a}$
• else predict ${b}$
• wait until the outcome is revealed
• for each ${i\in \{1,\ldots, n \}}$
• if ${i}$ was wrong then ${w^{t+1}_i := w^{t}_i/2}$
To analyze the algorithm, let ${m_i^t}$ be the indicator variable that expert ${i}$ was wrong at time ${t}$, that is, ${m_i^t=1}$ if the expert ${i}$ was wrong at time ${i}$ and ${m_i^t=0}$ otherwise. (Here ${m}$ stands for “mistake.”) Let ${m_i = \sum_{t=1}^T m_i^t}$ be the total number of mistakes made by expert ${i}$. Let ${m_A^t}$ be the indicator variable that our algorithm makes a mistake at time ${t}$, and ${m_A:= \sum_{t=1}^T m_A^t}$ be the total number of mistakes made by our algorithm.
We make the following two observations:
1. If the algorithm makes a mistake at time ${t}$, then the total weight of the experts that are mistaken at time ${t}$ is ${\geq w^{t}/2}$, and, at the following step, the weight of those experts is divided by two, and this means that, if we make a mistake at time ${t}$ then
$\displaystyle w^{t+1} \leq \frac 34 w^{t}$
Because the initial total weight is ${w^1 = n}$, we have that, at the end,
$\displaystyle w^{T+1} \leq \left( \frac 34 \right) ^{m_A} \cdot n$
2. For each expert ${i}$, the final weight is ${w^{T+1}_i = 2^{-m_i}}$, and, clearly,
$\displaystyle \frac 1 {2^{m_i}} = w_i^{T+1} \leq w^{T+1}$
Together, the two previous observations mean that, for every expert ${i}$,
$\displaystyle \frac 1 {2^{m_i}} \leq \left( \frac 34 \right) ^{m_A} \cdot n$
which means that, for every expert ${i}$,
$\displaystyle m_A \leq O(m_i + \log n)$
That is, the number of mistakes made by the algorithm is at most a constant times the number of mistakes of the best expert, plus an extra ${O(\log n)}$ mistakes.
We will now discuss an algorithm that improves the above result in two ways. We will show that, for every ${\epsilon}$, the improved algorithm we can make the number of mistakes be at most ${(1+\epsilon) m_i + O\left( \frac 1 \epsilon \log n \right)}$ for every ${\epsilon}$, which can be seen to be optimal for small ${n}$, and the improved algorithm will be able to handle a more general problem, in which the experts are suggesting arbitrary strategies, and the outcome of each strategy can be an arbitrary gain or loss.
2. The General Result
We now consider the following model. At each time step ${t}$, each expert ${i}$ suggests a certain strategy. We choose to follow the advice of expert ${i}$ with probability ${p^t_i}$, or, equivalently, we allocate a ${p^t_i}$ fraction of our resources in the way expert ${i}$ advised. Then we observe the outcome of the strategies suggested by the experts, and of our own strategy. We call ${m_i^t}$ the loss incurred by following the advice of expert ${i}$. The loss can be negative, in which case it is a gain, and we normalize losses and gains so that ${m_i^t \in [-1,1]}$ for every ${i}$ and every ${t}$. Our own loss for the time step ${t}$ will then be ${\sum_i p_i^t m_i^t}$. At the end, we would like to say that our own sum of losses is not much higher than the sum of losses of the best expert.
As before, our algorithm maintains a weight for each expert, corresponding to our confidence in the expert. The weights are initialized to 1. When an expert causes a loss, we reduce his weight, and when an expert causes a gain, we increase his weight. To express the weight updated in a single instruction, we have ${w^{t+1}_i := (1-\epsilon m_i^t) \cdot w^t_i}$, where ${0 < \epsilon < 1/2}$ is a parameter of our choice. Our probabilities ${p^t_i}$ are chosen proportionally to weights ${w^t_i}$.
• for each ${i\in \{1,\ldots, n\}}$ do ${w^1_i:= 1}$
• for each time ${t \in \{ 1 ,\ldots, T \}}$
• let ${w^t := \sum_i w_i^t}$
• let ${p^t_i := w_i^t/ w^t}$
• for each ${i}$, follow the strategy of expert ${i}$ with probability ${p^t_i}$
• wait until the outcome is revealed
• let ${m^t_i}$ be the loss of the strategy of expert ${i}$
• for each ${i\in \{1,\ldots, n \}}$
• ${w^{t+1}_i := (1 - \epsilon \cdot m_i^t ) \cdot w^t_i}$
To analyze the algorithm, we will need the following technical result.
Fact 1 For every ${\epsilon \in [-1/2,1/2]}$,
$\displaystyle e^{\epsilon - \epsilon^2} \leq 1+\epsilon \leq e^{\epsilon}$
Proof: We will use the Taylor expansion
$\displaystyle e^{x} = 1 + x + \frac {x^2}2 + \frac{x^3}{3!} + \cdots$
1. The upper bound. The Taylor expansion above can be seen as ${e^x = 1 + x + \sum_{t=1}^\infty x^{2t} \cdot \left( \frac {1}{(2t)!} + \frac{x}{(2t+1)!} \right)}$, that is, ${e^x}$ equals ${1+x}$ plus a sum of terms that are all non-negative when ${x\geq -1}$. Thus, in particular, we have ${1+\epsilon \leq e^{\epsilon}}$ for ${\epsilon \in [-1/2,1/2]}$.
2. The lower bound for positive ${\epsilon}$. We can also see that, for ${x\in [0,1]}$, we have
$\displaystyle e^x \leq 1 + x + x^2$
and so, for ${\epsilon \in [0,1]}$ we have
$\displaystyle e^{\epsilon-\epsilon^2} \leq 1 + \epsilon - \epsilon^2 + \epsilon^2 - 2\epsilon^3 + \epsilon^4 \leq 1 + \epsilon$
3. The lower bound for negative ${\epsilon}$. Finally, for ${x\in [-1,0]}$ we have
$\displaystyle e^x = 1 + x + \frac {x^2}2 + \sum_{t=1}^\infty x^{2t + 1} \left( \frac 1{(2t+1)!} + \frac x {(2t+2)!} \right) \leq 1 + x + \frac {x^2} 2$
and so, for ${\epsilon \in [-1/2,0]}$ we have
$\displaystyle e^{\epsilon - \epsilon^2} \leq 1 + \epsilon - \epsilon^2 + \frac 12 \epsilon^2 - \epsilon^3 + \frac 14 \epsilon^4 \leq 1 + \epsilon$
$\Box$
Now the analysis proceeds very similarly to the analysis in the previous section. We let
$\displaystyle m_A^t := \sum_i p_i^t m_i^t$
be the loss of the algorithm at time ${t}$, and ${m_A := \sum_{t=1}^T m_A^t}$ the total loss at the end. We denote by ${m_i:= \sum_{t=1}^T m_i^t}$ the total loss of expert ${i}$.
If we look at the total weight at time ${t+1}$, it is
$\displaystyle w^{t+1} = \sum_i w^{t+1}_i = \sum_i (1- \epsilon m_i^t) \cdot w^t_i$
and we can rewrite it as
$\displaystyle w^{t+1} = w^t - \sum_i \epsilon m_i^t w^t_i = w^t - w^t \epsilon \cdot \sum_i m_i^t p_i^t = w^t \cdot ( 1- \epsilon m_A^t)$
Recalling that, initially, ${w^1 = n}$, we have that the total weight at the end is
$\displaystyle w^{T+1} = n \cdot \prod_{t=1}^T (1 - \epsilon m_A^t)$
For each expert ${i}$, the weight of that expert at the end is
$\displaystyle w^{T+1}_i = \prod_{t=1}^T (1-\epsilon m_i^t)$
and, as before, we note that for every expert ${i}$ we have
$\displaystyle w^{T+1}_i \leq w^{T+1}$
Putting everything together, for every expert ${i}$ we have
$\displaystyle \prod_{t=1}^T (1-\epsilon m_i^t) \leq n \cdot \prod_{t=1}^T (1 - \epsilon m_A^t)$
Now it is just a matter of taking logarithms and of using the inequality that we proved before.
$\displaystyle \ln \prod_{t=1}^T (1 - \epsilon m_A^t) = \sum_{i=1}^T \ln 1 - \epsilon m_A^t \leq - \sum_{i=1}^T \epsilon m_A^t = - \epsilon m_A$
$\displaystyle \ln \prod_{t=1}^T (1-\epsilon m_i^t) = \sum_{t=1}^T \ln 1-\epsilon m_i^T \geq \sum_{t=1}^T - \epsilon m_i^t - \epsilon^2 (m_i^t)^2$
and, overall,
$\displaystyle m_A \leq + m_i + \epsilon \sum_{i=1}^T |m_i^t| +\frac{\ln n}{\epsilon} \ \ \ \ \ (1)$
In the model of the previous section, at every step the loss of each expert is either 0 or 1, and so the above expression simplifies to
$\displaystyle m_A \leq (1+ \epsilon) m_i + \frac {\ln n } {\epsilon}$
which shows that we can get arbitrarily close to the best expert.
In every case, (1) simplifies to
$\displaystyle m_A \leq m_i + \epsilon T + \frac {\ln n }{\epsilon}$
and, if we choose ${\epsilon = \sqrt{\ln n/T}}$, we have
$\displaystyle m_A \leq m_i + 2 \sqrt{T\ln n}$
which means that we come close to the optimum up to a small additive error.
