url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
http://mathhelpforum.com/algebra/145945-exponents-polynomials.html | # Math Help - Exponents on polynomials
1. ## Exponents on polynomials
I'm having trouble with this problem.
The only way I can get the answer the book gives is to illegally remove the exponent of 1/2 from the 1st term.
Problem: (2x+1)^(1/2) + (x+3)(2x+1)^(-1/2)
my attempt:
=(2x+1)^(1/2) + ((x+3) / ((2x + 1)^(1/2))
(I don't see any like terms here unless I remove the exponent)
Another question: I see that I should move
(2x+1)^(-1/2) to the denominator because of the negative exponent. When I do this, does it go in the denominator of all the other terms, or just in the den of (x+3)?
Please explain what I'm doing wrong. Thanks alot!!
2. $(2x+1)^{\frac{1}{2}} + (x+3)(2x+1)^{\frac{-1}{2}}=(2x+1)^{\frac{1}{2}} + \frac{(x+3)}{(2x+1)^{\frac{1}{2}}}=\frac{(2x+1) +(x+3)}{(2x+1)^{\frac{1}{2}}}
$
now simplify the numerator
3. Originally Posted by dwatkins741
I'm having trouble with this problem.
The only way I can get the answer the book gives is to illegally remove the exponent of 1/2 from the 1st term.
Problem: (2x+1)^(1/2) + (x+3)(2x+1)^(-1/2)
my attempt:
=(2x+1)^(1/2) + ((x+3) / ((2x + 1)^(1/2))
(I don't see any like terms here unless I remove the exponent)
Another question: I see that I should move (2x+1)^(-1/2) to the denominator because of the negative exponent. When I do this, does it go in the denominator of all the other terms, or just in the den of (x+3)?
Please explain what I'm doing wrong. Thanks alot!!
$\sqrt{2x+1}+\frac{x+3}{\sqrt{2x+1}}=\frac{2x+1+x+3 }{\sqrt{2x+1}}=\frac{3x+4}{\sqrt{2x+1}}$ , via $\sqrt{a}+\frac{1}{\sqrt{a}}=\frac{a+1}{\sqrt{a}}$ ... common denominator and stuff.
Tonio
4. ## That's new to me
That formula you gave, Tonio, was totally new to me. I never knew you could do that with radicals. I tested it out and it works. Thanks alot. Very helpful.
5. Originally Posted by dwatkins741
I'm having trouble with this problem.
The only way I can get the answer the book gives is to illegally remove the exponent of 1/2 from the 1st term.
Problem: (2x+1)^(1/2) + (x+3)(2x+1)^(-1/2)
my attempt:
=(2x+1)^(1/2) + ((x+3) / ((2x + 1)^(1/2))
(I don't see any like terms here unless I remove the exponent)
Another question: I see that I should move
(2x+1)^(-1/2) to the denominator because of the negative exponent. When I do this, does it go in the denominator of all the other terms, or just in the den of (x+3)?
Please explain what I'm doing wrong. Thanks alot!!
I expect you've been asked to simplify this... To simplify expressions involving fractions, you need a common denominator.
$\sqrt{2x + 1} + \frac{x + 3}{\sqrt{2x + 1}} = \frac{\sqrt{2x + 1}\sqrt{2x + 1}}{\sqrt{2x + 1}} + \frac{x + 3}{\sqrt{2x + 1}}$
$= \frac{2x + 1}{\sqrt{2x + 1}} + \frac{x + 3}{\sqrt{2x + 1}}$
$= \frac{2x + 1 + x + 3}{\sqrt{2x + 1}}$
$= \frac{3x + 4}{\sqrt{2x + 1}}$.
I choose to write fractions with rational denominators though, so to clean it up even more...
$= \frac{(3x + 4)\sqrt{2x + 1}}{2x + 1}$. | {"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": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9470720887184143, "perplexity": 1127.4634753261814}, "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-2014-23/segments/1405997894275.63/warc/CC-MAIN-20140722025814-00112-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://eccc.weizmann.ac.il/author/020146/ | Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > AUTHORS > PETER DIXON:
All reports by Author Peter Dixon:
TR22-160 | 31st October 2022
Jason Vander Woude, Peter Dixon, A. Pavan, Jamie Radcliffe, N. V. Vinodchandran
#### The Geometry of Rounding
Rounding has proven to be a fundamental tool in theoretical computer science. By observing that rounding and partitioning of $\mathbb{R}^d$ are equivalent, we introduce the following natural partition problem which we call the secluded hypercube partition problem: Given $k\in\mathbb{N}$ (ideally small) and $\epsilon>0$ (ideally large), is there a partition of ... more >>>
TR21-043 | 15th March 2021
Peter Dixon, A. Pavan, N. V. Vinodchandran
#### Promise Problems Meet Pseudodeterminism
The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high ... more >>>
ISSN 1433-8092 | Imprint | {"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": 0.9472333788871765, "perplexity": 2924.084930078811}, "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-2022-49/segments/1669446711376.47/warc/CC-MAIN-20221209011720-20221209041720-00129.warc.gz"} |
http://mathoverflow.net/questions/34520/a-differential-equation | # A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function" Assume that $0<\gamma<1$, $v$ is a positive number, and $$\frac{dg}{ds}+(v\gamma) g +(1-\gamma)(e^{\rho s}g)^{\frac{1}{\gamma-1}}g=0$$
Find a closed form for $g$?
-
Up to an implicit algebraic equation, yes. Ask Maple. The answer is large enough that I won't paste it here. But it's not so hard to do even by hand! – Jacques Carette Aug 4 '10 at 16:02
Please provide some context: why are you interested in this equation? Why do you particularly want a closed form (given that so many ODEs don't have closed forms)? What have you done already to try to find one? – Loop Space Aug 4 '10 at 16:20
If possible, please give more information in the title of your question. Titles on MO can be up to 240 characters --- almost two tweets. – Theo Johnson-Freyd Aug 4 '10 at 19:37
This reads like homework. I'm voting to close. I echo Theo's plea for a more descriptive title. – José Figueroa-O'Farrill Aug 4 '10 at 22:31
Hey guys, thanks for your comments, by letting $h=(e^{\rho s}g)^{\frac{1}{1-\gamma}}$ I got the solution for this ! – Lam Aug 5 '10 at 14:43
This seems to be a Bernoulli differential equation. Please cf. http://en.wikipedia.org/wiki/Bernoulli_differential_equation for the solution (in your case $n= \frac{\gamma}{\gamma-1}$). | {"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": 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": 0.8071134686470032, "perplexity": 735.2105426581815}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121561.0/warc/CC-MAIN-20160428161521-00164-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/determine-state-variable-description-circuit-shown-figure-kvl-1-x-t-r1-i1-c-0-kvl-2-c-r2-i-q2652267 | Determine the state-variable description for the circuit shown in the figure.
KVL 1
-x(t)+R1*I1+C=0
KVL 2
-C+R2*I2+L=0
Find A, b, c and D in terms of R1, R2, C, L. Note: If you have problem finding the required expressions, you may assume that R1, R2, C and L nominal values are 1 ?, 2 ?, 1 H and 2F respectively. | {"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": 0.8439192771911621, "perplexity": 1114.989804544644}, "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-2015-35/segments/1440644064445.47/warc/CC-MAIN-20150827025424-00081-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://www.achieversrule.com/2017/04/shortcut-tricks-to-solve-ratio-and-proportion.html | # Shortcut Tricks to Solve - Ratio and Proportion
## SHORTCUT RULES TO SOLVE PROBLEMS ON RATIO AND PROPORTION
### Effective for IBPS PO - SBI PO Exam
Here we will start a series of Quantitative Aptitude Shortcut Tricks for your upcoming SBI - IBPS - SSC and Other Government Competitive Exams. We will try to cover up all topics of the quantitative Aptitude Sections from which question was generally asked.
Note: The page may takes sometime to load the Quantitative formula's. If you face any problem just comment below the posts.
Trick - 1
• If two numbers are in the ratio of a:b and the sum of these numbers is x, then these numbers will be ax/a+b and bx/a+b respectively.
Trick - 2
• To find the number of coins, Number of each types of coin = Amount in rupees / value of coins in rupees
Tricks -3
• The contents of two vessels containing water and milk are in the ratio x1:y1 and x2:y2 are mixed in the ratio x:y. The resulting mixture will have water and milk in the ratio of
$x{{x}_{1}}({{x}_{2}}+{{y}_{2}})+y{{x}_{2}}\left( {{x}_{1}}+{{y}_{1}} \right):x{{y}_{1}}\left( {{x}_{2}}+{{y}_{2}} \right)+y{{y}_{2}}\left( {{x}_{1}}+{{y}_{1}} \right)$
Trick - 4
• If three numbers are in the ratio of a:b:c, and sum of these numbers is x, then these numbers will be ax/a+b+c, bx/a+b+c, and cx/a+b+c respectively.
Trick -5
• To find the strength of milk, Strength of milk in the mixture = Quantity of Milk / Total Quantity of Mixture
Trick -6
• If in a mixture of x liters, milk and water are in the ratio of a:b, then the quantities of milk and water in the mixture will be ax/a+b liters and bx/a+b liters respectively.
Trick - 7
• The ratio between two numbers is a:b. If each number be decreased by x, the ratio becomes c:d. Then sum of the two numbers
$=\frac{x\left( a+b \right)\left( d-c \right)}{ad-bc}$
Questions for Practice
Q1. Two numbers are in the ratio of 3;1. If sum of these two numbers is 440, find the numbers.
Q2. A bag contains an equal number of one rupee, 50 paise and 25 paise coins respectively. If the total value is Rs.35, how many coins of each type are there?
Q3. The contents of two vessels containing water and milk are in the ratio 1:2 and 2:5 are mixed in the ratio 1:4. The resulting mixture will have water and milk in the ratio.
Q4. An amount of Rs.750 is distributed among A, B and C in the ratio of 4:5:6. What is the share of B?
Q5. One man adds 3 liters of water to 12 liters of milk and another 4 liters of water to 10 liters of milk. What is the ratio of the strengths of milk in the two mixtures
Q6. In a mixture of 65 liters milk and water are in the ratio of 3:2 . What the quantities of milk and water in the mixture?
Q7. The ratio of two numbers is 7:9. If each number is decreased by 2, the ratio becomes 3:4. Find the sum of the two numbers. | {"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": 0.9338605999946594, "perplexity": 1045.5669328471301}, "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-2021-49/segments/1637964363134.25/warc/CC-MAIN-20211205005314-20211205035314-00322.warc.gz"} |
https://www.physicsforums.com/threads/reactive-force-on-radiation.598840/ | 1. Apr 21, 2012
### Philalethes
Hello,
In electrostatics, classical mechanics predicts that a charged particle's action-at-a-distance force (Coulomb's law) on other charges is accompanied by an equal and opposite reaction force on the particle.
Classically, radiant electromagnetic energy is a self-propagating electromagnetic field; therefore, must there likewise be a reaction force on this radiant energy resulting from its action on charged particles? If so, how would this reaction manifest? Would this mean a decrease in energy of the radiant EM energy? Does it slow down? Change its path? Is it explainable without significant explanation of special relativity or any QED?
Thanks,
Philalethes
2. Apr 21, 2012
### Bob S
You may be interesed in a simulation of the field lines of a moving point charge, using the Cal Tech simulation tool at http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html
The first panel shows the field lines of a charge moving at a velocity β = 0.5 from the left to the right. The field lines are straight, and slightly compressed toward the 90 degree line (due to length contraction along direction of motion). The lines move out away from the charge at β=1 in the reference frame of the observer.
In the second panel, a charge moving at β = 0.5 is suddenly decelerated to β = 0.225. This produced a kink in the field lines, which also radiates out away from the moving charge at β = 1 in the reference frame of the observer. Any observer at a distance $\ell$ from the particle will see a retarded signal at delayed time $\delta t = \ell / c \space$.
So the E field lines for a constant velocity charge are a longitudinal E field (i.e., radial), while the E field in the kink of the decelerated charge is transverse. Does this mean that the lost energy is being radiated as a pulsed TEM field?
Here is the caption to the applet
When the charge moves at relativistic speed, the electric field is concentrated near the pole, and consequently the field lines are shifted. The field lines always point to where the charge is at that instant, if we are within the current sphere of information. If that charge has changed speed or direction within a time t < r/c, where r is the distance away, and c is the speed of light, we will not know that charge has accelerated, and the field lines will still point to where the charge would be if it hadn't changed speed or direction.
Notice That when the charge is accelerated, because the field lines must be continuous, it is forced in a direction almost perpendicular to the the direction of propogation. As time goes on, the line becomes more and more perpendicular, the horizontal component increasing faster than the vertical component. Associated with this electric field is a magnetic field, perpendicular to the electric field and the direction of propogation, which describes light.
#### Attached Files:
• ###### Moving_charge1.jpg
File size:
26.1 KB
Views:
108
Last edited: Apr 21, 2012
3. Apr 21, 2012
### Bob S
The classical radiation (radiated power) from a decelerating charged particle is given by
$$\frac{dW}{dt}=-\frac{e^2 \dot v^2}{6 \pi \epsilon_o c^3}$$
where $\dot v$ is the acceleration. This radiation moves away from the charged particle at β = 1.
See Panofsky and Phillips Classical Electricity and Magnetism page 301.
4. Apr 21, 2012
### Philalethes
Thank you Bob, for pointing me toward the Larmor formula. I found this document from the Wikipedia article about it. In section 5.2, he uses the example of blue light scattering in the atmosphere as a demonstration of energy conservation as light passes by an atom, but declines to explain the exact mechanism behind the decrease in energy of the light! (See attachment screenshot of the PDF)
I'm looking over your first post again to see if the answer is contained therein....
#### Attached Files:
• ###### light.PNG
File size:
77.6 KB
Views:
111
5. Apr 21, 2012
### Bob S
Kinematically, scattering light off of an atom is just like Thomson scattering off of a single electron, which is the classical limit of Compton scattering. Look specifically at the derivation of this formula for the wavelength shift in http://en.wikipedia.org/wiki/Compton_scattering
$$\lambda'-\lambda=\frac{h}{mc}\left(1-\cos\theta \right)$$
where m is the mass of the electron and θ is the scattering angle. In Rayleigh scattering, the recoil mass is the entire atom, so let's cosider scattering at 90 degrees. We have
$$\lambda'-\lambda=\frac{h}{Mc}=\frac{hc}{Mc^2}$$
Using hc = 4.136x 10-15 eV-sec x 3 x 1010cm/sec = 1.24 x 10-4 eV -cm, and Mc2=1.3 x 1010 eV for a nitrogen atom, we get
$$\lambda'-\lambda=\frac{hc}{Mc^2} = \frac{1.24 x 10^{-4} eV \cdot cm}{1.3x10^{10} eV}=9.5 x 10^{-15}cm = 9.5 x 10^{-7} Angstroms$$
So there is a wavelength shift, but it is very small.
Incidentally, the author of the paper you referenced implied that the nucleus was moving and radiating. It is actually the electron cloud that is moving and radiating.
6. Apr 21, 2012
### Philalethes
Thanks! Thomson scattering describes what I was looking for.
Best,
Philalethes | {"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": 0.9339476227760315, "perplexity": 638.9683434511719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591332.73/warc/CC-MAIN-20180719222958-20180720002958-00218.warc.gz"} |
https://www.deepdyve.com/lp/ou_press/finite-region-boundedness-and-stabilization-for-2d-continuous-discrete-6OX6mSZdVT | Finite-region boundedness and stabilization for 2D continuous-discrete systems in Roesser model
Finite-region boundedness and stabilization for 2D continuous-discrete systems in Roesser model Abstract This paper investigates the finite-region boundedness (FRB) and stabilization problems for two-dimensional continuous-discrete linear Roesser models subject to two kinds of disturbances. For two-dimensional continuous-discrete system, we first put forward the concepts of finite-region stability and FRB. Then, by establishing special recursive formulas, sufficient conditions of FRB for two-dimensional continuous-discrete systems with two kinds of disturbances are formulated. Furthermore, we analyze the finite-region stabilization issues for the corresponding two-dimensional continuous-discrete systems and give generic sufficient conditions and sufficient conditions that can be verified by linear matrix inequalities for designing the state feedback controllers which ensure the closed-loop systems FRB. Finally, viable experimental results are demonstrated by illustrative examples. 1. Introduction The study of two-dimensional systems has a long history, with some meaningful works (Roesser, 1975; Fornasini & Marchesini, 1976; Lin & Bruton, 1989; Lu & Antoniou, 1992; Xu et al., 2005; Hu & Liu, 2006; Singh, 2014; Xie et al., 2015; Ahn et al., 2016) such as systems theory, stability properties and practical applications. Among all research topics of two-dimensional systems, stability as the most fundamental property has obtained fruitful achievements. To list some of these, Bachelier et al. (2016) proposed linear matrix inequality (LMI) stability criteria for two-dimensional systems by relaxing the polynomial-based texts of stability into that of LMIs. Bouzidi et al. (2015) presented computer algebra-based methods for testing the structural stability of n-dimensional discrete linear systems. Other works dedicated to discrete two-dimensional models are Bliman (2002) and Ebihara et al. (2006). Two-dimensional continuous-discrete systems arise naturally in several emerging application areas, for example, in analysis of repetitive processes (Rogers & Owens, 1992; Rogers et al., 2007) and in irrigation channels (Knorn & Middleton, 2013a). As such, the research on two-dimensional continuous-discrete systems has been a hot area in control field. Especially, the stability analysis for two-dimensional continuous-discrete systems has attracted much attention of some researchers over the last few decades, and several interesting findings in linear and non-linear frameworks have been obtained (see Xiao, 2001; Benton et al., 2002; Owens & Rogers, 2002; Rogers & Owens, 2002; Knorn & Middleton, 2013a;,Chesi & Middleton, 2014; Knorn & Middleton, 2016; Galkowski et al., 2016; Pakshin et al., 2016; Wang et al., 2017 and references therein). For example, in the linear setting, Xiao (2001) considered three models of two-dimensional continuous-discrete systems and gave sufficient and necessary conditions for their Lyapunov asymptotic stability (LAS)-based two-dimensional characteristic polynomial. In Owens & Rogers (2002), a stability analysis for differential linear repetitive processes, a class of two-dimensional continuous-discrete linear systems, was given in the presence of a general set of boundary conditions. These results further were extended to stability tests based on a one-dimensional Lyapunov equation and strictly bounded real lemma in Benton et al. (2002). Chesi & Middleton (2014) proposed necessary and sufficient conditions that can be checked with convex optimization for stability and performance analysis of two-dimensional continuous-discrete systems. For non-linear systems, Knorn & Middleton (2013b) modeled the homogeneous, unidirectional non-linear vehicle strings as a general two-dimensional continuous-discrete non-linear system and presented a sufficient condition for stability of this system. In Galkowski et al. (2016), the exponential stability conditions for non-linear differential repetitive processes were established by using a vector Lyapunov function-based approach. However, most of these results related to stability were focused on LAS or exponential stability. Apart from LAS or exponential stability, finite-time stability (FTS) is also a basic concept in the stability analysis. The concept of FTS was first introduced in Kamenkov (1953) and reintroduced by Dorato in Dorato (1961) which is related to dynamical systems whose state does not exceed some bound during the specified time interval. It is important to note that FTS and LAS are completely independent concepts. FTS aims at analyzing transient behavior of a system within a finite (possibly short) interval rather than the asymptotic behavior within a sufficiently long (in principle, infinite) time interval. In general, the characteristic of FTS does not guarantee stability in the sense of Lyapunov and vice versa. In addition, it should be noted that the FTS considered here is unrelated to the one adopted in some other works (Moulay & Perruquetti, 2008; Nersesov & Haddad, 2008), where the authors focus on the stability analysis of non-linear systems whose trajectories converge to an equilibrium point in finite time. Recent years have witnessed growing interests on FTS (see Amato et al., 2001; Amato & Ariola, 2005; Amato et al., 2010; Jammazi, 2010; Seo et al., 2011; Zhang et al., 2012, 201b, a; Haddad & L’Afflitto, 2016; Tan et al., 2016 and references therein) because it plays vital roles in many practical applications, for example, for many dynamic systems the state trajectories are required to stay within a desirable operative range over a certain time interval to fulfill hardware constraints or to maintain linearity of the system. What these literatures we mentioned here are about one-dimensional systems; little progress related to this problem has been made for two-dimensional systems due to their dynamical and structural complexity. Until recently, the authors in Zhang & Wang (2016a,b) studied the problems of finite-region stability (FRS) and finite-region boundedness (FRB) for two-dimensional discrete Roesser models and Fornasini-Marchesini second models. For two-dimensional continuous-discrete systems, it is worth mentioning that there are some results on the problems of exponential stability, weak stability and asymptotic stability over bounded region for repetitive processes (see Rogers et al., 2007; Galkowski et al., 2016; Pakshin et al., 2016 and references therein), where the variable $$x^h(t,k)$$ changes on finite interval [0, T]. Clearly, these problems are completely different from the finite-region control problem considered in Zhang & Wang (2016a,b) in that their methods can exhibit the transient performance of two-dimensional discrete systems over a given finite region by setting the state variables of the system less than a particular threshold. Therefore, it is necessary and important to consider the FRS and FRB problems for two-dimensional continuous-discrete systems, which motivates our present research. It is worth noting that though the two-dimensional system theory is developed from one-dimensional system theory, it is not a parallel promotion of one-dimensional system theory. There exist deep and substantial differences between one-dimensional case and two-dimensional case (Fan & Wen, 2002; Feng et al., 2012). In the two-dimensional case, the system depends on two variables and the initial conditions consist of infinite vectors, but for a one-dimensional system, the initial condition is a single vector. Moreover, most results obtained for one-dimensional systems cannot be straightforwardly extended to two-dimensional systems. For example, stability texts for one-dimensional systems are based on a simple calculation of the eigenvalues of a matrix or the roots of a polynomial, but this is not the case of two-dimensional systems, as stability conditions are given in terms of multidimensional polynomials. These factors make the analysis of FRS and FRB for two-dimensional continuous-discrete systems much more complicated and difficult than one-dimensional continuous or discrete systems. On the other hand, the existing results on two-dimensional discrete systems (Zhang & Wang, 2016a,b) cannot be immediately extended to two-dimensional continuous-discrete systems. This is because two-dimensional continuous-discrete systems are more complicated and technically more difficult to tackle than two-dimensional discrete systems. To fill this gap, it is necessary for us to further study the finite-region control problems of two-dimensional continuous-discrete systems deeply. In this paper, we deal with two-dimensional continuous-discrete linear Roesser models subject to two classes of disturbances. We first put forward the definitions of FRS and FRB for two-dimensional continuous-discrete system. Then, by establishing special recursive formulas, sufficient condition of FRB for two-dimensional continuous-discrete system with energy-bounded disturbances and sufficient condition of FRS for two-dimensional continuous-discrete system are given. Furthermore, by using the given sufficient conditions, generic sufficient conditions and sufficient conditions that are solvable by LMIs for the existence of state feedback controllers which ensure the corresponding closed-loop systems FRB or FRS are derived. We also show that, under stronger assumptions, our sufficient conditions for finite-region stabilization recover asymptotic stability. Finally, we address the FRB and finite-region stabilization problems for two-dimensional continuous-discrete system with disturbances generated by an external system and present sufficient condition of FRB and generic sufficient condition and sufficient condition that can be checked with LMIs of FRB via state feedback for the corresponding systems. The paper is organized as follows. In Section 2, the definitions of FRS and FRB for two-dimensional continuous-discrete linear system are proposed. In Section 3, the FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with the first disturbances are considered, and corresponding sufficient conditions and LMIs conditions are derived. Section 4 presents the results for the two-dimensional continuous-discrete system subject to the second disturbances. Numerical examples are given in Section 5 to show the effectiveness of the proposed approaches. Finally, conclusions are drawn in Section 6. Notations In this paper, we assume that vectors and matrices are real and have appropriate dimensions. $$N^+$$ denotes a set of positive integers, $$R^n$$ is the n-dimensional space with inner product $$x^Ty$$. A > 0 means that the matrix A is symmetric positive definite. $$A^T$$ denotes the transpose of matrix A, I represents the identity matrix. For a matrix A, its eigenvalue, maximum eigenvalue and minimum eigenvalue are denoted by $$\lambda (A)$$, $$\lambda _{\max }(A)$$ and $$\lambda _{\min }(A)$$, respectively. The symmetric terms in a matrix is represented by *. diag{⋅} denotes a black-diagonal matrix. 2. Preliminaries In this paper, we consider the following two-dimensional continuous-discrete linear system for Roesser model: $$x^{+}(t,k) = A x(t,k) + B u(t,k)+ Gw(t,k),$$ (2.1) where $$x^{+}(t,k)=\left [{\begin{array}{@{}c@{}} \frac{{\partial x^h(t,k)}}{{\partial t}} \\ x^v(t,k+1) \end{array}} \right ],$$ $$x(t,k)= \left [{\begin{array}{@{}c@{}} x^h(t,k) \\ x^v(t,k) \end{array}} \right ] \in R^n$$ is the state vector, $$u(t,k) \in R^p$$ is the two-dimensional control input, $$w(t,k)\in R^r$$ is the exogenous disturbance. t, k are the horizontal continuous variable and vertical discrete variable, respectively. $$A= \left [\begin{array}{@{}cc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right ],$$ $$B= \left [ {\begin{array}{@{}c@{}} B_1\\ B_2 \end{array}} \right ],$$ $$G= \left [ {\begin{array}{@{}c@{}} G_1\\ G_2 \end{array}} \right ]$$ are real matrices with appropriate dimensions. $$x_0 (t,k)= \left [ {\begin{array}{@{}c@{}} x^{h}(0,k) \\ x^{v}(t,0) \end{array}} \right ]$$ is used to represent the boundary condition. Define the finite region for two-dimensional continuous-discrete system (2.1) as follows: $$(T,N)=\{(t,k) | 0\leqslant t\leqslant T, 0\leqslant k\leqslant N; T>0, N\in N^{+}\}.$$ (2.2) The main aim of this note is to analyze the transient performance of system (2.1) over a given finite region by setting the state variables less than a particular threshold. Inspired by the definitions of FTS for one-dimensional systems in Amato et al. (2001) and Amato & Ariola (2005) and two-dimensional FRS for Roesser model in Zhang & Wang (2016b), the concept of FRS for the two-dimensional continuous-discrete linear system (2.1) in the uncontrolled case with no disturbances can be formalized. Definition 2.1 Given positive scalars $$c_1$$, $$c_2$$, T, N, with $$c_1<c_2$$, $$N\in N^{+}$$ and a matrix R > 0, where $$R=\textrm{diag}\ \{R_1, R_2\}$$, the two-dimensional continuous-discrete linear system $$x^{+}(t,k)=Ax(t,k)$$ (2.3) is said to be finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, if \begin{equation*} x^T_0(t,k) R x_0 (t,k) \leqslant c_1 \Rightarrow x^T (t,k) R x(t,k)< c_2,\quad \forall\ t\in [0,T], k \in \{1,\cdots,N\}. \end{equation*} Remark 2.1 Similar to one-dimensional continuous and discrete cases, LAS and FRS are complete independent concepts for two-dimensional continuous-discrete systems. A system which is FRS may not be LAS and vice versa. If we limit our attention to what happens within a finite region, we can consider Lyapunov stability as an ‘additional’ requirement. Under stronger assumptions, the conditions presented in this paper (see Remark 3.1) include a special case where a system being both finite-region stable and Lyapunov stable. Next, we consider the situation when the state is subject to some external signal disturbances $$\mathscr{W}(d)$$. This leads to the definition of FRB, which covers Definition 2.1 as a special case. In this paper, we will address two kinds of external signal disturbances: (i) energy-bounded disturbances, $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$; (ii) disturbances generated by an external system, $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, where F is a real matrix with appropriate dimensions. Definition 2.2 Given positive scalars $$c_1$$, $$c_2$$, d, T, N, with $$c_1 <c_2$$, $$N\in N^{+}$$ and a matrix R > 0, where $$R=\textrm{diag}\ \{R_1, R_2\}$$, the system $$x^{+}(t,k)= A x(t,k)+ Gw(t,k)$$ (2.4) is said to be finite-region bounded with respect to $$(c_1, c_2, T, N, R, d)$$, if \begin{equation*} x^T_0(t,k)R x_0(t,k)\leqslant c_1 \Rightarrow x^T (t,k) R x(t,k)< c_2,\quad \forall\ t\in [0,T], k\in \{1,\cdots,N\}, \end{equation*} for all $$w(t,k)\in \mathscr{W}(d)$$. Remark 2.2 When w(t, k) = 0, the concept of FRB given in Definition 2.2 is consistent with the definition of FRS in Definition 2.1. 