To see that this is essentially the best that we can hope for, consider a playing a fair roulette game as follows: for ${T}$ times, we either bet $1 on red or$1 on black. If we win we win $1, and if we lose we lose$1; we win and lose with probability 1/2 each at each step. Clearly, for every betting strategy, our expected win at the end is 0. We can think of the problem as there being two experts: the red expert always advises to bet red, and the black expert always advises to bet black. For each run of the game, the strategy of always following the best expert has a non-negative gain and, on average, following the best expert has a gain of ${\Omega( \sqrt{T})}$, because there is ${\Omega(1)}$ probability that the best expert has a gain of ${\Omega(\sqrt T)}$. This means that we cannot hope to always achieve at least the gain of the best expert minus ${o(\sqrt T)}$, even in a setting with 2 experts.
3. Applications
The general expert setting is very similar to a model of investments in which the experts correspond to stocks (or other investment vehicles) and the outcomes correspond to the variation in value of the stocks. The difference is that in our model we “invest” one unit of money at each step regardless of what happened in previous steps, while in investment strategies we compound our gains (and losses). If we look at the logarithm of the value of our investment, however, it is modeled correctly by the experts setting.
The multiplicative update algorithm that we described in the previous section arises in several other contexts, with a similar, or even identical, analysis. For example, it arises in the context of boosting in machine learning, and it leads to efficient approximate algorithms for certain special cases of linear programming.
## 2 thoughts on “CS261 Lecture 18: Using Expert Advice”
1. Very interesting article. As a logician I’m particularly interested in the case when T=infinity. In particular, I’m interested in the case when the “experts” are really expert trolls and are intentionally trying to confuse us. For example, say that, in addition to trying to determine whether it will rain tomorrow, we also have a “secondary mission” which is to determine which expert is the most reliable. For simplicity, assume there are only two experts, A and B, both who have perfect knowledge of the weather. Rather than perfectly predict it, though, they conspire together so that A makes correct predictions and B makes incorrect ones until we eventually suspect A is the best expert. Then, they switch, A making incorrect predictions and B making correct predictions, until we suspect that B is the best expert. This continues indefinitely, forcing us to change our minds infinitely often. And contrary to intuition, it does NOT require the experts be able to see our guesses, provided we’ve nailed down the multiplicative weight algorithm or any other particular algorithm. The experts can merely run that algorithm themselves, allowing them to confuse us even if our guesses are invisible to them. It would be interesting to compute exactly what predictions these mischievous experts would make (this is completely determined, assuming, say, that by coincidence it rains every day).
2. Some corrections:
1. In both the simplified and general settings, it seems w_{i}^{1} should be 1, not 0.
2. The inequality signs (\leq, \geq) just above the statement of Eq. (1) should be reversed.
[Thanks, fixed now. — L.] | {"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": 133, "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": 0.9320361614227295, "perplexity": 279.09684520203155}, "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/1642320300244.42/warc/CC-MAIN-20220116210734-20220117000734-00171.warc.gz"} |
https://davidfoxcroft.github.io/lsj-book/10-Categorical-data-analysis.html | # 10 Categorical data analysis
Now that we’ve covered the basic theory behind hypothesis testing it’s time to start looking at specific tests that are commonly used in psychology. So where should we start? Not every textbook agrees on where to start, but I’m going to start with “$$\chi^2$$ tests” (this chapter, pronounced “chi-square”1 and “t-tests” in Chapter 11). Both of these tools are very frequently used in scientific practice, and whilst they’re not as powerful as “regression” and “analysis of variance” which we cover in later chapters, they’re much easier to understand.
The term “categorical data” is just another name for “nominal scale data”. It’s nothing that we haven’t already discussed, it’s just that in the context of data analysis people tend to use the term “categorical data” rather than “nominal scale data”. I don’t know why. In any case, categorical data analysis refers to a collection of tools that you can use when your data are nominal scale. However, there are a lot of different tools that can be used for categorical data analysis, and this chapter covers only a few of the more common ones.
## 10.1 The $$\chi^2$$ (chi-square) goodness-of-fit test
The $$\chi^2$$ goodness-of-fit test is one of the oldest hypothesis tests around. It was invented by Karl Pearson around the turn of the century , with some corrections made later by Sir Ronald Fisher . It tests whether an observed frequency distribution of a nominal variable matches an expected frequency distribution. For example, suppose a group of patients has been undergoing an experimental treatment and have had their health assessed to see whether their condition has improved, stayed the same or worsened. A goodness-of-fit test could be used to determine whether the numbers in each category - improved, no change, worsened - match the numbers that would be expected given the standard treatment option. Let’s think about this some more, with some psychology.
### 10.1.1 The cards data
Over the years there have been many studies showing that humans find it difficult to simulate randomness. Try as we might to “act” random, we think in terms of patterns and structure and so, when asked to “do something at random”, what people actually do is anything but random. As a consequence, the study of human randomness (or non-randomness, as the case may be) opens up a lot of deep psychological questions about how we think about the world. With this in mind, let’s consider a very simple study. Suppose I asked people to imagine a shuffled deck of cards, and mentally pick one card from this imaginary deck “at random”. After they’ve chosen one card I ask them to mentally select a second one. For both choices what we’re going to look at is the suit (hearts, clubs, spades or diamonds) that people chose. After asking, say, $$N = 200$$ people to do this, I’d like to look at the data and figure out whether or not the cards that people pretended to select were really random. The data are contained in the randomness.csv file in which, when you open it up in jamovi and take a look at the spreadsheet view, you will see three variables. These are: an id variable that assigns a unique identifier to each participant, and the two variables choice_1 and choice_2 that indicate the card suits that people chose.
For the moment, let’s just focus on the first choice that people made. We’ll use the Frequency tables option under ‘Exploration’ - ‘Descriptives’ to count the number of times that we observed people choosing each suit. This is what we get (Table 10.1):
Table 10.1: Number of times each suit was chosen
35516450
That little frequency table is quite helpful. Looking at it, there’s a bit of a hint that people might be more likely to select hearts than clubs, but it’s not completely obvious just from looking at it whether that’s really true, or if this is just due to chance. So we’ll probably have to do some kind of statistical analysis to find out, which is what I’m going to talk about in the next section.
Excellent. From this point on, we’ll treat this table as the data that we’re looking to analyse. However, since I’m going to have to talk about this data in mathematical terms (sorry!) it might be a good idea to be clear about what the notation is. In mathematical notation, we shorten the human-readable word “observed” to the letter $$O$$, and we use subscripts to denote the position of the observation. So the second observation in our table is written as $$O_2$$ in maths. The relationship between the English descriptions and the mathematical symbols are illustrated in Table 10.2.
Table 10.2: Relationship between English descriptions and mathematical symbols
labelindex, imath. symbolthe value
clubs, $$\clubsuit$$1$$O_1$$35
diamonds, $$\diamondsuit$$2$$O_2$$51
hearts, $$\heartsuit$$3$$O_3$$64
spades, $$\spadesuit$$4$$O_4$$50
Hopefully that’s pretty clear. It’s also worth noting that mathematicians prefer to talk about general rather than specific things, so you’ll also see the notation $$O_i$$, which refers to the number of observations that fall within the i-th category (where i could be 1, 2, 3 or 4). Finally, if we want to refer to the set of all observed frequencies, statisticians group all observed values into a vector 2, which I’ll refer to as $$O$$.
$O = (O_1, O_2, O_3, O_4)$
Again, this is nothing new or interesting. It’s just notation. If I say that $$O = (35, 51, 64, 50)$$ all I’m doing is describing the table of observed frequencies (i.e., observed), but I’m referring to it using mathematical notation.
### 10.1.2 The null hypothesis and the alternative hypothesis
As the last section indicated, our research hypothesis is that “people don’t choose cards randomly”. What we’re going to want to do now is translate this into some statistical hypotheses and then construct a statistical test of those hypotheses. The test that I’m going to describe to you is Pearson’s $$\chi^2$$ (chi-square) goodness-of-fit test, and as is so often the case we have to begin by carefully constructing our null hypothesis. In this case, it’s pretty easy. First, let’s state the null hypothesis in words:
$H_0: \text{ All four suits are chosen with equal probability}$
Now, because this is statistics, we have to be able to say the same thing in a mathematical way. To do this, let’s use the notation $$P_j$$ to refer to the true probability that the j-th suit is chosen. If the null hypothesis is true, then each of the four suits has a 25% chance of being selected. In other words, our null hypothesis claims that $$P_1 = .25$$, $$P_2 = .25$$, $$P3 = .25$$ and finally that $$P_4 = .25$$ . However, in the same way that we can group our observed frequencies into a vector O that summarises the entire data set, we can use P to refer to the probabilities that correspond to our null hypothesis. So if I let the vector $$P = (P_1, P_2, P_3, P_4)$$ refer to the collection of probabilities that describe our null hypothesis, then we have:
$H_0: P =(.25, .25, .25, .25)$
In this particular instance, our null hypothesis corresponds to a vector of probabilities P in which all of the probabilities are equal to one another. But this doesn’t have to be the case. For instance, if the experimental task was for people to imagine they were drawing from a deck that had twice as many clubs as any other suit, then the null hypothesis would correspond to something like $$P = (.4, .2, .2, .2)$$. As long as the probabilities are all positive numbers, and they all sum to 1, then it’s a perfectly legitimate choice for the null hypothesis. However, the most common use of the goodness-of-fit test is to test a null hypothesis that all of the categories are equally likely, so we’ll stick to that for our example.