3. FRB and stabilization under the disturbances of the first case In this section, we focus on FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with the energy-bounded external disturbances $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$. Firstly, we present a sufficient condition of the system (2.1) in the uncontrolled case. Theorem 3.1 System (2.4) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag} \{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, $$\gamma _l$$, $$\alpha _2+\beta _1> 1$$ and matrices $$P_l>0$$, S > 0, where l = 1, 2, such that the following conditions hold: $$\qquad\qquad\qquad\quad\;\left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11}& - \alpha_1 P_1 P_1A_{12} &P_1G_1\\ * &-\beta_1 P_2& 0\\ * &* &-\gamma_1 S \end{array}}\right]<0,$$ (3.1a) $$\left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[4pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[4pt] * & * & G_2^T P_2 G_2 -\gamma_2 S \end{array}}\right]<0,$$ (3.1b) $${x^h}^T (0,k) R_1x^h(0,k) \leqslant \eta c_1,\quad{x^v}^T(t,0)R_2 x^v (t,0)\leqslant (1-\eta)c_1,$$ (3.1c) $$\qquad\qquad\quad\,\;\; \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_1 )} \eta c_1 + \frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_1)}\beta_1(1-\eta)c_2T+ \frac{\lambda_{\max}(S)} {\lambda_{\min}(\widetilde P_1 )} \gamma_1 d T < \eta c_2 e^{-\alpha_1 T},$$ (3.1d) $$\frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_2)} (1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_2)} N \alpha_0 \beta_2\eta c_2 + \frac{\lambda_{\max}(S)} {\lambda_{\min}(\widetilde P_2)}N \alpha_0 \gamma_2 d <(1- \eta) c_2,$$ (3.1e) where $$\alpha _0 =\textrm{max}\{1, \alpha _{2}^{N}\}$$, $$\widetilde{P}_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. For system (2.4) and $$P_1>0$$, $$P_2>0$$, define the following Lyapunov functions \begin{equation*} V_1(x^h(t,k))={x^h}^T (t,k)P_1 x^h(t,k),\ \ V_2(x^v(t,k))={x^v}^T(t,k)P_2x^v(t,k), \end{equation*} denote $$\psi (t,k)=[x^T(t,k)\ w^T(t,k)]^T$$, then, we have \begin{align} &\frac{\partial V_1 (x^h(t,k))}{\partial t}-\alpha_1 V_1(x^h(t,k))-\beta_1 V_2(x^v(t,k))- \gamma_1 w^T(t,k)Sw(t,k) \nonumber \\ &\quad= \psi^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1A_{12} & P_1G_1\\ * & -\beta_1P_2 & 0\\ * & *& -\gamma_1 S \end{array}}\right] \psi(t,k),\nonumber\\ &\quad\quad V_2(x^v(t,k+1))-\alpha_2 V_2(x^v(t,k))-\beta_2 V_1(x^h(t,k))-\gamma_2w^T(t,k)Sw(t,k) \nonumber \\ &\quad=\psi^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[4pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[4pt] * & * & G_2^T P_2 G_2 -\gamma_2 S \end{array}}\right]\psi(t,k).\nonumber \end{align} According to conditions (3.1a) and (3.1b), then $$\frac{\partial V_1 (x^h(t,k))}{\partial t} <\alpha_1 V_1(x^h(t,k))+\beta_1 V_2(x^v(t,k))+ \gamma_1 w^T(t,k)Sw(t,k),$$ (3.2) $$V_2(x^v(t,k+1))<\alpha_2 V_2(x^v(t,k))+\beta_2 V_1(x^h(t,k))+\gamma_2w^T(t,k)Sw(t,k).\quad$$ (3.3) Integrating from 0 to t for (3.2), with t ∈ [0, T], we obtain \begin{align} V_1 (x^h(t,k)) <& V_1(x^h(0,k)) e^{\alpha_1 t}+\beta_1 \int_{0}^{t} e^{\alpha_1( t- \tau)}{x^v}^T(\tau,k) P_2 x^v(\tau,k)\,\mathrm{d}\tau \nonumber \\ &+\gamma_1 \int_{0}^{t} e^{\alpha_1( t-\tau)}w^T(\tau,k)Sw(\tau,k)\,\mathrm{d}\tau \nonumber \\ \leqslant&\lambda_{\max}(\widetilde P_1){x^h}^T(0,k) R_1 x^h(0,k) e^{\alpha_1 T}+ \lambda_{\max}(\widetilde P_2) \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k)R_2 x^v(t,k)\,\mathrm{d}t \nonumber\\ &+ \lambda_{\max} (S) \gamma_1 e^{\alpha_1 T} d T . \end{align} (3.4) Fixing t ∈ [0, T], for k ∈ {1, ⋯ , N}, by iteration to (3.3), we have \begin{align} V_2(x^v(t,k)) <& \alpha_2 ^k V_2(x^v(t,0)) + \sum\limits_{l=0}^{k-1}\alpha_2^{k-l-1} \left[ \beta_2 {x^h}^T (t,l)P_1 x^h(t,l) + \gamma_2 w^T(t,l)Sw(t,l)\right]\nonumber \\ \leqslant& \lambda_{\max} (\widetilde P_2) {x^v}^T(t,0) R_2 x^v(t,0) \alpha_0 \nonumber \\ &+ \sum\limits_{l=0}^{k-1}\alpha_2^{k-l-1} \left[\lambda_{\max}(\widetilde P_1) \beta_2 {x^h}^T (t,l)R_1 x^h(t,l) + \lambda_{\max}(S) \gamma_2 d \right], \end{align} (3.5) where $$\alpha _0 ={\max }\{1, \alpha _2 ^N\}$$. On the other hand, $$V_1 (x^h(t,k))={x^h}^T (t,k)P_1 x^h(t,k) \geqslant \lambda_{\min}(\widetilde P_1) {x^h}^T(t,k) R_1 x^h(t,k),$$ (3.6) $$V_2 (x^v(t,k))={x^v}^T (t,k)P_2 x^v(t,k) \geqslant \lambda_{\min} (\widetilde P_2) {x^v}^T(t,k) R_2 x^v(t,k).$$ (3.7) It follows from condition (3.1c) and (3.4)–(3.7) that \begin{align} {x^h}^T(t,k) R_1 x^h(t,k) <& \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_1)}\eta c_1 e^{\alpha_1 T} + \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_1)}\beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T (t,k)R_2 x^v(t,k)\, \mathrm{d}t \nonumber \\ &+ \frac{ \lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_1)} \gamma_1 e^{\alpha_1 T} \textrm{d} T, \end{align} (3.8) \begin{align} {x^v}^T (t,k) R_2 x^v(t,k) <& \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1}\beta_2 {x^h}^T (t,l)R_1 x^h(t,l) \nonumber\\ & + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1}\gamma_2 \textrm{d}. \end{align} (3.9) Next, we will prove that the following inequalities hold for any t ∈ [0, T], k ∈{1, ⋯ , N}: $${x^h}^T(t,k)R_1 x^h(t,k) <\eta c_2,\ {x^v}^T(t,k)R_2 x^v(t,k) < (1-\eta) c_2.$$ (3.10) Noting that $$c_1<c_2$$, from condition (3.1c), we have $${x^h}^T(0,k)R_1 x^h(0,k) <\eta c_2,\ {x^v}^T(t,0)R_2 x^v(t,0) < (1-\eta) c_2.$$ (3.11) Setting k = 0 in (3.8) and using (3.11) and condition (3.1d), it is easy to obtain that $${x^h}^T(t,0) R_1 x^h(t,0) <\eta c_2.$$ (3.12) Now, we do the second mathematical induction for k to prove (3.10) holds for any t ∈ [0, T]. When k = 1, from (3.9), (3.12) and condition (3.1e), we get \begin{align} {x^v}^T (t,1) R_2 x^v(t,1) &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \beta_2 {x^h}^T (t,0)R_1 x^h(t,0) + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \gamma_2 \textrm{d} \nonumber \\ & < \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \gamma_2 \textrm{d}< (1-\eta)c_2. \end{align} (3.13) Similarly, setting k = 1 in (3.8), it is easy to obtain from (3.13) and condition (3.1d) that \begin{equation*} {x^h}^T(t,1) R_1 x^h(t,1) <\eta c_2. \end{equation*} Suppose that the conclusion (3.10) holds for k < N. When k = N, it follows from (3.9) and condition (3.1e) that \begin{align} {x^v}^T \!(t,N) R_2 x^v(t,N)\! &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{N-1} \alpha_2 ^{N-l-1}\beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{N-1} \alpha_2 ^{N-l-1}\gamma_2 \textrm{d} \nonumber \\ &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} N\alpha_0 \beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} N\alpha_0 \gamma_2 \textrm{d} < (1-\eta)c_2. \end{align} (3.14) Setting k = N in (3.8) and employing (3.14) and condition (3.1d), we have \begin{equation*} {x^h}^T(t,N) R_1 x^h(t,N) <\eta c_2. \end{equation*} By induction, the condition (3.10) is established for any t ∈ [0, T], k ∈ {1, ⋯ , N}. Therefore, for any t ∈ [0, T], k ∈ {1, ⋯ , N}, we have $$x^T(t,k) R x(t,k) < c_2$$, which implies that the system (2.4) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. For the simpler case of FRB, from Theorem 3.1, we can obtain the following corollary. Corollary 3.1 System (2.3) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$P_l>0$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\qquad\, \left[{\begin{array}{@{}cc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1 A_{12} \\ * & -\beta_1 P_2 \end{array}}\right]<0,$$ (3.15a) $$\left[{\begin{array}{@{}cc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} \\[2pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 \end{array}}\right]<0,$$ (3.15b) $$\quad\qquad\;\;\, \frac{\lambda_{\max}(\widetilde P_1 )} {\lambda_{\min}(\widetilde P_1 )} \eta c_1 + \frac{\lambda_{\max}(\widetilde P_2 )} {\lambda_{\min}(\widetilde P_1 )}\beta_1(1-\eta)c_2T < \eta c_2 e^{-\alpha_1 T},\\$$ (3.15c) $$\quad\;\; \frac{\lambda_{\max}(\widetilde P_2 )} {\lambda_{\min}(\widetilde P_2 )} (1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1 )} {\lambda_{\min}(\widetilde P_2 )}N \alpha_0 \beta_2\eta c_2 <(1- \eta) c_2,$$ (3.15d) where $$\alpha _0 ={\max }\{1, \alpha _2 ^N \}$$, $$\widetilde{P}_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. The proof can be obtained along the guidelines of Theorem 3.1. Remark 3.1 If conditions (3.15) in Corollary 3.1 are satisfied with $$\alpha _1=-\beta _2<0$$, $$\alpha _2+\beta _1=1$$, then system (2.3) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, and it is also asymptotically stable. Specifically, if $$\alpha _1=-\beta _2$$, $$\alpha _2+\beta _1=1$$, from the conditions(3.15a–3.15b), we derive \begin{equation*} \varPsi=\left[{\begin{array}{@{}cc@{}} A_{11}^T P_1 + P_1 A_{11}+ A_{21}^TP_2A_{21} &P_1 A_{12} +A_{21}^T P_2 A_{22} \\[2pt] * & A_{22}^T P_2 A_{22}-P_2 \end{array}}\right]<0. \end{equation*} By Theorem 2 in Bachelier et al. (2008), the system (2.3) is asymptotically stable. Similarly, if conditions (3.1) in Theorem 3.1 hold for $$\alpha _1=-\beta _2$$, $$\alpha _2+\beta _1=1$$, then system (2.4) is not only finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, but also asymptotically stable. It is worth noting that in the analysis of FRS, $$\varPsi$$ does not need to be negative definite but just less than $$diag\{(\alpha _1+\beta _2)P_1, (\alpha _2+\beta _1-1) P_2 \}$$. Remark 3.2 In the derivation of Theorem 3.1, we used constant Lyapunov function. It is well known that the use of constant Lyapunov function will lead to a certain conservatism. Recently, Bachelier et al. (2016) gave the solution to reduce this conservatism by relaxing the polynomial-based texts of asymptotic stability into that of LMIs. Chesi & Middleton (2014) provided the solutions to reduce or possibly cancel the conservatism by using the frequency domain method in the analysis of exponential stability. It is worth noting that the problem they considered is asymptotic stability for two-dimensional continuous-discrete systems. Though these methods reduce the conservatism, they do not apply to the analysis of FRS. This is because the matrix $$A_{11}$$ is not necessary Hurwitz and $$A_{22}$$ is not necessary Schur in the FRS analysis. The study on the reduction of conservatism will be discussed in the future. Next, we study the finite-region stabilization issue for two-dimensional continuous-discrete system (2.1) with energy-bounded external disturbances $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$. For given system (2.1), consider the following state feedback controller: $$u(t,k)=Kx(t,k),$$ (3.16) where $$K=[K_1,K_2]$$ is the constant real matrix with appropriate dimensions. Our goal is to find sufficient condition which guarantees the interconnection of (2.1) with the controller (3.16) $$x^{+}(t,k)= (A +BK) x(t,k)+Gw(t,k)$$ (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The following theorem gives the solution of this problem. Theorem 3.2 System (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, $$\gamma _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, M > 0, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} \varPsi- \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 & G_1M \\ * & -\beta_1 \widetilde{H}_2 & 0\\ * & *& -\gamma_1 M \end{array}}\right]<0,$$ (3.18a) $$\left[{\begin{array}{@{}cccc@{}} - \beta_2 \widetilde{H}_1 & 0 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[4pt] * & -\alpha_2 \widetilde{H}_2 & 0 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[4pt] * & * &-\gamma_2 M & MG_2^T \\[4pt] * & * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (3.18b) $$\frac{\eta c_1} {\lambda_{\min}( H_1 )} +\frac{\beta_1(1-\eta)c_2T} {\lambda_{\min}( H_2 )}+ \frac{\gamma_1 d T}{ \lambda_{\min}(M)} < \frac{\eta c_2e^{-\alpha_1 T} }{\lambda_{\max} ( H_1)},$$ (3.18c) $$\frac{(1-\eta) c_1\alpha_0}{\lambda_{\min}( H_2 )} + \frac{N \alpha_0 \beta_2\eta c_2}{\lambda_{\min}( H_1 )}+ \frac{N\alpha_0 \gamma_2 d}{ \lambda_{\min}(M)} <\frac{(1- \eta) c_2}{\lambda_{\max}(H_2)},$$ (3.18d) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. Let $$\widetilde{H}_l =P_l^{-1}$$, l = 1, 2 and $$M=S^{-1}$$ in Theorem 3.1. Noting that for symmetric positive definite matrices $$H_l=\widetilde{P}_l^{-1}$$, $$M=S^{-1}$$, their eigenvalues satisfy the following equations: \begin{equation*} \lambda_{\max}(H_l)=\frac{1}{\lambda_{\min}(\widetilde{P}_l)}, \lambda_{\min}( H_l )=\frac{1}{\lambda_{\max}(\widetilde{P}_l) }, \lambda_{\min}(M)=\frac{1}{\lambda_{\max}(S)}. \end{equation*} Clearly, conditions (3.1d)–(3.1e) can be rewritten as in (3.18c)–(3.18d). Now, let $$\widehat{A} =A+BK$$. If $$A_{1l}$$ and $$A_{2l}$$ in conditions (3.1a)–(3.1b) of Theorem 3.1 are replaced by $$\widehat{A}_{1l} =A_{1l}+B_1K_l$$ and $$\widehat{A}_{2l}=A_{2l}+B_2K_l$$, respectively, and in terms of $$Q_l=P_l$$, $$\widetilde{H}_l=P_l^{-1}$$ and $$M=S^{-1}$$, where l = 1, 2, we derive $$\qquad\qquad\qquad\;\,\,\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{11}^T \widetilde{H}_1^{-1} + \widetilde{H}_1^{-1} \widehat{A}_{11} - \alpha_1 \widetilde{H}_1^{-1} &\widetilde{H}_1^{-1} \widehat{A}_{12} & \widetilde{H}_1^{-1 }G_1\\[4pt] * & -\beta_1 \widetilde{H}_2^{-1} & 0\\[4pt] * & *& -\gamma_1 M^{-1} \end{array}}\right]<0,$$ (3.19) $$\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{21}^T \widetilde{H}_2^{-1}\widehat{A}_{21} - \beta_2 \widetilde{H}_1^{-1} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} \widehat{A}_{22} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} G_2\\[5pt] * & \widehat{A}_{22}^T \widetilde{H}_2^{-1} \widehat{A}_{22}-\alpha_2 \widetilde{H}_2^{-1} & \widehat{A}_{22}^T \widetilde{H}_2^{-1} G_2 \\[5pt] * & * & G_2^T \widetilde{H}_2^{-1} G_2 -\gamma_2 M^{-1} \end{array}}\right]<0.$$ (3.20) Next, we will prove that the conditions (3.19) and (3.20) are equivalent to (3.18a) and (3.18b), respectively. Pre- and post-multiplying (3.19) by the symmetric matrix $$diag\{\widetilde{H}_1,\widetilde{H}_2,M\}$$, we obtain the following equivalent condition $$\left[{\begin{array}{@{}ccc@{}} \widetilde{H}_1 \widehat{A}_{11}^T + \widehat{A}_{11} \widetilde{H}_1- \alpha_1 \widetilde{H}_1 & \widehat{A}_{12}\widetilde{H}_2 & G_1 M\\[4pt] * & -\beta_1 \widetilde{H}_2 & 0\\[4pt] * & *& -\gamma_1 M \end{array}}\right]<0.$$ (3.21) Recalling that $$\widehat{A}_{1l} =A_{1l}+B_1K_l$$, l = 1, 2, and letting $$L_l = K_l \widetilde{H}_l$$, l = 1, 2, we obtain that the condition (3.21) is equivalent to (3.18a). Applying Schur complement lemma (Boyd et al., 1994) twice to (3.20) produces $$\left[\begin{array}{@{}ccccc@{}} - \beta_2 {\widetilde{H}}_1^{-1} & 0 & {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1}G_2 & 0 & {\widehat{A}}_{21}^T\\[5pt] * & -\alpha_2 {\widetilde{H}}_2^{-1} & {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} G_2 & 0 & {\widehat{A}}_{22}^T \\[5pt] * & * & -\gamma_2 M^{-1} & G_2^T & 0 \\[5pt] * & * & * & -{\widetilde{H}}_2 & 0 \\ * & * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.22) Pre-multiplying (3.22) by $$\left[\begin{array}{@{}ccccc@{}} {\widetilde{H}}_1 & 0 & 0 & -{\widetilde{H}}_1 {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1} & {\widetilde{H}}_1 {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & {\widetilde{H}}_2 & 0 & -{\widetilde{H}}_2 {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} & {\widetilde{H}}_2 {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & 0 & M & 0 & -M G_2^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & 0 & 0 & I & 0 \\[5pt] 0 & 0 & 0 & 0 & I \end{array}\right]$$ (3.23) and post-multiplying it by the transpose of (3.23), we have the following equivalent condition: $$\left[\begin{array}{@{}ccccc@{}} -\beta_2 {\widetilde{H}}_1 & 0 & 0 & {\widetilde{H}}_1 {\widehat{A}}_{21}^T & 0 \\[5pt] * & -\alpha_2 {\widetilde{H}}_2 & 0 & {\widetilde{H}}_2 {\widehat{A}}_{22}^T & 0 \\[5pt] * & * & -\gamma_2 M -MG_2^T {\widetilde{H}}_2^{-1}G_2M & MG_2^T & M G_2^T \\[5pt] * & * & * & -{\widetilde{H}}_2 & 0 \\[5pt] * & * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.24) Applying Schur complement lemma (Boyd et al., 1994) to (3.24) yields the following equivalent condition: $$\left[\begin{array}{@{}cccc@{}} -\beta_2 {\widetilde{H}}_1 & 0 & 0 & {\widetilde{H}}_1 {\widehat{A}}_{21}^T \\[5pt] * & -\alpha_2 {\widetilde{H}}_2 & 0 & {\widetilde{H}}_2 {\widehat{A}}_{22}^T \\[5pt] * & * & -\gamma_2 M & MG_2^T \\[5pt] * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.25) Noting that $$\widehat{A}_{2l} =A_{2l}+B_2K_l$$, l = 1, 2, and letting $$L_l = K_l\widetilde{H}_l$$, l = 1, 2, we obtain that the condition (3.25) is equivalent to (3.18b). From Theorem 3.1, we obtain that the system (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The following corollary of Theorem 3.2 allows us to find a state feedback controller K such that closed-loop system $$x^{+}(t,k)=(A+BK)x(t,k)$$ (3.26) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$. Corollary 3.2 System (3.26) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}cc@{}} \varPsi - \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 \\[3pt] * & -\beta_1 \widetilde{H}_2 \end{array}}\right]<0,$$ (3.27a) $$\left[{\begin{array}{@{}ccc@{}} - \beta_2 \widetilde{H}_1 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[5pt] * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (3.27b) $$\frac{\eta c_1} {\lambda_{\min}( H_1 )} + \frac{\beta_1(1-\eta)c_2T} {\lambda_{\min}( H_2 )} <\frac{\eta c_2e^{-\alpha_1 T}}{\lambda_{\max} (H_1)},$$ (3.27c) $$\frac{(1-\eta) c_1 \alpha_0} {\lambda_{\min}( H_2 )} + \frac{N \alpha_0 \beta_2\eta c_2} {\lambda_{\min}( H_1 )} <\frac{(1- \eta) c_2}{\lambda_{\max}( H_2 )},$$ (3.27d) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1 ^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. The proof can be obtained as in Theorem 3.2, applying the results of Corollary 3.1 to system (3.26). From a computational point of view, it is important to point out that the conditions (3.18c), (3.18d) in Theorem 3.2 and the conditions (3.27c), (3.27d) in Corollary 3.2 are difficult to solve. Besides, the conditions in Theorem 3.2 and Corollary 3.2 involve the unknown finite-region scalar $$\alpha _l, \beta _l, \gamma _l, l=1,2$$, which lead to Theorem 3.2 and Corollary 3.2 are difficult to solve by means of LMI Toolbox. In the following, we will show that once we have fixed values of $$\alpha _l, \beta _l, \gamma _l, l=1,2$$, the feasibility of Theorem 3.2 and Corollary 3.2 can be turned into LMIs-based feasibility problems (Boyd et al., 1994) using procedures proposed in Amato et al. (2001). Clearly, the conditions (3.18c) and (3.18d) are guaranteed by imposing additional conditions $$\, \lambda_{l1}I_l < H_l < \lambda_{l2}I_l,\ \lambda_{31}I_r <M,\quad l=1,2,\quad$$ (3.28a) $$\qquad\quad\;\, \frac{\eta c_1}{\lambda_{11}} + \frac{\beta_1(1-\eta)c_2T}{\lambda_{21}} + \frac{\gamma_1 d T}{\lambda_{31}} < \frac{\eta c_2e^{-\alpha_1T}}{\lambda_{12}},$$ (3.28b) $$\frac{(1-\eta) c_1\alpha_0}{\lambda_{21}} +\frac{ N \alpha_0 \beta_2\eta c_2}{\lambda_{11}} + \frac{N \alpha_0 \gamma_2 d}{\lambda_{31}} <\frac{(1- \eta) c_2}{\lambda_{22}}$$ (3.28c) for some positive numbers $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$. Using Schur complement (Boyd et al., 1994) to inequalities (3.28b) and (3.28c) produces $$\qquad\qquad\qquad \left[{\begin{array}{@{}cccc@{}} \lambda_{12}\eta c_2 e^{-\alpha_1T}& \lambda_{12}\sqrt{ \beta_1 (1-\eta)c_2T} & \lambda_{12}\sqrt{ \eta c_1 } &\lambda_{12} \sqrt{\gamma_1 d T} \\ * & \lambda_{21} & 0 &0\\ * & * & \lambda_{11}& 0\\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (3.29a) $$\left[{\begin{array}{@{}cccc@{}} \lambda_{22}(1-\eta) c_2 & \lambda_{22}\sqrt{ N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} & \lambda_{22} \sqrt{N \alpha_0 \gamma_2 d} \\ * & \lambda_{11} & 0 & 0 \\ * & * & \lambda_{21} & 0 \\ * & * & * & \lambda_{31} \end{array}}\right]>0.$$ (3.29b) The following theorem gives the LMI feasibility problem of Theorem 3.2. Theorem 3.3 Given system (3.17) and $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$\gamma _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, M > 0, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$ satisfying (3.1c) and the LMIs (3.18a), (3.18b), (3.28a) and (3.29), where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller K given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$ renders system (3.17) finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. Similarly, LMI feasibility problem can be derived from Corollary 3.2. Corollary 3.3 Given system (3.26) and $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$ satisfying (3.1c), the LMIs (3.27a), (3.27b) and $$\lambda_{l1}I_l < H_l < \lambda_{l2}I_l,\quad l=1,2,$$ (3.30a) $$\left[{\begin{array}{@{}ccc@{}} \lambda_{12}\eta c_2 e^{-\alpha_1T}& \lambda_{12}\sqrt{ \beta_1 (1-\eta)c_2T} & \lambda_{12} \sqrt{\eta c_1 } \\ * & \lambda_{21} & 0\\ * & * & \lambda_{11} \end{array}}\right]>0,$$ (3.30b) $$\left[{\begin{array}{@{}ccc@{}} \lambda_{22}(1-\eta) c_2 & \lambda_{22}\sqrt{ N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} \\ * & \lambda_{11} & 0\\ * & * & \lambda_{21} \end{array}}\right]>0,$$ (3.30c) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2 ^{-1}]$$ renders system (3.26) finite-region stable with respect to $$(c_1,c_2,T,N,R)$$. 4. FRB and stabilization under the second case of disturbances In this section, we will study the FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$. Firstly, we consider the FRB issue for the two-dimensional continuous-discrete system in the form $$x^{+}(t,k) = A x(t,k)+ Gw(k),$$ (4.1a) $$w(k+1)=Fw(k).\qquad\qquad\quad$$ (4.1b) Theorem 4.1 System (4.1) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$P_l>0$$, S > 0, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1A_{12} & P_1G_1\\ * & -\beta_1 P_2 & 0\\ * & *& -\alpha_1 S \end{array}}\right]<0,$$ (4.2a) $$\left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[5pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[5pt] * & * & G_2^T P_2 G_2 + F^T SF -\alpha_2 S \end{array}}\right]<0,$$ (4.2b) $$\quad\, \frac{\lambda_{\max}(\widetilde P_1) \eta c_1 + \lambda_{\max}(S) \gamma d} {\lambda_{\min}(\widetilde P_1 )} + \frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_1)}\beta_1(1-\eta)c_2T < \eta c_2 e^{-\alpha_1 T},$$ (4.2c) $$\frac{\lambda_{\max}(\widetilde P_2) (1-\eta) c_1 + \lambda_{\max}(S) d} {\lambda_{\min}(\widetilde P_2)} \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_2)} N \alpha_0 \beta_2\eta c_2 <(1- \eta) c_2,$$ (4.2d) where $$\gamma = {\max }\ \{1, \|F^k\|^2 _{k=1,2,\cdots , N}\}$$, $$\alpha _0 ={\max }\ \{1, \alpha _2 ^N \}$$, $$\widetilde P_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. Firstly, we derive the recursive relations of the weights of state variables of system (4.1a). For simple mark, let $$z^+(t,k)=\left [{\begin{array}{@{}c@{}} x^+(t,k)\\ w(k+1)\end{array}} \right ],$$ $$z(t,k)=\left [{\begin{array}{@{}c@{}} x(t,k)\\ w(k)\end{array}} \right ],$$ then the system (4.1) is equivalent to $$z^+(t,k)= \left[{\begin{array}{@{}cc@{}} A & G\\ 0 & F \end{array}}\right] z(t,k).$$ (4.3) We define the Lyapunov functions of system (4.3) as follows \begin{equation*} V_1(z^h(t,k))= {z^h}^T(t,k) \left[{\begin{array}{@{}cc@{}} P_1 & 0\\ * & S \end{array}} \right] z^h(t,k), V_2\left(z^v(t,k)\right)= {z^v}^T(t,k) \left[{\begin{array}{@{}cc@{}} P_2 & 0\\ * & S \end{array}} \right] z^v(t,k), \nonumber \end{equation*} where $$z^h (t,k)=\left [{\begin{array}{c} x^h(t,k)\\ w(k)\end{array}} \right ], z^v (t,k)=\left [{\begin{array}{c} x^v(t,k)\\ w(k)\end{array}} \right ],$$ then it follows that \begin{align} \frac{\partial V_1 (z^h(t,k))}{\partial t} &- \alpha_1 V_1 (z^h(t,k)) -\beta_1 {x^v}^T (t,k)P_2 x^v(t,k) \nonumber\\ &\quad= z^T (t,k) \left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 +P_1 A_{11} -\alpha_1 P_1 & P_1 A_{12} & P_1 G_1\\ * & -\beta_1 P_2 & 0 \\ * & * & -\alpha_1 S \end{array}}\right] z(t,k). \end{align} (4.4) Similarly, we can obtain the following equation \begin{align} V_2(z^v(t,k+1))&-\alpha_2 V_2 (z^v(t,k)) -\beta_2 {x^h}^T (t,k) P_1 x^h(t,k) \nonumber \\ &\quad=z^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{21} ^T P_2 A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^ T P_2 G_2 \\[5pt] * & A_{22}^T P_2 A_{22} -\alpha_2 P_2 & A_{22}^T P_2 A_{22} \\[5pt] * & * & G_2^T P_2 G_2 + F^T S F - \alpha_2 S \end{array}}\right] z(t,k). \end{align} (4.5) Using conditions (4.2a) and (4.2b) for (4.4) and (4.5), we derive $$\frac{\partial V_1 (z^h(t,k))}{\partial t} < \alpha_1 V_1(z^h(t,k))+\beta_1 {x^v}^T(t,k) P_2 x^v(t,k),$$ (4.6) $$V_2 (z^v(t,k+1))< \alpha_2 V_2(z^v(t,k))+\beta_2 {x^h}^T(t,k) P_1 x^h(t,k).\;\;\;$$ (4.7) When k = 0, integrating from 0 to t for (4.6), with t ∈ [0, T], we derive \begin{align} V_1 (z^h(t,0)) &< V_1(z^h(0,0))e ^{\alpha_1 t}+\beta_1 \int_{0}^{t}e ^{\alpha_1( t-\tau)} {x^v}^T(\tau,0)P_2 x^v(\tau,0)\,\mathrm{d}\tau \nonumber\\ &\leqslant \left[{x^h}^T (0,0)P_1x^h(0,0) +w^T(0)Sw(0)\right]e^{\alpha_1 T}+ \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,0) P_2 x^v(t,0)\,\mathrm{d}t \nonumber \\ &\leqslant \left[ \lambda_{\max}(\widetilde P_1 ) \eta c_1 + \lambda_{\max}(S) d \right]e^{\alpha_1 T}+ \lambda_{\max}(\widetilde P_2 )\beta_1 e^{\alpha_1 T}\int_{0}^{T} {x^v}^T(t,0) R_2 x^v(t,0)\,\mathrm{d}t, \end{align} (4.8) for fixed k ∈ {1, ⋯ , N}, integrating from 0 to t for (4.6), with t ∈ [0, T], we obtain \begin{align} V_1 (z^h(t,k)) &< V_1(z^h(0,k))e ^{\alpha_1 t}+\beta_1 \int_{0}^{t}e ^{\alpha_1( t-\tau)} {x^v}^T(\tau,k) P_2 x^v(\tau,k)\,\mathrm{d}\tau \nonumber\\ &\leqslant \left[ \lambda_{\max}(\widetilde P_1 ) \eta c_1 + \lambda_{\max}(S) \|F^k\|^2 d\right]e^{\alpha_1 T} + \lambda_{\max}(\widetilde P_2 )\beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k) R_2 x^v(t,k)\,\mathrm{d}t. \end{align} (4.9) Fixing t ∈ [0, T], by iteration to (4.7), we have \begin{align} V_2 (z^v(t,k)) &< \alpha_2^k V_2(x^v(t,0))+ \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l)P_1 x^h(t,l)\nonumber \\ &\leqslant \left[{x^v}^T (t,0)P_2x^v(t,0)+ w^T(0)Sw(0) \right] \alpha_0 + \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l)P_1 x^h(t,l)\nonumber \\ &\leqslant \left[\lambda_{\max} (\widetilde P_2 ) (1-\eta)c_1 + \lambda_{\max}(S) d \right] \alpha_0 + \lambda_{\max} (\widetilde P_1 ) \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l) R_1 x^h(t,l), \end{align} (4.10) where $$\alpha _0=\max \{1, \alpha _2^N\}$$. On the other hand, $$V_1 (z^h(t,k))= {x^h}^T(t,k) P_1 x^h(t,k) + w^T(k)Sw(k) \geqslant \lambda_{\min} (\widetilde P_1) {x^h}^T(t,k) R_1 x^h(t,k),$$ (4.11) $$V_2 (z^v(t,k))= {x^v}^T(t,k) P_2 x^v(t,k) + w^T(k)Sw(k) \geqslant \lambda_{\min} (\widetilde P_2) {x^v}^T(t,k) R_2 x^v(t,k).$$ (4.12) Putting together (4.8), (4.9) with (4.11) and (4.10) with (4.12), respectively, we obtain \begin{align} {x^h}^T(t,k) R_1 x^h(t,k) <& \frac{\lambda_{\max}(\widetilde P_1 )\eta c_1 + \lambda_{\max}(S) \gamma d}{\lambda_{\min} (\widetilde P_1 )} e^{\alpha_1 T}\nonumber \\ &+ \frac{ \lambda_{\max}(\widetilde P_2 ) }{\lambda_{\min} (\widetilde P_1 )} \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k) R_2 x^v(t,k)\,\mathrm{d}t, \end{align} (4.13) \begin{align} {x^v}^T(t,k) R_2 x^v(t,k) <& \frac{\lambda_{\max} (\widetilde P_2 ) (1-\eta)c_1 +\lambda_{\max}(S) d}{\lambda_{\min} (\widetilde P_2 ) }\alpha_0 \nonumber \\ &+ \frac{\lambda_{\max} (\widetilde P_1 )}{\lambda_{\min} (\widetilde P_2 )} \sum\limits_{l=0}^{k-1}\alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l) R_1 x^h(t,l), \end{align} (4.14) where $$\gamma = \textrm{max}\ \{1, \|F^k\|^2 _{k=1,2,\cdots , N}\}$$. Similar to the proof of Theorem 3.1, we can get that for any t ∈ [0, T], k ∈ {1, ⋯ , N}, $$x^T(t,k) R x (t,k) <c_2$$ holds. This implies that the two-dimensional continuous-discrete system (4.1) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. Remark 4.1 Let F = I and $$\gamma =1$$ in Theorem 4.1, the sufficient condition of the FRB for system (4.1a) subject to exogenous unknown constant disturbances can be derived. In Theorem 4.1, when the external system (4.1b) is a one-dimensional continuous system or a two-dimensional continuous-discrete system, the corresponding results can also be obtained. Remark 4.2 The sufficient condition which ensures system (2.3) finite-region stable with respect to $$(c_1,c_2,T,N,R)$$ by Theorem 4.1 is consistent with Corollary 3.1. Next, we investigate the finite-region stabilization issue for the system (2.1) with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, with the aim to find sufficient condition that ensures the system $$x^{+}(t,k)= (A +BK) x(t,k)+Gw(k),$$ (4.15a) $$w(k+1)=Fw(k)\qquad\qquad\qquad\qquad\;\,$$ (4.15b) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The solution of this issue is given by the following theorem. Theorem 4.2 System (4.15) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, M > 0, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} \varPsi - \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 & G_1M \\ * & -\beta_1 \widetilde{H}_2 & 0\\ * & *& -\alpha_1 M \end{array}}\right]<0,$$ (4.16a) $$\left[{\begin{array}{@{}ccccc@{}} - \beta_2 \widetilde{H}_1 & 0 & 0 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & 0 & 0 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[5pt] * & * &-\alpha_2 M & MF^T & MG_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (4.16b) $$\frac{\eta c_1 }{\lambda_{\min}( H_1 )} + \frac{\gamma d }{ \lambda_{\min}(M)} + \frac{\beta_1(1-\eta)c_2T}{\lambda_{\min}( H_2 )} < \frac{\eta c_2e^{-\alpha_1 T}}{\lambda_{\max} ( H_1)},$$ (4.16c) $$\frac{(1-\eta) c_1 \alpha_0}{\lambda_{\min}( H_2 )} + \frac{d \alpha_0}{ \lambda_{\min}(M)} + \frac{N \alpha_0 \beta_2\eta c_2}{\lambda_{\min}( H_1 )} <\frac{(1- \eta) c_2}{\lambda_{\max}(H_2)} ,$$ (4.16d) where $$\varPsi = \widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\gamma = \textrm{max}\ \{1, \|F^k\|^2_{ k=1,2,\cdots , N} \}$$, $$\alpha _0 ={\max }\ \{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. Let $$\widetilde{H}_l =P_l^{-1}$$, l = 1, 2 and $$M=S^{-1}$$ in Theorem 4.1. Similar to the proof of Theorem 3.2, applying the results of Theorem 4.1 to system (4.15), the proof can be obtained. Here, we only give the proof of condition (4.16b). If $$A_{2l}$$ in condition (4.2b) of Theorem 4.1 is replaced by $$\widehat{A}_{2l}=A_{2l}+B_2K_l$$, l = 1, 2, we derive $$\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{21}^T \widetilde{H}_2^{-1}\widehat{A}_{21} - \beta_2 \widetilde{H}_1^{-1} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} \widehat{A}_{22} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} G_2\\[5pt] * & \widehat{A}_{22}^T \widetilde{H}_2^{-1} \widehat{A}_{22}-\alpha_2 \widetilde{H}_2^{-1} & \widehat{A}_{22}^T \widetilde{H}_2^{-1} G_2 \\[5pt] * & * & G_2^T \widetilde{H}_2^{-1} G_2 + F^T M^{-1}F -\alpha_2 M^{-1} \end{array}}\right]<0.$$ (4.17) Applying Schur complement lemma (Boyd et al., 1994) to (4.17) produces $$\left[{\begin{array}{@{}ccccc@{}} - \beta_2 \widetilde{H}_1^{-1} & 0 & 0 & 0 & \widehat{A}_{21}^T\\[5pt] * & -\alpha_2 \widetilde{H}_2^{-1} & 0 & 0 & \widehat{A}_{22}^T \\[5pt] * & * & -\alpha_2 M^{-1} & F^T & G_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0.$$ (4.18) Pre- and post-multiplying (4.18) by $$diag\{\widetilde{H}_1, \widetilde{H}_2, M,I,I\}$$, we have the following equivalent condition: $$\left[{\begin{array}{@{}ccccc@{}} -\beta_2 \widetilde{H}_1 & 0 & 0 & 0 & \widetilde{H}_1 \widehat{A}_{21}^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & 0 & 0 & \widetilde{H}_2 \widehat{A}_{22}^T \\[5pt] * & * & -\alpha_2 M & MF^T & MG_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0.$$ (4.19) Letting $$L_l = K_l\widetilde{H}_l$$, l = 1, 2, we finally obtain that the condition (4.19) is equivalent to (4.16b). Remark 4.3 Let F = I and $$\gamma =1$$ in Theorem 4.2, the sufficient condition of the FRB via state feedback for system (4.15a) subject to unknown constant disturbances can be derived. Similarly, when the external system is a one-dimensional continuous-discrete system or a two-dimensional continuous-discrete system, the corresponding conclusions can also be obtained. Remark 4.4 When setting $$\eta =0$$, $$\eta =1$$ in Theorems 4.1 and 4.2, respectively, we can obtain the corresponding conclusions for one-dimensional discrete linear system (Amato & Ariola, 2005) and continuous linear system (Amato et al., 2001), respectively. Similarly, Theorem 4.2 can be reducible to the following LMIs-based feasibility problem. Theorem 4.3 Given system (4.15) and $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, M > 0, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$ satisfying (3.1c) and the LMIs (4.16a), (4.16b), (3.28a) and $$\qquad\;\;\left[{\begin{array}{cccc} \lambda_{12}\eta c_2e^{-\alpha_1T} &\lambda_{12} \sqrt{ \beta_1 (1-\eta)c_2T} &\lambda_{12} \sqrt{ \eta c_1 } &\lambda_{12} \sqrt{\gamma d} \\ * & \lambda_{21} & 0 &0\\ * & * & \lambda_{11}& 0\\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (4.20a) $$\left[{\begin{array}{cccc} \lambda_{22} (1-\eta) c_2 & \lambda_{22} \sqrt{N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} & \lambda_{22}\sqrt{d \alpha_0} \\ * & \lambda_{11} & 0 & 0 \\ * & * & \lambda_{21} & 0 \\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (4.20b) where $$\varPsi = \widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$ renders system (4.15) finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. 5. Numerical examples In this section, numerical examples are used to illustrate the effectiveness of the proposed methods. Example 5.1 It is well known that some dynamical processes in gas absorption, water stream heating and air drying can be described by the Darboux equation (Marszalek, 1984): $$\frac{\partial^2 s(t,\tau)}{\partial t \partial \tau}=a_1 \frac{\partial s(t,\tau)}{\partial \tau}+a_2 \frac{\partial s(t,\tau)}{\partial t}+ a_0 s(t,\tau)+bf(t,\tau),$$ (5.1) where $$s(t,\tau )$$ is an unknown vector function at $$[0,t_f]$$ and $$\tau \in [0,\infty ]$$, $$a_0,a_1,a_2,b$$ are real constants and $$f(t,\tau )$$ is the input function. Taking $$r(t,\tau )=\partial s(t,\tau ) / \partial \tau -a_2 s(t,\tau )$$, $$x^h(t,k)=r(t,k)=r(t,k\Delta \tau )$$ and $$x^v(t,k)=s(t,k)=s(t,k\Delta \tau )$$, we can write the Equation (5.1) in the two-dimensional continuous-discrete system of the form (2.1). Via appropriate selection of the parameters $$a_0,a_1,a_2,b$$, we consider the system (2.1) subject to the energy-bounded external disturbances with \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -2.1 & 0.5 \\ 1.5 & 2.5 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.5 \end{array}}\right], G=\left[{\begin{array}{@{}c@{}} 0.3 \\ 0 \end{array}}\right], \end{equation*} where $$c_1=2.5$$, $$c_2=10$$, T = 5, N = 20, d = 1, R = I, $$\eta =0.7$$. Given above positive constants, the positive definite matrix R and the initial condition $$x^h(0,k)=-1.3$$, $$x^v(t,0)=0.85$$. When external disturbances satisfy $$w^T(t,k)w(t,k)\leqslant 1$$, we have considered the problem of FRB via state feedback and designed the state feedback controller by solving the feasibility problem in Theorem 3.3. When control input u(t, k) = 0, Fig. 1 shows that the open-loop system (2.4) is not finite-region bounded with respect to (2.5, 10, 5, 20, I, 1). Using LMI toolbox of MATLAB and Theorem 3.3, the LMIs (3.18a), (3.18b), (3.28a) and (3.29) are feasible with $$\alpha _1=0.03$$, $$\beta _1=0.05$$, $$\gamma _1=0.1$$, $$\alpha _2=1.03$$, $$\beta _2=0.01$$, $$\gamma _2=0.15$$, the solution is given below \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 11.6622 & 0\\ * & 1.8365 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} -11.7054\\ -3.0609 \end{array}}\right],\ M=14.5931. \end{equation*} Then, we find the state feedback controller \begin{equation*} K=[-1.0037, -1.6667]. \end{equation*} The weighted-state values $$x^T(t,k)Rx(t,k)$$ are limited by the given bound 10 for the closed-loop system (3.17) obtained after stabilization (see Fig. 2). Let us take the dynamical process in gas absorption as an example to explain the practical significance of state feedback. In the process of absorption of a gas, $$s(t,\tau )$$ in Darboux equation (5.1) denotes the quantity of gas absorbed by unit volume of the absorbent (Tichonov & Samarsky, 1963). The kinetics of absorption is represented by $$\partial s(t,\tau ) / \partial \tau$$. Let $$a_2=-\gamma$$, where $$\frac{1}{\gamma }$$ is Henry’s coefficient, then $$\partial s(t,\tau ) / \partial \tau -a_2 s(t,\tau )$$ denotes the concentration of gas in the pores of the absorbent in the layer t. In practice, due to the material constraints, the concentration of gas in the pores of the absorbent in the layer t and the quantity of gas absorbed by unit volume of the absorbent are required to stay within a desirable threshold range during the specified absorbing layer and time interval, that is, the states in system (2.1) are required to stay within a particular threshold range over a given finite-region. Therefore, when the established system is not finite-region stable, we need to use state feedback to make the system states not exceed the particular threshold. Fig. 1. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.4). Fig. 1. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.4). Fig. 2. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.17). Fig. 2. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.17). Example 5.2 Consider the system (2.1) with w(t, k) = 0, where \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -0.5 & 2.5 \\ 1.1& 2.5 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.2 \end{array}}\right]. \end{equation*} Assume that $$c_1=2$$, $$c_2=10$$, T = 5, N = 10, R = I, $$\eta =0.7$$ and $$x^h(0,k)=1.1$$, $$x^v(t,0)=-0.2$$. We have considered the FRS via state feedback and designed the state feedback controller by solving the feasibility problem in Corollary 3.3. When the system has control input u(t, k) = 0, the weighted-state values $$x^T(t,k)Rx(t,k)$$ of system (2.3) are as shown in Fig. 3, obviously, the open-loop system (2.3) is not finite-region stable with respect to (2, 10, 5, 10, I) before stabilization. Using LMI control toolbox and Corollary 3.3, the LMIs (3.27a), (3.27b) and (3.30) are feasible with $$\alpha _1=0.03$$, $$\beta _1=0.1$$, $$\alpha _2=1.1$$, $$\beta _2=0.01$$. The solution is given below \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 1.3719 & 0\\ * & 0.9058 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}c@{}} -1.1670\\ -1.8870 \end{array}}\right]. \end{equation*} Then, we obtain the state feedback controller \begin{equation*}K=[-0.8506, -2.0833].\end{equation*} The weighted-state values $$x^T(t,k)Rx(t,k)$$ of the closed-loop system (3.26) are depicted in Fig. 4. It can be seen that the closed-loop system (3.26) is finite-region stable with respect to (2, 10, 5, 10, I) but not asymptotically stable. Fig. 3. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.3). Fig. 3. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.3). Fig. 4. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.26). Fig. 4. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.26). Example 5.3 Let us consider the system (2.1) with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, where \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -2.1 & 0.5 \\ 1 & 1.2 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.5 \end{array}}\right],\ G=\left[{\begin{array}{@{}c@{}} 0.6 \\ 0.1 \end{array}}\right], \ F=0.5. \end{equation*} Given $$c_1=2.5$$, $$c_2=10$$, T = 5, N = 15, d = 1, R = I, $$\eta =0.9$$, $$\alpha _1=0.01$$, $$\beta _1=0.05$$, $$\alpha _2=1.1$$, $$\beta _2=0.01$$ and the initial condition $$x^h(0,k)=1.5$$, $$x^v(t,0)=0.5$$. Using LMI toolbox of MATLAB and Theorem 4.3, a feasible solution of the LMIs (4.16a), (4.16b), (3.28a) and (4.20) can be derived as follows: \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 23.4079 & 0\\ * & 2.2412 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}c@{}} -15.6388\\ -1.7929 \end{array}}\right],\ M= 1.7078. \end{equation*} Moreover, the state feedback controller is given by K = [−0.6681, −0.8000]. Figures 5 and 6 show the weighted-state values $$x^T(t,k)Rx(t,k)$$ of systems (4.1) and (4.15) with the same initial condition, respectively. Fig. 5. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.1). Fig. 5. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.1). Fig. 6. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.15). Fig. 6. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.15). 6. Conclusions In this paper, we have investigated the FRB and finite-region stabilization problems for two-dimensional continuous-discrete linear Roesser models subject to two kinds of disturbances. First, the definitions of FRS and FRB for two-dimensional continuous-discrete linear system were put forward. Next, sufficient condition of FRB for two-dimensional continuous-discrete system subject to energy-bounded disturbances and sufficient condition of FRS for two-dimensional continuous-discrete system were established. By employing the given conditions, the sufficient conditions for the finite-region stabilization via state feedback were obtained. The conditions then were turned into optimization problems involving LMIs. Moreover, the sufficient conditions of FRB and finite-region stabilization for two-dimensional continuous-discrete system with disturbances generated by an external system were presented. Finally, numerical examples were provided to illustrate the proposed results. It should be pointed out that the future research topics may include the robust finite-region control synthesis of two-dimensional systems subject to uncertain time-varying parameters. Funding National Natural Science Foundation of China (61573007, 61603188). References Ahn , C. K. , Wu , L. & Shi , P. ( 2016 ) Stochastic stability analysis for 2-D Roesser systems with multiplicative noise . Automatica , 69 , 356 – 363 . Google Scholar CrossRef Search ADS Amato , F. & Ariola , M. ( 2005 ) Finite-time control of discrete-time linear systems . IEEE Trans. Automat. Control , 50 , 724 – 729 . Google Scholar CrossRef Search ADS Amato , F. , Ariola , M. & Cosentino , C. ( 2010 ) Finite-time stability of linear time-varying systems: analysis and controller design . IEEE Trans. Automat. Control , 55 , 1003 – 1008 . Google Scholar CrossRef Search ADS Amato , F. , Ariola , M. & Dorato , P. ( 2001 ) Finite-time control of linear systems subject to parametric uncertainties and disturbances . Automatica , 37 , 1459 – 1463 . Google Scholar CrossRef Search ADS Bachelier , O. , Paszke , W. & Mehdi , D. ( 2008 ) On the Kalman-Yakubovich-Popov lemma and the multidimensional models . Multidimens. Syst. Signal Process. , 19 , 425 – 447 . Google Scholar CrossRef Search ADS Bachelier , O. , Paszke , W. , Yeganefar , N. , Mehdi , D. & Cherifi , A. ( 2016 ) LMI stability conditions for 2D Roesser models . IEEE Trans. Automat. Control , 61 , 766 – 770 . Google Scholar CrossRef Search ADS Benton , S. E. , Rogers , E. & Owens , D. H. ( 2002 ) Stability conditions for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Int. J. Control , 75 , 55 – 60 . Google Scholar CrossRef Search ADS Bliman , P. A. ( 2002 ) Lyapunov equation for the stability of 2-D systems . Multidimens. Syst. Signal Process. , 13 , 202 – 222 . Google Scholar CrossRef Search ADS Bouzidi , Y. , Quadrat , A. & Rouillier , F. ( 2015 ) Computer algebra methods for testing the structural stability of multidimensional systems . Proc. 9th International Workshop on Multidimensional Systems. Vila Real, Portugal . Boyd , S. , Ghaoui , L. El , Feron , E. & Balakrishnan , V. ( 1994 ) Linear Matrix Inequalities in System and Control Theory . Philadelphia : SIAM . Google Scholar CrossRef Search ADS Chesi , G. & Middleton , R. H. ( 2014 ) Necessary and sufficient LMI conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems . IEEE Trans. Automat. Control , 59 , 996 – 1007 . Google Scholar CrossRef Search ADS Dorato , P. ( 1961 ) Short time stability in linear time-varying systems . Proc. IRE Int. Convention Record Pt . 4, 83 – 87 . Ebihara , Y. , Ito , Y. & Hagiwara , T. ( 2006 ) Exact stability analysis of 2-D systems using LMIs . IEEE Trans. Automat. Control , 51 , 1509 – 1513 . Google Scholar CrossRef Search ADS Fan , H. & Wen , C. ( 2002 ) A sufficient condition on the exponential stability of two-dimensional (2-D) shift-variant systems . IEEE Trans. Automat. Control , 47 , 647 – 655 . Google Scholar CrossRef Search ADS Feng , Z. , Wu , Q. & Xu , L. ( 2012 ) $$H_\infty$$ control of linear multidimensional discrete systems . Multidimens. Syst. Signal Process. , 23 , 381 – 411 . Google Scholar CrossRef Search ADS Fornasini , E. & Marchesini , G. ( 1976 ) State-space realization theory of two-dimensional filters . IEEE Trans. Automat. Control , 21 , 484 – 492 . Google Scholar CrossRef Search ADS Galkowski , K. , Emelianov , M. A. , Pakshin , P. V. & Rogers , E. ( 2016 ) Vector Lyapunov functions for stability and stabilization of differential repetitive processes . J. Comput. Syst. Sci. Int. , 55 , 503 – 514 . Google Scholar CrossRef Search ADS Haddad , W. M. & L’Afflitto , A. ( 2016 ) Finite-time stabilization and optimal feedback control . IEEE Trans. Automat. Control , 61 , 1069 – 1074 . Google Scholar CrossRef Search ADS Hu , G. D. & Liu , M. ( 2006 ) Simple criteria for stability of two-dimensional linear systems . IEEE Trans. Signal Process. , 53 , 4720 – 4723 . Jammazi , C. ( 2010 ) On a sufficient condition for finite-time partial stability and stabilization: applications . IMA J. Math. Control Inform. , 27 , 29 – 56 . Google Scholar CrossRef Search ADS Kamenkov , G. V. ( 1953 ) On stability of motion over a finite interval of time . J. Appl. Math. Mech. USSR , 17 , 529 – 540 (in Russian) . Knorn , S. & Middleton , R. H. ( 2013a ) Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs . IEEE Trans. Automat. Control , 58 , 2579 – 2590 . Google Scholar CrossRef Search ADS Knorn , S. & Middleton , R. H. ( 2013b ) Two-dimensional analysis of string stability of nonlinear vehicle strings . IEEE Conference on Decision and Control , pp. 5864 – 5869 . Knorn , S. & Middleton , R. H. ( 2016 ) Asymptotic and exponential stability of nonlinear two-dimensional continuous-discrete Roesser models . Syst. Control Lett. , 93 , 35 – 42 . Google Scholar CrossRef Search ADS Lin , Z. & Bruton , L.T. ( 1989 ) BIBO stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind . IEEE Trans. Circuits Syst. , 36 , 244 – 254 . Google Scholar CrossRef Search ADS Lu , W. S. & Antoniou , A. ( 1992 ) Two-Dimensional Digital Filters . New York, NY, USA : Marcel Dekker, Inc . Marszalek , W. ( 1984 ) Two-dimensional state-space discrete models for hyperbolic partial differential equations . Appl. Math. Model. , 8 , 11 – 14 . Google Scholar CrossRef Search ADS Moulay , E. & Perruquetti , W. ( 2008 ) Finite time stability conditions for non-autonomous continuous systems . Int. J. Control , 81 , 797 – 803 . Google Scholar CrossRef Search ADS Nersesov , S. G. & Haddad , W. M. ( 2008 ) Finite-time stabilization of nonlinear impulsive dynamical systems . Nonlinear Anal. Hybrid Syst. , 2 , 832 – 845 . Google Scholar CrossRef Search ADS Owens , D. H. & Rogers , E. ( 2002 ) Stability analysis for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Syst. Control Lett. , 37 , 55 – 60 . Google Scholar CrossRef Search ADS Pakshin , P. , Emelianova , J. , Emelianov , M. , Galkowski , K. & Rogers , E. ( 2016 ) Dissipativity and stabilization of nonlinear repetitive processes . Syst. Control Lett. , 91 , 14 – 20 . Google Scholar CrossRef Search ADS Roesser , R. ( 1975 ) A discrete state-space model for linear image processing . IEEE Trans. Automat. Control , 20 , 1 – 10 . Google Scholar CrossRef Search ADS Rogers , E. , Galkowski , K. & Owens , D.H. ( 2007 ) Control Systems Theory and Applications for Linear Repetitive Processes . Berlin Heidelberg : Springer . Rogers , E. & Owens , D. H. ( 1992 ) Stability Analysis for Linear Repetitive Processes . Berlin Heidelberg : Springer . Google Scholar CrossRef Search ADS Rogers , E. & Owens , D. H. ( 2002 ) Kronecker product based stability tests and performance bounds for a class of 2D continuous–discrete linear systems . Linear Algebra Appl. , 353 , 33 – 52 . Google Scholar CrossRef Search ADS Seo , S. , Shim , H. & Jin , H. S. ( 2011 ) Finite-time stabilizing dynamic control of uncertain multi-input linear systems . IMA J. Math. Control Inform. , 28 , 525 – 537 . Google Scholar CrossRef Search ADS Singh , V. ( 2014 ) Stability analysis of 2-D linear discrete systems based on the Fornasini-Marchesini second model: stability with asymmetric Lyapunov matrix . Digit. Signal Process. , 26 , 183 – 186 . Google Scholar CrossRef Search ADS Tan , F. , Zhou , B. & Duan , G. R. ( 2016 ) Finite-time stabilization of linear time-varying systems by piecewise constant feedback . Automatica , 68 , 277 – 285 . Google Scholar CrossRef Search ADS Tichonov , A. N. & Samarsky , A. A. ( 1963 ) Equations of Mathematical Physics . Warsaw : PWN (in Polish) . Wang , L. , Wang , W. , Gao , J. & Chen , W. ( 2017 ) Stability and robust stabilization of 2-D continuous-discrete systems in Roesser model based on KYP lemma . Multidimens. Syst. Signal Process. , 28 , 251 – 264 . Google Scholar CrossRef Search ADS Xiao , Y. ( 2001 ) Stability test for 2-D continuous-discrete systems . IEEE Conference on Decision and Control , pp. 3649 – 3654 . Xie , X. , Zhang , Z. & Hu , S. ( 2015 ) Control synthesis of Roesser type discrete-time 2-D T-S fuzzy systems via a multi-instant fuzzy state-feedback control scheme . Neurocomputing , 151 , 1384 – 1391 . Google Scholar CrossRef Search ADS Xu , L. , Wu , L. , Wu , Q. , Lin , Z. & Xiao , Y. ( 2005 ) On realization of 2D discrete systems by Fornasini-Marchesini model . Int. J. Control Autom. , 3 , 631 – 639 . Zhang , G. & Wang , W. ( 2016a ) Finite-region stability and boundedness for discrete 2-D Fornasini-Marchesini second models . Int. J. Systems Sci. , 48 , 778 – 787 . Google Scholar CrossRef Search ADS Zhang , G. & Wang , W. ( 2016b ) Finite-region stability and finite-region boundedness for 2-D Roesser models . Math. Method Appl. Sci. , 39 , 5757 – 5769 . Google Scholar CrossRef Search ADS Zhang , Y. , Liu , C. & Mu , X. ( 2012 ) Robust finite-time stabilization of uncertain singular Markovian jump systems . Appl. Math. Model. , 36 , 5109 – 5121 . Google Scholar CrossRef Search ADS Zhang , Y. , Shi , P. , Nguang , S. K. & Karimi , H. R. ( 2014 ) Observer-based finite-time fuzzy $$H_\infty$$ control for discrete-time systems with stochastic jumps and time-delays . Signal Process. , 97 , 252 – 261 . Google Scholar CrossRef Search ADS Zhang , Y. , Shi , P. , Nguang , S. K. & Song , Y. ( 2014 ) Robust finite-time $$H_\infty$$ control for uncertain discrete-time singular systems with Markovian jumps . IET Control Theory Appl. , 8 , 1105 – 1111 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press
Finite-region boundedness and stabilization for 2D continuous-discrete systems in Roesser model
, Volume Advance Article – May 19, 2018
25 pages
/lp/ou_press/finite-region-boundedness-and-stabilization-for-2d-continuous-discrete-6OX6mSZdVT
Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dny017
Publisher site
See Article on Publisher Site
Abstract
Abstract This paper investigates the finite-region boundedness (FRB) and stabilization problems for two-dimensional continuous-discrete linear Roesser models subject to two kinds of disturbances. For two-dimensional continuous-discrete system, we first put forward the concepts of finite-region stability and FRB. Then, by establishing special recursive formulas, sufficient conditions of FRB for two-dimensional continuous-discrete systems with two kinds of disturbances are formulated. Furthermore, we analyze the finite-region stabilization issues for the corresponding two-dimensional continuous-discrete systems and give generic sufficient conditions and sufficient conditions that can be verified by linear matrix inequalities for designing the state feedback controllers which ensure the closed-loop systems FRB. Finally, viable experimental results are demonstrated by illustrative examples. 1. Introduction The study of two-dimensional systems has a long history, with some meaningful works (Roesser, 1975; Fornasini & Marchesini, 1976; Lin & Bruton, 1989; Lu & Antoniou, 1992; Xu et al., 2005; Hu & Liu, 2006; Singh, 2014; Xie et al., 2015; Ahn et al., 2016) such as systems theory, stability properties and practical applications. Among all research topics of two-dimensional systems, stability as the most fundamental property has obtained fruitful achievements. To list some of these, Bachelier et al. (2016) proposed linear matrix inequality (LMI) stability criteria for two-dimensional systems by relaxing the polynomial-based texts of stability into that of LMIs. Bouzidi et al. (2015) presented computer algebra-based methods for testing the structural stability of n-dimensional discrete linear systems. Other works dedicated to discrete two-dimensional models are Bliman (2002) and Ebihara et al. (2006). Two-dimensional continuous-discrete systems arise naturally in several emerging application areas, for example, in analysis of repetitive processes (Rogers & Owens, 1992; Rogers et al., 2007) and in irrigation channels (Knorn & Middleton, 2013a). As such, the research on two-dimensional continuous-discrete systems has been a hot area in control field. Especially, the stability analysis for two-dimensional continuous-discrete systems has attracted much attention of some researchers over the last few decades, and several interesting findings in linear and non-linear frameworks have been obtained (see Xiao, 2001; Benton et al., 2002; Owens & Rogers, 2002; Rogers & Owens, 2002; Knorn & Middleton, 2013a;,Chesi & Middleton, 2014; Knorn & Middleton, 2016; Galkowski et al., 2016; Pakshin et al., 2016; Wang et al., 2017 and references therein). For example, in the linear setting, Xiao (2001) considered three models of two-dimensional continuous-discrete systems and gave sufficient and necessary conditions for their Lyapunov asymptotic stability (LAS)-based two-dimensional characteristic polynomial. In Owens & Rogers (2002), a stability analysis for differential linear repetitive processes, a class of two-dimensional continuous-discrete linear systems, was given in the presence of a general set of boundary conditions. These results further were extended to stability tests based on a one-dimensional Lyapunov equation and strictly bounded real lemma in Benton et al. (2002). Chesi & Middleton (2014) proposed necessary and sufficient conditions that can be checked with convex optimization for stability and performance analysis of two-dimensional continuous-discrete systems. For non-linear systems, Knorn & Middleton (2013b) modeled the homogeneous, unidirectional non-linear vehicle strings as a general two-dimensional continuous-discrete non-linear system and presented a sufficient condition for stability of this system. In Galkowski et al. (2016), the exponential stability conditions for non-linear differential repetitive processes were established by using a vector Lyapunov function-based approach. However, most of these results related to stability were focused on LAS or exponential stability. Apart from LAS or exponential stability, finite-time stability (FTS) is also a basic concept in the stability analysis. The concept of FTS was first introduced in Kamenkov (1953) and reintroduced by Dorato in Dorato (1961) which is related to dynamical systems whose state does not exceed some bound during the specified time interval. It is important to note that FTS and LAS are completely independent concepts. FTS aims at analyzing transient behavior of a system within a finite (possibly short) interval rather than the asymptotic behavior within a sufficiently long (in principle, infinite) time interval. In general, the characteristic of FTS does not guarantee stability in the sense of Lyapunov and vice versa. In addition, it should be noted that the FTS considered here is unrelated to the one adopted in some other works (Moulay & Perruquetti, 2008; Nersesov & Haddad, 2008), where the authors focus on the stability analysis of non-linear systems whose trajectories converge to an equilibrium point in finite time. Recent years have witnessed growing interests on FTS (see Amato et al., 2001; Amato & Ariola, 2005; Amato et al., 2010; Jammazi, 2010; Seo et al., 2011; Zhang et al., 2012, 201b, a; Haddad & L’Afflitto, 2016; Tan et al., 2016 and references therein) because it plays vital roles in many practical applications, for example, for many dynamic systems the state trajectories are required to stay within a desirable operative range over a certain time interval to fulfill hardware constraints or to maintain linearity of the system. What these literatures we mentioned here are about one-dimensional systems; little progress related to this problem has been made for two-dimensional systems due to their dynamical and structural complexity. Until recently, the authors in Zhang & Wang (2016a,b) studied the problems of finite-region stability (FRS) and finite-region boundedness (FRB) for two-dimensional discrete Roesser models and Fornasini-Marchesini second models. For two-dimensional continuous-discrete systems, it is worth mentioning that there are some results on the problems of exponential stability, weak stability and asymptotic stability over bounded region for repetitive processes (see Rogers et al., 2007; Galkowski et al., 2016; Pakshin et al., 2016 and references therein), where the variable $$x^h(t,k)$$ changes on finite interval [0, T]. Clearly, these problems are completely different from the finite-region control problem considered in Zhang & Wang (2016a,b) in that their methods can exhibit the transient performance of two-dimensional discrete systems over a given finite region by setting the state variables of the system less than a particular threshold. Therefore, it is necessary and important to consider the FRS and FRB problems for two-dimensional continuous-discrete systems, which motivates our present research. It is worth noting that though the two-dimensional system theory is developed from one-dimensional system theory, it is not a parallel promotion of one-dimensional system theory. There exist deep and substantial differences between one-dimensional case and two-dimensional case (Fan & Wen, 2002; Feng et al., 2012). In the two-dimensional case, the system depends on two variables and the initial conditions consist of infinite vectors, but for a one-dimensional system, the initial condition is a single vector. Moreover, most results obtained for one-dimensional systems cannot be straightforwardly extended to two-dimensional systems. For example, stability texts for one-dimensional systems are based on a simple calculation of the eigenvalues of a matrix or the roots of a polynomial, but this is not the case of two-dimensional systems, as stability conditions are given in terms of multidimensional polynomials. These factors make the analysis of FRS and FRB for two-dimensional continuous-discrete systems much more complicated and difficult than one-dimensional continuous or discrete systems. On the other hand, the existing results on two-dimensional discrete systems (Zhang & Wang, 2016a,b) cannot be immediately extended to two-dimensional continuous-discrete systems. This is because two-dimensional continuous-discrete systems are more complicated and technically more difficult to tackle than two-dimensional discrete systems. To fill this gap, it is necessary for us to further study the finite-region control problems of two-dimensional continuous-discrete systems deeply. In this paper, we deal with two-dimensional continuous-discrete linear Roesser models subject to two classes of disturbances. We first put forward the definitions of FRS and FRB for two-dimensional continuous-discrete system. Then, by establishing special recursive formulas, sufficient condition of FRB for two-dimensional continuous-discrete system with energy-bounded disturbances and sufficient condition of FRS for two-dimensional continuous-discrete system are given. Furthermore, by using the given sufficient conditions, generic sufficient conditions and sufficient conditions that are solvable by LMIs for the existence of state feedback controllers which ensure the corresponding closed-loop systems FRB or FRS are derived. We also show that, under stronger assumptions, our sufficient conditions for finite-region stabilization recover asymptotic stability. Finally, we address the FRB and finite-region stabilization problems for two-dimensional continuous-discrete system with disturbances generated by an external system and present sufficient condition of FRB and generic sufficient condition and sufficient condition that can be checked with LMIs of FRB via state feedback for the corresponding systems. The paper is organized as follows. In Section 2, the definitions of FRS and FRB for two-dimensional continuous-discrete linear system are proposed. In Section 3, the FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with the first disturbances are considered, and corresponding sufficient conditions and LMIs conditions are derived. Section 4 presents the results for the two-dimensional continuous-discrete system subject to the second disturbances. Numerical examples are given in Section 5 to show the effectiveness of the proposed approaches. Finally, conclusions are drawn in Section 6. Notations In this paper, we assume that vectors and matrices are real and have appropriate dimensions. $$N^+$$ denotes a set of positive integers, $$R^n$$ is the n-dimensional space with inner product $$x^Ty$$. A > 0 means that the matrix A is symmetric positive definite. $$A^T$$ denotes the transpose of matrix A, I represents the identity matrix. For a matrix A, its eigenvalue, maximum eigenvalue and minimum eigenvalue are denoted by $$\lambda (A)$$, $$\lambda _{\max }(A)$$ and $$\lambda _{\min }(A)$$, respectively. The symmetric terms in a matrix is represented by *. diag{⋅} denotes a black-diagonal matrix. 