What about our alternative hypothesis, $$H_1$$? All we’re really interested in is demonstrating that the probabilities involved aren’t all identical (that is, people’s choices weren’t completely random). As a consequence, the “human friendly” versions of our hypotheses look like this:
$$H_0: \text{ All four suits are chosen with equal probability}$$ $$H_1: \text{ At least one of the suit-choice probabilities isn’t 0.25}$$
…and the “mathematician friendly” version is:
$$H_0: P= (.25, .25, .25, .25)$$ $$H_1: P \neq (.25, .25, .25, .25)$$
### 10.1.3 The “goodness-of-fit” test statistic
At this point, we have our observed frequencies O and a collection of probabilities P corresponding to the null hypothesis that we want to test. What we now want to do is construct a test of the null hypothesis. As always, if we want to test $$H_0$$ against $$H_1$$, we’re going to need a test statistic. The basic trick that a goodness-of-fit test uses is to construct a test statistic that measures how “close” the data are to the null hypothesis. If the data don’t resemble what you’d “expect” to see if the null hypothesis were true, then it probably isn’t true. Okay, if the null hypothesis were true, what would we expect to see? Or, to use the correct terminology, what are the expected frequencies. There are $$N = 200$$ observations, and (if the null is true) the probability of any one of them choosing a heart is $$P_3 = .25$$, so I guess we’re expecting $$200 \times .25 = 50$$ hearts, right? Or, more specifically, if we let Ei refer to “the number of category i responses that we’re expecting if the null is true”, then
$E_i=N \times P_i$
This is pretty easy to calculate.If there are 200 observations that can fall into four categories, and we think that all four categories are equally likely, then on average we’d expect to see 50 observations in each category, right?
Now, how do we translate this into a test statistic? Clearly, what we want to do is compare the expected number of observations in each category ($$E_i$$) with the observed number of observations in that category ($$O_i$$). And on the basis of this comparison we ought to be able to come up with a good test statistic. To start with, let’s calculate the difference between what the null hypothesis expected us to find and what we actually did find. That is, we calculate the “observed minus expected” difference score, $$O_i - E_i$$ . This is illustrated in Table 10.3.
Table 10.3: Expected and observed frequencies
$$\clubsuit$$$$\diamondsuit$$$$\heartsuit$$$$\spadesuit$$
expected frequency $$E_i$$50505050
observed frequency $$O_i$$35516450
difference score $$O_i-E_i$$-151140
So, based on our calculations, it’s clear that people chose more hearts and fewer clubs than the null hypothesis predicted. However, a moment’s thought suggests that these raw differences aren’t quite what we’re looking for. Intuitively, it feels like it’s just as bad when the null hypothesis predicts too few observations (which is what happened with hearts) as it is when it predicts too many (which is what happened with clubs). So it’s a bit weird that we have a negative number for clubs and a positive number for hearts. One easy way to fix this is to square everything, so that we now calculate the squared differences, $$(E_i - O_i)^2$$ . As before, we can do this by hand (Table 10.4).
Table 10.4: Squaring the difference scores
$$\clubsuit$$$$\diamondsuit$$$$\heartsuit$$$$\spadesuit$$
22511960
Now we’re making progress. What we’ve got now is a collection of numbers that are big whenever the null hypothesis makes a bad prediction (clubs and hearts), but are small whenever it makes a good one (diamonds and spades). Next, for some technical reasons that I’ll explain in a moment, let’s also divide all these numbers by the expected frequency Ei , so we’re actually calculating $$\frac{(E_i-O_i)^2}{E_i}$$ . Since $$E_i = 50$$ for all categories in our example, it’s not a very interesting calculation, but let’s do it anyway (Table 10.5).
Table 10.5: Dividing the squared difference scores by the expected frequency to provide an ‘error’ score
$$\clubsuit$$$$\diamondsuit$$$$\heartsuit$$$$\spadesuit$$
4.500.023.920.00
In effect, what we’ve got here are four different “error” scores, each one telling us how big a “mistake” the null hypothesis made when we tried to use it to predict our observed frequencies. So, in order to convert this into a useful test statistic, one thing we could do is just add these numbers up. The result is called the goodness-of-fit statistic, conventionally referred to either as $$\chi^2$$ (chi-square) or GOF. We can calculate it as in Table 10.6.
$\sum( (observed - expected)^2 / expected )$
This gives us a value of 8.44.
[Additional technical detail 3]
As we’ve seen from our calculations, in our cards data set we’ve got a value of $$\chi^2$$ = 8.44. So now the question becomes is this a big enough value to reject the null?
### 10.1.4 The sampling distribution of the GOF statistic
To determine whether or not a particular value of $$\chi^2$$ is large enough to justify rejecting the null hypothesis, we’re going to need to figure out what the sampling distribution for $$\chi^2$$ would be if the null hypothesis were true. So that’s what I’m going to do in this section. I’ll show you in a fair amount of detail how this sampling distribution is constructed, and then, in the next section, use it to build up a hypothesis test. If you want to cut to the chase and are willing to take it on faith that the sampling distribution is a $$\chi^2$$ (chi-square) distribution with $$k - 1$$ degrees of freedom, you can skip the rest of this section. However, if you want to understand why the goodness-of-fit test works the way it does, read on.
Okay, let’s suppose that the null hypothesis is actually true. If so, then the true probability that an observation falls in the i-th category is $$P_i$$ . After all, that’s pretty much the definition of our null hypothesis. Let’s think about what this actually means. This is kind of like saying that “nature” makes the decision about whether or not the observation ends up in category i by flipping a weighted coin (i.e., one where the probability of getting a head is $$P_j$$ ). And therefore we can think of our observed frequency $$O_i$$ by imagining that nature flipped N of these coins (one for each observation in the data set), and exactly $$O_i$$ of them came up heads. Obviously, this is a pretty weird way to think about the experiment. But what it does (I hope) is remind you that we’ve actually seen this scenario before. It’s exactly the same set up that gave rise to Section 7.4 in Chapter 7. In other words, if the null hypothesis is true, then it follows that our observed frequencies were generated by sampling from a binomial distribution:
$O_i \sim Binomial(P_i,N)$
Now, if you remember from our discussion of Section 8.3.3 the binomial distribution starts to look pretty much identical to the normal distribution, especially when $$N$$ is large and when $$P_i$$ isn’t too close to 0 or 1. In other words as long as $$N^P_i$$ is large enough. Or, to put it another way, when the expected frequency Ei is large enough then the theoretical distribution of $$O_i$$ is approximately normal. Better yet, if $$O_i$$ is normally distributed, then so is $$(O_i-E_i)/\sqrt{(E_i)}$$ . Since $$E_i$$ is a fixed value, subtracting off Ei and dividing by ? Ei changes the mean and standard deviation of the normal distribution but that’s all it does. Okay, so now let’s have a look at what our goodness-of-fit statistic actually is. What we’re doing is taking a bunch of things that are normally-distributed, squaring them, and adding them up. Wait. We’ve seen that before too! As we discussed in the section on Section 7.6, when you take a bunch of things that have a standard normal distribution (i.e., mean 0 and standard deviation 1), square them and then add them up, the resulting quantity has a chi-square distribution. So now we know that the null hypothesis predicts that the sampling distribution of the goodness-of-fit statistic is a chi-square distribution. Cool.
There’s one last detail to talk about, namely the degrees of freedom. If you remember back to Section 7.6, I said that if the number of things you’re adding up is k, then the degrees of freedom for the resulting chi-square distribution is k. Yet, what I said at the start of this section is that the actual degrees of freedom for the chi-square goodness-of-fit test is $$k - 1$$. What’s up with that? The answer here is that what we’re supposed to be looking at is the number of genuinely independent things that are getting added together. And, as I’ll go on to talk about in the next section, even though there are k things that we’re adding only $$k - 1$$ of them are truly independent, and so the degrees of freedom is actually only $$k - 1$$. That’s the topic of the next section4.
### 10.1.5 Degrees of freedom
When I introduced the chi-square distribution in Section 7.6, I was a bit vague about what “degrees of freedom” actually means. Obviously, it matters. Looking at Figure 10.1, you can see that if we change the degrees of freedom then the chi-square distribution changes shape quite substantially. But what exactly is it? Again, when I introduced the distribution and explained its relationship to the normal distribution, I did offer an answer: it’s the number of “normally distributed variables” that I’m squaring and adding together. But, for most people, that’s kind of abstract and not entirely helpful. What we really need to do is try to understand degrees of freedom in terms of our data. So here goes.
The basic idea behind degrees of freedom is quite simple. You calculate it by counting up the number of distinct “quantities” that are used to describe your data and then subtracting off all of the “constraints” that those data must satisfy.5 This is a bit vague, so let’s use our cards data as a concrete example. We describe our data using four numbers, $$O1, O2, O3$$ and O4 corresponding to the observed frequencies of the four different categories (hearts, clubs, diamonds, spades). These four numbers are the random outcomes of our experiment. But my experiment actually has a fixed constraint built into it: the sample size $$N$$. 6 That is, if we know
how many people chose hearts, how many chose diamonds and how many chose clubs, then we’d be able to figure out exactly how many chose spades. In other words, although our data are described using four numbers, they only actually correspond to $$4 - 1 = 3$$ degrees of freedom. A slightly different way of thinking about it is to notice that there are four probabilities that we’re interested in (again, corresponding to the four different categories), but these probabilities must sum to one, which imposes a constraint. Therefore the degrees of freedom is $$4 - 1 = 3$$. Regardless of whether you want to think about it in terms of the observed frequencies or in terms of the probabilities, the answer is the same. In general, when running the $$\chi^2$$(chi-square) goodness-of-fit test for an experiment involving $$k$$ groups, then the degrees of freedom will be $$k - 1$$.
### 10.1.6 Testing the null hypothesis
The final step in the process of constructing our hypothesis test is to figure out what the rejection region is. That is, what values of $$\chi^2$$ would lead us to reject the null hypothesis. As we saw earlier, large values of $$\chi^2$$ imply that the null hypothesis has done a poor job of predicting the data from our experiment, whereas small values of $$\chi^2$$ imply that it’s actually done pretty well. Therefore, a pretty sensible strategy would be to say there is some critical value such that if $$\chi^2$$ is bigger than the critical value we reject the null, but if $$\chi^2$$ is smaller than this value we retain the null. In other words, to use the language we introduced in Chapter 9 the chi-square goodness-of-fit test is always a one-sided test. Right, so all we have to do is figure out what this critical value is. And it’s pretty straightforward. If we want our test to have significance level of $$\alpha = .05$$ (that is, we are willing to tolerate a Type I error rate of $$5%$$), then we have to choose our critical value so that there is only a 5% chance that $$\chi^2$$ could get to be that big if the null hypothesis is true. This is illustrated in Figure 10.2.
Ah but, I hear you ask, how do I find the critical value of a chi-square distribution with $$k-1$$ degrees of freedom? Many many years ago when I first took a psychology statistics class we used to look up these critical values in a book of critical value tables, like the one in Figure 10.3. Looking at this Figure, we can see that the critical value for a $$\chi^2$$ distribution with 3 degrees of freedom, and p=0.05 is 7.815.
So, if our calculated $$\chi^2$$ statistic is bigger than the critical value of $$7.815$$, then we can reject the null hypothesis (remember that the null hypothesis, $$H_0$$, is that all four suits are chosen with equal probability). Since we actually already calculated that before (i.e., $$\chi^2$$ = 8.44) we can reject the null hypothesis. And that’s it, basically. You now know “Pearson’s $$\chi^2$$ test for the goodness-of-fit”. Lucky you.
### 10.1.7 Doing the test in jamovi
Not surprisingly, jamovi provides an analysis that will do these calculations for you. Let’s use the Randomness.omv file. From the main ‘Analyses’ toolbar select ‘Frequencies’ - ‘One Sample Proportion Tests’ - ‘$$N$$ Outcomes’. Then in the analysis window that appears move the variable you want to analyse (choice 1 across into the ‘Variable’ box. Also, click on the ‘Expected counts’ check box so that these are shown on the results table. When you have done all this, you should see the analysis results in jamovi as in Figure 10.4. No surprise then that jamovi provides the same expected counts and statistics that we calculated by hand above, with a $$\chi^2$$ value of $$(8.44$$ with $$3$$ d.f. and $$p=0.038$$. Note that we don’t need to look up a critical p-value threshold value any more, as jamovi gives us the actual p-value of the calculated $$\chi^2$$ for $$3$$ d.f.
### 10.1.8 Specifying a different null hypothesis
At this point you might be wondering what to do if you want to run a goodness-of-fit test but your null hypothesis is not that all categories are equally likely. For instance, let’s suppose that someone had made the theoretical prediction that people should choose red cards $$60\%$$ of the time, and black cards $$40\%$$ of the time (I’ve no idea why you’d predict that), but had no other preferences. If that were the case, the null hypothesis would be to expect $$30\%$$ of the choices to be hearts, $$30\%$$ to be diamonds, $$20\%$$ to be spades and $$20\%$$ to be clubs. In other words we would expect hearts and diamonds to appear 1.5 times more often than spades and clubs (the ratio $$30\%$$ : $$20\%$$ is the same as 1.5 : 1). This seems like a silly theory to me, and it’s pretty easy to test this explicitly specified null hypothesis with the data in our jamovi analysis. In the analysis window (labelled ‘Proportion Test (N Outcomes)’ in Figure 10.4 you can expand the options for ‘Expected Proportions’. When you do this, there are options for entering different ratio values for the variable you have selected, in our case this is choice 1. Change the ratio to reflect the new null hypothesis, as in Figure 10.5, and see how the results change.
The expected counts are now shown in Table 10.6.
Table 10.6: Expected counts for a different null hypothesis
$$\clubsuit$$$$\diamondsuit$$$$\heartsuit$$$$\spadesuit$$
expected frequency $$E_i$$40606040
and the $$\chi^2$$ statistic is 4.74, 3 d.f., $$p = 0.182$$. Now, the results of our updated hypotheses and the expected frequencies are different from what they were last time. As a consequence our $$\chi^2$$ test statistic is different, and our p-value is different too. Annoyingly, the p-value is $$.182$$, so we can’t reject the null hypothesis (look back at Section 9.5 to remind yourself why). Sadly, despite the fact that the null hypothesis corresponds to a very silly theory, these data don’t provide enough evidence against it.
### 10.1.9 How to report the results of the test
So now you know how the test works, and you know how to do the test using a wonderful jamovi flavoured magic computing box. The next thing you need to know is how to write up the results. After all, there’s no point in designing and running an experiment and then analysing the data if you don’t tell anyone about it! So let’s now talk about what you need to do when reporting your analysis. Let’s stick with our card-suits example. If I wanted to write this result up for a paper or something, then the conventional way to report this would be to write something like this:
Of the 200 participants in the experiment, 64 selected hearts for their first choice, 51 selected diamonds, 50 selected spades, and 35 selected clubs. A chi-square goodness-of-fit test was conducted to test whether the choice probabilities were identical for all four suits. The results were significant ($$\chi^2(3) = 8.44, p< .05)$$, suggesting that people did not select suits purely at random.
This is pretty straightforward and hopefully it seems pretty unremarkable. That said, there’s a few things that you should note about this description:
• The statistical test is preceded by the descriptive statistics. That is, I told the reader something about what the data look like before going on to do the test. In general, this is good practice. Always remember that your reader doesn’t know your data anywhere near as well as you do. So, unless you describe it to them properly, the statistical tests won’t make any sense to them and they’ll get frustrated and cry.
• The description tells you what the null hypothesis being tested is. To be honest, writers don’t always do this but it’s often a good idea in those situations where some ambiguity exists, or when you can’t rely on your readership being intimately familiar with the statistical tools that you’re using. Quite often the reader might not know (or remember) all the details of the test that your using, so it’s a kind of politeness to “remind” them! As far as the goodness-of-fit test goes, you can usually rely on a scientific audience knowing how it works (since it’s covered in most intro stats classes). However, it’s still a good idea to be explicit about stating the null hypothesis (briefly!) because the null hypothesis can be different depending on what you’re using the test for. For instance, in the cards example my null hypothesis was that all the four suit probabilities were identical (i.e., $$P1 = P2 = P3 = P4 = 0.25$$), but there’s nothing special about that hypothesis. I could just as easily have tested the null hypothesis that $$P_1 = 0.7$$ and $$P2 = P3 = P4 = 0.1$$ using a goodness-of-fit test. So it’s helpful to the reader if you explain to them what your null hypothesis was. Also, notice that I described the null hypothesis in words, not in maths. That’s perfectly acceptable. You can describe it in maths if you like, but since most readers find words easier to read than symbols, most writers tend to describe the null using words if they can.
• A “stat block” is included. When reporting the results of the test itself, I didn’t just say that the result was significant, I included a “stat block” (i.e., the dense mathematical looking part in the parentheses) which reports all the “key” statistical information. For the chi-square goodness-of-fit test, the information that gets reported is the test statistic (that the goodness-of-fit statistic was 8.44), the information about the distribution used in the test ($$\chi^2$$ with 3 degrees of freedom which is usually shortened to $$\chi^2$$(3)), and then the information about whether the result was significant (in this case $$p< .05$$). The particular information that needs to go into the stat block is different for every test, and so each time I introduce a new test I’ll show you what the stat block should look like.7 However the general principle is that you should always provide enough information so that the reader could check the test results themselves if they really wanted to.
• The results are interpreted. In addition to indicating that the result was significant, I provided an interpretation of the result (i.e., that people didn’t choose randomly). This is also a kindness to the reader, because it tells them something about what they should believe about what’s going on in your data. If you don’t include something like this, it’s really hard for your reader to understand what’s going on.8
As with everything else, your overriding concern should be that you explain things to your reader. Always remember that the point of reporting your results is to communicate to another human being. I cannot tell you just how many times I’ve seen the results section of a report or a thesis or even a scientific article that is just gibberish, because the writer has focused solely on making sure they’ve included all the numbers and forgotten to actually communicate with the human reader.
Satan delights equally in statistics and in quoting scripture9 – H.G. Wells
## 10.2 The $$\chi^2$$ test of independence (or association)
GUARDBOT 1: Halt!
GUARDBOT 2: Be you robot or human?
LEELA: Robot…we be.
FRY: Uh, yup! Just two robots out roboting it up! Eh?
GUARDBOT 1: Administer the test.
GUARDBOT 2: Which of the following would you most prefer? A: A puppy, B: A pretty flower from your sweetie, or C: A large properly-formatted data file?
GUARDBOT 1: Choose!
Futurama, “Fear of a Bot Planet”
The other day I was watching an animated documentary examining the quaint customs of the natives of the planet Chapek 9. Apparently, in order to gain access to their capital city a visitor must prove that they’re a robot, not a human. In order to determine whether or not a visitor is human, the natives ask whether the visitor prefers puppies, flowers, or large, properly formatted data files. “Pretty clever,” I thought to myself “but what if humans and robots have the same preferences? That probably wouldn’t be a very good test then, would it?” As it happens, I got my hands on the testing data that the civil authorities of Chapek 9 used to check this. It turns out that what they did was very simple. They found a bunch of robots and a bunch of humans and asked them what they preferred. I saved their data in a file called chapek9.omv, which we can now load into jamovi. As well as the ID variable that identifies individual people, there are two nominal text variables, species and choice. In total there are 180 entries in the data set, one for each person (counting both robots and humans as “people”) who was asked to make a choice. Specifically, there are 93 humans and 87 robots, and overwhelmingly the preferred choice is the data file. You can check this yourself by asking jamovi for Frequency Tables, under the ‘Exploration’ - ‘Descriptives’ button. However, this summary does not address the question we’re interested in. To do that, we need a more detailed description of the data. What we want to do is look at the choices broken down by species. That is, we need to cross-tabulate the data (see Section 6.1). In jamovi we do this using the ‘Frequencies’ - ‘Contingency Tables’ - ‘Independent Samples’ analysis, and we should get a table something like Table 10.7.