2. Preliminaries In this paper, we consider the following two-dimensional continuous-discrete linear system for Roesser model: $$x^{+}(t,k) = A x(t,k) + B u(t,k)+ Gw(t,k),$$ (2.1) where $$x^{+}(t,k)=\left [{\begin{array}{@{}c@{}} \frac{{\partial x^h(t,k)}}{{\partial t}} \\ x^v(t,k+1) \end{array}} \right ],$$ $$x(t,k)= \left [{\begin{array}{@{}c@{}} x^h(t,k) \\ x^v(t,k) \end{array}} \right ] \in R^n$$ is the state vector, $$u(t,k) \in R^p$$ is the two-dimensional control input, $$w(t,k)\in R^r$$ is the exogenous disturbance. t, k are the horizontal continuous variable and vertical discrete variable, respectively. $$A= \left [\begin{array}{@{}cc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array} \right ],$$ $$B= \left [ {\begin{array}{@{}c@{}} B_1\\ B_2 \end{array}} \right ],$$ $$G= \left [ {\begin{array}{@{}c@{}} G_1\\ G_2 \end{array}} \right ]$$ are real matrices with appropriate dimensions. $$x_0 (t,k)= \left [ {\begin{array}{@{}c@{}} x^{h}(0,k) \\ x^{v}(t,0) \end{array}} \right ]$$ is used to represent the boundary condition. Define the finite region for two-dimensional continuous-discrete system (2.1) as follows: $$(T,N)=\{(t,k) | 0\leqslant t\leqslant T, 0\leqslant k\leqslant N; T>0, N\in N^{+}\}.$$ (2.2) The main aim of this note is to analyze the transient performance of system (2.1) over a given finite region by setting the state variables less than a particular threshold. Inspired by the definitions of FTS for one-dimensional systems in Amato et al. (2001) and Amato & Ariola (2005) and two-dimensional FRS for Roesser model in Zhang & Wang (2016b), the concept of FRS for the two-dimensional continuous-discrete linear system (2.1) in the uncontrolled case with no disturbances can be formalized. Definition 2.1 Given positive scalars $$c_1$$, $$c_2$$, T, N, with $$c_1<c_2$$, $$N\in N^{+}$$ and a matrix R > 0, where $$R=\textrm{diag}\ \{R_1, R_2\}$$, the two-dimensional continuous-discrete linear system $$x^{+}(t,k)=Ax(t,k)$$ (2.3) is said to be finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, if \begin{equation*} x^T_0(t,k) R x_0 (t,k) \leqslant c_1 \Rightarrow x^T (t,k) R x(t,k)< c_2,\quad \forall\ t\in [0,T], k \in \{1,\cdots,N\}. \end{equation*} Remark 2.1 Similar to one-dimensional continuous and discrete cases, LAS and FRS are complete independent concepts for two-dimensional continuous-discrete systems. A system which is FRS may not be LAS and vice versa. If we limit our attention to what happens within a finite region, we can consider Lyapunov stability as an ‘additional’ requirement. Under stronger assumptions, the conditions presented in this paper (see Remark 3.1) include a special case where a system being both finite-region stable and Lyapunov stable. Next, we consider the situation when the state is subject to some external signal disturbances $$\mathscr{W}(d)$$. This leads to the definition of FRB, which covers Definition 2.1 as a special case. In this paper, we will address two kinds of external signal disturbances: (i) energy-bounded disturbances, $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$; (ii) disturbances generated by an external system, $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, where F is a real matrix with appropriate dimensions. Definition 2.2 Given positive scalars $$c_1$$, $$c_2$$, d, T, N, with $$c_1 <c_2$$, $$N\in N^{+}$$ and a matrix R > 0, where $$R=\textrm{diag}\ \{R_1, R_2\}$$, the system $$x^{+}(t,k)= A x(t,k)+ Gw(t,k)$$ (2.4) is said to be finite-region bounded with respect to $$(c_1, c_2, T, N, R, d)$$, if \begin{equation*} x^T_0(t,k)R x_0(t,k)\leqslant c_1 \Rightarrow x^T (t,k) R x(t,k)< c_2,\quad \forall\ t\in [0,T], k\in \{1,\cdots,N\}, \end{equation*} for all $$w(t,k)\in \mathscr{W}(d)$$. Remark 2.2 When w(t, k) = 0, the concept of FRB given in Definition 2.2 is consistent with the definition of FRS in Definition 2.1. 3. FRB and stabilization under the disturbances of the first case In this section, we focus on FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with the energy-bounded external disturbances $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$. Firstly, we present a sufficient condition of the system (2.1) in the uncontrolled case. Theorem 3.1 System (2.4) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag} \{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, $$\gamma _l$$, $$\alpha _2+\beta _1> 1$$ and matrices $$P_l>0$$, S > 0, where l = 1, 2, such that the following conditions hold: $$\qquad\qquad\qquad\quad\;\left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11}& - \alpha_1 P_1 P_1A_{12} &P_1G_1\\ * &-\beta_1 P_2& 0\\ * &* &-\gamma_1 S \end{array}}\right]<0,$$ (3.1a) $$\left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[4pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[4pt] * & * & G_2^T P_2 G_2 -\gamma_2 S \end{array}}\right]<0,$$ (3.1b) $${x^h}^T (0,k) R_1x^h(0,k) \leqslant \eta c_1,\quad{x^v}^T(t,0)R_2 x^v (t,0)\leqslant (1-\eta)c_1,$$ (3.1c) $$\qquad\qquad\quad\,\;\; \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_1 )} \eta c_1 + \frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_1)}\beta_1(1-\eta)c_2T+ \frac{\lambda_{\max}(S)} {\lambda_{\min}(\widetilde P_1 )} \gamma_1 d T < \eta c_2 e^{-\alpha_1 T},$$ (3.1d) $$\frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_2)} (1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_2)} N \alpha_0 \beta_2\eta c_2 + \frac{\lambda_{\max}(S)} {\lambda_{\min}(\widetilde P_2)}N \alpha_0 \gamma_2 d <(1- \eta) c_2,$$ (3.1e) where $$\alpha _0 =\textrm{max}\{1, \alpha _{2}^{N}\}$$, $$\widetilde{P}_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. For system (2.4) and $$P_1>0$$, $$P_2>0$$, define the following Lyapunov functions \begin{equation*} V_1(x^h(t,k))={x^h}^T (t,k)P_1 x^h(t,k),\ \ V_2(x^v(t,k))={x^v}^T(t,k)P_2x^v(t,k), \end{equation*} denote $$\psi (t,k)=[x^T(t,k)\ w^T(t,k)]^T$$, then, we have \begin{align} &\frac{\partial V_1 (x^h(t,k))}{\partial t}-\alpha_1 V_1(x^h(t,k))-\beta_1 V_2(x^v(t,k))- \gamma_1 w^T(t,k)Sw(t,k) \nonumber \\ &\quad= \psi^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1A_{12} & P_1G_1\\ * & -\beta_1P_2 & 0\\ * & *& -\gamma_1 S \end{array}}\right] \psi(t,k),\nonumber\\ &\quad\quad V_2(x^v(t,k+1))-\alpha_2 V_2(x^v(t,k))-\beta_2 V_1(x^h(t,k))-\gamma_2w^T(t,k)Sw(t,k) \nonumber \\ &\quad=\psi^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[4pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[4pt] * & * & G_2^T P_2 G_2 -\gamma_2 S \end{array}}\right]\psi(t,k).\nonumber \end{align} According to conditions (3.1a) and (3.1b), then $$\frac{\partial V_1 (x^h(t,k))}{\partial t} <\alpha_1 V_1(x^h(t,k))+\beta_1 V_2(x^v(t,k))+ \gamma_1 w^T(t,k)Sw(t,k),$$ (3.2) $$V_2(x^v(t,k+1))<\alpha_2 V_2(x^v(t,k))+\beta_2 V_1(x^h(t,k))+\gamma_2w^T(t,k)Sw(t,k).\quad$$ (3.3) Integrating from 0 to t for (3.2), with t ∈ [0, T], we obtain \begin{align} V_1 (x^h(t,k)) <& V_1(x^h(0,k)) e^{\alpha_1 t}+\beta_1 \int_{0}^{t} e^{\alpha_1( t- \tau)}{x^v}^T(\tau,k) P_2 x^v(\tau,k)\,\mathrm{d}\tau \nonumber \\ &+\gamma_1 \int_{0}^{t} e^{\alpha_1( t-\tau)}w^T(\tau,k)Sw(\tau,k)\,\mathrm{d}\tau \nonumber \\ \leqslant&\lambda_{\max}(\widetilde P_1){x^h}^T(0,k) R_1 x^h(0,k) e^{\alpha_1 T}+ \lambda_{\max}(\widetilde P_2) \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k)R_2 x^v(t,k)\,\mathrm{d}t \nonumber\\ &+ \lambda_{\max} (S) \gamma_1 e^{\alpha_1 T} d T . \end{align} (3.4) Fixing t ∈ [0, T], for k ∈ {1, ⋯ , N}, by iteration to (3.3), we have \begin{align} V_2(x^v(t,k)) <& \alpha_2 ^k V_2(x^v(t,0)) + \sum\limits_{l=0}^{k-1}\alpha_2^{k-l-1} \left[ \beta_2 {x^h}^T (t,l)P_1 x^h(t,l) + \gamma_2 w^T(t,l)Sw(t,l)\right]\nonumber \\ \leqslant& \lambda_{\max} (\widetilde P_2) {x^v}^T(t,0) R_2 x^v(t,0) \alpha_0 \nonumber \\ &+ \sum\limits_{l=0}^{k-1}\alpha_2^{k-l-1} \left[\lambda_{\max}(\widetilde P_1) \beta_2 {x^h}^T (t,l)R_1 x^h(t,l) + \lambda_{\max}(S) \gamma_2 d \right], \end{align} (3.5) where $$\alpha _0 ={\max }\{1, \alpha _2 ^N\}$$. On the other hand, $$V_1 (x^h(t,k))={x^h}^T (t,k)P_1 x^h(t,k) \geqslant \lambda_{\min}(\widetilde P_1) {x^h}^T(t,k) R_1 x^h(t,k),$$ (3.6) $$V_2 (x^v(t,k))={x^v}^T (t,k)P_2 x^v(t,k) \geqslant \lambda_{\min} (\widetilde P_2) {x^v}^T(t,k) R_2 x^v(t,k).$$ (3.7) It follows from condition (3.1c) and (3.4)–(3.7) that \begin{align} {x^h}^T(t,k) R_1 x^h(t,k) <& \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_1)}\eta c_1 e^{\alpha_1 T} + \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_1)}\beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T (t,k)R_2 x^v(t,k)\, \mathrm{d}t \nonumber \\ &+ \frac{ \lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_1)} \gamma_1 e^{\alpha_1 T} \textrm{d} T, \end{align} (3.8) \begin{align} {x^v}^T (t,k) R_2 x^v(t,k) <& \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1}\beta_2 {x^h}^T (t,l)R_1 x^h(t,l) \nonumber\\ & + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1}\gamma_2 \textrm{d}. \end{align} (3.9) Next, we will prove that the following inequalities hold for any t ∈ [0, T], k ∈{1, ⋯ , N}: $${x^h}^T(t,k)R_1 x^h(t,k) <\eta c_2,\ {x^v}^T(t,k)R_2 x^v(t,k) < (1-\eta) c_2.$$ (3.10) Noting that $$c_1<c_2$$, from condition (3.1c), we have $${x^h}^T(0,k)R_1 x^h(0,k) <\eta c_2,\ {x^v}^T(t,0)R_2 x^v(t,0) < (1-\eta) c_2.$$ (3.11) Setting k = 0 in (3.8) and using (3.11) and condition (3.1d), it is easy to obtain that $${x^h}^T(t,0) R_1 x^h(t,0) <\eta c_2.$$ (3.12) Now, we do the second mathematical induction for k to prove (3.10) holds for any t ∈ [0, T]. When k = 1, from (3.9), (3.12) and condition (3.1e), we get \begin{align} {x^v}^T (t,1) R_2 x^v(t,1) &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \beta_2 {x^h}^T (t,0)R_1 x^h(t,0) + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \gamma_2 \textrm{d} \nonumber \\ & < \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \gamma_2 \textrm{d}< (1-\eta)c_2. \end{align} (3.13) Similarly, setting k = 1 in (3.8), it is easy to obtain from (3.13) and condition (3.1d) that \begin{equation*} {x^h}^T(t,1) R_1 x^h(t,1) <\eta c_2. \end{equation*} Suppose that the conclusion (3.10) holds for k < N. When k = N, it follows from (3.9) and condition (3.1e) that \begin{align} {x^v}^T \!(t,N) R_2 x^v(t,N)\! &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{N-1} \alpha_2 ^{N-l-1}\beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} \sum\limits_{l=0}^{N-1} \alpha_2 ^{N-l-1}\gamma_2 \textrm{d} \nonumber \\ &< \frac{\lambda_{\max}(\widetilde P_2)}{\lambda_{\min}(\widetilde P_2)}(1-\eta) c_1\alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)}{\lambda_{\min}(\widetilde P_2)} N\alpha_0 \beta_2 \eta c_2 + \frac{\lambda_{\max}(S)}{\lambda_{\min}(\widetilde P_2)} N\alpha_0 \gamma_2 \textrm{d} < (1-\eta)c_2. \end{align} (3.14) Setting k = N in (3.8) and employing (3.14) and condition (3.1d), we have \begin{equation*} {x^h}^T(t,N) R_1 x^h(t,N) <\eta c_2. \end{equation*} By induction, the condition (3.10) is established for any t ∈ [0, T], k ∈ {1, ⋯ , N}. Therefore, for any t ∈ [0, T], k ∈ {1, ⋯ , N}, we have $$x^T(t,k) R x(t,k) < c_2$$, which implies that the system (2.4) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. For the simpler case of FRB, from Theorem 3.1, we can obtain the following corollary. Corollary 3.1 System (2.3) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$P_l>0$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\qquad\, \left[{\begin{array}{@{}cc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1 A_{12} \\ * & -\beta_1 P_2 \end{array}}\right]<0,$$ (3.15a) $$\left[{\begin{array}{@{}cc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} \\[2pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 \end{array}}\right]<0,$$ (3.15b) $$\quad\qquad\;\;\, \frac{\lambda_{\max}(\widetilde P_1 )} {\lambda_{\min}(\widetilde P_1 )} \eta c_1 + \frac{\lambda_{\max}(\widetilde P_2 )} {\lambda_{\min}(\widetilde P_1 )}\beta_1(1-\eta)c_2T < \eta c_2 e^{-\alpha_1 T},\\$$ (3.15c) $$\quad\;\; \frac{\lambda_{\max}(\widetilde P_2 )} {\lambda_{\min}(\widetilde P_2 )} (1-\eta) c_1 \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1 )} {\lambda_{\min}(\widetilde P_2 )}N \alpha_0 \beta_2\eta c_2 <(1- \eta) c_2,$$ (3.15d) where $$\alpha _0 ={\max }\{1, \alpha _2 ^N \}$$, $$\widetilde{P}_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. The proof can be obtained along the guidelines of Theorem 3.1. Remark 3.1 If conditions (3.15) in Corollary 3.1 are satisfied with $$\alpha _1=-\beta _2<0$$, $$\alpha _2+\beta _1=1$$, then system (2.3) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, and it is also asymptotically stable. Specifically, if $$\alpha _1=-\beta _2$$, $$\alpha _2+\beta _1=1$$, from the conditions(3.15a–3.15b), we derive \begin{equation*} \varPsi=\left[{\begin{array}{@{}cc@{}} A_{11}^T P_1 + P_1 A_{11}+ A_{21}^TP_2A_{21} &P_1 A_{12} +A_{21}^T P_2 A_{22} \\[2pt] * & A_{22}^T P_2 A_{22}-P_2 \end{array}}\right]<0. \end{equation*} By Theorem 2 in Bachelier et al. (2008), the system (2.3) is asymptotically stable. Similarly, if conditions (3.1) in Theorem 3.1 hold for $$\alpha _1=-\beta _2$$, $$\alpha _2+\beta _1=1$$, then system (2.4) is not only finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, but also asymptotically stable. It is worth noting that in the analysis of FRS, $$\varPsi$$ does not need to be negative definite but just less than $$diag\{(\alpha _1+\beta _2)P_1, (\alpha _2+\beta _1-1) P_2 \}$$. Remark 3.2 In the derivation of Theorem 3.1, we used constant Lyapunov function. It is well known that the use of constant Lyapunov function will lead to a certain conservatism. Recently, Bachelier et al. (2016) gave the solution to reduce this conservatism by relaxing the polynomial-based texts of asymptotic stability into that of LMIs. Chesi & Middleton (2014) provided the solutions to reduce or possibly cancel the conservatism by using the frequency domain method in the analysis of exponential stability. It is worth noting that the problem they considered is asymptotic stability for two-dimensional continuous-discrete systems. Though these methods reduce the conservatism, they do not apply to the analysis of FRS. This is because the matrix $$A_{11}$$ is not necessary Hurwitz and $$A_{22}$$ is not necessary Schur in the FRS analysis. The study on the reduction of conservatism will be discussed in the future. Next, we study the finite-region stabilization issue for two-dimensional continuous-discrete system (2.1) with energy-bounded external disturbances $$\mathscr{W}(d) =\{w(t,k) |w^T(t,k)w(t,k)\leqslant d\}$$. For given system (2.1), consider the following state feedback controller: $$u(t,k)=Kx(t,k),$$ (3.16) where $$K=[K_1,K_2]$$ is the constant real matrix with appropriate dimensions. Our goal is to find sufficient condition which guarantees the interconnection of (2.1) with the controller (3.16) $$x^{+}(t,k)= (A +BK) x(t,k)+Gw(t,k)$$ (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The following theorem gives the solution of this problem. Theorem 3.2 System (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, $$\gamma _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, M > 0, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} \varPsi- \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 & G_1M \\ * & -\beta_1 \widetilde{H}_2 & 0\\ * & *& -\gamma_1 M \end{array}}\right]<0,$$ (3.18a) $$\left[{\begin{array}{@{}cccc@{}} - \beta_2 \widetilde{H}_1 & 0 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[4pt] * & -\alpha_2 \widetilde{H}_2 & 0 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[4pt] * & * &-\gamma_2 M & MG_2^T \\[4pt] * & * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (3.18b) $$\frac{\eta c_1} {\lambda_{\min}( H_1 )} +\frac{\beta_1(1-\eta)c_2T} {\lambda_{\min}( H_2 )}+ \frac{\gamma_1 d T}{ \lambda_{\min}(M)} < \frac{\eta c_2e^{-\alpha_1 T} }{\lambda_{\max} ( H_1)},$$ (3.18c) $$\frac{(1-\eta) c_1\alpha_0}{\lambda_{\min}( H_2 )} + \frac{N \alpha_0 \beta_2\eta c_2}{\lambda_{\min}( H_1 )}+ \frac{N\alpha_0 \gamma_2 d}{ \lambda_{\min}(M)} <\frac{(1- \eta) c_2}{\lambda_{\max}(H_2)},$$ (3.18d) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. Let $$\widetilde{H}_l =P_l^{-1}$$, l = 1, 2 and $$M=S^{-1}$$ in Theorem 3.1. Noting that for symmetric positive definite matrices $$H_l=\widetilde{P}_l^{-1}$$, $$M=S^{-1}$$, their eigenvalues satisfy the following equations: \begin{equation*} \lambda_{\max}(H_l)=\frac{1}{\lambda_{\min}(\widetilde{P}_l)}, \lambda_{\min}( H_l )=\frac{1}{\lambda_{\max}(\widetilde{P}_l) }, \lambda_{\min}(M)=\frac{1}{\lambda_{\max}(S)}. \end{equation*} Clearly, conditions (3.1d)–(3.1e) can be rewritten as in (3.18c)–(3.18d). Now, let $$\widehat{A} =A+BK$$. If $$A_{1l}$$ and $$A_{2l}$$ in conditions (3.1a)–(3.1b) of Theorem 3.1 are replaced by $$\widehat{A}_{1l} =A_{1l}+B_1K_l$$ and $$\widehat{A}_{2l}=A_{2l}+B_2K_l$$, respectively, and in terms of $$Q_l=P_l$$, $$\widetilde{H}_l=P_l^{-1}$$ and $$M=S^{-1}$$, where l = 1, 2, we derive $$\qquad\qquad\qquad\;\,\,\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{11}^T \widetilde{H}_1^{-1} + \widetilde{H}_1^{-1} \widehat{A}_{11} - \alpha_1 \widetilde{H}_1^{-1} &\widetilde{H}_1^{-1} \widehat{A}_{12} & \widetilde{H}_1^{-1 }G_1\\[4pt] * & -\beta_1 \widetilde{H}_2^{-1} & 0\\[4pt] * & *& -\gamma_1 M^{-1} \end{array}}\right]<0,$$ (3.19) $$\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{21}^T \widetilde{H}_2^{-1}\widehat{A}_{21} - \beta_2 \widetilde{H}_1^{-1} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} \widehat{A}_{22} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} G_2\\[5pt] * & \widehat{A}_{22}^T \widetilde{H}_2^{-1} \widehat{A}_{22}-\alpha_2 \widetilde{H}_2^{-1} & \widehat{A}_{22}^T \widetilde{H}_2^{-1} G_2 \\[5pt] * & * & G_2^T \widetilde{H}_2^{-1} G_2 -\gamma_2 M^{-1} \end{array}}\right]<0.$$ (3.20) Next, we will prove that the conditions (3.19) and (3.20) are equivalent to (3.18a) and (3.18b), respectively. Pre- and post-multiplying (3.19) by the symmetric matrix $$diag\{\widetilde{H}_1,\widetilde{H}_2,M\}$$, we obtain the following equivalent condition $$\left[{\begin{array}{@{}ccc@{}} \widetilde{H}_1 \widehat{A}_{11}^T + \widehat{A}_{11} \widetilde{H}_1- \alpha_1 \widetilde{H}_1 & \widehat{A}_{12}\widetilde{H}_2 & G_1 M\\[4pt] * & -\beta_1 \widetilde{H}_2 & 0\\[4pt] * & *& -\gamma_1 M \end{array}}\right]<0.$$ (3.21) Recalling that $$\widehat{A}_{1l} =A_{1l}+B_1K_l$$, l = 1, 2, and letting $$L_l = K_l \widetilde{H}_l$$, l = 1, 2, we obtain that the condition (3.21) is equivalent to (3.18a). Applying Schur complement lemma (Boyd et al., 1994) twice to (3.20) produces $$\left[\begin{array}{@{}ccccc@{}} - \beta_2 {\widetilde{H}}_1^{-1} & 0 & {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1}G_2 & 0 & {\widehat{A}}_{21}^T\\[5pt] * & -\alpha_2 {\widetilde{H}}_2^{-1} & {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} G_2 & 0 & {\widehat{A}}_{22}^T \\[5pt] * & * & -\gamma_2 M^{-1} & G_2^T & 0 \\[5pt] * & * & * & -{\widetilde{H}}_2 & 0 \\ * & * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.22) Pre-multiplying (3.22) by $$\left[\begin{array}{@{}ccccc@{}} {\widetilde{H}}_1 & 0 & 0 & -{\widetilde{H}}_1 {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1} & {\widetilde{H}}_1 {\widehat{A}}_{21}^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & {\widetilde{H}}_2 & 0 & -{\widetilde{H}}_2 {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} & {\widetilde{H}}_2 {\widehat{A}}_{22}^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & 0 & M & 0 & -M G_2^T {\widetilde{H}}_2^{-1} \\[5pt] 0 & 0 & 0 & I & 0 \\[5pt] 0 & 0 & 0 & 0 & I \end{array}\right]$$ (3.23) and post-multiplying it by the transpose of (3.23), we have the following equivalent condition: $$\left[\begin{array}{@{}ccccc@{}} -\beta_2 {\widetilde{H}}_1 & 0 & 0 & {\widetilde{H}}_1 {\widehat{A}}_{21}^T & 0 \\[5pt] * & -\alpha_2 {\widetilde{H}}_2 & 0 & {\widetilde{H}}_2 {\widehat{A}}_{22}^T & 0 \\[5pt] * & * & -\gamma_2 M -MG_2^T {\widetilde{H}}_2^{-1}G_2M & MG_2^T & M G_2^T \\[5pt] * & * & * & -{\widetilde{H}}_2 & 0 \\[5pt] * & * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.24) Applying Schur complement lemma (Boyd et al., 1994) to (3.24) yields the following equivalent condition: $$\left[\begin{array}{@{}cccc@{}} -\beta_2 {\widetilde{H}}_1 & 0 & 0 & {\widetilde{H}}_1 {\widehat{A}}_{21}^T \\[5pt] * & -\alpha_2 {\widetilde{H}}_2 & 0 & {\widetilde{H}}_2 {\widehat{A}}_{22}^T \\[5pt] * & * & -\gamma_2 M & MG_2^T \\[5pt] * & * & * & -{\widetilde{H}}_2 \end{array}\right]<0.$$ (3.25) Noting that $$\widehat{A}_{2l} =A_{2l}+B_2K_l$$, l = 1, 2, and letting $$L_l = K_l\widetilde{H}_l$$, l = 1, 2, we obtain that the condition (3.25) is equivalent to (3.18b). From Theorem 3.1, we obtain that the system (3.17) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The following corollary of Theorem 3.2 allows us to find a state feedback controller K such that closed-loop system $$x^{+}(t,k)=(A+BK)x(t,k)$$ (3.26) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$. Corollary 3.2 System (3.26) is finite-region stable with respect to $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}cc@{}} \varPsi - \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 \\[3pt] * & -\beta_1 \widetilde{H}_2 \end{array}}\right]<0,$$ (3.27a) $$\left[{\begin{array}{@{}ccc@{}} - \beta_2 \widetilde{H}_1 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[5pt] * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (3.27b) $$\frac{\eta c_1} {\lambda_{\min}( H_1 )} + \frac{\beta_1(1-\eta)c_2T} {\lambda_{\min}( H_2 )} <\frac{\eta c_2e^{-\alpha_1 T}}{\lambda_{\max} (H_1)},$$ (3.27c) $$\frac{(1-\eta) c_1 \alpha_0} {\lambda_{\min}( H_2 )} + \frac{N \alpha_0 \beta_2\eta c_2} {\lambda_{\min}( H_1 )} <\frac{(1- \eta) c_2}{\lambda_{\max}( H_2 )},$$ (3.27d) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1 ^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. The proof can be obtained as in Theorem 3.2, applying the results of Corollary 3.1 to system (3.26). From a computational point of view, it is important to point out that the conditions (3.18c), (3.18d) in Theorem 3.2 and the conditions (3.27c), (3.27d) in Corollary 3.2 are difficult to solve. Besides, the conditions in Theorem 3.2 and Corollary 3.2 involve the unknown finite-region scalar $$\alpha _l, \beta _l, \gamma _l, l=1,2$$, which lead to Theorem 3.2 and Corollary 3.2 are difficult to solve by means of LMI Toolbox. In the following, we will show that once we have fixed values of $$\alpha _l, \beta _l, \gamma _l, l=1,2$$, the feasibility of Theorem 3.2 and Corollary 3.2 can be turned into LMIs-based feasibility problems (Boyd et al., 1994) using procedures proposed in Amato et al. (2001). Clearly, the conditions (3.18c) and (3.18d) are guaranteed by imposing additional conditions $$\, \lambda_{l1}I_l < H_l < \lambda_{l2}I_l,\ \lambda_{31}I_r <M,\quad l=1,2,\quad$$ (3.28a) $$\qquad\quad\;\, \frac{\eta c_1}{\lambda_{11}} + \frac{\beta_1(1-\eta)c_2T}{\lambda_{21}} + \frac{\gamma_1 d T}{\lambda_{31}} < \frac{\eta c_2e^{-\alpha_1T}}{\lambda_{12}},$$ (3.28b) $$\frac{(1-\eta) c_1\alpha_0}{\lambda_{21}} +\frac{ N \alpha_0 \beta_2\eta c_2}{\lambda_{11}} + \frac{N \alpha_0 \gamma_2 d}{\lambda_{31}} <\frac{(1- \eta) c_2}{\lambda_{22}}$$ (3.28c) for some positive numbers $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$. Using Schur complement (Boyd et al., 1994) to inequalities (3.28b) and (3.28c) produces $$\qquad\qquad\qquad \left[{\begin{array}{@{}cccc@{}} \lambda_{12}\eta c_2 e^{-\alpha_1T}& \lambda_{12}\sqrt{ \beta_1 (1-\eta)c_2T} & \lambda_{12}\sqrt{ \eta c_1 } &\lambda_{12} \sqrt{\gamma_1 d T} \\ * & \lambda_{21} & 0 &0\\ * & * & \lambda_{11}& 0\\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (3.29a) $$\left[{\begin{array}{@{}cccc@{}} \lambda_{22}(1-\eta) c_2 & \lambda_{22}\sqrt{ N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} & \lambda_{22} \sqrt{N \alpha_0 \gamma_2 d} \\ * & \lambda_{11} & 0 & 0 \\ * & * & \lambda_{21} & 0 \\ * & * & * & \lambda_{31} \end{array}}\right]>0.$$ (3.29b) The following theorem gives the LMI feasibility problem of Theorem 3.2. Theorem 3.3 Given system (3.17) and $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$\gamma _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, M > 0, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$ satisfying (3.1c) and the LMIs (3.18a), (3.18b), (3.28a) and (3.29), where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller K given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$ renders system (3.17) finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. Similarly, LMI feasibility problem can be derived from Corollary 3.2. Corollary 3.3 Given system (3.26) and $$(c_1,c_2,T,N,R)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$ satisfying (3.1c), the LMIs (3.27a), (3.27b) and $$\lambda_{l1}I_l < H_l < \lambda_{l2}I_l,\quad l=1,2,$$ (3.30a) $$\left[{\begin{array}{@{}ccc@{}} \lambda_{12}\eta c_2 e^{-\alpha_1T}& \lambda_{12}\sqrt{ \beta_1 (1-\eta)c_2T} & \lambda_{12} \sqrt{\eta c_1 } \\ * & \lambda_{21} & 0\\ * & * & \lambda_{11} \end{array}}\right]>0,$$ (3.30b) $$\left[{\begin{array}{@{}ccc@{}} \lambda_{22}(1-\eta) c_2 & \lambda_{22}\sqrt{ N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} \\ * & \lambda_{11} & 0\\ * & * & \lambda_{21} \end{array}}\right]>0,$$ (3.30c) where $$\varPsi =\widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\alpha _0 =\textrm{max}\{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2 ^{-1}]$$ renders system (3.26) finite-region stable with respect to $$(c_1,c_2,T,N,R)$$. 4. FRB and stabilization under the second case of disturbances In this section, we will study the FRB and finite-region stabilization issues for two-dimensional continuous-discrete system with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$. Firstly, we consider the FRB issue for the two-dimensional continuous-discrete system in the form $$x^{+}(t,k) = A x(t,k)+ Gw(k),$$ (4.1a) $$w(k+1)=Fw(k).\qquad\qquad\quad$$ (4.1b) Theorem 4.1 System (4.1) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$P_l>0$$, S > 0, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 + P_1 A_{11} - \alpha_1 P_1 & P_1A_{12} & P_1G_1\\ * & -\beta_1 P_2 & 0\\ * & *& -\alpha_1 S \end{array}}\right]<0,$$ (4.2a) $$\left[{\begin{array}{@{}ccc@{}} A_{21}^TP_2A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^T P_2 G_2\\[5pt] * & A_{22}^T P_2 A_{22}-\alpha_2 P_2 & A_{22}^T P_2 G_2 \\[5pt] * & * & G_2^T P_2 G_2 + F^T SF -\alpha_2 S \end{array}}\right]<0,$$ (4.2b) $$\quad\, \frac{\lambda_{\max}(\widetilde P_1) \eta c_1 + \lambda_{\max}(S) \gamma d} {\lambda_{\min}(\widetilde P_1 )} + \frac{\lambda_{\max}(\widetilde P_2)} {\lambda_{\min}(\widetilde P_1)}\beta_1(1-\eta)c_2T < \eta c_2 e^{-\alpha_1 T},$$ (4.2c) $$\frac{\lambda_{\max}(\widetilde P_2) (1-\eta) c_1 + \lambda_{\max}(S) d} {\lambda_{\min}(\widetilde P_2)} \alpha_0 + \frac{\lambda_{\max}(\widetilde P_1)} {\lambda_{\min}(\widetilde P_2)} N \alpha_0 \beta_2\eta c_2 <(1- \eta) c_2,$$ (4.2d) where $$\gamma = {\max }\ \{1, \|F^k\|^2 _{k=1,2,\cdots , N}\}$$, $$\alpha _0 ={\max }\ \{1, \alpha _2 ^N \}$$, $$\widetilde P_l =R_l^{-\frac{1}{2}} P_l R_l^{-\frac{1}{2}}$$, l = 1, 2. Proof. Firstly, we derive the recursive relations of the weights of state variables of system (4.1a). For simple mark, let $$z^+(t,k)=\left [{\begin{array}{@{}c@{}} x^+(t,k)\\ w(k+1)\end{array}} \right ],$$ $$z(t,k)=\left [{\begin{array}{@{}c@{}} x(t,k)\\ w(k)\end{array}} \right ],$$ then the system (4.1) is equivalent to $$z^+(t,k)= \left[{\begin{array}{@{}cc@{}} A & G\\ 0 & F \end{array}}\right] z(t,k).$$ (4.3) We define the Lyapunov functions of system (4.3) as follows \begin{equation*} V_1(z^h(t,k))= {z^h}^T(t,k) \left[{\begin{array}{@{}cc@{}} P_1 & 0\\ * & S \end{array}} \right] z^h(t,k), V_2\left(z^v(t,k)\right)= {z^v}^T(t,k) \left[{\begin{array}{@{}cc@{}} P_2 & 0\\ * & S \end{array}} \right] z^v(t,k), \nonumber \end{equation*} where $$z^h (t,k)=\left [{\begin{array}{c} x^h(t,k)\\ w(k)\end{array}} \right ], z^v (t,k)=\left [{\begin{array}{c} x^v(t,k)\\ w(k)\end{array}} \right ],$$ then it follows that \begin{align} \frac{\partial V_1 (z^h(t,k))}{\partial t} &- \alpha_1 V_1 (z^h(t,k)) -\beta_1 {x^v}^T (t,k)P_2 x^v(t,k) \nonumber\\ &\quad= z^T (t,k) \left[{\begin{array}{@{}ccc@{}} A_{11}^T P_1 +P_1 A_{11} -\alpha_1 P_1 & P_1 A_{12} & P_1 G_1\\ * & -\beta_1 P_2 & 0 \\ * & * & -\alpha_1 S \end{array}}\right] z(t,k). \end{align} (4.4) Similarly, we can obtain the following equation \begin{align} V_2(z^v(t,k+1))&-\alpha_2 V_2 (z^v(t,k)) -\beta_2 {x^h}^T (t,k) P_1 x^h(t,k) \nonumber \\ &\quad=z^T(t,k) \left[{\begin{array}{@{}ccc@{}} A_{21} ^T P_2 A_{21} - \beta_2 P_1 & A_{21}^T P_2 A_{22} & A_{21}^ T P_2 G_2 \\[5pt] * & A_{22}^T P_2 A_{22} -\alpha_2 P_2 & A_{22}^T P_2 A_{22} \\[5pt] * & * & G_2^T P_2 G_2 + F^T S F - \alpha_2 S \end{array}}\right] z(t,k). \end{align} (4.5) Using conditions (4.2a) and (4.2b) for (4.4) and (4.5), we derive $$\frac{\partial V_1 (z^h(t,k))}{\partial t} < \alpha_1 V_1(z^h(t,k))+\beta_1 {x^v}^T(t,k) P_2 x^v(t,k),$$ (4.6) $$V_2 (z^v(t,k+1))< \alpha_2 V_2(z^v(t,k))+\beta_2 {x^h}^T(t,k) P_1 x^h(t,k).\;\;\;$$ (4.7) When k = 0, integrating from 0 to t for (4.6), with t ∈ [0, T], we derive \begin{align} V_1 (z^h(t,0)) &< V_1(z^h(0,0))e ^{\alpha_1 t}+\beta_1 \int_{0}^{t}e ^{\alpha_1( t-\tau)} {x^v}^T(\tau,0)P_2 x^v(\tau,0)\,\mathrm{d}\tau \nonumber\\ &\leqslant \left[{x^h}^T (0,0)P_1x^h(0,0) +w^T(0)Sw(0)\right]e^{\alpha_1 T}+ \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,0) P_2 x^v(t,0)\,\mathrm{d}t \nonumber \\ &\leqslant \left[ \lambda_{\max}(\widetilde P_1 ) \eta c_1 + \lambda_{\max}(S) d \right]e^{\alpha_1 T}+ \lambda_{\max}(\widetilde P_2 )\beta_1 e^{\alpha_1 T}\int_{0}^{T} {x^v}^T(t,0) R_2 x^v(t,0)\,\mathrm{d}t, \end{align} (4.8) for fixed k ∈ {1, ⋯ , N}, integrating from 0 to t for (4.6), with t ∈ [0, T], we obtain \begin{align} V_1 (z^h(t,k)) &< V_1(z^h(0,k))e ^{\alpha_1 t}+\beta_1 \int_{0}^{t}e ^{\alpha_1( t-\tau)} {x^v}^T(\tau,k) P_2 x^v(\tau,k)\,\mathrm{d}\tau \nonumber\\ &\leqslant \left[ \lambda_{\max}(\widetilde P_1 ) \eta c_1 + \lambda_{\max}(S) \|F^k\|^2 d\right]e^{\alpha_1 T} + \lambda_{\max}(\widetilde P_2 )\beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k) R_2 x^v(t,k)\,\mathrm{d}t. \end{align} (4.9) Fixing t ∈ [0, T], by iteration to (4.7), we have \begin{align} V_2 (z^v(t,k)) &< \alpha_2^k V_2(x^v(t,0))+ \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l)P_1 x^h(t,l)\nonumber \\ &\leqslant \left[{x^v}^T (t,0)P_2x^v(t,0)+ w^T(0)Sw(0) \right] \alpha_0 + \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l)P_1 x^h(t,l)\nonumber \\ &\leqslant \left[\lambda_{\max} (\widetilde P_2 ) (1-\eta)c_1 + \lambda_{\max}(S) d \right] \alpha_0 + \lambda_{\max} (\widetilde P_1 ) \sum\limits_{l=0}^{k-1} \alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l) R_1 x^h(t,l), \end{align} (4.10) where $$\alpha _0=\max \{1, \alpha _2^N\}$$. On the other hand, $$V_1 (z^h(t,k))= {x^h}^T(t,k) P_1 x^h(t,k) + w^T(k)Sw(k) \geqslant \lambda_{\min} (\widetilde P_1) {x^h}^T(t,k) R_1 x^h(t,k),$$ (4.11) $$V_2 (z^v(t,k))= {x^v}^T(t,k) P_2 x^v(t,k) + w^T(k)Sw(k) \geqslant \lambda_{\min} (\widetilde P_2) {x^v}^T(t,k) R_2 x^v(t,k).$$ (4.12) Putting together (4.8), (4.9) with (4.11) and (4.10) with (4.12), respectively, we obtain \begin{align} {x^h}^T(t,k) R_1 x^h(t,k) <& \frac{\lambda_{\max}(\widetilde P_1 )\eta c_1 + \lambda_{\max}(S) \gamma d}{\lambda_{\min} (\widetilde P_1 )} e^{\alpha_1 T}\nonumber \\ &+ \frac{ \lambda_{\max}(\widetilde P_2 ) }{\lambda_{\min} (\widetilde P_1 )} \beta_1 e^{\alpha_1 T} \int_{0}^{T} {x^v}^T(t,k) R_2 x^v(t,k)\,\mathrm{d}t, \end{align} (4.13) \begin{align} {x^v}^T(t,k) R_2 x^v(t,k) <& \frac{\lambda_{\max} (\widetilde P_2 ) (1-\eta)c_1 +\lambda_{\max}(S) d}{\lambda_{\min} (\widetilde P_2 ) }\alpha_0 \nonumber \\ &+ \frac{\lambda_{\max} (\widetilde P_1 )}{\lambda_{\min} (\widetilde P_2 )} \sum\limits_{l=0}^{k-1}\alpha_2 ^{k-l-1} \beta_2 {x^h}^T (t,l) R_1 x^h(t,l), \end{align} (4.14) where $$\gamma = \textrm{max}\ \{1, \|F^k\|^2 _{k=1,2,\cdots , N}\}$$. Similar to the proof of Theorem 3.1, we can get that for any t ∈ [0, T], k ∈ {1, ⋯ , N}, $$x^T(t,k) R x (t,k) <c_2$$ holds. This implies that the two-dimensional continuous-discrete system (4.1) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. Remark 4.1 Let F = I and $$\gamma =1$$ in Theorem 4.1, the sufficient condition of the FRB for system (4.1a) subject to exogenous unknown constant disturbances can be derived. In Theorem 4.1, when the external system (4.1b) is a one-dimensional continuous system or a two-dimensional continuous-discrete system, the corresponding results can also be obtained. Remark 4.2 The sufficient condition which ensures system (2.3) finite-region stable with respect to $$(c_1,c_2,T,N,R)$$ by Theorem 4.1 is consistent with Corollary 3.1. Next, we investigate the finite-region stabilization issue for the system (2.1) with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, with the aim to find sufficient condition that ensures the system $$x^{+}(t,k)= (A +BK) x(t,k)+Gw(k),$$ (4.15a) $$w(k+1)=Fw(k)\qquad\qquad\qquad\qquad\;\,$$ (4.15b) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. The solution of this issue is given by the following theorem. Theorem 4.2 System (4.15) is finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, if there exist positive scalars $$0<\eta <1$$, $$\alpha _l$$, $$\beta _l$$, where $$\alpha _2+\beta _1> 1$$, and matrices $$H_l>0$$, M > 0, $$L_l$$, where l = 1, 2, such that the condition (3.1c) and following inequalities hold: $$\left[{\begin{array}{@{}ccc@{}} \varPsi - \alpha_1 \widetilde{H}_1 & A_{12} \widetilde{H}_2 +B_1L_2 & G_1M \\ * & -\beta_1 \widetilde{H}_2 & 0\\ * & *& -\alpha_1 M \end{array}}\right]<0,$$ (4.16a) $$\left[{\begin{array}{@{}ccccc@{}} - \beta_2 \widetilde{H}_1 & 0 & 0 & 0 & \widetilde{H}_1 A_{21}^T + L_1^T B_2^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & 0 & 0 & \widetilde{H}_2 A_{22}^T + L_2^TB_2^T\\[5pt] * & * &-\alpha_2 M & MF^T & MG_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0,$$ (4.16b) $$\frac{\eta c_1 }{\lambda_{\min}( H_1 )} + \frac{\gamma d }{ \lambda_{\min}(M)} + \frac{\beta_1(1-\eta)c_2T}{\lambda_{\min}( H_2 )} < \frac{\eta c_2e^{-\alpha_1 T}}{\lambda_{\max} ( H_1)},$$ (4.16c) $$\frac{(1-\eta) c_1 \alpha_0}{\lambda_{\min}( H_2 )} + \frac{d \alpha_0}{ \lambda_{\min}(M)} + \frac{N \alpha_0 \beta_2\eta c_2}{\lambda_{\min}( H_1 )} <\frac{(1- \eta) c_2}{\lambda_{\max}(H_2)} ,$$ (4.16d) where $$\varPsi = \widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\gamma = \textrm{max}\ \{1, \|F^k\|^2_{ k=1,2,\cdots , N} \}$$, $$\alpha _0 ={\max }\ \{1, \alpha _2 ^N \}$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. In this case, the controller K is given by $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$. Proof. Let $$\widetilde{H}_l =P_l^{-1}$$, l = 1, 2 and $$M=S^{-1}$$ in Theorem 4.1. Similar to the proof of Theorem 3.2, applying the results of Theorem 4.1 to system (4.15), the proof can be obtained. Here, we only give the proof of condition (4.16b). If $$A_{2l}$$ in condition (4.2b) of Theorem 4.1 is replaced by $$\widehat{A}_{2l}=A_{2l}+B_2K_l$$, l = 1, 2, we derive $$\left[{\begin{array}{@{}ccc@{}} \widehat{A}_{21}^T \widetilde{H}_2^{-1}\widehat{A}_{21} - \beta_2 \widetilde{H}_1^{-1} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} \widehat{A}_{22} & \widehat{A}_{21}^T \widetilde{H}_2^{-1} G_2\\[5pt] * & \widehat{A}_{22}^T \widetilde{H}_2^{-1} \widehat{A}_{22}-\alpha_2 \widetilde{H}_2^{-1} & \widehat{A}_{22}^T \widetilde{H}_2^{-1} G_2 \\[5pt] * & * & G_2^T \widetilde{H}_2^{-1} G_2 + F^T M^{-1}F -\alpha_2 M^{-1} \end{array}}\right]<0.$$ (4.17) Applying Schur complement lemma (Boyd et al., 1994) to (4.17) produces $$\left[{\begin{array}{@{}ccccc@{}} - \beta_2 \widetilde{H}_1^{-1} & 0 & 0 & 0 & \widehat{A}_{21}^T\\[5pt] * & -\alpha_2 \widetilde{H}_2^{-1} & 0 & 0 & \widehat{A}_{22}^T \\[5pt] * & * & -\alpha_2 M^{-1} & F^T & G_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0.$$ (4.18) Pre- and post-multiplying (4.18) by $$diag\{\widetilde{H}_1, \widetilde{H}_2, M,I,I\}$$, we have the following equivalent condition: $$\left[{\begin{array}{@{}ccccc@{}} -\beta_2 \widetilde{H}_1 & 0 & 0 & 0 & \widetilde{H}_1 \widehat{A}_{21}^T \\[5pt] * & -\alpha_2 \widetilde{H}_2 & 0 & 0 & \widetilde{H}_2 \widehat{A}_{22}^T \\[5pt] * & * & -\alpha_2 M & MF^T & MG_2^T \\ * & * & * & -M & 0 \\ * & * & * & * & -\widetilde{H}_2 \end{array}}\right]<0.$$ (4.19) Letting $$L_l = K_l\widetilde{H}_l$$, l = 1, 2, we finally obtain that the condition (4.19) is equivalent to (4.16b). Remark 4.3 Let F = I and $$\gamma =1$$ in Theorem 4.2, the sufficient condition of the FRB via state feedback for system (4.15a) subject to unknown constant disturbances can be derived. Similarly, when the external system is a one-dimensional continuous-discrete system or a two-dimensional continuous-discrete system, the corresponding conclusions can also be obtained. Remark 4.4 When setting $$\eta =0$$, $$\eta =1$$ in Theorems 4.1 and 4.2, respectively, we can obtain the corresponding conclusions for one-dimensional discrete linear system (Amato & Ariola, 2005) and continuous linear system (Amato et al., 2001), respectively. Similarly, Theorem 4.2 can be reducible to the following LMIs-based feasibility problem. Theorem 4.3 Given system (4.15) and $$(c_1,c_2,T,N,R,d)$$, where $$R=\textrm{diag}\{R_1, R_2\}$$, fix $$\alpha _l>0$$, $$\beta _l>0$$, $$0<\eta <1$$, where $$\alpha _2+\beta _1> 1$$, and find matrices $$H_l>0$$, M > 0, $$L_l$$ and positive scalars $$\lambda _{l1}$$, $$\lambda _{l2}$$, $$\lambda _{31}$$ satisfying (3.1c) and the LMIs (4.16a), (4.16b), (3.28a) and $$\qquad\;\;\left[{\begin{array}{cccc} \lambda_{12}\eta c_2e^{-\alpha_1T} &\lambda_{12} \sqrt{ \beta_1 (1-\eta)c_2T} &\lambda_{12} \sqrt{ \eta c_1 } &\lambda_{12} \sqrt{\gamma d} \\ * & \lambda_{21} & 0 &0\\ * & * & \lambda_{11}& 0\\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (4.20a) $$\left[{\begin{array}{cccc} \lambda_{22} (1-\eta) c_2 & \lambda_{22} \sqrt{N \alpha_0 \beta_2 \eta c_2 } & \lambda_{22}\sqrt{ (1-\eta )c_1 \alpha_0} & \lambda_{22}\sqrt{d \alpha_0} \\ * & \lambda_{11} & 0 & 0 \\ * & * & \lambda_{21} & 0 \\ * & * & * & \lambda_{31} \end{array}}\right]>0,$$ (4.20b) where $$\varPsi = \widetilde{H}_1 A_{11}^T + L_1 ^T B_1^T + A_{11}\widetilde{H}_1 +B_1 L_1$$, $$\widetilde{H}_l =R_l^{-\frac{1}{2}} H_l R_l^{-\frac{1}{2}}$$, l = 1, 2. If the problem is feasible, the controller $$K=[L_1 \widetilde{H}_1^{-1}, L_2 \widetilde{H}_2^{-1}]$$ renders system (4.15) finite-region bounded with respect to $$(c_1,c_2,T,N,R,d)$$. 5. Numerical examples In this section, numerical examples are used to illustrate the effectiveness of the proposed methods. Example 5.1 It is well known that some dynamical processes in gas absorption, water stream heating and air drying can be described by the Darboux equation (Marszalek, 1984): $$\frac{\partial^2 s(t,\tau)}{\partial t \partial \tau}=a_1 \frac{\partial s(t,\tau)}{\partial \tau}+a_2 \frac{\partial s(t,\tau)}{\partial t}+ a_0 s(t,\tau)+bf(t,\tau),$$ (5.1) where $$s(t,\tau )$$ is an unknown vector function at $$[0,t_f]$$ and $$\tau \in [0,\infty ]$$, $$a_0,a_1,a_2,b$$ are real constants and $$f(t,\tau )$$ is the input function. Taking $$r(t,\tau )=\partial s(t,\tau ) / \partial \tau -a_2 s(t,\tau )$$, $$x^h(t,k)=r(t,k)=r(t,k\Delta \tau )$$ and $$x^v(t,k)=s(t,k)=s(t,k\Delta \tau )$$, we can write the Equation (5.1) in the two-dimensional continuous-discrete system of the form (2.1). Via appropriate selection of the parameters $$a_0,a_1,a_2,b$$, we consider the system (2.1) subject to the energy-bounded external disturbances with \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -2.1 & 0.5 \\ 1.5 & 2.5 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.5 \end{array}}\right], G=\left[{\begin{array}{@{}c@{}} 0.3 \\ 0 \end{array}}\right], \end{equation*} where $$c_1=2.5$$, $$c_2=10$$, T = 5, N = 20, d = 1, R = I, $$\eta =0.7$$. Given above positive constants, the positive definite matrix R and the initial condition $$x^h(0,k)=-1.3$$, $$x^v(t,0)=0.85$$. When external disturbances satisfy $$w^T(t,k)w(t,k)\leqslant 1$$, we have considered the problem of FRB via state feedback and designed the state feedback controller by solving the feasibility problem in Theorem 3.3. When control input u(t, k) = 0, Fig. 1 shows that the open-loop system (2.4) is not finite-region bounded with respect to (2.5, 10, 5, 20, I, 1). Using LMI toolbox of MATLAB and Theorem 3.3, the LMIs (3.18a), (3.18b), (3.28a) and (3.29) are feasible with $$\alpha _1=0.03$$, $$\beta _1=0.05$$, $$\gamma _1=0.1$$, $$\alpha _2=1.03$$, $$\beta _2=0.01$$, $$\gamma _2=0.15$$, the solution is given below \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 11.6622 & 0\\ * & 1.8365 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} -11.7054\\ -3.0609 \end{array}}\right],\ M=14.5931. \end{equation*} Then, we find the state feedback controller \begin{equation*} K=[-1.0037, -1.6667]. \end{equation*} The weighted-state values $$x^T(t,k)Rx(t,k)$$ are limited by the given bound 10 for the closed-loop system (3.17) obtained after stabilization (see Fig. 2). Let us take the dynamical process in gas absorption as an example to explain the practical significance of state feedback. In the process of absorption of a gas, $$s(t,\tau )$$ in Darboux equation (5.1) denotes the quantity of gas absorbed by unit volume of the absorbent (Tichonov & Samarsky, 1963). The kinetics of absorption is represented by $$\partial s(t,\tau ) / \partial \tau$$. Let $$a_2=-\gamma$$, where $$\frac{1}{\gamma }$$ is Henry’s coefficient, then $$\partial s(t,\tau ) / \partial \tau -a_2 s(t,\tau )$$ denotes the concentration of gas in the pores of the absorbent in the layer t. In practice, due to the material constraints, the concentration of gas in the pores of the absorbent in the layer t and the quantity of gas absorbed by unit volume of the absorbent are required to stay within a desirable threshold range during the specified absorbing layer and time interval, that is, the states in system (2.1) are required to stay within a particular threshold range over a given finite-region. Therefore, when the established system is not finite-region stable, we need to use state feedback to make the system states not exceed the particular threshold. Fig. 1. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.4). Fig. 1. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.4). Fig. 2. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.17). Fig. 2. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.17). Example 5.2 Consider the system (2.1) with w(t, k) = 0, where \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -0.5 & 2.5 \\ 1.1& 2.5 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.2 \end{array}}\right]. \end{equation*} Assume that $$c_1=2$$, $$c_2=10$$, T = 5, N = 10, R = I, $$\eta =0.7$$ and $$x^h(0,k)=1.1$$, $$x^v(t,0)=-0.2$$. We have considered the FRS via state feedback and designed the state feedback controller by solving the feasibility problem in Corollary 3.3. When the system has control input u(t, k) = 0, the weighted-state values $$x^T(t,k)Rx(t,k)$$ of system (2.3) are as shown in Fig. 3, obviously, the open-loop system (2.3) is not finite-region stable with respect to (2, 10, 5, 10, I) before stabilization. Using LMI control toolbox and Corollary 3.3, the LMIs (3.27a), (3.27b) and (3.30) are feasible with $$\alpha _1=0.03$$, $$\beta _1=0.1$$, $$\alpha _2=1.1$$, $$\beta _2=0.01$$. The solution is given below \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 1.3719 & 0\\ * & 0.9058 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}c@{}} -1.1670\\ -1.8870 \end{array}}\right]. \end{equation*} Then, we obtain the state feedback controller \begin{equation*}K=[-0.8506, -2.0833].\end{equation*} The weighted-state values $$x^T(t,k)Rx(t,k)$$ of the closed-loop system (3.26) are depicted in Fig. 4. It can be seen that the closed-loop system (3.26) is finite-region stable with respect to (2, 10, 5, 10, I) but not asymptotically stable. Fig. 3. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.3). Fig. 3. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (2.3). Fig. 4. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.26). Fig. 4. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (3.26). Example 5.3 Let us consider the system (2.1) with disturbances generated by an external system $$\mathscr{W}(d) =\{w(k) |w(k+1)=Fw(k), w^T(0)w(0)\leqslant d\}$$, where \begin{equation*} A=\left[{\begin{array}{@{}cc@{}} -2.1 & 0.5 \\ 1 & 1.2 \end{array}}\right],\ B=\left[{\begin{array}{@{}c@{}} -0.2 \\ 1.5 \end{array}}\right],\ G=\left[{\begin{array}{@{}c@{}} 0.6 \\ 0.1 \end{array}}\right], \ F=0.5. \end{equation*} Given $$c_1=2.5$$, $$c_2=10$$, T = 5, N = 15, d = 1, R = I, $$\eta =0.9$$, $$\alpha _1=0.01$$, $$\beta _1=0.05$$, $$\alpha _2=1.1$$, $$\beta _2=0.01$$ and the initial condition $$x^h(0,k)=1.5$$, $$x^v(t,0)=0.5$$. Using LMI toolbox of MATLAB and Theorem 4.3, a feasible solution of the LMIs (4.16a), (4.16b), (3.28a) and (4.20) can be derived as follows: \begin{equation*} \left[{\begin{array}{@{}cc@{}} \widetilde{H}_1 & 0\\ * & \widetilde{H}_2 \end{array}}\right]= \left[{\begin{array}{@{}cc@{}} 23.4079 & 0\\ * & 2.2412 \end{array}}\right],\ \left[{\begin{array}{@{}c@{}} L_1\\ L_2 \end{array}}\right]= \left[{\begin{array}{@{}c@{}} -15.6388\\ -1.7929 \end{array}}\right],\ M= 1.7078. \end{equation*} Moreover, the state feedback controller is given by K = [−0.6681, −0.8000]. Figures 5 and 6 show the weighted-state values $$x^T(t,k)Rx(t,k)$$ of systems (4.1) and (4.15) with the same initial condition, respectively. Fig. 5. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.1). Fig. 5. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.1). Fig. 6. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.15). Fig. 6. View largeDownload slide $$x^T(t,k)Rx(t,k)$$ of system (4.15). 6. Conclusions In this paper, we have investigated the FRB and finite-region stabilization problems for two-dimensional continuous-discrete linear Roesser models subject to two kinds of disturbances. First, the definitions of FRS and FRB for two-dimensional continuous-discrete linear system were put forward. Next, sufficient condition of FRB for two-dimensional continuous-discrete system subject to energy-bounded disturbances and sufficient condition of FRS for two-dimensional continuous-discrete system were established. By employing the given conditions, the sufficient conditions for the finite-region stabilization via state feedback were obtained. The conditions then were turned into optimization problems involving LMIs. Moreover, the sufficient conditions of FRB and finite-region stabilization for two-dimensional continuous-discrete system with disturbances generated by an external system were presented. Finally, numerical examples were provided to illustrate the proposed results. It should be pointed out that the future research topics may include the robust finite-region control synthesis of two-dimensional systems subject to uncertain time-varying parameters. Funding National Natural Science Foundation of China (61573007, 61603188). References Ahn , C. K. , Wu , L. & Shi , P. ( 2016 ) Stochastic stability analysis for 2-D Roesser systems with multiplicative noise . Automatica , 69 , 356 – 363 . Google Scholar CrossRef Search ADS Amato , F. & Ariola , M. ( 2005 ) Finite-time control of discrete-time linear systems . IEEE Trans. Automat. Control , 50 , 724 – 729 . Google Scholar CrossRef Search ADS Amato , F. , Ariola , M. & Cosentino , C. ( 2010 ) Finite-time stability of linear time-varying systems: analysis and controller design . IEEE Trans. Automat. Control , 55 , 1003 – 1008 . Google Scholar CrossRef Search ADS Amato , F. , Ariola , M. & Dorato , P. ( 2001 ) Finite-time control of linear systems subject to parametric uncertainties and disturbances . Automatica , 37 , 1459 – 1463 . Google Scholar CrossRef Search ADS Bachelier , O. , Paszke , W. & Mehdi , D. ( 2008 ) On the Kalman-Yakubovich-Popov lemma and the multidimensional models . Multidimens. Syst. Signal Process. , 19 , 425 – 447 . Google Scholar CrossRef Search ADS Bachelier , O. , Paszke , W. , Yeganefar , N. , Mehdi , D. & Cherifi , A. ( 2016 ) LMI stability conditions for 2D Roesser models . IEEE Trans. Automat. Control , 61 , 766 – 770 . Google Scholar CrossRef Search ADS Benton , S. E. , Rogers , E. & Owens , D. H. ( 2002 ) Stability conditions for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Int. J. Control , 75 , 55 – 60 . Google Scholar CrossRef Search ADS Bliman , P. A. ( 2002 ) Lyapunov equation for the stability of 2-D systems . Multidimens. Syst. Signal Process. , 13 , 202 – 222 . Google Scholar CrossRef Search ADS Bouzidi , Y. , Quadrat , A. & Rouillier , F. ( 2015 ) Computer algebra methods for testing the structural stability of multidimensional systems . Proc. 9th International Workshop on Multidimensional Systems. Vila Real, Portugal . Boyd , S. , Ghaoui , L. El , Feron , E. & Balakrishnan , V. ( 1994 ) Linear Matrix Inequalities in System and Control Theory . Philadelphia : SIAM . Google Scholar CrossRef Search ADS Chesi , G. & Middleton , R. H. ( 2014 ) Necessary and sufficient LMI conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems . IEEE Trans. Automat. Control , 59 , 996 – 1007 . Google Scholar CrossRef Search ADS Dorato , P. ( 1961 ) Short time stability in linear time-varying systems . Proc. IRE Int. Convention Record Pt . 4, 83 – 87 . Ebihara , Y. , Ito , Y. & Hagiwara , T. ( 2006 ) Exact stability analysis of 2-D systems using LMIs . IEEE Trans. Automat. Control , 51 , 1509 – 1513 . Google Scholar CrossRef Search ADS Fan , H. & Wen , C. ( 2002 ) A sufficient condition on the exponential stability of two-dimensional (2-D) shift-variant systems . IEEE Trans. Automat. Control , 47 , 647 – 655 . Google Scholar CrossRef Search ADS Feng , Z. , Wu , Q. & Xu , L. ( 2012 ) $$H_\infty$$ control of linear multidimensional discrete systems . Multidimens. Syst. Signal Process. , 23 , 381 – 411 . Google Scholar CrossRef Search ADS Fornasini , E. & Marchesini , G. ( 1976 ) State-space realization theory of two-dimensional filters . IEEE Trans. Automat. Control , 21 , 484 – 492 . Google Scholar CrossRef Search ADS Galkowski , K. , Emelianov , M. A. , Pakshin , P. V. & Rogers , E. ( 2016 ) Vector Lyapunov functions for stability and stabilization of differential repetitive processes . J. Comput. Syst. Sci. Int. , 55 , 503 – 514 . Google Scholar CrossRef Search ADS Haddad , W. M. & L’Afflitto , A. ( 2016 ) Finite-time stabilization and optimal feedback control . IEEE Trans. Automat. Control , 61 , 1069 – 1074 . Google Scholar CrossRef Search ADS Hu , G. D. & Liu , M. ( 2006 ) Simple criteria for stability of two-dimensional linear systems . IEEE Trans. Signal Process. , 53 , 4720 – 4723 . Jammazi , C. ( 2010 ) On a sufficient condition for finite-time partial stability and stabilization: applications . IMA J. Math. Control Inform. , 27 , 29 – 56 . Google Scholar CrossRef Search ADS Kamenkov , G. V. ( 1953 ) On stability of motion over a finite interval of time . J. Appl. Math. Mech. USSR , 17 , 529 – 540 (in Russian) . Knorn , S. & Middleton , R. H. ( 2013a ) Stability of two-dimensional linear systems with singularities on the stability boundary using LMIs . IEEE Trans. Automat. Control , 58 , 2579 – 2590 . Google Scholar CrossRef Search ADS Knorn , S. & Middleton , R. H. ( 2013b ) Two-dimensional analysis of string stability of nonlinear vehicle strings . IEEE Conference on Decision and Control , pp. 5864 – 5869 . Knorn , S. & Middleton , R. H. ( 2016 ) Asymptotic and exponential stability of nonlinear two-dimensional continuous-discrete Roesser models . Syst. Control Lett. , 93 , 35 – 42 . Google Scholar CrossRef Search ADS Lin , Z. & Bruton , L.T. ( 1989 ) BIBO stability of inverse 2-D digital filters in the presence of nonessential singularities of the second kind . IEEE Trans. Circuits Syst. , 36 , 244 – 254 . Google Scholar CrossRef Search ADS Lu , W. S. & Antoniou , A. ( 1992 ) Two-Dimensional Digital Filters . New York, NY, USA : Marcel Dekker, Inc . Marszalek , W. ( 1984 ) Two-dimensional state-space discrete models for hyperbolic partial differential equations . Appl. Math. Model. , 8 , 11 – 14 . Google Scholar CrossRef Search ADS Moulay , E. & Perruquetti , W. ( 2008 ) Finite time stability conditions for non-autonomous continuous systems . Int. J. Control , 81 , 797 – 803 . Google Scholar CrossRef Search ADS Nersesov , S. G. & Haddad , W. M. ( 2008 ) Finite-time stabilization of nonlinear impulsive dynamical systems . Nonlinear Anal. Hybrid Syst. , 2 , 832 – 845 . Google Scholar CrossRef Search ADS Owens , D. H. & Rogers , E. ( 2002 ) Stability analysis for a class of 2D continuous-discrete linear systems with dynamic boundary conditions . Syst. Control Lett. , 37 , 55 – 60 . Google Scholar CrossRef Search ADS Pakshin , P. , Emelianova , J. , Emelianov , M. , Galkowski , K. & Rogers , E. ( 2016 ) Dissipativity and stabilization of nonlinear repetitive processes . Syst. Control Lett. , 91 , 14 – 20 . Google Scholar CrossRef Search ADS Roesser , R. ( 1975 ) A discrete state-space model for linear image processing . IEEE Trans. Automat. Control , 20 , 1 – 10 . Google Scholar CrossRef Search ADS Rogers , E. , Galkowski , K. & Owens , D.H. ( 2007 ) Control Systems Theory and Applications for Linear Repetitive Processes . Berlin Heidelberg : Springer . Rogers , E. & Owens , D. H. ( 1992 ) Stability Analysis for Linear Repetitive Processes . Berlin Heidelberg : Springer . Google Scholar CrossRef Search ADS Rogers , E. & Owens , D. H. ( 2002 ) Kronecker product based stability tests and performance bounds for a class of 2D continuous–discrete linear systems . Linear Algebra Appl. , 353 , 33 – 52 . Google Scholar CrossRef Search ADS Seo , S. , Shim , H. & Jin , H. S. ( 2011 ) Finite-time stabilizing dynamic control of uncertain multi-input linear systems . IMA J. Math. Control Inform. , 28 , 525 – 537 . Google Scholar CrossRef Search ADS Singh , V. ( 2014 ) Stability analysis of 2-D linear discrete systems based on the Fornasini-Marchesini second model: stability with asymmetric Lyapunov matrix . Digit. Signal Process. , 26 , 183 – 186 . Google Scholar CrossRef Search ADS Tan , F. , Zhou , B. & Duan , G. R. ( 2016 ) Finite-time stabilization of linear time-varying systems by piecewise constant feedback . Automatica , 68 , 277 – 285 . Google Scholar CrossRef Search ADS Tichonov , A. N. & Samarsky , A. A. ( 1963 ) Equations of Mathematical Physics . Warsaw : PWN (in Polish) . Wang , L. , Wang , W. , Gao , J. & Chen , W. ( 2017 ) Stability and robust stabilization of 2-D continuous-discrete systems in Roesser model based on KYP lemma . Multidimens. Syst. Signal Process. , 28 , 251 – 264 . Google Scholar CrossRef Search ADS Xiao , Y. ( 2001 ) Stability test for 2-D continuous-discrete systems . IEEE Conference on Decision and Control , pp. 3649 – 3654 . Xie , X. , Zhang , Z. & Hu , S. ( 2015 ) Control synthesis of Roesser type discrete-time 2-D T-S fuzzy systems via a multi-instant fuzzy state-feedback control scheme . Neurocomputing , 151 , 1384 – 1391 . Google Scholar CrossRef Search ADS Xu , L. , Wu , L. , Wu , Q. , Lin , Z. & Xiao , Y. ( 2005 ) On realization of 2D discrete systems by Fornasini-Marchesini model . Int. J. Control Autom. , 3 , 631 – 639 . Zhang , G. & Wang , W. ( 2016a ) Finite-region stability and boundedness for discrete 2-D Fornasini-Marchesini second models . Int. J. Systems Sci. , 48 , 778 – 787 . Google Scholar CrossRef Search ADS Zhang , G. & Wang , W. ( 2016b ) Finite-region stability and finite-region boundedness for 2-D Roesser models . Math. Method Appl. Sci. , 39 , 5757 – 5769 . Google Scholar CrossRef Search ADS Zhang , Y. , Liu , C. & Mu , X. ( 2012 ) Robust finite-time stabilization of uncertain singular Markovian jump systems . Appl. Math. Model. , 36 , 5109 – 5121 . Google Scholar CrossRef Search ADS Zhang , Y. , Shi , P. , Nguang , S. K. & Karimi , H. R. ( 2014 ) Observer-based finite-time fuzzy $$H_\infty$$ control for discrete-time systems with stochastic jumps and time-delays . Signal Process. , 97 , 252 – 261 . Google Scholar CrossRef Search ADS Zhang , Y. , Shi , P. , Nguang , S. K. & Song , Y. ( 2014 ) Robust finite-time $$H_\infty$$ control for uncertain discrete-time singular systems with Markovian jumps . IET Control Theory Appl. , 8 , 1105 – 1111 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. [email protected]
Journal
IMA Journal of Mathematical Control and InformationOxford University Press
Published: May 19, 2018
DeepDyve is your personal research library
It’s your single place to instantly
that matters to you.
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year
Save searches from
PubMed
Create lists to
Export lists, citations | {"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": 42, "equation": 207, "x-ck12": 0, "texerror": 0, "math_score": 0.9763083457946777, "perplexity": 1721.6624455459817}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583517628.91/warc/CC-MAIN-20181024001232-20181024022732-00395.warc.gz"} |
http://mathhelpforum.com/calculus/185103-finding-equation-curve-given-length-integral.html | # Math Help - Finding equation for a curve, given length integral
1. ## Finding equation for a curve, given length integral
Hey,
So I'm a bit confused on how an answer was reached. The question states:
Find a curve through the point (1,1) whose length integral is given by:
$L = \int_1^4 \sqrt{1+\frac{1}{4x}}dx$. From here I try to use point-slope formula, where m = $\frac{1}{2\sqrt{x}}$ (which comes from taking the square root of $\frac{1}{4x}$. Now, I apply the point-slope formula to try and get the equation of the curve, using:
y - y1 = m(x-x1) ==> y - (1) = $\frac{1}{2\sqrt{x}} (x - (1))$
After adding 1 to both sides and distributing the $\frac{1}{2\sqrt{x}}$ to the (x - 1), I end up with something like:
$y = \frac{x+2\sqrt{x}-1}{2\sqrt{x}}$... which isn't anything like the y = $\sqrt{x}$ answer it SHOULD be.
Can someone point out where I went wrong?
2. ## Re: Finding equation for a curve, given length integral
Originally Posted by Calcme
Hey,
So I'm a bit confused on how an answer was reached. The question states:
Find a curve through the point (1,1) whose length integral is given by:
$L = \int_1^4 \sqrt{1+\frac{1}{4x}}dx$. From here I try to use point-slope formula, where m = $\frac{1}{2\sqrt{x}}$ (which comes from taking the square root of $\frac{1}{4x}$. Now, I apply the point-slope formula to try and get the equation of the curve, using:
y - y1 = m(x-x1) ==> y - (1) = $\frac{1}{2\sqrt{x}} (x - (1))$
After adding 1 to both sides and distributing the $\frac{1}{2\sqrt{x}}$ to the (x - 1), I end up with something like:
$y = \frac{x+2\sqrt{x}-1}{2\sqrt{x}}$... which isn't anything like the y = $\sqrt{x}$ answer it SHOULD be.
Can someone point out where I went wrong?
1. The length of a curve with the equation y = f(x) is calculated by:
$L=\int(\sqrt{1+(y')^2})dx$
2. Therefore $(y')^2=\frac1{4x}~\implies~y' = \sqrt{\frac1{4x}} = \frac1{2\sqrt{x}}$
Thus $y = \int\left(\frac1{2\sqrt{x}} \right)dx = \sqrt{x} + c$
3. ## Re: Finding equation for a curve, given length integral
Ahh, I see... we're just working all the way backwards until we get our original curve/function.
Thanks a lot. | {"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": 17, "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": 0.9568077325820923, "perplexity": 367.0110654685777}, "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/1466783396538.42/warc/CC-MAIN-20160624154956-00006-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://highnoongmt.wordpress.com/2010/12/03/recovery_of_sparse_signals_with_from_compressive_measurements_now_with_other_flavors/ | # Recovery of Sparse Signals from Compressive Measurements: Now with Other Flavors
Continuing my experiments from the previous few days, I have now coded and experimented with Probabilistic OMP (PrOMP), and the convex relaxation of the strict sparsity measure to an $$\ell_1$$-norm of the solution, a.k.a, the principle known as Basis Pursuit (BP for short).
The problem set-up is the same as before. I am not including any noise. PrOMP turned out to be a real hassle to tune, but I finally settled on $$m=2$$, $$\epsilon = 0.001$$, and stopped the search when either $$||\vr||^2 \ge 10^{-22}$$ or 10,000 iterations have elapsed. (An explanation of these values is here.) For BP I used CVX, with the following
cvx_begin
variable xhatBP(dimensions);
minimize( norm(xhatBP,1) );
subject to
y == Phi*xhatBP;
cvx_end
where Phi is the same kind of matrix used before, and y are the measurements.
This time, under time constraints, I only looked at sparsity values up to 0.4,
and I only performed 500 runs of each algorithm (each with the same sensing matrix and sparse vector).
Below we see a comparison of the probability of “exact recovery” as a function of sparsity of these methods with MP and OMP.
First, it is immediately clear that MP is now making a fool of itself. Why does it keep showing up to this party? BP, however, is surprisingly poor! (Unless I have made some error in the code.) I don’t know why this is, but there was little change when I forced the sensing matrix to have a coherence of less than 0.5. OMP does well, and PrOMP does a little better.