Table 10.7: Cross-tabulating the data
RobotHumanTotal
Puppy131528
Flower301343
Data4465109
Total8793180
From this, it’s quite clear that the vast majority of the humans chose the data file, whereas the robots tended to be a lot more even in their preferences. Leaving aside the question of why the humans might be more likely to choose the data file for the moment (which does seem quite odd, admittedly), our first order of business is to determine if the discrepancy between human choices and robot choices in the data set is statistically significant.
### 10.2.1 Constructing our hypothesis test
How do we analyse this data? Specifically, since my research hypothesis is that “humans and robots answer the question in different ways”, how can I construct a test of the null hypothesis that “humans and robots answer the question the same way”? As before, we begin by establishing some notation to describe the data (Table 10.8).
Table 10.8: Notation to describe the data
RobotHumanTotal
Puppy$$O_{11}$$$$O_{12}$$$$R_{1}$$
Flower$$O_{21}$$$$O_{22}$$$$R_{2}$$
Data$$O_{31}$$$$O_{32}$$$$R_{3}$$
Total$$C_{1}$$$$C_{2}$$N
In this notation we say that $$O_{ij}$$ is a count (observed frequency) of the number of respondents that are of species j (robots or human) who gave answer i (puppy, flower or data) when asked to make a choice. The total number of observations is written $$N$$, as usual. Finally, I’ve used $$R_i$$ to denote the row totals (e.g., $$R_1$$ is the total number of people who chose the flower), and $$C_j$$ to denote the column totals (e.g., $$C_1$$ is the total number of robots).10
So now let’s think about what the null hypothesis says. If robots and humans are responding in the same way to the question, it means that the probability that “a robot says puppy” is the same as the probability that “a human says puppy”, and so on for the other two possibilities. So, if we use $$P_{ij}$$ to denote “the probability that a member of species j gives response i” then our null hypothesis is that:
\begin{aligned} H_0 &: \text{All of the following are true:} \\ &P_{11} = P_{12}\text{ (same probability of saying “puppy”),} \\ &P_{21} = P_{22}\text{ (same probability of saying “flower”), and} \\ &P_{31} = P_{32}\text{ (same probability of saying “data”).} \end{aligned}
And actually, since the null hypothesis is claiming that the true choice probabilities don’t depend on the species of the person making the choice, we can let Pi refer to this probability, e.g., P1 is the true probability of choosing the puppy.
Next, in much the same way that we did with the goodness-of-fit test, what we need to do is calculate the expected frequencies. That is, for each of the observed counts $$O_{ij}$$ , we need to figure out what the null hypothesis would tell us to expect. Let’s denote this expected frequency by $$E_{ij}$$. This time, it’s a little bit trickier. If there are a total of $$C_j$$ people that belong to species $$j$$, and the true probability of anyone (regardless of species) choosing option $$i$$ is $$P_i$$ , then the expected frequency is just:
$E_{ij}=C_j \times P_i$
Now, this is all very well and good, but we have a problem. Unlike the situation we had with the goodness-of-fit test, the null hypothesis doesn’t actually specify a particular value for Pi .
It’s something we have to estimate (see Chapter 8) from the data! Fortunately, this is pretty easy to do. If 28 out of 180 people selected the flowers, then a natural estimate for the probability of choosing flowers is $$\frac{28}{180}$$, which is approximately $$.16$$. If we phrase this in mathematical terms, what we’re saying is that our estimate for the probability of choosing option i is just the row total divided by the total sample size:
$\hat{P}_{i}= \frac{R_i}{N}$
Therefore, our expected frequency can be written as the product (i.e. multiplication) of the row total and the column total, divided by the total number of observations:11
$\hat{E}_{ij}= \frac{R_i \times C_j}{N}$
[Additional technical detail 12]
As before, large values of $$X^2$$ indicate that the null hypothesis provides a poor description of the data, whereas small values of $$X^2$$ suggest that it does a good job of accounting for the data. Therefore, just like last time, we want to reject the null hypothesis if $$X^2$$ is too large.
Not surprisingly, this statistic is $$\chi^2$$ distributed. All we need to do is figure out how many degrees of freedom are involved, which actually isn’t too hard. As I mentioned before, you can (usually) think of the degrees of freedom as being equal to the number of data points that you’re analysing, minus the number of constraints. A contingency table with r rows and c columns contains a total of $$r^{c}$$ observed frequencies, so that’s the total number of observations. What about the constraints? Here, it’s slightly trickier. The answer is always the same
$df=(r-1)(c-1)$
but the explanation for why the degrees of freedom takes this value is different depending on the experimental design. For the sake of argument, let’s suppose that we had honestly intended to survey exactly 87 robots and 93 humans (column totals fixed by the experimenter), but left the row totals free to vary (row totals are random variables). Let’s think about the constraints that apply here. Well, since we deliberately fixed the column totals by Act of Experimenter, we have $$c$$ constraints right there. But, there’s actually more to it than that. Remember how our null hypothesis had some free parameters (i.e., we had to estimate the Pi values)? Those matter too. I won’t explain why in this book, but every free parameter in the null hypothesis is rather like an additional constraint. So, how many of those are there? Well, since these probabilities have to sum to 1, there’s only $$r - 1$$ of these. So our total degrees of freedom is:
$\begin{split} df & = \text{(number of observations) - (number of constraints)} \\\\ & = (r \times c) - (c + (r - 1)) \\\\ & = rc - c - r + 1 \\\\ & = (r - 1)(c - 1) \end{split}$
Alternatively, suppose that the only thing that the experimenter fixed was the total sample size N. That is, we quizzed the first 180 people that we saw and it just turned out that 87 were robots and 93 were humans. This time around our reasoning would be slightly different, but would still lead us to the same answer. Our null hypothesis still has $$r - 1$$ free parameters corresponding to the choice probabilities, but it now also has $$c - 1$$ free parameters corresponding to the species probabilities, because we’d also have to estimate the probability that a randomly sampled person turns out to be a robot.13 Finally, since we did actually fix the total number of observations N, that’s one more constraint. So, now we have rc observations, and $$(c-1)+(r-1)+1$$ constraints. What does that give?
$\begin{split} df & = \text{(number of observations) - (number of constraints)} \\\\ & = (r \times c) - ((c-1) + (r - 1)+1) \\\\ & = (r - 1)(c - 1) \end{split}$ Amazing.
### 10.2.2 Doing the test in jamovi
Okay, now that we know how the test works let’s have a look at how it’s done in jamovi. As tempting as it is to lead you through the tedious calculations so that you’re forced to learn it the long way, I figure there’s no point. I already showed you how to do it the long way for the goodness-of-fit test in the last section, and since the test of independence isn’t conceptually any different, you won’t learn anything new by doing it the long way. So instead I’ll go straight to showing you the easy way. After you have run the test in jamovi (‘Frequencies’ - ‘Contingency Tables’ - ‘Independent Samples’), all you have to do is look underneath the contingency table in the jamovi results window and there is the $$\chi^2$$ statistic for you. This shows a $$\chi^2$$ statistic value of 10.72, with 2 d.f. and p-value = 0.005.
That was easy, wasn’t it! You can also ask jamovi to show you the expected counts - just click on the check box for ‘Counts’ - ‘Expected’ in the ‘Cells’ options and the expected counts will appear in the contingency table. And whilst you are doing that, an effect size measure would be helpful. We’ll choose Cramér’s $$V$$, and you can specify this from a check box in the ‘Statistics’ options, and it gives a value for Cramér’s $$V$$ of $$0.24$$. See Figure 10.6. We will talk about this some more in just a moment.
This output gives us enough information to write up the result:
Pearson’s $$\chi^2$$ revealed a significant association between species and choice ($$\chi^2(2) = 10.7, p< .01)$$. Robots appeared to be more likely to say that they prefer flowers, but the humans were more likely to say they prefer data.
Notice that, once again, I provided a little bit of interpretation to help the human reader understand what’s going on with the data. Later on in my discussion section I’d provide a bit more context. To illustrate the difference, here’s what I’d probably say later on:
The fact that humans appeared to have a stronger preference for raw data files than robots is somewhat counter-intuitive. However, in context it makes some sense, as the civil authority on Chapek 9 has an unfortunate tendency to kill and dissect humans when they are identified. As such it seems most likely that the human participants did not respond honestly to the question, so as to avoid potentially undesirable consequences. This should be considered to be a substantial methodological weakness.
This could be classified as a rather extreme example of a reactivity effect, I suppose. Obviously, in this case the problem is severe enough that the study is more or less worthless as a tool for understanding the difference preferences among humans and robots. However, I hope this illustrates the difference between getting a statistically significant result (our null hypothesis is rejected in favour of the alternative), and finding something of scientific value (the data tell us nothing of interest about our research hypothesis due to a big methodological flaw).
## 10.3 The continuity correction
Okay, time for a little bit of a digression. I’ve been lying to you a little bit so far. There’s a tiny change that you need to make to your calculations whenever you only have 1 degree of freedom. It’s called the “continuity correction”, or sometimes the Yates correction. Remember what I pointed out earlier: the $$\chi^2$$ test is based on an approximation, specifically on the assumption that the binomial distribution starts to look like a normal distribution for large $$N$$. One problem with this is that it often doesn’t quite work, especially when you’ve only got 1 degree of freedom (e.g., when you’re doing a test of independence on a $$2 \times 2$$ contingency table). The main reason for this is that the true sampling distribution for the $$X^{2}$$ statistic is actually discrete (because you’re dealing with categorical data!) but the $$\chi^2$$ distribution is continuous. This can introduce systematic problems. Specifically, when N is small and when $$df = 1$$, the goodness-of-fit statistic tends to be “too big”, meaning that you actually have a bigger α value than you think (or, equivalently, the p values are a bit too small).