PrOMP selects atoms at random based on a distribution that places most probability mass on the atoms highly coherent with the residual, and less on all the others (and zero on those atoms already selected, or with zero length projections). PrOMP terminates when the residual norm is exactly zero, at which time it serves up the solution. But, even though the norm residual is zero, we are not guaranteed the solution is correct! There are infinitely many solutions $$\vx$$ such that $$||\vy – \mathbf{\Phi}\vx|| = 0$$ by virtue of $$\mathbf{\Phi}$$ being fat and full rank. It has a null space into which we can point the sparsest solution and still have zero norm residual. And there cometh the spark of $$\mathbf{\Phi}$$ being that much less sparse than the sparsest solution. (In my experiments, the null space of the dictionaries have at least dimension 200, so that is a low spark.) Thus, there is no guarantee PrOMP serves up the correct solution. Yet still! PrOMP does a fine job of outperforming OMP and BP.
My code is attached. Next I shall experiment with re-weighted $$\ell_1$$ and $$\ell_2$$ methods, iterative hard thresholding, CoSaMP, StOMP, regularized OMP, A*OMP, two-stage thresholding, etc.
## 2 thoughts on “Recovery of Sparse Signals from Compressive Measurements: Now with Other Flavors”
1. Alejandro says:
“BP, however, is surprisingly poor!”
That’s strange. Michael Elad, in his book “Sparse and Redundant Representations”, does a comparison between BP and OMP, and BP is better than OMP. Quoting page 52:
“It is clear that the relaxation methods are performing much better”
The code he used to generate these results is available here: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8311917781829834, "perplexity": 1104.6971384580222}, "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-2017-51/segments/1512948512121.15/warc/CC-MAIN-20171211033436-20171211053436-00176.warc.gz"} |
http://en.wikipedia.org/wiki/Classical_field_theory | Classical field theory
A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate quantum mechanics (quantum field theories).
A physical field can be thought of as the assignment of a physical quantity at each point of space and time. For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents the direction of the movement of air at that point. As the day progresses, the directions in which the vectors point change as the directions of the wind change. From the mathematical viewpoint, classical fields are described by sections of fiber bundles (covariant classical field theory). The term 'classical field theory' is commonly reserved for describing those physical theories that describe electromagnetism and gravitation, two of the fundamental forces of nature.
Descriptions of physical fields were given before the advent of relativity theory and then revised in light of this theory. Consequently, classical field theories are usually categorised as non-relativistic and relativistic.
Non-relativistic field theories
Some of the simplest physical fields are vector force fields. Historically, the first time fields were taken seriously was with Faraday's lines of force when describing the electric field. The gravitational field was then similarly described.
Newtonian gravitation
A classical field theory describing gravity is Newtonian gravitation, which describes the gravitational force as a mutual interaction between two masses.
Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on a small test mass m located at r, and then dividing by m:[1]
$\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m}.$
Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on the behavior of M.
According to Newton's law of universal gravitation, F(r) is given by[1]
$\mathbf{F}(\mathbf{r}) = -\frac{G M m}{r^2}\hat{\mathbf{r}},$
where $\hat{\mathbf{r}}$ is a unit vector lying along the line joining M and m and pointing from m to M. Therefore, the gravitational field of M is[1]
$\mathbf{g}(\mathbf{r}) = \frac{\mathbf{F}(\mathbf{r})}{m} = -\frac{G M}{r^2}\hat{\mathbf{r}}.$
The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to the identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle, which leads to general relativity.
Because the gravitational force F is conservative, the gravitational field g can be rewritten in terms of the gradient of a gravitational potential Φ(r):
$\mathbf{g}(\mathbf{r}) = -\nabla \Phi(\mathbf{r}).$
Electromagnetism
Electrostatics
Main article: Electrostatics
A charged test particle with charge q experiences a force F based solely on its charge. We can similarly describe the electric field E so that F = qE. Using this and Coulomb's law tells us that the electric field due to a single charged particle as
$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}.$
The electric field is conservative, and hence can be described by a scalar potential, V(r):
$\mathbf{E}(\mathbf{r}) = -\nabla V(\mathbf{r}).$
Magnetostatics
Main article: Magnetostatics
A steady current I flowing along a path will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is
$\mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}),$
where B(r) is the magnetic field, which is determined from I by the Biot–Savart law:
$\mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int \frac{d\boldsymbol{\ell} \times d\hat{\mathbf{r}}}{r^2}.$
The magnetic field is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential, A(r):
$\mathbf{B}(\mathbf{r}) = \boldsymbol{\nabla} \times \mathbf{A}(\mathbf{r})$
Electrodynamics
Main article: Electrodynamics
In general, in the presence of both a charge density ρ(r, t) and current density J(r, t), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations, a set of differential equations which directly relate E and B to ρ and J.[2]
Alternatively, one can describe the system in terms of its scalar and vector potentials V and A. A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J,[note 1] and from there the electric and magnetic fields are determined via the relations[3]
$\mathbf{E} = -\boldsymbol{\nabla} V - \frac{\partial \mathbf{A}}{\partial t}$
$\mathbf{B} = \boldsymbol{\nabla} \times \mathbf{A}.$
Hydrodynamics
Main article: Hydrodynamics
Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation and Newton's laws connect the density, pressure, and velocity fields:
$\dot \bold{u} = \mathbf{F} - {\nabla p \over \rho}$${\displaystyle \ \dot{\rho} + \nabla \cdot (\rho \bold u) = 0 }$
Relativistic field theory
Modern formulations of classical field theories generally require Lorentz covariance as this is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using Lagrangians. This is a function that, when subjected to an action principle, gives rise to the field equations and a conservation law for the theory.
We use units where c=1 throughout.
Lagrangian dynamics
Main article: Lagrangian
Given a field tensor $\phi$, a scalar called the Lagrangian density $\mathcal{L}(\phi,\partial\phi,\partial\partial\phi, ...,x)$ can be constructed from $\phi$ and its derivatives.
From this density, the functional action can be constructed by integrating over spacetime
$\mathcal{S} = \int{\mathcal{L} \mathrm{d}^4x}.$
Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all space.
Then by enforcing the action principle, the Euler–Lagrange equations are obtained
$\frac{\delta \mathcal{S}}{\delta\phi}=\frac{\partial\mathcal{L}}{\partial\phi} -\partial_\mu \left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\right)+.~.~.+(-1)^m\partial_{\mu_1} \partial_{\mu_2}.~.~.\partial_{\mu_{m-1}} \partial_{\mu_m} \left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu_1} \partial_{\mu_2}...\partial_{\mu_{m-1}}\partial_{\mu_m} \phi)}\right)=0.$
Relativistic fields
Two of the most well-known Lorentz-covariant classical field theories are now described.
Electromagnetism
Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a better (and more consistent with mechanics) formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together is used.
We have the electromagnetic potential, $A_a=\left(-\phi, \vec{A} \right)$, and the electromagnetic four-current $j_a=\left(-\rho, \vec{j}\right)$. The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor
$F_{ab} = \partial_a A_b - \partial_b A_a.$
The Lagrangian
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have $\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab}.$ We can use gauge field theory to get the interaction term, and this gives us
$\mathcal{L} = \frac{-1}{4\mu_0}F^{ab}F_{ab} + j^aA_a.$
The Equations
This, coupled with the Euler–Lagrange equations, gives us the desired result, since the E-L equations say that
$\partial_b\left(\frac{\partial\mathcal{L}}{\partial\left(\partial_b A_a\right)}\right)=\frac{\partial\mathcal{L}}{\partial A_a}.$
It is easy to see that $\partial\mathcal{L}/\partial A_a = \mu_0 j^a$. The left hand side is trickier. Being careful with factors of $F^{ab}$, however, the calculation gives $\partial\mathcal{L}/\partial(\partial_b A_a) = F^{ab}$. Together, then, the equations of motion are:
$\partial_b F^{ab}=\mu_0j^a.$
This gives us a vector equation, which are Maxwell's equations in vacuum. The other two are obtained from the fact that F is the 4-curl of A:
$6F_{[ab,c]} \, = F_{ab,c} + F_{ca,b} + F_{bc,a} = 0.$
where the comma indicates a partial derivative.
Gravitation
Main articles: Gravitation and General Relativity
After Newtonian gravitation was found to be inconsistent with special relativity, Albert Einstein formulated a new theory of gravitation called general relativity. This treats gravitation as a geometric phenomenon ('curved spacetime') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. The field equations may be derived by using the Einstein–Hilbert action. Varying the Lagrangian
$\mathcal{L} = \, R \sqrt{-g}$,
where $R \, =R_{ab}g^{ab}$ is the Ricci scalar written in terms of the Ricci tensor $\, R_{ab}$ and the metric tensor $\, g_{ab}$, will yield the vacuum field equations:
$G_{ab}\, =0$,
where $G_{ab} \, =R_{ab}-\frac{R}{2}g_{ab}$ is the Einstein tensor. | {"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": 36, "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": 0.9742878079414368, "perplexity": 481.60232564878163}, "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-2015-06/segments/1422115856115.9/warc/CC-MAIN-20150124161056-00163-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://juleann.blogspot.com/2011/09/conversations-with-valerie.html?showComment=1321437032813 | ## Friday, September 23, 2011
### Conversations with Valerie
(It's 10:00 a.m. and we are on our way to a sporting goods store.)
V: Where are we going?
JA: We're going to buy goggles for Mommy.
V: And then what are we going to do?
JA: Then we are going to go to the gym so Mommy can swim and you can play in the treehouse.
V: And then what are we going to do?
JA: Then we're going to go to Miss Stephanie's house.
V: And then what are we going to do?
JA: Then we'll go back to our house.
V: And then what are we going to do?
JA: Then we'll eat dinner.
V: And then what are we going to do?
JA: Then it will be tubby time.
V: And then what are we going to do?
JA: Then we'll read some stories.
V: And then what are we going to do?
JA: Then it will be time for night night.
V: (crying) But I'm not acting tired!
V: Can you put on the Go Diego Go with the chinchilla?
JA: Okay. Do you know where the clicker is?
V: Yeah, I'll show it to you. (Comes over to couch.) Oh, it's not there. I put it somewhere Dorothy can't get it. (Looks around for another minute.) I know, I put it on the changing table, I'll go get it. (Climbs over couch to changing table, where clicker is not.) Mommy, do you know where I put the clicker?
JA: You sure are related to your father! | {"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": 0.8461786508560181, "perplexity": 2051.5797199298577}, "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/1368698203920/warc/CC-MAIN-20130516095643-00078-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/412676/compute-slide-number-in-beamer | # Compute slide number in beamer
The MWE below is to insert some constant content in all odd slides.
Since the number of even slides could change, how to insert the odd contents automatically?
For example, if the last even slide is \only<10>{This is slide \thepage.} the command for constant content should be automatically computed to \only<1,3,...,11>{This is slide \thepage.}
MWE
\documentclass{beamer}
\begin{document}
\begin{frame}
\only<1,3,5>{This is slide \thepage. ODD.}
\only<2> {This is slide \thepage.}
\only<4> {This is slide \thepage.}
\end{frame}
\end{document}
• I'm not sure if you want 'odd in absolute terms' or rather 'relative to a currently even slide', e.g. is the \only<2> line the 'anchor'. I ask as this feels to me like a case where you want relative slide specifiers. – Joseph Wright Jan 29 '18 at 12:14
• @JosephWright, thanks for comment. But I'm afraid I don't understand what you said. For example, what I want is to insert song lyrics, and after each part, show a chorus. I think I will use a single frame, so odd/even works well. But it would be nice to extend and apply the rule for first, first+2, fisrt+4,... where first is the 1st one in the frame. – Sigur Jan 29 '18 at 12:18
\documentclass{beamer}
\makeatletter
\newcommand{\insertonodd}{%
\ifodd\the\beamer@slideinframe
This is slide \insertslideinframe. ODD.
\else
\only<\numexpr\insertslideinframe+1\relax>{}
\fi
}
\newcommand{\insertslideinframe}{\the\beamer@slideinframe}
\makeatother
\begin{document}
\begin{frame}
\insertonodd
\only<2> {This is slide \insertslideinframe.}
\only<4> {This is slide \insertslideinframe.}
\end{frame}
\end{document}
With an up-to-date beamer version, this can be simplified to:
\documentclass{beamer}
\newcommand{\insertonodd}{%
\ifodd\insertoverlaynumber
This is slide \insertoverlaynumber. ODD.
\else
\only<\numexpr\insertoverlaynumber+1\relax>{}
\fi
}
\begin{document}
\begin{frame}
\insertonodd
\only<2> {This is slide \insertoverlaynumber.}
\only<4> {This is slide \insertoverlaynumber.}
\end{frame}
\end{document}
• Nice. Your code works as expected, but I don't understand the logical rule. Since there is no loop within insertonodd definition, how it inserts content in slide number 3? I suppose it is executed only once at beginning. Could you give me an idea? – Sigur Jan 29 '18 at 12:10
• @Sigur The command is executed once on every overlay – samcarter_is_at_topanswers.xyz Jan 29 '18 at 12:13
• Is the \if else fi responsible for this? – Sigur Jan 29 '18 at 12:22
• @Sigur No, every command in the body of a frame is executed once per overlay. You can test this by replacing \insertonodd with a counter. – samcarter_is_at_topanswers.xyz Jan 29 '18 at 12:34
• Oh, I didn't know that. Nice to know. Thanks. – Sigur Jan 29 '18 at 12:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9576437473297119, "perplexity": 2485.6297164674816}, "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-2021-04/segments/1610703538082.57/warc/CC-MAIN-20210123125715-20210123155715-00663.warc.gz"} |
https://en.universaldenker.org/formulas/751 | # Formula Wave in Vacuum Phase velocity Frequency Wavelength
## Phase velocity
Unit
Phase velocity is the speed of propagation of the same phase of a (monochromatic) wave.
In the case of electromagnetic waves, the phase velocity is the speed of light: $$c = \lambda \, f$$, with the value $$c = 299 \, 792 \, 458 \, \frac{\mathrm m}{\mathrm s}$$.
## Frequency
Unit
Frequency of the wave. Indicates how often the wave (e.g. light wave) oscillates per second.
## Wavelength
Unit
Wavelength is the smallest distance between two points of the same phase. For example, the distance between two wave crests. | {"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": 0.99483323097229, "perplexity": 1157.5979448474943}, "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/1679296948765.13/warc/CC-MAIN-20230328042424-20230328072424-00777.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/157473-f2-binary-field-print.html | # F2 binary field
• September 26th 2010, 08:29 AM
Suy
F2 binary field
A system of m equations over F2 in n unknowns has exactly 2^(n−m) solutions. • A. True • B. False?
I don't understand why
please also explain the relationship between power of 2 and F2 binary field
thx!
• September 26th 2010, 09:04 AM
HappyJoe
The question doesn't assume that the equations are independent, which might add redundancy when solving them, and it doesn't assume that the equations are linear, in which case you are not guaranteed a solution at all.
Anyway, the equations are probably assumed implicitly to be linear, and you need to think in terms of free variables. If you are solving your system of linear equations, and you end up with, say, 7 free variables, then for each combination of choices of values for these free variables, the set of linear equations has a solution. But since each variable can take values in F_2 (giving 2 choices for each variable), this makes a total of 2^7 choices.
• September 26th 2010, 09:59 AM
Suy
I see, thank you so much...
Can you also explain
__________________________________
"general rule
2^(# of variable - # of equation)
--------
x+y =0
y+z=0
(0,0,1)
(1,1,0)
2 solution
2^1=2
---------
"
________________________________
It was in my note, but it's not very clear..
• September 26th 2010, 10:31 AM
tonio
Quote:
Originally Posted by Suy
I see, thank you so much...
Can you also explain
__________________________________
"general rule
2^(# of variable - # of equation)
--------
x+y =0
y+z=0
(0,0,1)
(1,1,0)
2 solution
None of the two above are solutions to that system: neither (0,0,1) nor (1,1,0) are asolution to $y+z=0$...
Solutions are ,for example, $(0,0,0),\,(1,1,1)$...can you find more?
Tonio
2^1=2
---------
"
________________________________
It was in my note, but it's not very clear..
.
• September 26th 2010, 10:41 AM
HappyJoe
Quote:
Originally Posted by Suy
I see, thank you so much...
Can you also explain
__________________________________
"general rule
2^(# of variable - # of equation)
--------
x+y =0
y+z=0
(0,0,1)
(1,1,0)
2 solution
2^1=2
---------
"
________________________________
It was in my note, but it's not very clear..
Sure. But I'm not certain what the deal is with the triples (0,0,1) and (1,1,0). Maybe they are supposed to be the two solutions of the system, but they are not. Instead, the two solutions are (0,0,0) and (1,1,1).
To see this, just treat the system like any old system of linear equations. If you like, you may plug it into a matrix and start row reducing it. Usually, one has the last variable (here z) as the free variable, so add the equation y+z = 0 to the equation x+y = 0 to obtain x+2y+z = 0. Since we are in F_2, we have 2=0, whereas this last equation is x+z=0. So you have now the two equations x+z=0 and y+z=0, where z is the free variable. The equations are equivalent to x = -z and y = -z. But remember that in F_2, we have -z=z (signs don't matter in F_2), so we have x = z and y = z.
So both x and y are completely determined by z. In other words, whenever you choose a value for z, this will automatically force some value on both x and y. There are two possible choices for z (either 0 or 1).
Letting z = 0 gives x=0 and y=0, and the solution (0,0,0).
Letting z = 1 gives x=1 and y=1, and the solution (1,1,1).
So a total number of 2 solutions.
Notice that this is consistent with the "general rule". You have 3 variables and 2 equations, so # of variables - # of equations = 1, and 2^1 = 2, which was exactly the number of solutions.
In general, if you have m linear _independent_ equation in n variables, then the number of free variables will be n-m. Assigning a value to each of these free variables completely determines a solutions - and each of the free variables can be chosen in 2 ways.
• September 26th 2010, 10:45 AM
Suy
sorry , I typed wrong
it should be
x+y =0
y+z=1
(0,0,1)
(1,1,0)
2 solution | {"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": 0.8563546538352966, "perplexity": 792.9713295720073}, "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-2015-11/segments/1424936461266.22/warc/CC-MAIN-20150226074101-00075-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/169273-ploynomial-print.html | # Ploynomial...
• Jan 25th 2011, 05:43 AM
breitling
Ploynomial...
Not sure if im going mad but struggling with this (Thinking)
(a) Write down the polynomial P in x, y such that:
x^6 y^6 = (x^2 y^2)*P.
• Jan 25th 2011, 05:59 AM
FernandoRevilla
Hint :
Denote $t=x^2$ and $a=y^2$ then,
$x^6-y^6=t^3-a^3$
so,
$a$ is a root of $q(t)=t^3-a^3$
and you can decompose it using Ruffini's rule.
Fernando Revilla
• Jan 25th 2011, 06:10 AM
Soroban
Hello, breitling!
Quote:
$\text{Write down the polynomial }P\text{ in }x, y\text{ such that:}$
. . $x^6 - y^6 \:=\: (x^2 - y^2)\cdot P$
Don't you recognize that difference of cubes?
. . . . $a^3 - b^3 \:=\:(a-b)(a^2 + ab + b^2)$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
The very least you can do is some Long Division . . .
. . $\begin{array}{cccccccccc} &&&& x^4 & + & x^2y^2 &+& y^4 \\ && -- & -- & -- & -- & -- & -- & -- \\ x^2-y^2 & | & x^6 &&&&& - & y^6 \\ && x^6 &-& x^4y^2 \\ && -- & -- & -- \\ &&&& x^4y^2 \\ &&&& x^4y^2 &-& x^2y^4 \\ &&&& -- & -- & -- \\ &&&&&& x^2y^4 &-& y^6 \\ &&&&&& x^2y^4 &-& y^6 \\ &&&&&& -- & -- & -- \end{array}$
• Jan 25th 2011, 07:32 AM
Prove It
A better solution:
$\displaystyle x^6 - y^6 = (x^2)^3 - (y^2)^3$
$\displaystyle = (x^2 - y^2)[(x^2)^2 + x^2y^2 + (y^2)^2]$ by the Difference of Two Cubes rule...
$\displaystyle = (x^2 - y^2)(x^4 + x^2y^2 + y^4)$.
So what is $\displaystyle P$?
• Jan 31st 2011, 05:03 AM
breitling
The question actually asked x^6-y^6=(x^2+y^2)*p got the sign wrong.
• Jan 31st 2011, 05:25 AM
HallsofIvy
Quote:
Originally Posted by breitling
The question actually asked x^6-y^6=(x^2+y^2)*p got the sign wrong.
Then do exactly what people suggested you do before! | {"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": 13, "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": 0.9788981080055237, "perplexity": 2209.1124158363805}, "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/1490218189686.31/warc/CC-MAIN-20170322212949-00102-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/114131/relating-two-notions-of-geometric-realization | # Relating two notions of geometric realization
Let $K$ be an abstract simplicial complex on the (finite) vertex set $V$. The geometric realization $|K|$ is typically defined (see Spanier's book for instance) as the collection of functions $\alpha:V \to \mathbb{R}$ so that (a) the support of each $\alpha$ is a simplex, and (b) the sum $\sum_{v \in V}\alpha(v)$ equals $1$. Now each (closed) simplex $\sigma$ is realized as the collection of $\alpha \in K$ so that $\alpha(v) \neq 0$ implies $v \in \sigma$. From this one knows the star of each simplex.
A simplicial approximation of $f:|K| \to |L|$ is a simplicial map $g:K \to L$ so that $f(\text{star }\sigma) \subset \text{star }g(\sigma)$ for each simplex $\sigma \in K$. It is a standard result that the Piecewise Linear map induced by $g$ is homotopy equivalent to $f$
Now consider the case where $K$ is not abstract, but rather $V$ is an open cover of some topological space $X$. So, each simplex corresponds to an actual topological space, i.e., a non-empty intersection of some finite open sets in $X$. Let's call this $X_\sigma$.
My question is this:
What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X_\sigma$ for each simplex $\sigma \in K$?
Here is some idea of what type of answer I am hoping for:
In the case where $X$ is paracompact and $V$ is a contractible cover, the nerve theorem applies and I know that $X$ and $|K|$ are homotopy equivalent. But is there a more general relationship between these two notions of realization of which the Nerve theorem is a consequence?
Furthermore, is there some functoriality to the nerve theorem? That is, assume you are given contractible covers $U$ and $V$ of $X$ and $Y$ generating the nerves $K$ and $L$. Given a function $f : X \to Y$ and a simplicial map $g:K \to L$, is there some magic analogoue of the star condition like $f(X_\sigma) \subset Y_{g(\sigma)}$ that makes $g$ induce a map homotopy equivalent to the composite $|K| \to X \to Y \to |L|$ where the maps on the edge come from the nerve theorem and the map in the middle is $f$?
-
Regarding
What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X_\sigma$ for each simplex $\sigma \in K$?
It seems to me that one can build an intermediate space $Y$ and a diagram $$|K| \leftarrow Y \rightarrow X$$ which is natural in both the cover and in $X$ (where if we have a map $X\to X'$ the covering for $X$ should be the inverse image of the covering elements for $X'$). The space $Y$ is given by the realization of the nerve of the topological poset $\cal P$ whose elements are pairs $(U,x)$ in which $U$ is a finite collection of intersection of open sets in the covering $V$, and $x$ is a point of $U$.
There are then forgetful maps $Y \to |K|$ as well as $Y \to X$.
To see this, note that $X$ can be regarded as a topological category (or poset) whose objects are points of $X$ and only identity morphisms. Then ${\cal P} \to X$ is just the forgetful functor and it induces the map $Y \to X$ on realization (the realization of $X$ when considered as a topological poset is $X$ as a space). There is also a forgetful functor from ${\cal P}$ to the nerve of the covering which induces the map $Y \to |K|$.
If every non-empty finite intersection of members of $V$ is contractible, then the maps $Y \to |K|$, $Y \to X$ are weak equivalences.
-
In your pairs $(U,x)$ do you require $x \in U$? – Pinying Nov 22 '12 at 11:40
yes. I've edited to take that into account. – John Klein Nov 22 '12 at 12:36
The map $Y\to X$ looks like it's going to be a weak equivalence in general, so that (in an up-to-homotopy way) you have an indirect map $X\to |K|$ in any case. – Tom Goodwillie Nov 22 '12 at 14:20
@Tom: That's a cofinality type argument, right? – John Klein Nov 23 '12 at 4:50
Thank you. If you have any ideas about the functoriality question, please also write them down. In any case, I will accept this nice answer! – Pinying Nov 26 '12 at 2:12 | {"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": 0.9610024690628052, "perplexity": 98.92954511957379}, "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-2015-11/segments/1424936462573.55/warc/CC-MAIN-20150226074102-00293-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/13492/ternary-relations-that-are-not-binary-functions/13495 | # Ternary relations that are not binary functions
By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions.
"Elementary" shall mean "part of the signature of a first-order theory".
The most prominent ternary relation that comes to my mind is the betweenness relation in geometry.
I am looking for examples from all over mathematics of elementary ternary relations that are not binary functions (resp. the corresponding theories).
Examples of elementary quaternary relations are also welcome!
-
Coprimeness, as in there is no $p$ dividing all three of $a$, $b$, and $c$. – Dan Piponi Jan 31 '10 at 0:17
Geometry seems like a natural source, e.g., colinearity of three points. Or how about, for three non-colinear points, clockwise(p,q,r) if the path going through p, q, and then r runs clockwise on the circle whose circumference they lie on? It seems unintuitive to me to try to break up this relation into binary functions and relations.
-
Majority/median functions such as $$f(x,y,z) = (x \vee y) \wedge (y \vee z) \wedge (z \vee x)$$ are very natural. (This also makes sense for more than three arguments, but it is degenerate in the case of two arguments.)
-
Isn't this a 4-ary relation in 3-ary function's disguise? – Hans Stricker Jan 30 '10 at 20:45
Every 3-ary function is a 4-ary relation. Is that a problem? – François G. Dorais Jan 30 '10 at 21:00
Of course it's not a problem per se, but it's not what I am looking for: relations that are NOT functions. – Hans Stricker Jan 30 '10 at 22:41
I'm sorry, I thought the word "binary" in your question was important. – François G. Dorais Jan 30 '10 at 22:52
You might be willing to count this as a quaternary relation, from the Wikipedia article:
In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces.
http://en.wikipedia.org/wiki/Uniform_space
-
This may be too specific for what you're going for, but this question immediately brought to mind to triality on $D_4$. Here's triality in a nutshell:
View $\mathbb{R}^8 = \mathbb{O}$ as the inner product space collection of all Cayley numbers (where $\{1,i,j,k,l,li,lj,lk\}$ forms an orthonormal basis.). The collection of all orientation preserving isometries of $\mathbb{O}$ is simply $SO(8)$ (whose Lie algebra is $D_4$).
For $A$, $B$, and $C$ in $SO(8)$ and $x$ and $y$ in $\mathbb{O}$, consider the equation $$A(x)B(y) = C(xy)$$
Triality is the following claim: Given $A$, there exists a $B$ and $C$ making this equation hold for all Cayley numbers. The choice of the pair (B,C) is ALMOST unique - the only ambiguity is in replacing $(B,C)$ with $(-B,-C)$. Likewise, given $B$ or $C$, the other two matrices exist and are unique up to simultaneously changing the sign of both.
This gives rise to a natural relation $R\subseteq SO(8)\times SO(8)\times SO(8)$ with $(A,B,C)\in R$ iff $A(x)B(y) = C(xy)$ for all Cayley numbers $x$ and $y$. The ambiguity in sign shows that this is not a 2-ary function.
- | {"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": 0.8402848839759827, "perplexity": 261.68855768587014}, "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-2015-06/segments/1422121934081.85/warc/CC-MAIN-20150124175214-00079-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.semanticscholar.org/paper/INVARIANT-OPERATORS-OF-THE-FIRST-ORDER-ON-MANIFOLDS-Slov%C3%A1k-Sou%C4%8Dek/9bb1ae3a627a4248ed3c01067495f3ffade3c375 | • Corpus ID: 40420678
# INVARIANT OPERATORS OF THE FIRST ORDER ON MANIFOLDS WITH A GIVEN PARABOLIC STRUCTURE
```@inproceedings{Slovk2000INVARIANTOO,
title={INVARIANT OPERATORS OF THE FIRST ORDER ON MANIFOLDS WITH A GIVEN PARABOLIC STRUCTURE},
year={2000}
}```
• Published 2000
• Mathematics
The goal of this paper is to describe explicitly all invariant first order operators on manifolds equipped with parabolic geometries. Both the results and the methods present an essential generalization of Fegan's description of the first order invariant operators on conformal Riemannian manifolds. On the way to the results, we present a short survey on basic structures and properties of parabolic geometries, together with links to further literature. Resume (Operateurs invariants d'ordre 1 sur…
Invariant Bilinear Differential Pairings on Parabolic Geometries
This thesis is concerned with the theory of invariant bilinear differential pairings on parabolic geometries. It introduces the concept formally with the help of the jet bundle formalism and provides
Symplectic spinors and Hodge theory
Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are
Representation Theory in Clifford Analysis
This chapter introduces contemporary Clifford analysis as a local function theory of first-order systems of PDEs invariant under various Lie groups. A concept of a symmetry of a system of partial
Explicit resolutions for the complex of several Fueter operators
• Mathematics
• 2007
Contact Symplectic Geometry in Parabolic Invariant Theory and Symplectic Dirac Operator
The authors of (Stein, Weiss) introduced the general method of construction of first order differential operators based on covariant derivatives composed with projections onto irreducible components
Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables
In this paper, we analyze differential operators of first order acting between vector bundles associated to G/P where G = Spin(n+2, 2) and P is a parabolic subgroup. The operators in question are
## References
SHOWING 1-10 OF 41 REFERENCES
Invariants and calculus for projective geometries
An understanding of the local invariants is essential in any study of a differential geometry with local structure. In his ground-breaking paper "Parabolic invariant theory in complex analysis" [F2]
Conformally convariant equations on differential forms
Let M be a pseudo-Riemannian manifold o f dimension n≧3. A second-order linear differential operator , which is the sum of a variant of the Laplace-Beltrami operator on k-forms (obtained by weighting
Differential operators cononically associated to a conformal structure.
Soit (M,g) une variete pseudoriemannienne de dimension n. On donne des formules explicites pour un operateur d'ordre 4. D 4 ,k sur les k-formes dans M, n¬=1, 2, 4 pour un operateur D6 d'ordre 6 sur
Complex Paraconformal Manifolds – their Differential Geometry and Twistor Theory
• Mathematics
• 1991
A complex paraconformal manifold is a/^-dimensional complex manifold (/?, q > 2) whose tangent bündle factors äs a tensor product of two bundles of ranks p and q. We also assume that we are given a
The Penrose Transform: Its Interaction with Representation Theory
• Mathematics
• 1990
Part 1 Lie algebras and flag manifolds: some structure theory borel and parabolic subalgebras generalized flag varieties fibrations of generalized flag varieties. Part 2 Homogeneous vector bundles on
Real hypersurfaces in complex manifolds
• Mathematics
• 1974
Whether one studies the geometry or analysis in the complex number space C a + l , or more generally, in a complex manifold, one will have to deal with domains. Their boundaries are real
Tractor calculi for parabolic geometries
• Mathematics
• 2001
Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are
On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections
paper we have omitted the proof of Theorem 1 there, because it is essentially achieved by Tanaka [11] and the theorem is now familiar.) Let Mi (i=1, 2) be a real hypersurf ace of a complex manifold
Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators
• Mathematics
• 1998
This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. | {"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": 0.8891363143920898, "perplexity": 1459.0707026986574}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103331729.20/warc/CC-MAIN-20220627103810-20220627133810-00092.warc.gz"} |
http://www.ck12.org/book/CK-12-Geometry-Concepts/r2/section/8.4/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 8.4: 45-45-90 Right Triangles
Difficulty Level: At Grade Created by: CK-12
Estimated6 minsto complete
%
Progress
Practice 45-45-90 Right Triangles
Progress
Estimated6 minsto complete
%
What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides? After completing this Concept, you'll be able to use the 45-45-90 Theorem to solve problems like this one.
### Watch This
Watch the second half of this video.
Watch the second half of this video.
### Guidance
There are two types of special right triangles, based on their angle measures. The first is an isosceles right triangle. Here, the legs are congruent and, by the Base Angles Theorem, the base angles will also be congruent. Therefore, the angle measures will be \begin{align*}90^\circ, 45^\circ,\end{align*} and \begin{align*}45^\circ\end{align*}. You will also hear an isosceles right triangle called a 45-45-90 triangle. Because the three angles are always the same, all isosceles right triangles are similar.
##### Investigation: Properties of an Isosceles Right Triangle
Tools Needed: Pencil, paper, compass, ruler, protractor
1. Construct an isosceles right triangle with 2 in legs. Use the SAS construction that you learned in Chapter 4.
2. Find the measure of the hypotenuse. What is it? Simplify the radical.
3. Now, let’s say the legs are of length \begin{align*}x\end{align*} and the hypotenuse is \begin{align*}h\end{align*}. Use the Pythagorean Theorem to find the hypotenuse. What is it? How is this similar to your answer in #2?
\begin{align*}x^2 + x^2 & = h^2\\ 2x^2 & = h^2\\ x \sqrt{2} & = h\end{align*}
45-45-90 Corollary: If a triangle is an isosceles right triangle, then its sides are in the extended ratio \begin{align*}x : x : x \sqrt{2}\end{align*}.
Step 3 in the above investigation proves the 45-45-90 Triangle Theorem. So, anytime you have a right triangle with congruent legs or congruent angles, then the sides will always be in the ratio \begin{align*}x : x : x \sqrt{2}\end{align*}. The hypotenuse is always \begin{align*}x \sqrt{2}\end{align*} because that is the longest length. This is a specific case of the Pythagorean Theorem, so it will still work, if for some reason you forget this corollary.
#### Example A
Find the length of the missing sides.
Use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio.
\begin{align*}TV = 6\end{align*} because it is equal to \begin{align*}ST\end{align*}. So, \begin{align*}SV = 6 \sqrt{2}\end{align*} .
#### Example B
Find the length of \begin{align*}x\end{align*}.
Again, use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio. We are given the hypotenuse, so we need to solve for \begin{align*}x\end{align*} in the ratio.
\begin{align*}x \sqrt{2} &= 16\\ x &= \frac{16}{ \sqrt{2}} \cdot \frac{ \sqrt{2}}{\sqrt{2}} \\ x&== \frac{16 \sqrt{2}}{2} \\x&= 8 \sqrt{2}\end{align*}
Note that we rationalized the denominator. Whenever there is a radical in the denominator of a fraction, multiply the top and bottom by that radical. This will cancel out the radical from the denominator and reduce the fraction.
#### Example C
A square has a diagonal with length 10, what are the lengths of the sides?
Draw a picture.
We know half of a square is a 45-45-90 triangle, so \begin{align*}10=s \sqrt{2}\end{align*}.
\begin{align*}s \sqrt{2} &= 10\\ s &= \frac{10}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}= \frac{10 \sqrt{2}}{2}=5 \sqrt{2}\end{align*}
Watch this video for help with the Examples above.
### Vocabulary
A right triangle is a triangle with a \begin{align*}90^\circ\end{align*} angle. A 45-45-90 triangle is a right triangle with angle measures of \begin{align*}45^\circ, 45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}.
### Guided Practice
1. Find the length of the missing sides.
2. Find the length of \begin{align*}x\end{align*}.
3. \begin{align*}x\end{align*} is the hypotenuse of a 45-45-90 triangle with leg lengths of \begin{align*}5\sqrt{3}\end{align*}.
1. Use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio. \begin{align*}AB = 9 \sqrt{2}\end{align*} because it is equal to \begin{align*}AC\end{align*}. So, \begin{align*}BC = 9 \sqrt{2} \cdot \sqrt{2} = 9 \cdot 2 = 18\end{align*}.
2. Use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio. We need to solve for \begin{align*}x\end{align*} in the ratio.
\begin{align*}12 \sqrt{2} &= x \sqrt{2}\\ 12 &= x\end{align*}
3. \begin{align*}x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\end{align*}
### Practice
1. In an isosceles right triangle, if a leg is \begin{align*}x\end{align*}, then the hypotenuse is __________.
2. In an isosceles right triangle, if the hypotenuse is \begin{align*}x\end{align*}, then each leg is __________.
3. A square has sides of length 15. What is the length of the diagonal?
4. A square’s diagonal is 22. What is the length of each side?
5. A square has sides of length \begin{align*}6\sqrt{2}\end{align*}. What is the length of the diagonal?
6. A square has sides of length \begin{align*}4 \sqrt{3}\end{align*}. What is the length of the diagonal?
7. A baseball diamond is a square with 90 foot sides. What is the distance from home base to second base? (HINT: It’s the length of the diagonal).
8. Four isosceles triangles are formed when both diagonals are drawn in a square. If the length of each side in the square is \begin{align*}s\end{align*}, what are the lengths of the legs of the isosceles triangles?
Find the lengths of the missing sides. Simplify all radicals.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
45-45-90 Theorem
For any isosceles right triangle, if the legs are x units long, the hypotenuse is always x$\sqrt{2}$.
45-45-90 Triangle
A 45-45-90 triangle is a special right triangle with angles of $45^\circ$, $45^\circ$, and $90^\circ$.
Hypotenuse
The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle
The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
The $\sqrt{}$, or square root, sign.
Show Hide Details
Description
Difficulty Level:
Authors:
Tags:
Subjects: | {"extraction_info": {"found_math": true, "script_math_tex": 37, "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": 5, "texerror": 0, "math_score": 0.9765698313713074, "perplexity": 1050.4190386637663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738659680.65/warc/CC-MAIN-20160924173739-00046-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/88128/alternative-to-text-inside-an-align-environment?answertab=votes | # Alternative to \text inside an align environment
Is there a way to write texts without resorting to using the \text macro every time I write something in the align enviroment?
I am trying to write a proof and pretty much 90% requires me to write inside the align environment.
Here is an example
\documentclass[11pt]{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
E is {\bf open} & \iff \textit{every} point is an {\bf interior point}
& \iff \forall x \in E, there exists a neighbourhood of x such that each N is disjoint
& \iff \forall x \in E, x is not a limit point of E^c
& E^c contains all its limit points
\end{align*}
\end{document}
Basically if I were to include the text, I have to wrap each sentence with \text{...}
-
Do the remaining 10% actually have to be aligned (to each other)? There’s also \intertext and with the mathtools package \shortintertext for longer passages that get typeset as normal text (not on the equation line). Can you make an example? What would the alternative to \text be? Another macro? – Qrrbrbirlbel Dec 25 '12 at 1:08
I'll make the example of what I kinda what to do, just a sec – sidht Dec 25 '12 at 1:16
Please post a minimal working example (MWE), one that begins with \documentclass and ends with \end{document}. It would be easier to help if the solvers know the context that you are working in. Also, see Does it matter if I use \textit or \it, \bfseries or \bf, etc – hpesoj626 Dec 25 '12 at 1:26
No it doesn't truly matter if you use \it or \textit or \bf, though it would be easier on me if you are consistent with what I use. Also I know my current example does not work and I explained why and what I want to get at. I'll update immediately. Thanks. – sidht Dec 25 '12 at 1:31
Not that related to your problem actually, but if you read through the link in my comment, you will see that it is considered bad practice these days to use \it and \bf in LaTeX2e. I suggest that you use \textit and \textbf instead. :) – hpesoj626 Dec 25 '12 at 1:48
The short answer to your question is yes. If you're going to place text within a math environment that you don't want treated like part of a mathematical expression, it needs to be within \text{...}.
But, since the majority of your proof is text, I would use a tabular environment.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{tabularx}
\usepackage{amsmath}
\usepackage{lipsum}
\pagestyle{empty}
\begin{document}
\lipsum[1]
\vspace{2ex}
\hspace*{\fill}%
\begin{tabularx}{0.9\linewidth}{l@{}c@{}X}
$E$ is \textbf{open} & $\iff$ & \textit{every} point is an \textbf{interior point} \\
& $\iff$ & $\forall x \in E$, there exists a neighbourhood of $x$ such that each $N$ is disjoint \\
& $\iff$ & $\forall x \in E, x$is not a limit point of $E^c$ \\
& & $E^c$ contains all its limit points \\
\end{tabularx}%
\hspace*{\fill}
\vspace{2ex}
\lipsum[2]
\vspace{2ex}
\hspace*{\fill}%
\begin{tabular}{l@{}c@{}p{3in}}
$E$ is \textbf{open} & $\iff$ & \textit{every} point is an \textbf{interior point} \\
& $\iff$ & $\forall x \in E$, there exists a neighbourhood of $x$ such that each $N$ is disjoint \\
& $\iff$ & $\forall x \in E, x$is not a limit point of $E^c$ \\
& & $E^c$ contains all its limit points \\
\end{tabular}%
\hspace*{\fill}
\vspace{2ex}
\end{document}
To get rid of the extra space around \iff I've pre/post-pended @{} to the column type to eliminate all intercolumn spacing around that column.
I've provided two approaches to formatting the last column. I'm sure you don't want it running into the right hand margin. If your margins are small enough, everything should work out fine. Nevertheless, you can use the package tabularx which defines a new table environment tabularx allowing you to create an expandable column type X. For this to work correctly, you have to specify the width of the entire table: I've set it to 0.9\linewidth. But it is not necessary to use tabularx. You can manually set the last column to be a paragraph of a prespecified width: I've done this in the second example where I set the column to p{3in}. This isprobably much narrower than you would really want, but I want you to be able to see the effect.
As already mentioned in the comments to your posting, using \bf and its ilk are looked upon disapprovingly. Instead you should use the LaTeX equivalents. See Does it matter if I use \textit or \it, \bfseries or \bf, etc.
While I've maintained your use of \textbf and \textit, if your purpose is for emphasis, it would be better to use \emph for both. Using too many different font styles is not highly recommended. See Why are these commands considered as bad practice?. While I don't follow it to the letter (I mostly make small documents like handouts and quizzes), I think it is good to be familiar with what the general consensus of the community is: particularly when you're writing something that you might want to share with a larger community.
-
Is there a way to center all of that? Including that text inside the center environment has zero effect – sidht Dec 25 '12 at 2:09
Perhaps \begin{tabular}{l@{\;}c@{\;}l} will make the horizontal spacing better but that is just my preference. :) – hpesoj626 Dec 25 '12 at 2:12
There is a high probability that this may be on my part of formatting. But i've noticed that longer sentences gets "cut off" and sent to the second line. For instance, in our example "is disjoint" is sent to the second line. My document set up, I will update – sidht Dec 25 '12 at 2:21
play with the parameter of the third column. – A.Ellett Dec 25 '12 at 2:24
Hmm if I were to give more space for my sentences. I sacrifice my centering. There's gotta be a way to get both right? – sidht Dec 25 '12 at 2:29 | {"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": 0.9289036393165588, "perplexity": 821.0654163826546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500831098.94/warc/CC-MAIN-20140820021351-00207-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/130810/what-is-the-schwarzschild-metric-with-proper-radial-distance | # What is the Schwarzschild metric with proper radial distance?
Reading the marvellous book "The Membrane Paradigm" I stumbled upon a suggested change of variable that I'm not able to deal with. Starting with the usual Schwarzschild metric for the spatial 3-geometry
$$ds^2 = \frac{1}{f(r)} dr^2 + r^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$
where $f(r) = 1-\frac{2M}{r}$ (that is simply Schwarzschild metric without time element $-f(r)\ dt^2$) they suggest a new radial coordinate $R$ with the "property that $R-2M$ measures the proper radial distance outward from the horizon". This new variable is
$$R= 2M + \sqrt{r(r-2M)}+\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right].\ \ \ \ \ \ \ \ \ \ (2)$$
I'd like to see the new metric but I'm not able to invert (2) (namely find $r(R)$) to perform the coordinate change $$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}\frac{\partial x^{\mu}}{\partial x'^{\rho}}\frac{\partial x^{\nu}}{\partial x'^{\sigma}}dx'^{\rho}dx'^{\sigma}=g'_{\rho\sigma}dx'^{\rho}dx'^{\sigma}$$
I tried also to use the inverse of the Jacobian as suggested in https://physics.stackexchange.com/a/43084 but at the end I always need at least to change the variable $r^2$ in the angular part of (1).
Do you have any idea on how to deal with it or how to invert (2) ?
• You have certainly an error in $(2)$ : there should be a length term $[L]$ multiplying the $ln$ term. Aug 13, 2014 at 13:29
• @Trimok I see your point. Actually it is correctly copied from the book, so the error should be in the book. BTW here $c=1$ everywhere.
– DDd
Aug 13, 2014 at 14:25
$$R= 2M + \sqrt{r(r-2M)}+2M\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right] \tag{1}$$
You have (if no error) : $dR = \dfrac{dr}{\sqrt{ 1 - \dfrac{2M}{r}}}$, so $dR^2 = \dfrac{dr^2}{f(r)}$, and this simplifies your metrics.
However, you cannot invert the formula $(1)$ to get $r$ as an explicit function of $R$. You will simply write the metrics :
$$ds^2 = dR^2 + (r(R))^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right) \tag{2}$$ where $r(R)$ is implicitely defined by the equation $(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": 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": 0.9026517271995544, "perplexity": 319.6298943058747}, "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/1664030335469.40/warc/CC-MAIN-20220930113830-20220930143830-00028.warc.gz"} |
http://projecteuclid.org/euclid.aoms/1177706720 | ## The Annals of Mathematical Statistics
### A Multivariate Tchebycheff Inequality
#### Abstract
A multivariate Tchebycheff inequality is given, in terms of the covariances of the random variables in question, and it is shown that the inequality is sharp, i.e., the bound given can be achieved. This bound is obtained from the solution of a certain matrix equation and cannot be computed easily in general. Some properties of the solution are given, and the bound is given explicitly for some special cases. A Less sharp but easily computed and useful bound is also given.
#### Article information
Source
Ann. Math. Statist. Volume 29, Number 1 (1958), 226-234.
Dates
First available: 27 April 2007
Permanent link to this document
http://projecteuclid.org/euclid.aoms/1177706720
JSTOR | {"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": 0.9702017903327942, "perplexity": 490.48967711236224}, "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-2014-10/segments/1394011176878/warc/CC-MAIN-20140305091936-00018-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://electowiki.org/w/index.php?title=Kemeny%E2%80%93Young_method&oldid=1084&mobileaction=toggle_view_mobile | # Kemeny–Young method
Each possible complete ranking of the candidates is given a "distance" score. For each pair of candidates, find the number of ballots that order them the the opposite way as the given ranking. The distance is the sum across all such pairs. The ranking with the least distance wins.
The winning candidate is the top candidate in the winning ranking.
## Strategic Vulnerability
Kemeny-Young is vulnerable to compromising, burying, and crowding.
## Example
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters
The preferences of the voters would be divided like this:
42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis
Consider the ranking Nashville>Chattanooga>Knoxville>Memphis. This ranking contains 6 orderings of pairs of candidates:
• Nashville>Chattanooga, for which 32% of the voters disagree.
• Nashville>Knoxville, for which 32% of the voters disagree.
• Nashville>Memphis, for which 42% of the voters disagree.
• Chattanooga>Knoxville, for which 17% of the voters disagree.
• Chattanooga>Memphis, for which 42% of the voters disagree.
• Knoxville>Memphis, for which 42% of the voters disagree.
The distance score for this ranking is 32+32+42+17+42+42=207.
It can be shown that this ranking is the one with the lowest distance score. Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville. | {"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": 0.8375771045684814, "perplexity": 4965.942618563449}, "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-2021-31/segments/1627046154214.63/warc/CC-MAIN-20210801154943-20210801184943-00406.warc.gz"} |
http://kmaeda.net/kmaeda/en/demo/gershgorin/ | # Gershgorin's theorem
Complex Plane
$t=$
$M(t)=D+tA=$, $\lambda(t)=$
$M=$
## What's this?
Gershgorin's theorem tells us a region that contains all the eigenvalues of a complex square matrix.
Let $M=(m_{i, j})$ be a complex matrix of order $n$, and $R_i=\sum_{k \ne i} |m_{i, k}|,\quad i=1, 2, \dots, n.$ Let $D_i$ be the closed disc centered at $m_{i, i}$ with radius $R_i$: $D_i=\{\, z \in \mathbb C: |z-m_{i, i}|\le R_i \,\}.$ Then, (i) all the eigenvalues of the matrix $M$ are in $\cup_{i=1}^n D_i$. Furthermore, (ii) if a union of $k$ discs is disjoint from the other $n-k$ discs, then the union contains exactly $k$ eigenvalues.
This program draws the Gershgorin discs (yellow) and the eigenvalues (red) on the complex plane for a given real matrix of order 4. You can observe a continuous deformation used in the proof of the statement (ii) of the theorem and see its correctness.
The proof of (ii) is given as follows. Let $D=\mathop{\mathrm{diag}}(m_{1, 1}, m_{2, 2}, \dots, m_{n, n})$ and $A=M-D$, which are the diagonal part and the non-diagonal part of $M$, respectively. Since $D$ satisfies the statement (ii), the eigenvalues of $M(t)=D+tA$ are continuous in $t$, and the radius $R_i(t)$ increases monotonically and continuously as $t$ varies from $0$ to $1$ continuously, (ii) is true for $M(t)$, $t \in [0, 1]$, from (i). This means that $M=M(1)$ satisfies the statement (ii). The program enables us to understand the above proof in a visual way.
## Libraries
This program uses the following Javascript libraries: | {"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": 0.9425276517868042, "perplexity": 154.88645273737066}, "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-2018-43/segments/1539583513804.32/warc/CC-MAIN-20181021073314-20181021094814-00115.warc.gz"} |
http://www.researchgate.net/researcher/71070948_Xinlong_Wang | # Xinlong Wang
Nanjing University, Nanjing, Jiangsu Sheng, China
Are you Xinlong Wang?
## Publications (20)43.94 Total impact
• ##### Article: Acoustical “transparency” induced by local resonance in Bragg bandgaps
Gaokun Yu, Xinlong Wang
[Hide abstract]
ABSTRACT: We show that sound waves can resonantly transmit through Bragg bandgaps in an acoustical duct periodically attached with an array of Helmholtz resonators, forming within the normally forbidden band a transparency window with group velocity smaller than the normal speed of sound. The transparency occurs for the locally resonant frequency so much close to the Bragg one that both the local-resonance-induced bandgap and the Bragg one heavily overlap with each other. The phenomenon seems an acoustical analog of the well-known electromagnetically induced transparency by quantum interference. Different from the Fano-like interference explanation, we also provide a mechanism for the transparency window phenomenon which makes it possible to extend the phenomenon in general.
Journal of Applied Physics 01/2014; 115(4):044913-044913-6. · 2.21 Impact Factor
• ##### Article: Extraordinary sound tunneling through a barred horn via subwavelength hole resonance
Gaokun Yu, Xinlong Wang
[Hide abstract]
ABSTRACT: Tuning the extraordinary acoustical transmission is shown to be practically feasible simply by controlling acoustical impedances induced by surface evanescent waves. We demonstrate this idea with an example of making a sound tunnel in an acoustical waveguide with a subwavelength short throat and a catenoid horn working below its cutoff frequency. The throat acting as a resonant aperture assists sound waves effectively tunneling through the normally barred horn, leading to resonant transmission of sound waves within an adjustable narrow band. The example may find its applications for highly efficient acoustical filters and transmitters.
Applied Physics Letters 12/2011; 99(25). · 3.52 Impact Factor
• ##### Article: Resonant emission of solitons from impurity-induced localized waves in nonlinear lattices.
Gaokun Yu, Xinlong Wang, Zhiyong Tao
[Hide abstract]
ABSTRACT: We propose a mechanism for soliton creation from resonantly excited localized waves via supratransmission in band gaps of nonlinear lattices. A nonlinear localized wave, which is formed by and vibrates around an impurity with an intrinsic frequency, is found to undergo a local resonance when subject to an external forcing. Under the resonance, an instability develops that leads to the efficient emission of solitons at a much lower rate than that in uniform lattices with no impurity.
Physical Review E 02/2011; 83(2 Pt 2):026605. · 2.31 Impact Factor
• ##### Article: Extraction of time varying information from noisy signals: An approach based on the empirical mode decomposition
[Hide abstract]
ABSTRACT: A windowed average technique is designed as an efficient assistance of empirical mode decomposition, aimed especially at extracting components with temporally variant frequencies from heavily noisy signals. Unlike those relying on detection of such points as local extrema that are highly sensitive to noise interference, the present method evaluates a local mean curve that reflects the slow variation of a signal in longer time scales by locally integral average over a sliding window. It adapts to variation of signal component in a broad frequency range by making the window width variable in response to the variation. The enhanced performance and robustness of the new algorithm with respect to noise resistance are demonstrated in comparison with other EMD-based methods, and examples of processing both speech and underwater acoustic signals are given to show the success of extracting time varying information.
Mechanical Systems and Signal Processing 01/2011; · 1.91 Impact Factor
• ##### Article: Detection of Dynamic Structures of Speech Fundamental Frequency in Tonal Languages
[Hide abstract]
ABSTRACT: An approach is proposed specially for capturing fine dynamic structures of speech fundamental frequency F <sub>0</sub> that may vary in such a nonmonotonic way as those of the third tones in Chinese speech. It first estimates the rough trend of variation of a F <sub>0</sub> contour by means of the cepstrum technique, and then, utilizes the trend as a reference to track the variation and calculates the detailed contour from a few of intrinsic mode functions that are decomposed by the ensemble empirical mode decomposition. Intensive evaluation and direct comparisons with existing methods are conducted with the standard Chinese Mandarin database, showing the effectiveness of the proposed method in acquiring accurate and reliable F <sub>0</sub> contours from speech signals even heavily contaminated with noise.
IEEE Signal Processing Letters 11/2010; · 1.67 Impact Factor
• Source
##### Article: Theory of resonant sound transmission through small apertures on periodically perforated slabs
Xinlong Wang
[Hide abstract]
ABSTRACT: An analytical theory for sound transmitting through apertures that are slits or holes periodically pored on one- or two-dimensional rigid panels is developed in small-aperture approximation, with all coefficients of reflection and transmission given explicitly in concise and easily calculable forms. We utilize acoustical impedance to quantitatively describe the effect of sound diffraction by both surfaces of a perforated slab on the aperture resonance. We show that diffraction induced reactance X<sub> a </sub> , which is acoustically inertant (X<sub> a </sub>>0) for incident wavelength λ longer than the period Λ of the perforated slab, can become infinitely large as λ approaches to Λ . We further show that the singularity of X<sub> a </sub> not only causes the already known full reflection of acoustic waves at λ=Λ , but also drastically changes the aperture resonance leading to the extraordinary acoustical transmission that was observed in recent experiments. With this understanding, tuning the resonant transmission becomes practically feasible in applications of the resonant transmission phenomenon.
Journal of Applied Physics 10/2010; · 2.21 Impact Factor
• ##### Article: Ship classification using nonlinear features of radiated sound: an approach based on empirical mode decomposition.
[Hide abstract]
ABSTRACT: Classification for ship-radiated underwater sound is one of the most important and challenging subjects in underwater acoustical signal processing. An approach to ship classification is proposed in this work based on analysis of ship-radiated acoustical noise in subspaces of intrinsic mode functions attained via the ensemble empirical mode decomposition. It is shown that detection and acquisition of stable and reliable nonlinear features become practically feasible by nonlinear analysis of the time series of individual decomposed components, each of which is simple enough and well represents an oscillatory mode of ship dynamics. Surrogate and nonlinear predictability analysis are conducted to probe and measure the nonlinearity and regularity. The results of both methods, which verify each other, substantiate that ship-radiated noises contain components with deterministic nonlinear features well serving for efficient classification of ships. The approach perhaps opens an alternative avenue in the direction toward object classification and identification. It may also import a new view of signals as complex as ship-radiated sound.
The Journal of the Acoustical Society of America 07/2010; 128(1):206-14. · 1.65 Impact Factor
• Source
##### Article: Acoustical mechanism for the extraordinary sound transmission through subwavelength apertures
Xinlong Wang
[Hide abstract]
ABSTRACT: An acoustical theory is developed for sound transmitting through subwavelength apertures. We show that the excitation of evanescent high-order modes induces, on each end of an aperture (either a slit or a hole), an additional acoustical reactance, which is singular as incident wavelength approaches one of cutoff wavelengths of high modes. The anomaly of the induced reactance greatly changes the resonant behaviors of the aperture, and makes the conjugate impedance matching possible at a wavelength slightly longer than the cutoff, thus leading to the extraordinary full transmission.
Applied Physics Letters 04/2010; · 3.52 Impact Factor
• ##### Article: EMD-based extraction of modulated cavitation noise
[Hide abstract]
ABSTRACT: The empirical mode decomposition (EMD) is applied to extract information of modulation from signals contaminated by noise. The EMD method is capable of recovering the amplitude-modulated components from strong background noise in an adaptive way, and achieves better performance than traditional methods. We further propose a modified EMD technique by estimating the local mean of a signal via windowed average. This method alleviates the unfavorable influence of noise disturbance effectively in the process of sifting and yields a remarkable improvement for the modulation extraction. Finally, by utilizing this novel technique, we achieve the adaptive and effective extraction of the modulated cavitation noise from ship-radiated noise.
Mechanical Systems and Signal Processing 01/2010; · 1.91 Impact Factor
• Source
##### Article: Adaptive extraction of modulation for cavitation noise.
[Hide abstract]
ABSTRACT: Modulation analysis is an important issue in target classification and identification for ship-radiated noise. However, the modulated cavitation noise sought for analyzing is always submerged under strong ambient noise and difficult to be separated out. In this paper, an approach is proposed to extract the modulated cavitation noise adaptively by combining empirical mode decomposition and singular value decomposition. The results for both synthetical and practical signals demonstrate the practicability and effectivity of the approach.
The Journal of the Acoustical Society of America 12/2009; 126(6):3106-13. · 1.65 Impact Factor
• Source
##### Article: Local Integral Mean-Based Sifting for Empirical Mode Decomposition
Hong Hong, Xinlong Wang, Zhiyong Tao
[Hide abstract]
ABSTRACT: A novel sifting method based on the concept of local integral mean of a signal is developed for empirical mode decomposition (EMD), aiming at decomposing those modes whose frequencies are within an octave. Instead of averaging the upper and lower envelopes, the proposed technique computes the local mean curve of a signal by interpolating data points that are local integral averages over segments between successive extrema of the signal. With the sifting method, EMD can separate intrinsic modes of oscillations with frequency ratios up to 0.8, thus considerably improving the frequency resolving power. Also, it is shown that the integral property of the sifting considerably accelerates the convergence of the sifting iteration and remarkably enhances the robustness of EMD against noise disturbance.
IEEE Signal Processing Letters 11/2009; · 1.67 Impact Factor
• ##### Article: Sound transmission within the Bragg gap via the high-order modes in a waveguide with periodically corrugated walls
[Hide abstract]
ABSTRACT: The well-known Bragg resonance in periodic waveguides always leads to the creation of the so-called Bragg gap, within which sound propagations are effectively forbidden. Here we report the possibility of sound energy transmission in the Bragg gap via the high-order transverse modes, which penetrate through the forbidden band due to the interactions between different sound modes in an acoustic duct with periodically corrugated walls. The theoretical analysis indicates that in the waveguides with transverse scales comparable to its period, the guided wave modes can interact with the Bragg gap so that the forbidden band undergoes an abnormal change, giving rise to both a considerable compression in the band width and a sharp descent of the transmission loss on the upper edge of the stopband. The experiment confirms the existence and the significance of the interacting effect, and the measurements of the transmission loss and the radial distribution of sound fields agree quite well with the theoretical predictions.
Journal of Applied Physics 07/2009; · 2.21 Impact Factor
• ##### Article: Non-Bragg resonance of surface water waves in a trough with periodic walls.
Yumeng Xiao, Zhiyong Tao, Weiyu He, Xinlong Wang
[Hide abstract]
ABSTRACT: The non-Bragg resonance of surface water waves is investigated, both theoretically and experimentally, in a trough with square-wave corrugated sidewalls. Unlike the familiar Bragg resonances, the non-Bragg resonances occur far from the edges of the Brillouin zone and open additional forbidden bands. The experimental observations confirm the existence of these resonances, and the measurements for the transmission properties showing both Bragg and non-Bragg band gaps agree fairly well with the theoretical predictions obtained by the plane-wave expansion method. It is shown that both Bragg and non-Bragg resonances highly depend on the symmetry of the corrugations on the opposite sidewall. As the relative shift between the two corrugations increases from zero to the half period of the corrugations, the Bragg gap shrinks and vanishes, but the non-Bragg gap varies in the opposite way, reaching its maximum, which is impressively wide and much more efficient in reflecting water waves.
Physical Review E 08/2008; 78(1 Pt 2):016311. · 2.31 Impact Factor
• ##### Article: Wide forbidden band induced by the interference of different transverse acoustic standing-wave modes
Zhiyong Tao, Weiyu He, Yumeng Xiao, Xinlong Wang
[Hide abstract]
ABSTRACT: A non-Bragg nature forbidden band is experimentally observed in an axially symmetric hard-walled duct with a periodically varying cross section. Unlike the familiar Bragg ones, the observed bandgap is found to result from the interference of sound wave modes having different transverse standing-wave profiles, the so-called non-Bragg resonance. The experiments also show that the non-Bragg band can be comparably wider than the Bragg one; furthermore, the sound transmission loss within the band can be much more effective, exhibiting the great significance of the non-Bragg resonance in wave propagation in periodic waveguides.
Applied Physics Letters 03/2008; 92(12):121920-121920-3. · 3.52 Impact Factor
• ##### Article: Resonance-induced band gaps in a periodic waveguide
Zhi-Yong Tao, Wei-Yu He, Xinlong Wang
[Hide abstract]
ABSTRACT: The band gaps in periodic structures are usually regarded as being induced by the Bragg resonances. Until the recent years, the non-Bragg nature resonances were not taken into account in analysing and computing the band gaps, though it can exist in all kinds of waveguides with periodic structures. Here, the resonance-induced band gaps in a periodic acoustic duct are investigated extensively and a graphical method is introduced to analyse the dependence of these resonances on the duct geometry. With this method, it becomes quite easy to estimate the frequency band gaps of the waveguide and shape the band structures by choosing the proper geometric parameters. Our analysis show that the location and the width of the band gap are closely related to the wavenumber and the amplitude of the wall corrugations, and the non-Bragg resonance can result in the obvious band gap when the wall wavenumber is close to the cut-off frequency of the first mode.
Journal of Sound and Vibration 01/2008; · 1.61 Impact Factor
• Source
##### Article: Spatiotemporal bifurcations of a parametrically excited solitary wave.
Likun Zhang, Xinlong Wang, Zhiyong Tao
[Hide abstract]
ABSTRACT: The bifurcation behaviors of a parametrically excited solitary wave are investigated via Faraday's water tank experiment. It is observed that, as the driving frequency fd is decreased or/and the driving amplitude Ad is increased, the standing (but vertically oscillatory) solitary wave becomes modulationally unstable, leading to the temporal modulation of the vertical oscillation and the emergence of very low subharomic components on the frequency spectrum. Further lowering fd or/and increasing Ad will cause the modulational oscillation unstable and then, the peak of the solitary wave becomes rocking along the trough in the longitudinal direction. These bifurcations also give rise to the emission of continuous waves resulting in complex wave patterns and complicated fluctuations, especially for the quite low fd and large Ad. A possible route from solitary waves to chaos via bifurcations and mode competitions is therefore suggested on the basis of these observations.
Physical Review E 04/2007; 75(3 Pt 2):036602. · 2.31 Impact Factor
• ##### Article: Enhancement of Chinese speech based on nonlinear dynamics
Junfeng Sun, Nengheng Zheng, Xinlong Wang
[Hide abstract]
ABSTRACT: Based on recently observed nonlinear dynamic features of human speech, the local projection (LP) method, originally developed for noisy chaotic time series, is generalized and adapted to the enhancement of Chinese speech. The analysis of minimum embedding dimensions estimated by the false nearest neighbor algorithm shows that all the basic phonemes and syllables in Chinese can be faithfully embedded in some low-dimensional phase space. Over-embedding is applied to reconstruct the dynamics of continuous speech in some extended phase space of higher dimension, thus solving the problem of nonstationarity in continuous speech. A generalization of the LP method, named the local subspace method, is presented for speech enhancement in the phase space. It is demonstrated that, the local subspace method is essentially an extension of the well-known linear subspace technique in the local phase space, and the LP method is the least square case of this generalization. Noise reduction is then carried out in the local phase space. Results show that the LP method, with 2 or 3 iterations, achieves better performances than the local subspace method. For both isolated and continuous speech with additive white noise, experiments show the superiority of the LP method over two other popular algorithms.
Signal Processing 01/2007; · 2.24 Impact Factor
• ##### Article: Acoustical diffraction tomography in a finite form based on the Rytov transform.
Zhi-Yong Tao, Zhen-Qiu Lu, Xinlong Wang
[Hide abstract]
ABSTRACT: A new reconstruction algorithm in a finite form based on the Rytov transform is presented for acoustical diffraction tomography. Applying the Rytov transform to the governing differential wave equation necessarily introduces the so-called generalized scattering. Our analysis shows that the generalized scattered wave is asymptotically equivalent to the physically scattered wave, and also satisfies the Sommerfeld radiation condition in the far field. Using the method of formal parameter expansion, we further find that all other terms in the expansion of the object function vanish except the first- and second-order ones, and thus reach a finite form solution to the diffraction tomography. Our computer simulation confirms the effectiveness of the algorithm in the case of the scattering objects with cylindrical symmetry, also shows its limitations when it applies to the strong scattering.
IEEE Transactions on Image Processing 06/2006; 15(5):1264-9. · 3.20 Impact Factor
• Source
##### Article: On solitary waves. Part 2 A unified perturbation theory for higher-order waves
[Hide abstract]
ABSTRACT: A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on (i) the accuracy of the theoretically predicted wave properties and (ii) the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property (e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy), on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ(n) (the relative mean-square difference between successive orders n of wave elevations) reaching a minimum, δ m , at a specific order, n=n m , both depending on the base adopted, e.g. n m , α =11-12 based on parameter α (wave amplitude), n m , β =15 on β (amplitude-speed square ratio), and n m , ∈ =17 on ∈ ( wave number squared). The asymptotic range is brought to completion by the highest order of n=18 reached in this work.
Acta Mechanica Sinica 01/2006; 21(6):515-530. · 0.69 Impact Factor
• ##### Article: Integral convergence of the higher-order theory for solitary waves
Xinlong Wang, Theodore Yaotsu Wu
[Hide abstract]
ABSTRACT: An exact analytic solution for a solitary wave of arbitrary height is attained by series expansions of flow variables based on parameter ε=k2h2, (k being the wave number of the solitary wave on water of uniform depth h) by orders in O(εn) up to n=25. Its convergence behavior is found first to yield a set of asymptotic representations for all the flow variables, each and every becoming highest in accuracy at O(ε17). For n>17, the field variables and wave parameters, e.g., wave amplitude, have their errors continue increasing with n, but, in sharp contrast, all the wave integral properties including the excess mass first undergo finite fluctuations from O(ε17) to O(ε20), then all converge uniformly beyond O(ε20) in a group of tight bundle within the range 0ε0.283, with ε=0.283 corresponding to the highest solitary wave with a 120° vertex angle. This remarkable behavior of series convergence seems to have no precedent, and furthermore, is unique in ε, not shared by the exact solutions based on all other parameters examined here.
Physics Letters A 01/2006; 350(1):44-50. · 1.63 Impact Factor
#### Publication Stats
54 Citations 43.94 Total Impact Points
#### Institutions
• ###### Nanjing University
• • State Key Laboratory of Modern Acoustics
• • Institute of Acoustics
Nanjing, Jiangsu Sheng, China | {"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": 0.8784172534942627, "perplexity": 2199.9631887559326}, "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/1418802764752.1/warc/CC-MAIN-20141217075244-00101-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Set_Difference_is_Subset/Proof_1 | # Set Difference is Subset/Proof 1
## Theorem
$S \setminus T \subseteq S$
## Proof
$\displaystyle x \in S \setminus T$ $\implies$ $\displaystyle x \in S \land x \notin T$ $\quad$ Definition of Set Difference $\quad$ $\displaystyle$ $\implies$ $\displaystyle x \in S$ $\quad$ Rule of Simplification $\quad$
The result follows from the definition of subset.