As far as I can tell from reading Yates’ paper14, the correction is basically a hack. It’s not derived from any principled theory. Rather, it’s based on an examination of the behaviour of the test, and observing that the corrected version seems to work better. You can specify this correction in jamovi from a check box in the ‘Statistics’ options, where it is called ‘$$\chi^2$$ continuity correction’.
## 10.4 Effect size
As we discussed earlier in Section 9.8, it’s becoming commonplace to ask researchers to report some measure of effect size. So, let’s suppose that you’ve run your chi-square test, which turns out to be significant. So you now know that there is some association between your variables (independence test) or some deviation from the specified probabilities (goodness-of-fit test). Now you want to report a measure of effect size. That is, given that there is an association or deviation, how strong is it?
There are several different measures that you can choose to report, and several different tools that you can use to calculate them. I won’t discuss all of them but will instead focus on the most commonly reported measures of effect size.
By default, the two measures that people tend to report most frequently are the $$\phi$$ statistic and the somewhat superior version, known as Cramér’s $$V$$ .
[Additional technical detail 15]
And you’re done. This seems to be a fairly popular measure, presumably because it’s easy to calculate, and it gives answers that aren’t completely silly. With Cramér’s $$V$$, you know that the value really does range from 0 (no association at all) to 1 (perfect association).
## 10.5 Assumptions of the test(s)
All statistical tests make assumptions, and it’s usually a good idea to check that those assumptions are met. For the chi-square tests discussed so far in this chapter, the assumptions are:
• Expected frequencies are sufficiently large. Remember how in the previous section we saw that the $$\chi^2$$ sampling distribution emerges because the binomial distribution is pretty similar to a normal distribution? Well, like we discussed in Chapter 7 this is only true when the number of observations is sufficiently large. What that means in practice is that all of the expected frequencies need to be reasonably big. How big is reasonably big? Opinions differ, but the default assumption seems to be that you generally would like to see all your expected frequencies larger than about 5, though for larger tables you would probably be okay if at least 80% of the the expected frequencies are above 5 and none of them are below 1. However, from what I’ve been able to discover (e.g., Cochran (1954)) these seem to have been proposed as rough guidelines, not hard and fast rules, and they seem to be somewhat conservative .
• Data are independent of one another. One somewhat hidden assumption of the chi-square test is that you have to genuinely believe that the observations are independent. Here’s what I mean. Suppose I’m interested in proportion of babies born at a particular hospital that are boys. I walk around the maternity wards and observe 20 girls and only 10 boys. Seems like a pretty convincing difference, right? But later on, it turns out that I’d actually walked into the same ward 10 times and in fact I’d only seen 2 girls and 1 boy. Not as convincing, is it? My original 30 observations were massively non-independent, and were only in fact equivalent to 3 independent observations. Obviously this is an extreme (and extremely silly) example, but it illustrates the basic issue. Non-independence “stuffs things up”. Sometimes it causes you to falsely reject the null, as the silly hospital example illustrates, but it can go the other way too. To give a slightly less stupid example, let’s consider what would happen if I’d done the cards experiment slightly differently Instead of asking 200 people to try to imagine sampling one card at random, suppose I asked 50 people to select 4 cards. One possibility would be that everyone selects one heart, one club, one diamond and one spade (in keeping with the “representativeness heuristic” . This is highly non-random behaviour from people, but in this case I would get an observed frequency of 50 for all four suits. For this example the fact that the observations are non-independent (because the four cards that you pick will be related to each other) actually leads to the opposite effect, falsely retaining the null.
If you happen to find yourself in a situation where independence is violated, it may be possible to use the McNemar test (which we’ll discuss) or the Cochran test (which we won’t). Similarly, if your expected cell counts are too small, check out the Fisher exact test. It is to these topics that we now turn.
## 10.6 The Fisher exact test
What should you do if your cell counts are too small, but you’d still like to test the null hypothesis that the two variables are independent? One answer would be “collect more data”, but that’s far too glib There are a lot of situations in which it would be either infeasible or unethical do that. If so, statisticians have a kind of moral obligation to provide scientists with better tests. In this instance, Fisher (1922) kindly provided the right answer to the question. To illustrate the basic idea let’s suppose that we’re analysing data from a field experiment looking at the emotional status of people who have been accused of Witchcraft, some of whom are currently being burned at the stake.16 Unfortunately for the scientist (but rather fortunately for the general populace), it’s actually quite hard to find people in the process of being set on fire, so the cell counts are awfully small in some cases. A contingency table of the salem.csv data illustrates the point (Table 10.9).
Table 10.9: Contingency table of the salem.csv data
happyFALSETRUE
on.fireFALSE310
TRUE30
Looking at this data, you’d be hard pressed not to suspect that people not on fire are more likely to be happy than people on fire. However, the chi-square test makes this very hard to test because of the small sample size. So, speaking as someone who doesn’t want to be set on fire, I’d really like to be able to get a better answer than this. This is where Fisher’s exact test comes in very handy.
The Fisher exact test works somewhat differently to the chi-square test (or in fact any of the other hypothesis tests that I talk about in this book) insofar as it doesn’t have a test statistic, but it calculates the p-value “directly”. I’ll explain the basics of how the test works for a $$2 \times 2$$ contingency table. As before, let’s have some notation (Table 10.10).
Table 10.10: Notation for the Fisher exact test
Set on fire$$O_{11}$$$$O_{12}$$$$R_{1}$$
Not set on fire$$O_{21}$$$$O_{22}$$$$R_{2}$$
Total$$C_{1}$$$$C_{2}$$$$N$$
In order to construct the test Fisher treats both the row and column totals $$(R_1, R_2, C_1 \text{ and } C_2)$$ as known, fixed quantities and then calculates the probability that we would have obtained the observed frequencies that we did $$(O_{11}, O_{12}, O_{21} \text{ and } O_{22})$$ given those totals. In the notation that we developed in Chapter 7 this is written:
$P(O_{11}, O_{12}, O_{21}, O_{22} \text{ | } R_1, R_2, C_1, C_2)$ and as you might imagine, it’s a slightly tricky exercise to figure out what this probability is. But it turns out that this probability is described by a distribution known as the hypergeometric distribution. What we have to do to calculate our p-value is calculate the probability of observing this particular table or a table that is “more extreme”. 17 Back in the 1920s, computing this sum was daunting even in the simplest of situations, but these days it’s pretty easy as long as the tables aren’t too big and the sample size isn’t too large. The conceptually tricky issue is to figure out what it means to say that one contingency table is more “extreme” than another. The easiest solution is to say that the table with the lowest probability is the most extreme. This then gives us the p-value.
You can specify this test in jamovi from a check box in the ‘Statistics’ options of the ‘Contingency Tables’ analysis. When you do this with the data from the salem.csv file, the Fisher exact test statistic is shown in the results. The main thing we’re interested in here is the p-value, which in this case is small enough (p = .036) to justify rejecting the null hypothesis that people on fire are just as happy as people not on fire. See Figure 10.7.
## 10.7 The McNemar test
Suppose you’ve been hired to work for the Australian Generic Political Party (AGPP), and part of your job is to find out how effective the AGPP political advertisements are. So you decide to put together a sample of $$N = 100$$ people and ask them to watch the AGPP ads. Before they see anything, you ask them if they intend to vote for the AGPP, and then after showing the ads you ask them again to see if anyone has changed their minds. Obviously, if you’re any good at your job, you’d also do a whole lot of other things too, but let’s consider just this one simple experiment. One way to describe your data is via the contingency table shown in Table 10.11.
Table 10.11: Contingency table with data on AGPP political advertisements
BeforeAfterTotal
Yes301040
No7090160
Total100100200
At first pass, you might think that this situation lends itself to the Pearson $$\chi^2$$ test of independence (as per The $$\chi^2$$ test of independence (or association)). However, a little bit of thought reveals that we’ve got a problem. We have 100 participants but 200 observations. This is because each person has provided us with an answer in both the before column and the after column. What this means is that the 200 observations aren’t independent of each other. If voter A says “yes” the first time and voter B says “no”, then you’d expect that voter A is more likely to say “yes” the second time than voter B! The consequence of this is that the usual $$\chi^2$$ test won’t give trustworthy answers due to the violation of the independence assumption. Now, if this were a really uncommon situation, I wouldn’t be bothering to waste your time talking about it. But it’s not uncommon at all. This is a standard repeated measures design, and none of the tests we’ve considered so far can handle it. Eek.
The solution to the problem was published by McNemar (1947). The trick is to start by tabulating your data in a slightly different way (Table 10.12).
Table 10.12: Tabulate the data in a different way when you have repeated measures data
Before: YesBefore: NoTotal
After: Yes5510
After: No256590
Total3070100
Next, let’s think about what our null hypothesis is: it’s that the “before” test and the “after” test have the same proportion of people saying “Yes, I will vote for AGPP”. Because of the way that we have rewritten the data, it means that we’re now testing the hypothesis that the row totals and column totals come from the same distribution. Thus, the null hypothesis in McNemar’s test is that we have “marginal homogeneity”. That is, the row totals and column totals have the same distribution: $$P_a + P_b = P_a + P_c$$ and similarly that $$P_c + P_d = P_b + P_d$$. Notice that this means that the null hypothesis actually simplifies to Pb = Pc. In other words, as far as the McNemar test is concerned, it’s only the off-diagonal entries in this table (i.e., b and c) that matter! After noticing this, the McNemar test of marginal homogeneity is no different to a usual $$\chi^2$$ test. After applying the Yates correction, our test statistic becomes:
$\chi^2=\frac{(|b-c|-0.5)^2}{b+c}$ or, to revert to the notation that we used earlier in this chapter:
$\chi^2=\frac{(|O_{12}-O_{21}|-0.5)^2}{O_{12}+O_{21}}$ and this statistic has a $$\chi^2$$ distribution (approximately) with df = 1. However, remember that just like the other $$\chi^2$$ tests it’s only an approximation, so you need to have reasonably large expected cell counts for it to work.
### 10.7.1 Doing the McNemar test in jamovi
Now that you know what the McNemar test is all about, lets actually run one. The agpp.csv file contains the raw data that I discussed previously. The agpp data set contains three variables, an id variable that labels each participant in the data set (we’ll see why that’s useful in a moment), a response_before variable that records the person’s answer when they were asked the question the first time, and a response_after variable that shows the answer that they gave when asked the same question a second time. Notice that each participant appears only once in this data set. Go to the ‘Analyses’ - ‘Frequencies’ - ‘Contingency Tables’ - ‘Paired Samples’ analysis in jamovi, and move response_before into the ‘Rows’ box, and response_after into the ‘Columns’ box. You will then get a contingency table in the results window, with the statistic for the McNemar test just below it, see Figure 10.8.
And we’re done. We’ve just run a McNemar’s test to determine if people were just as likely to vote AGPP after the ads as they were before hand. The test was significant ($$\chi^2(1)= 12.03, p< .001)$$, suggesting that they were not. And, in fact it looks like the ads had a negative effect: people were less likely to vote AGPP after seeing the ads. Which makes a lot of sense when you consider the quality of a typical political advertisement.
## 10.8 What’s the difference between McNemar and independence?
Let’s go all the way back to the beginning of the chapter and look at the cards data set again. If you recall, the actual experimental design that I described involved people making two choices. Because we have information about the first choice and the second choice that everyone made, we can construct the following contingency table that cross-tabulates the first choice against the second choice (Table 10.13).
Table 10.13: Cross-tabulating first against second choice with the Randomness.omv (cards) data
Before: YesBefore: NoTotal
After: Yes$$a$$$$b$$$$a + b$$
After: No$$c$$$$d$$$$c + d$$
Total$$a+c$$$$b+d$$$$n$$
Suppose I wanted to know whether the choice you make the second time is dependent on the choice you made the first time. This is where a test of independence is useful, and what we’re trying to do is see if there’s some relationship between the rows and columns of this table.
Alternatively, suppose I wanted to know if on average, the frequencies of suit choices were different the second time than the first time. In that situation, what I’m really trying to see is if the row totals are different from the column totals. That’s when you use the McNemar test.
The different statistics produced by these different analyses are shown in Figure 10.9. Notice that the results are different! These aren’t the same test.
## 10.9 Summary
The key ideas discussed in this chapter are:
• The $$\chi^2$$ (chi-square) goodness-of-fit test is used when you have a table of observed frequencies of different categories, and the null hypothesis gives you a set of “known” probabilities to compare them to.
• The $$\chi^2$$ test of independence (or association) is used when you have a contingency table (cross-tabulation) of two categorical variables. The null hypothesis is that there is no relationship or association between the variables.
• Effect size for a contingency table can be measured in several ways. In particular we noted the Cramér’s $$V$$ statistic.
• Both versions of the Pearson test rely on two assumptions: that the expected frequencies are sufficiently large, and that the observations are independent (Assumptions of the test(s). The Fisher exact test can be used when the expected frequencies are small. The McNemar test can be used for some kinds of violations of independence.
If you’re interested in learning more about categorical data analysis a good first choice would be Agresti (1996) which, as the title suggests, provides an Introduction to Categorical Data Analysis. If the introductory book isn’t enough for you (or can’t solve the problem you’re working on) you could consider Agresti (2002), Categorical Data Analysis. The latter is a more advanced text, so it’s probably not wise to jump straight from this book to that one.
1. Also sometimes referred to as “chi-squared”.↩︎
2. A vector is a sequence of data elements of the same basic type.↩︎
3. If we let k refer to the total number of categories (i.e., k = 4 for our cards data), then the $$\chi^2$$ statistic is given by: $\chi^2 = \sum_{i=1}^{k} \frac{(O_i-E_i)^2}{E_i}$ Intuitively, it’s clear that if $$chi^2$$ is small, then the observed data Oi are very close to what the null hypothesis predicted $$E_i$$, so we’re going to need a large $$\chi^2$$ statistic in order to reject the null.↩︎
4. If you rewrite the equation for the goodness-of-fit statistic as a sum over k - 1 independent things you get the “proper” sampling distribution, which is chi-square with k - 1 degrees of freedom. It’s beyond the scope of an introductory book to show the maths in that much detail. All I wanted to do is give you a sense of why the goodness-of-fit statistic is associated with the chi-square distribution.↩︎
5. I feel obliged to point out that this is an over-simplification. It works nicely for quite a few situations, but every now and then we’ll come across degrees of freedom values that aren’t whole numbers. Don’t let this worry you too much; when you come across this just remind yourself that “degrees of freedom” is actually a bit of a messy concept, and that the nice simple story that I’m telling you here isn’t the whole story. For an introductory class it’s usually best to stick to the simple story, but I figure it’s best to warn you to expect this simple story to fall apart. If I didn’t give you this warning you might start getting confused when you see $$df = 3.4$$ or something, (incorrectly) thinking that you had misunderstood something that I’ve taught you rather than (correctly) realising that there’s something that I haven’t told you.↩︎
6. In practice, the sample size isn’t always fixed. For example, we might run the experiment over a fixed period of time and the number of people participating depends on how many people show up. That doesn’t matter for the current purposes↩︎
7. Well, sort of. The conventions for how statistics should be reported tend to differ somewhat from discipline to discipline. I’ve tended to stick with how things are done in psychology, since that’s what I do. But the general principle of providing enough information to the reader to allow them to check your results is pretty universal, I think.↩︎
8. To some people, this advice might sound odd, or at least in conflict with the “usual” advice on how to write a technical report. Very typically, students are told that the “results” section of a report is for describing the data and reporting statistical analysis, and the “discussion” section is for providing interpretation. That’s true as far as it goes, but I think people often interpret it way too literally. The way I usually approach it is to provide a quick and simple interpretation of the data in the results section, so that my reader understands what the data are telling us. Then, in the discussion, I try to tell a bigger story about how my results fit with the rest of the scientific literature. In short, don’t let the “interpretation goes in the discussion” advice turn your results section into incomprehensible garbage. Being understood by your reader is much more important.↩︎
9. If you’ve been reading very closely, and are as much of a mathematical pedant as I am, there is one thing about the way I wrote up the chi-square test in the last section that might be bugging you a little bit. There’s something that feels a bit wrong with writing “$$\chi^2(3) = 8.44$$”, you might be thinking. After all, it’s the goodness-of-fit statistic that is equal to 8.44, so shouldn’t I have written $$X^2 = 8.44$$ or maybe $$GOF = 8.44$$? This seems to be conflating the sampling distribution (i.e., $$\chi^2$$ with df = 3) with the test statistic (i.e., $$X^2$$). Odds are you figured it was a typo, since $$\chi$$ and X look pretty similar. Oddly, it’s not. Writing $$\chi^2$$(3)= 8.44 is essentially a highly condensed way of writing “the sampling distribution of the test statistic is $$\chi^2$$(3), and the value of the test statistic is 8.44” In one sense, this is kind of stupid. There are lots of different test statistics out there that turn out to have a chi-square sampling distribution. The $$X^2$$ statistic that we’ve used for our goodness-of-fit test is only one of many (albeit one of the most commonly encountered ones). In a sensible, perfectly organised world we’d always have a separate name for the test statistic and the sampling distribution. That way, the stat block itself would tell you exactly what it was that the researcher had calculated. Sometimes this happens. For instance, the test statistic used in the Pearson goodness-of-fit test is written $$X^2$$ , but there’s a closely related test known as the G-test$$^a$$ , in which the test statistic is written as $$G$$. As it happens, the Pearson goodness-of-fit test and the G-test both test the same null hypothesis, and the sampling distribution is exactly the same (i.e., chi-square with $$k - 1$$ degrees of freedom). If I’d done a G-test for the cards data rather than a goodness-of-fit test, then I’d have ended up with a test statistic of $$G = 8.65$$, which is slightly different from the $$X^2 = 8.44$$ value that I got earlier and which produces a slightly smaller p-value of $$p = .034$$. Suppose that the convention was to report the test statistic, then the sampling distribution, and then the p-value. If that were true, then these two situations would produce different stat blocks: my original result would be written $$X^2 = 8.44$$, $$\chi^2(3)$$, $$p = .038$$, whereas the new version using the G-test would be written as $$G = 8.65$$, $$\chi^2(3)$$, $$p = .034$$. However, using the condensed reporting standard, the original result is written $$\chi^2(3) = 8.44, p =.038$$, and the new one is written $$\chi^2(3) = 8.65, p = .034$$, and so it’s actually unclear which test I actually ran. So why don’t we live in a world in which the contents of the stat block uniquely specifies what tests were ran? The deep reason is that life is messy. We (as users of statistical tools) want it to be nice and neat and organised. We want it to be designed, as if it were a product, but that’s not how life works. Statistics is an intellectual discipline just as much as any other one, and as such it’s a massively distributed, partly-collaborative and partly-competitive project that no-one really understands completely. The things that you and I use as data analysis tools weren’t created by an Act of the Gods of Statistics. They were invented by lots of different people, published as papers in academic journals, implemented, corrected and modified by lots of other people, and then explained to students in textbooks by someone else. As a consequence, there’s a lot of test statistics that don’t even have names, and as a consequence they’re just given the same name as the corresponding sampling distribution. As we’ll see later, any test statistic that follows a $$\chi^2$$ distribution is commonly called a “chi-square statistic”, anything that follows a $$t$$-distribution is called a “t-statistic”, and so on. But, as the $$\chi^2$$ versus $$G$$ example illustrates, two different things with the same sampling distribution are still, well, different. As a consequence, it’s sometimes a good idea to be clear about what the actual test was that you ran, especially if you’re doing something unusual. If you just say “chi-square test” it’s not actually clear what test you’re talking about. Although, since the two most common chi-square tests are the goodness-of-fit test and the independence test, most readers with stats training can probably guess. Nevertheless, it’s something to be aware of. – $$^a$$ Complicating matters, the G-test is a special case of a whole class of tests that are known as likelihood ratio tests. I don’t cover LRTs in this book, but they are quite handy things to know about.↩︎
10. A technical note. The way I’ve described the test pretends that the column totals are fixed (i.e., the researcher intended to survey 87 robots and 93 humans) and the row totals are random (i.e., it just turned out that 28 people chose the puppy). To use the terminology from my mathematical statistics textbook I should technically refer to this situation as a chi-square test of homogeneity and reserve the term chi-square test of independence for the situation where both the row and column totals are random outcomes of the experiment. In the initial drafts of this book that’s exactly what I did. However, it turns out that these two tests are identical, and so I’ve collapsed them together.↩︎
11. Technically, $$E_{ij}$$ here is an estimate, so I should probably write it $$\hat{E_{ij}}$$ . But since no-one else does, I won’t either.↩︎
12. Now that we’ve figured out how to calculate the expected frequencies, it’s straightforward to define a test statistic, following the exact same strategy that we used in the goodness-of-fit test. In fact, it’s pretty much the same statistic. For a contingency table with r rows and c columns, the equation that defines our $$X^2$$ statistic is $X^2=\sum_{i=1}^{r}\sum_{j=1}^{c} \frac{(E_{ij}-O_{ij})^2}{E_{ij}}$ The only difference is that I have to include two summation signs (i.e., $$\sum$$ ) to indicate that we’re summing over both rows and columns.↩︎
13. A problem many of us worry about in real life.↩︎
14. Yates (1934) suggested a simple fix, in which you redefine the goodness-of-fit statistic as: $\chi^{2}=\sum_{i}\frac{(|E_i-O_i|-0.5)^2}{E_i}$ Basically, he just subtracts off 0.5 everywhere.↩︎
15. Mathematically, they’re very simple. To calculate the $$\phi$$ statistic, you just divide your $$X^2$$ value by the sample size, and take the square root: $\phi=\sqrt{\frac{X^2}{N}}$ The idea here is that the $$\phi$$ statistic is supposed to range between 0 (no association at all) and 1 (perfect association), but it doesn’t always do this when your contingency table is bigger than $$2 \times 2$$ , which is a total pain. For bigger tables it’s actually possible to obtain $$\phi > 1$$, which is pretty unsatisfactory. So, to correct for this, people usually prefer to report the $$V$$ statistic proposed by Cramer (1946). It’s a pretty simple adjustment to $$\phi$$. If you’ve got a contingency table with r rows and c columns, then define $$k = min(r, c)$$ to be the smaller of the two values. If so, then Cramér’s $$V$$ statistic is $V=\sqrt{\frac{X^2}{N(k-1)}}$↩︎
16. This example is based on a joke article published in the Journal of Irreproducible Results↩︎
17. Not surprisingly, the Fisher exact test is motivated by Fisher’s interpretation of a p-value, not Neyman’s! See Section 9.5.↩︎ | {"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": 0.8449850678443909, "perplexity": 361.94464037262634}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2022-49/segments/1669446711016.32/warc/CC-MAIN-20221205100449-20221205130449-00850.warc.gz"} |
https://www.raxlab.science/publication/carrasco-2018-osa-latin/ | # The online set aggregation problem
### Abstract
We introduce the online Set Aggregation Problem, which is a natural generalization of the Multi-Level Aggregation Problem, which in turn generalizes the TCP Acknowledgment Problem and the Joint Replenishment Problem. We give a deterministic online algorithm, and show that its competitive ratio is logarithmic in the number of requests. We also give a matching lower bound on the competitive ratio of any randomized online algorithm.
Type
Publication
Lecture Notes in Computer Science | {"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": 0.9559231400489807, "perplexity": 472.73694851016006}, "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/1593657140746.69/warc/CC-MAIN-20200713002400-20200713032400-00235.warc.gz"} |
https://brilliant.org/discussions/thread/a-ball-thrown-in-earth/ | ×
# A ball thrown in earth
A tunnel is dug through the earth and a point mass m falls from rest in it. I do know other methods but i want to do it using calculus , i tried it , but the answer i get is not matching . and also briefly tell how to accurately solve 2nd degree differential equations, i have take i account the shrinking of gravity while going down !
$$\This \note \is \now \closed!$$ Doubt resolved all thanks to Mark Heninngs sir !
Note by Shubham Dhull
1 week ago
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Assuming that the tunnel is radial...
At a distance $$x$$ from the centre of the earth, the gravitational attraction is proportional to $$x$$. Thus we obtain a differential equation $\frac{d^2x}{dt^2} \; = \; -k^2x$ lumping all the constants together into $$k$$. The simplest way to solve this DE is to look for solutions of the form $$x = e^{imt}$$. This function will be a solution provided that $$m^2 = k^2$$, so we have solutions $$e^{\pm ikt}$$. Since we can choose the initial speed and position of the particle, the general solution is $x \; = \; Ae^{ikt} + Be^{-ikt}$ Changing the shape of the arbitrary constants, this solution becomes $x \; = \; C \cos kt + D \sin kt$ The particle performs SHM.
You can integrate the DE in the way you tried (except that you missed out the minus sign in the DE), obtaining $\left(\frac{dx}{dt}\right)^2 + k^2x^2 \; = \; c$ If we start with the particle at rest at the earth's surface then, on the way down: $\frac{dx}{dt} \; = \; -k\sqrt{R^2 - x^2}$ We can separate variables and integrate to obtain $$x = R\cos kt$$. The downside of this method, compared with the previous one, is that you need to keep on solving different DEs' depending on the direction of motion of the particle (and the consequent choice of sign for the square root). · 1 week ago
Comment deleted 1 week ago
The integral of $$\frac{1}{\sqrt{R^2-x^2}}$$ is $$\sin^{-1}\frac{x}{R}$$, giving $$\sin^{-1}\frac{x}{R} = t\sqrt{k} + c$$.
The first integral of the DE, with $$\dot{x}^2 + x^2$$ constant, is a statement of conservation of energy. · 1 week ago
but sir i have changed it to cos inverse yet the answer is coming wrong ! · 1 week ago
i think the answer is give wrong sir ! · 1 week ago
yes sir the answer given is wrong the answer i got was right ! · 1 week ago
sir i knew that i was missing a negative sign (as x is decreasing) but since i had to square root every term after integrating once , i thought the minus is just creating difficulties so i dropped it for a second :( and this time i will do my method very clearly so that you can check it . also, i know no words how to thank you for the knowledge you impart me ! $$Thank\ you\ very\ much\ Sir$$ · 1 week ago | {"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": 0.9774710536003113, "perplexity": 305.5863982633843}, "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/1480698540698.78/warc/CC-MAIN-20161202170900-00091-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/56634/bibtex-ordering-by-appearence-of-the-bib-file-instead-of-the-tex-file | # Bibtex ordering by appearence of the .bib file instead of the .tex file
I'm using TexMaker + BibTex + XeLaTeX and everything was fine, until I noticed that it was ordering the references by the .bib file order instead of the .tex file order. So it starts at [5] instead of [1] because my first citation it's the 5th element in the bib file.
This are the packages which I'm using:
\usepackage{graphicx} % images
\usepackage{fontspec} %font to show accents in xelatex
\usepackage{url} %for urls
\usepackage[portuguese]{babel} %portuguese documment
\usepackage{color} %for text color
I put the references before the end document:
\bibliographystyle{IEEEtran} %for IEEE references style
\bibliography{references}
\end{document}
Also I always do the F1-F11-F1-F1 to compile.
-
Do you have a \nocite{*} command in your document? – egreg May 20 '12 at 14:48
Welcome to TeX.sx! Please add a minimal working example (MWE) that illustrates your problem. In the manual you will find: IEEEtran.bst: The standard IEEEtran BIBTEX style file (unsorted, i.e., references will appear in the order in which they are cited). Recommended for work that is to be submitted to the IEEE. – Marco Daniel May 20 '12 at 14:50
Thanks egreg, you made my day. You are right I had a \nocite{*} %% This should output the entire bib file command hided in my document that was the cause of the wrong order. – RandomGuy May 20 '12 at 14:52
When a \nocite{*} appears in the document, all entries in the .bib database are automatically inserted in the bibliography, with the same order as specified by the chosen bibliography style, in the case of IEEEtran it's "unsorted".
Find the offending command and remove it, or place it at the end, where it won't influence the "unsorted" order of explicit citations. Of course, using \nocite{*} in such a document requires organizing the .bib file in some sensible order, so I'd recommend not using it. If you want to have some entry in the bibliography that wasn't cited in the document, use
\nocite{<key1>,<key2>,...}
just before the \bibliography{command}, where the keys are ordered as you wish.
That indeed solved the problem. I have just commented the \nocite{*} it's enough for me. Thank you very much! – RandomGuy May 20 '12 at 21:19 | {"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": 0.9665268063545227, "perplexity": 2484.950440138384}, "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/1454701157012.30/warc/CC-MAIN-20160205193917-00122-ip-10-236-182-209.ec2.internal.warc.gz"} |
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