$\blacksquare$ | {"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": 0.9869868159294128, "perplexity": 660.5881698454293}, "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/1526794866938.68/warc/CC-MAIN-20180525024404-20180525044404-00003.warc.gz"} |
https://consequently.org/class/2023/py1012/ | # PY1012: Reasoning
## January 2023 NOW ON
### at the University of St Andrews
py1012: Reasoning introduces the essential concepts and techniques of critical reasoning, formal propositional logic, and basic predicate logic. Among the central questions are these: what distinguishes an argument from a mere rhetorical ploy? What makes an argument a good one? How can we formally prove that a conclusion follows from some premises? In addressing these questions, we will also cover topics such as ambiguity, argument forms and analyses, induction compared to deduction, counterexamples, truth-tables, natural deduction, and quantification. | {"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": 0.970001220703125, "perplexity": 1723.5790498896536}, "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-2023-06/segments/1674764500041.18/warc/CC-MAIN-20230202200542-20230202230542-00230.warc.gz"} |
http://mathhelpforum.com/algebra/96946-horizontal-asymptote.html | # Math Help - horizontal asymptote
1. ## horizontal asymptote
The horizontal asymptote of $y= \frac{3x-7}{5x+6}$ is $\frac{-7}{6}$
am i correct?
2. Originally Posted by william
The horizontal asymptote of $y= \frac{3x-7}{5x+6}$ is $\frac{-7}{6}$
am i correct?
no ... it's $y = \frac{3}{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": 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": 0.998976469039917, "perplexity": 4489.989063043942}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042988317.67/warc/CC-MAIN-20150728002308-00318-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.zora.uzh.ch/23078/ | # Extreme values of Markov population processes
Barbour, A D (1983). Extreme values of Markov population processes. Stochastic Processes and their Applications, 14(3):297-313. | {"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": 0.9671527147293091, "perplexity": 3152.9862470842513}, "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-2016-40/segments/1474738662058.98/warc/CC-MAIN-20160924173742-00284-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://ned.ipac.caltech.edu/level5/March01/Battaner/node20.html | ## 4.1 The nature of galactic dark matter
Galaxies are born out of primordial fluctuations with an evolution probably driven by gravitation as the dominant effect. Gravitation, as a geometric concept, has the same effect on the different types of particles. Some forces other than gravitation, such as the interaction with photons, dissipative effects, magnetic fields, etc., could also have an influence and act on the involved particles differentially, but an overall trend for galaxies and clusters to have a similar composition to the general composition of the Universe is to be expected.
Our knowledge about the composition of the Universe has changed in recent times with respect to the classical view, summarized, for instance, by Schramm (1992). This new conception has been reviewed, for instance, by Turner (1999a, b). The dominant matter is considered to be cold dark matter (CDM), consisting of particles moving slowly, so that the CDM energy density is mainly due to the particle's rest mass, there being a large series of candidates for CDM particles, but axions and neutralinos being the most attractive possibilities.
Big Bang nucleosynthesis studies have been able to accurately determine the baryon density as (0.019±0.0012)h-2. The cluster baryon density has also been accurately determined by X-ray and the Sunyaev-Zeldovich effect to be fB = (0.07±0.007)h-3/2 and, assuming that rich clusters provide a fair sample of matter in the Universe, also / = fB, from which, it follows = (0.27±0.05)h-1/2. The Universe is however flat, = 1, with the CMB spectrum being a sensitive indicator. Therefore = 1 = + , where ~ 0.7 represents the contribution of the vacuum energy, or rather, the contribution of the cosmological term . With this high value of the Universe should be in accelerating expansion, which has been confirmed by the study of high-redshift supernovae, which also suggest ~ 0.7 (Perlmutter, Turner and White, 1999; Perlmutter et al. 1999). The stellar or visible matter is estimated to be = 0.003 - 0.006. All these values can be written in a list easier to remember, with values compatible with the above figures, adopting the values of H0 = 65kms-1Mpc-1; h=0.65:
= 0.003
= 0.03
= 0.3
= 1
less precise but useful for exploratory fast calculations.
A large cluster should have more or less this composition, including the halo of course, even if a halo could contain several baryonic concentrations or simply none. Therefore, a first direct approach to the problem suggests that halos are non baryonic, with baryonic matter being a minor constituent.
This is also the point of view assumed by most current theoretical models (this will be considered later, in Section 4.2.2), which follow the seminal papers by Press and Schechter (1974) and White and Rees (1978). We advance the comment that, in these models, a dominant collisionless non dissipative cold dark matter is the main ingredient of halos while baryons, probably simply gas, constitute the dissipative component, able to cool, concentrate, fragment and star-producing. Some gas can be retained mixed in the halo, and therefore halos would be constituted of non-baryonic matter plus small quantities of gas, its fraction decreasing with time, while mergers and accretion would provide increasing quantities to the visible disks and bulges. Therefore, a first approach suggests that galactic dark matter is mainly non-baryonic, which would be considered as the standard description. Baryons, and therefore visible matter, may not have condensed completely within a large DM halo, and therefore the baryon/DM ratio should be similar in the largest halos and in the whole Universe, although this ratio could be different in normal galaxies.
However, other interesting possibilities have also been proposed. The galactic visible/dark matter fraction depends very much on the type of galaxy, but a typical value could be 0.1. This is also approximately the visible/baryon matter fraction in the Universe, which has led some authors to think that the galactic dark matter is baryonic (e.g. Freeman, 1997) in which case the best candidates would be gas clouds, stellar remnants or substellar objects. The stellar remnants present some problems: white dwarfs require unjustified initial mass functions; neutron stars and black holes would have produced much more metal enrichment. We cannot account for the many different possibilities explored. Substellar objects, like brown dwarfs, are an interesting identification of MACHOs, the compact objects producing microlensing of foreground stars. Alcock et al. (1993), Aubourg et al. (1993) and others have suggested that MACHOSs could provide a substantial amount of the halo dark matter, as much as 50-60% for masses of about 0.25 M, but the results very much depend on the model assumed for the visible and dark matter components, and are still uncertain. Honma and Kan-ya (1998) argued that if the Milky Way does not have a flat rotation curve out to 50 kpc, brown dwarfs could account for the whole halo, and in this case the Milky Way mass is only 1.1 × 1011M.
Let us then briefly comment on the possibility of dark gas clouds, as defended by Pfenniger and Combes (1994), Pfenniger, Combes and Martinet (1994) and Pfenniger (1997). They have proposed that spiral galaxies evolve from Sd to Sa, i.e. the bulge and the disk both increase and at the same time the M/L ratio decreases. Sd are gas-richer than Sa. It is then tempting to conclude that dark matter gradually transforms into visible matter, i.e. into stars. Then, the dark matter should be identified with gas. Why, then, cannot we see that gas? Such a scenario could be the case if molecular clouds possessed a fractal structure from 0.01 to 100 pc. Clouds would be fragmented into smaller, denser and colder sub-clumps, with the fractal dimension being 1.6-2. Available millimeter radiotelescopes are unable to detect such very small clouds. This hypothesis would also explain Bosma's relation between dark matter and gas (Section 2.3), because dark matter would, in fact, be gas (the observable HI disk could be the observable atmosphere of the dense molecular clouds). In this case, the dark matter should have a disk distribution.
The identification of disk gas as galactic dark matter was first proposed by Valentijn (1991) and was later analyzed by González-Serrano and Valentijn (1991), Lequeux, Allen and Guilloteau (1993), Pfenniger, Combes and Martinet (1994), Gerhard and Silk (1996) and others. H2 could be associated to dust, producing a colour dependence of the radial scale length compatible with large amounts of H2. Recently, Valentijn and van der Werf (1999) detected rotational lines of H2 at 28.2 and 17.0 m in NGC 891 on board ISO, which are compatible with the required dark matter. If confirmed, this experiment would be crucial, demonstrating that a disk baryonic visible component is responsible for the anomalous rotation curve and the fragility of apparently solid theories. Confirmation in other galaxies could be difficult as H2 in NGC 891 seems to be exceptionally warm (80-90 K).
A disk distribution is, indeed, the most audacious statement of this scenario. Olling (1996) has deduced that the galaxy NGC 4244 has a flaring that requires a flattened halo. However, this analysis needs many theoretical assumptions; for example, the condition of vertical hydrostatic equilibrium requires further justification, particularly considering that NGC 4244 is a Scd galaxy, with vertical outflows being more important in late type galaxies. Warps have also been used to deduce the shape of the halo. Again, Hofner and Sparke (1994) found that only one galaxy NGC 2903, out of the five studied, had a flattened halo. In this paper, a particular model of warps is assumed (Sparke and Casertano, 1988), but there are other alternatives (Binney 1991, 1992). The Sparke and Casertano model seems to fail once the response of the halo to the precession of the disk is taken into account (Nelson and Tremaine, 1995; Dubinski and Kuijken, 1995). Kuijken (1997) concludes that "perhaps the answer lies in the magnetic generation of warps" (Battaner, Florido and Sanchez-Saavedra 1990). On the other hand, if warps are a deformation of that part of the disk that is already gravitationally dominated by the halo, the deformation of the disk would be a consequence of departures from symmetry in the halo. To isolate disk perturbations embedded in a perfect unperturbed halo is unrealistic. Many other proposals have been made to study the shape of the halo, most of which are reviewed in the cited papers by Olling, and in Ashman (1982), but very different shapes have been reported (see section 3.4).
There is also the possibility that a visible halo component could have been observed (Sackett et al. 1994; Rausher et al. 1997) but due to the difficulty of working at these faint levels, this finding has yet to be confirmed.
Many other authors propose that the halo is baryonic, even if new models of galactic formation and evolution should be developed (de Paolis et al. 1997). This is in part based on the fact that all dark matter "observed" in galaxies and clusters could be accounted for by baryonic matter alone. Under the interpretation of de Paolis et al. (1995) small dense clouds of H2 could also be identified with dark matter, and even be responsible for microlensing, but instead of being distributed in the disk, they would lie in a spherical halo. | {"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": 0.9206738471984863, "perplexity": 1259.8275418221447}, "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-2017-22/segments/1495463607726.24/warc/CC-MAIN-20170524001106-20170524021106-00195.warc.gz"} |
http://www.cs.columbia.edu/~jebara/htmlpapers/UTHESIS/node21.html | Next: Real-Time Implementation Up: Real-Time Symmetry Transform Previous: Lines of Symmetry
## Intersection of Lines of Symmetry
Sela proposes that the intersection of the lines of symmetry in generates an interest point. This interest point has a magnitude depending on the configuration and strength of the lines of symmetry that generated it. We utilize the magnitude of the interest point as a measure of the level of symmetric enclosure at that point. Since perpendicular lines of symmetry generate the strongest sense of enclosure [42], the greater the level of orthogonality between two lines of symmetry, the stronger their contribution to the interest magnitude at point p. The contribution of each pair of lines of symmetry intersecting point p is summed to generate an interest value I(p) which is defined as
(2.5)
where are the orientation values of a pair of symmetry lines intersecting point p whose symmetry magnitudes are . The effect of orthogonality is included in the term which is maximal when the lines of symmetry are perpendicular. The weight w2 is used to tune the sensitivity of the computation to the orthogonality of the intersecting lines of symmetry. A value of w2=5 is typically used so that orthogonal lines of symmetry dominate the response of the interest operator.
Next: Real-Time Implementation Up: Real-Time Symmetry Transform Previous: Lines of Symmetry
Tony Jebara
2000-06-23 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9087937474250793, "perplexity": 653.9806773651428}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662527626.15/warc/CC-MAIN-20220519105247-20220519135247-00484.warc.gz"} |
https://biz.libretexts.org/Bookshelves/Finance/Book%3A_Finance_Banking_and_Money/05%3A_The_Economics_of_Interest-Rate_Fluctuations | # 5: The Economics of Interest-Rate Fluctuations
• Anonymous
• LibreTexts
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
Learning Objectives
By the end of this chapter, students should be able to
• Describe, at the first level of analysis, the factors that cause changes in the interest rate.
• List and explain four major factors that determine the quantity demanded of an asset.
• List and explain three major factors that cause shifts in the bond supply curve.
• Explain why the Fisher Equation holds; that is, explain why the expectation of higher inflation leads to a higher nominal interest rate.
• Predict, in a general way, what will happen to the interest rate during an economic expansion or contraction and explain why.
• Discuss how changes in the money supply may affect interest rates.
Thumbnail: Image by Mediamodifier from Pixabay
5: The Economics of Interest-Rate Fluctuations is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | {"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": 0.8918574452400208, "perplexity": 1230.965289562407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036077.8/warc/CC-MAIN-20220625160220-20220625190220-00232.warc.gz"} |
http://kgslab.org/papers/paper/speedy | Quantifying condition-dependent intracellular protein levels enables high-precision fitness estimates
Geiler-Samerotte K, Hashimoto T, Dion M, Airoldi E, Drummond DA, PLOS ONE 8 :75320 (2013).
Full text
DOI
Share
tweet
# Abstract
Countless studies monitor the growth rate of microbial populations as a measure of fitness. However, an enormous gap separates growth-rate differences measurable in the laboratory from those that natural selection can distinguish efficiently. Taking advantage of the recent discovery that transcript and protein levels in budding yeast closely track growth rate, we explore the possibility that growth rate can be more sensitively inferred by monitoring the proteomic response to growth, rather than growth itself. We find a set of proteins whose levels, in aggregate, enable prediction of growth rate to a higher precision than direct measurements. However, we find little overlap between these proteins and those that closely track growth rate in other studies. These results suggest that, in yeast, the pathways that set the pace of cell division can differ depending on the growth-altering stimulus. Still, with proper validation, protein measurements can provide high-precision growth estimates that allow extension of phenotypic growth-based assays closer to the limits of evolutionary selection. | {"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": 0.9745937585830688, "perplexity": 4346.490399167341}, "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/1600400219691.59/warc/CC-MAIN-20200924163714-20200924193714-00075.warc.gz"} |
https://casmusings.wordpress.com/tag/derivative/ | Tag Archives: derivative
A recent post in the AP Calculus Community expressed some confusion about different ways to compute $\displaystyle \frac{dy}{dx}$ at (0,4) for the function $x=2ln(y-3)$. I share below the two approaches suggested in the original post, proffer two more, and a slightly more in-depth activity I’ve used in my calculus classes for years. I conclude with an alternative to derivatives of inverses.
Two Approaches Initially Proposed
1 – Accept the function as posed and differentiate implicitly.
$\displaystyle \frac{d}{dx} \left( x = 2 ln(y-3) \right)$
$\displaystyle 1 = 2*\frac{1}{y-3} * \frac{dy}{dx}$
$\displaystyle \frac{dy}{dx} = \frac{y-3}{2}$
Which gives $\displaystyle \frac{dy}{dx} = \frac{1}{2}$ at (x,y)=(0,4).
2 – Solve for y and differentiate explicitly.
$\displaystyle x = 2ln(y-3) \longrightarrow y = 3 + e^{x/2}$
$\displaystyle \frac{dy}{dx} = e^{x/2} * \frac{1}{2}$
Evaluating this at (x,y)=(0,4) gives $\displaystyle \frac{dy}{dx} = \frac{1}{2}$ .
Two Alternative Approaches
3 – Substitute early.
The question never asked for an algebraic expression of $\frac{dy}{dx}$, only the numerical value of this slope. Because students tend to make more silly mistakes manipulating algebraic expressions than numeric ones, the additional algebra steps are unnecessary, and potentially error-prone. Admittedly, the manipulations are pretty straightforward here, in more algebraically complicated cases, early substitutions could significantly simplify work. Using approach #1 and substituting directly into the second line gives
$\displaystyle 1 = 2 * \frac{1}{y-3} * \frac{dy}{dx}$.
At (x,y)=(0,4), this is
$\displaystyle 1 = 2 * \frac{1}{4-3}*\frac{dy}{dx}$
$\displaystyle \frac{dy}{dx} = \frac{1}{2}$
The numeric manipulations on the right side are obviously easier than the earlier algebra.
4 – Solve for $\frac{dx}{dy}$ and reciprocate.
There’s nothing sacred about solving for $\frac{dy}{dx}$ directly. Why not compute the derivative of the inverse and reciprocate at the end? Differentiating first with respect to y eventually leads to the same solution.
$\displaystyle \frac{d}{dy} \left( x = 2 ln(y-3) \right)$
$\displaystyle \frac{dx}{dy} = 2 * \frac{1}{y-3}$
At (x,y)=(0,4), this is
$\displaystyle \frac{dx}{dy} = \frac{2}{4-3} = 2$, so
$\displaystyle \frac{dy}{dx} = \frac{1}{2}$.
Equivalence = A fundamental mathematical concept
I sometimes wonder if teachers should place much more emphasis on equivalence. We spend so much time manipulating expressions in mathematics classes at all levels, changing mathematical objects (shapes, expressions, equations, etc.) into a different, but equivalent objects. Many times, these manipulations are completed under the guise of “simplification.” (Here is a brilliant Dan Teague post cautioning against taking this idea too far.)
But it is critical for students to recognize that proper application of manipulations creates equivalent expressions, even if when the resulting expressions don’t look the same. The reason we manipulate mathematical objects is to discover features about the object in one form that may not be immediately obvious in another.
For the function $x = 2 ln(y-3)$, the slope at (0,4) must be the same, no matter how that slope is calculated. If you get a different looking answer while using correct manipulations, the final answers must be equivalent.
Another Example
A similar question appeared on the AP Calculus email list-server almost a decade ago right at the moment I was introducing implicit differentiation. A teacher had tried to find $\displaystyle \frac{dy}{dx}$ for
$\displaystyle x^2 = \frac{x+y}{x-y}$
using implicit differentiation on the quotient, manipulating to a product before using implicit differentiation, and finally solving for y in terms of x to use an explicit derivative.
1 – Implicit on a quotient
Take the derivative as given:$$\displaystyle \frac{d}{dx} \left( x^2 = \frac{x+y}{x-y} \right)$ $\displaystyle 2x = \frac{(x-y) \left( 1 + \frac{dy}{dx} \right) - (x+y) \left( 1 - \frac{dy}{dx} \right) }{(x-y)^2}$ $\displaystyle 2x * (x-y)^2 = (x-y) + (x-y)*\frac{dy}{dx} - (x+y) + (x+y)*\frac{dy}{dx}$ $\displaystyle 2x * (x-y)^2 = -2y + 2x * \frac{dy}{dx}$ $\displaystyle \frac{dy}{dx} = \frac{-2x * (x-y)^2 + 2y}{2x}$ 2 – Implicit on a product Multiplying the original equation by its denominator gives $x^2 * (x - y) = x + y$ . Differentiating with respect to x gives $\displaystyle 2x * (x - y) + x^2 * \left( 1 - \frac{dy}{dx} \right) = 1 + \frac{dy}{dx}$ $\displaystyle 2x * (x-y) + x^2 - 1 = x^2 * \frac{dy}{dx} + \frac{dy}{dx}$ $\displaystyle \frac{dy}{dx} = \frac{2x * (x-y) + x^2 - 1}{x^2 + 1}$ 3 – Explicit Solving the equation at the start of method 2 for y gives $\displaystyle y = \frac{x^3 - x}{x^2 + 1}$. Differentiating with respect to x gives $\displaystyle \frac{dy}{dx} = \frac {\left( x^2+1 \right) \left( 3x^2 - 1\right) - \left( x^3 - x \right) (2x+0)}{\left( x^2 + 1 \right) ^2}$ Equivalence Those 3 forms of the derivative look VERY DIFFERENT. Assuming no errors in the algebra, they MUST be equivalent because they are nothing more than the same derivative of different forms of the same function, and a function’s rate of change doesn’t vary just because you alter the look of its algebraic representation. Substituting the y-as-a-function-of-x equation from method 3 into the first two derivative forms converts all three into functions of x. Lots of by-hand algebra or a quick check on a CAS establishes the suspected equivalence. Here’s my TI-Nspire CAS check. Here’s the form of this investigation I gave my students. Final Example I’m not a big fan of memorizing anything without a VERY GOOD reason. My teachers telling me to do so never held much weight for me. I memorized as little as possible and used that information as long as I could until a scenario arose to convince me to memorize more. One thing I managed to avoid almost completely were the annoying derivative formulas for inverse trig functions. For example, find the derivative of $y = arcsin(x)$ at $x = \frac{1}{2}$. Since arc-trig functions annoy me, I always rewrite them. Taking sine of both sides and then differentiating with respect to x gives. $sin(y) = x$ $\displaystyle cos(y) * \frac{dy}{dx} = 1$ I could rewrite this equation to give $\frac{dy}{dx} = \frac{1}{cos(y)}$, a perfectly reasonable form of the derivative, albeit as a less-common expression in terms of y. But I don’t even do that unnecessary algebra. From the original function, $x=\frac{1}{2} \longrightarrow y=\frac{\pi}{6}$, and I substitute that immediately after the differentiation step to give a much cleaner numeric route to my answer. $\displaystyle cos \left( \frac{\pi}{6} \right) * \frac{dy}{dx} = 1$ $\displaystyle \frac{\sqrt{3}}{2} * \frac{dy}{dx} = 1$ $\displaystyle \frac{dy}{dx} = \frac{2}{\sqrt{3}}$ And this is the same result as plugging $x = \frac{1}{2}$ into the memorized version form of the derivative of arcsine. If you like memorizing, go ahead, but my mind remains more nimble and less cluttered. One final equivalent approach would have been differentiating $sin(y) = x$ with respect to y and reciprocating at the end. CONCLUSION There are MANY ways to compute derivatives. For any problem or scenario, use the one that makes sense or is computationally easiest for YOU. If your resulting algebra is correct, you know you have a correct answer, even if it looks different. Be strong! Advertisements Base-x Numbers and Infinite Series In my previous post, I explored what happened when you converted a polynomial from its variable form into a base-x numerical form. That is, what are the computational implications when polynomial $3x^3-11x^2+2$ is represented by the base-x number $3(-11)02_x$, where the parentheses are used to hold the base-x digit, -11, for the second power of x? So far, I’ve explored only the Natural number equivalents of base-x numbers. In this post, I explore what happens when you allow division to extend base-x numbers into their Rational number counterparts. Level 5–Infinite Series: Numbers can have decimals, so what’s the equivalence for base-x numbers? For starters, I considered trying to get a “decimal” form of $\displaystyle \frac{1}{x+2}$. It was “obvious” to me that $12_x$ won’t divide into $1_x$. There are too few “places”, so some form of decimals are required. Employing division as described in my previous post somewhat like you would to determine the rational number decimals of $\frac{1}{12}$ gives Remember, the places are powers of x, so the decimal portion of $\displaystyle \frac{1}{x+2}$ is $0.1(-2)4(-8)..._x$, and it is equivalent to $\displaystyle 1x^{-1}-2x^{-2}+4x^{-3}-8x^{-4}+...=\frac{1}{x}-\frac{2}{x^2}+\frac{4}{x^3}-\frac{8}{x^4}+...$. This can be seen as a geometric series with first term $\displaystyle \frac{1}{x}$ and ratio $\displaystyle r=\frac{-2}{x}$. It’s infinite sum is therefore $\displaystyle \frac{\frac{1}{x}}{1-\frac{-2}{x}}$ which is equivalent to $\displaystyle \frac{1}{x+2}$, confirming the division computation. Of course, as a geometric series, this is true only so long as $\displaystyle |r|=\left | \frac{-2}{x} \right |<1$, or $2<|x|$. I thought this was pretty cool, and it led to lots of other cool series. For example, if $x=8$,you get $\frac{1}{10}=\frac{1}{8}-\frac{2}{64}+\frac{4}{512}-...$. Likewise, $x=3$ gives $\frac{1}{5}=\frac{1}{3}-\frac{2}{9}+\frac{4}{27}-\frac{8}{81}+...$. I found it quite interesting to have a “polynomial” defined with a rational expression. Boundary Convergence: As shown above, $\displaystyle \frac{1}{x+2}=\frac{1}{x}-\frac{2}{x^2}+\frac{4}{x^3}-\frac{8}{x^4}+...$ only for $|x|>2$. At $x=2$, the series is obviously divergent, $\displaystyle \frac{1}{4} \ne \frac{1}{2}-\frac{2}{4}+\frac{4}{8}-\frac{8}{16}+...$. For $x=-2$, I got $\displaystyle \frac{1}{0} = \frac{1}{-2}-\frac{2}{4}+\frac{4}{-8}-\frac{8}{16}+...=-\frac{1}{2}-\frac{1}{2}-\frac{1}{2}-\frac{1}{2}-...$ which is properly equivalent to $-\infty$ as $x \rightarrow -2$ as defined by the convergence domain and the graphical behavior of $\displaystyle y=\frac{1}{x+2}$ just to the left of $x=-2$. Nice. I did find it curious, though, that $\displaystyle \frac{1}{x}-\frac{2}{x^2}+\frac{4}{x^3}-\frac{8}{x^4}+...$ is a solid approximation for $\displaystyle \frac{1}{x+2}$ to the left of its vertical asymptote, but not for its rotationally symmetric right side. I also thought it philosophically strange (even though I understand mathematically why it must be) that this series could approximate function behavior near a vertical asymptote, but not near the graph’s stable and flat portion near $x=0$. What a curious, asymmetrical approximator. Maclaurin Series: Some quick calculus gives the Maclaurin series for $\displaystyle \frac{1}{x+2}$ : $\displaystyle \frac{1}{2}-\frac{x}{4}+\frac{x^2}{8}-\frac{x^3}{16}+...$, a geometric series with first term $\frac{1}{2}$ and ratio $\frac{-x}{2}$. Interestingly, the ratio emerging from the Maclaurin series is the reciprocal of the ratio from the “rational polynomial” resulting from the base-x division above. As a geometric series, the interval of convergence is $\displaystyle |r|=\left | \frac{-x}{2} \right |<1$, or $|x|<2$. Excluding endpoint results, the Maclaurin interval is the complete Real number complement to the base-x series. For the endpoints, $x=-2$ produces the right-side vertical asymptote divergence to $+ \infty$ that $x=-2$ did for the left side of the vertical asymptote in the base-x series. Again, $x=2$ is divergent. It’s lovely how these two series so completely complement each other to create clean approximations of $\displaystyle \frac{1}{x+2}$ for all $x \ne 2$. Other base-x “rational numbers” Because any polynomial divided by another is absolutely equivalent to a base-x rational number and thereby a base-x decimal number, it will always be possible to create a “rational polynomial” using powers of $\displaystyle \frac{1}{x}$ for non-zero denominators. But, the decimal patterns of rational base-x numbers don’t apply in the same way as for Natural number bases. Where $\displaystyle \frac{1}{12}$ is guaranteed to have a repeating decimal pattern, the decimal form of $\displaystyle \frac{1}{x+2}=\frac{1_x}{12_x}=0.1(-2)4(-8)..._x$ clearly will not repeat. I’ve not explored the full potential of this, but it seems like another interesting field. CONCLUSIONS and QUESTIONS Once number bases are understood, I’d argue that using base-x multiplication might be, and base-x division definitely is, a cleaner way to compute products and quotients, respectively, for polynomials. The base-x division algorithm clearly is accessible to Algebra II students, and even opens the doors to studying series approximations to functions long before calculus. Is there a convenient way to use base-x numbers to represent horizontal translations as cleanly as polynomials? How difficult would it be to work with a base-$(x-h)$ number for a polynomial translated h units horizontally? As a calculus extension, what would happen if you tried employing division of non-polynomials by replacing them with their Taylor series equivalents? I’ve played a little with proving some trig identities using base-x polynomials from the Maclaurin series for sine and cosine. What would happen if you tried to compute repeated fractions in base-x? It’s an open question from my perspective when decimal patterns might terminate or repeat when evaluating base-x rational numbers. I’d love to see someone out there give some of these questions a run! Calculus Derivative Rules Over the past few days I’ve been rethinking my sequencing of introducing derivative rules for the next time I teach calculus. The impetus for this was an approach I encountered in a Coursera MOOC in Calculus I’m taking this summer to see how a professor would run a Taylor Series-centered calculus class. Historically, I’ve introduced my high school calculus classes to the product and quotient rules before turing to the chain rule. I’m now convinced the chain rule should be first because of how beautifully it sets up the other two. Why the chain rule should be first Assuming you know the chain rule, check out these derivations of the product and quotient rules. For each of these, $g_1$ and $g_2$ can be any differentiable functions of x. PRODUCT RULE: Let $P(x)=g_1(x) \cdot g_2(x)$. Applying a logarithm gives, $ln(P)=ln \left( g_1 \cdot g_2 \right) = ln(g_1)+ln(g_2)$. Now differentiate and rearrange. $\displaystyle \frac{P'}{P} = \frac{g_1'}{g_1}+\frac{g_2'}{g_2}$ $\displaystyle P' = P \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$ $\displaystyle P' = g_1 \cdot g_2 \cdot \left( \frac{g_1'}{g_1}+\frac{g_2'}{g_2} \right)$ $P' = g_1' \cdot g_2+g_1 \cdot g_2'$ QUOTIENT RULE: Let $Q(x)=\displaystyle \frac{g_1(x)}{g_2(x)}$. As before, apply a logarithm, differentiate, and rearrange. $\displaystyle ln(Q)=ln \left( \frac{g_1}{g_2} \right) = ln(g_1)-ln(g_2)$ $\displaystyle \frac{Q'}{Q} = \frac{g_1'}{g_1}-\frac{g_2'}{g_2}$ $\displaystyle Q' = Q \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$ $\displaystyle Q' = \frac{g_1}{g_2} \cdot \left( \frac{g_1'}{g_1}-\frac{g_2'}{g_2} \right)$ $\displaystyle Q' = \frac{g_1'}{g_2}-\frac{g_1 \cdot g_2'}{\left( g_2 \right)^2} = \frac{g_1'g_2-g_1g_2'}{\left( g_2 \right)^2}$ The exact same procedure creates both rules. (I should have seen this long ago.) Proposed sequencing I’ve always emphasized the Chain Rule as the critical algebra manipulation rule for calculus students, but this approach makes it the only rule required. That completely fits into my overall teaching philosophy: learn a limited set of central ideas and use them as often as possible. With this, I’ll still introduce power, exponential, sine, and cosine derivative rules first, but then I’ll follow with the chain rule. After that, I think everything else required for high school calculus will be a variation on what is already known. That’s a lovely bit of simplification. I need to rethink my course sequencing, but I think it’ll be worth it. Polar Derivatives on TI-Nspire CAS The following question about how to compute derivatives of polar functions was posted on the College Board’s AP Calculus Community bulletin board today. From what I can tell, there are no direct ways to get derivative values for polar functions. There are two ways I imagined to get the polar derivative value, one graphically and the other CAS-powered. The CAS approach is much more accurate, especially in locations where the value of the derivative changes quickly, but I don’t think it’s necessarily more intuitive unless you’re comfortable using CAS commands. For an example, I’ll use $r=2+3sin(\theta )$ and assume you want the derivative at $\theta = \frac{\pi }{6}$. METHOD 1: Graphical Remember that a derivative at a point is the slope of the tangent line to the curve at that point. So, finding an equation of a tangent line to the polar curve at the point of interest should find the desired result. Create a graphing window and enter your polar equation (menu –> 3:Graph Entry –> 4:Polar). Then drop a tangent line on the polar curve (menu –> 8:Geometry –> 1:Points&Lines –> 7:Tangent). You would then click on the polar curve once to select the curve and a second time to place the tangent line. Then press ESC to exit the Tangent Line command. To get the current coordinates of the point and the equation of the tangent line, use the Coordinates & Equation tool (menu –> 1:Actions –> 8:Coordinates and Equations). Click on the point and the line to get the current location’s information. After each click, you’ll need to click again to tell the nSpire where you want the information displayed. To get the tangent line at $\theta =\frac{\pi }{6}$, you could drag the point, but the graph settings seem to produce only Cartesian coordinates. Converting $\theta =\frac{\pi }{6}$ on $r=2+3sin(\theta )$ to Cartesian gives $\left( x,y \right) = \left( r \cdot cos(\theta ), r \cdot sin(\theta ) \right)=\left( \frac{7\sqrt{3}}{4},\frac{7}{4} \right)$ . So the x-coordinate is $\frac{7\sqrt{3}}{4} \approx 3.031$. Drag the point to find the approximate slope, $\frac{dy}{dx} \approx 8.37$. Because the slope of the tangent line changes rapidly at this location on this polar curve, this value of 8.37 will be shown in the next method to be a bit off. Unfortunately, I tried to double-click the x-coordinate to set it to exactly $\frac{7\sqrt{3}}{4}$, but that property is also disabled in polar mode. METHOD 2: CAS Using the Chain Rule, $\displaystyle \frac{dy}{dx} = \frac{dy}{d\theta }\cdot \frac{d\theta }{dx} = \frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$. I can use this and the nSpire’s ability to define user-created functions to create a $\displaystyle \frac{dy}{dx}$ polar differentiator for any polar function $r=a(\theta )$. On a Calculator page, I use the Define function (menu –> 1:Actions –> 1:Define) to make the polar differentiator. All you need to do is enter the expression for a as shown in line 2 below. This can be evaluated exactly or approximately at $\theta=\frac{\pi }{6}$ to show $\displaystyle \frac{dy}{dx} = 5\sqrt{3}=\approx 8.660$. Conclusion: As with all technologies, getting the answers you want often boils down to learning what questions to ask and how to phrase them. Controlling graphs and a free online calculator When graphing functions with multiple local features, I often find myself wanting to explain a portion of the graph’s behavior independent of the rest of the graph. When I started teaching a couple decades ago, the processor on my TI-81 was slow enough that I could actually watch the pixels light up sequentially. I could see HOW the graph was formed. Today, processors obviously are much faster. I love the problem-solving power that has given my students and me, but I’ve sometimes missed being able to see function graphs as they develop. Below, I describe the origins of the graph control idea, how the control works, and then provide examples of polynomials with multiple roots, rational functions with multiple intercepts and/or vertical asymptotes, polar functions, parametric collision modeling, and graphing derivatives of given curves. BACKGROUND: A colleague and I were planning a rational function unit after school last week wanting to be able to create graphs in pieces so that we could discuss the effect of each local feature. In the past, we “rigged” calculator images by graphing the functions parametrically and controlling the input values of t. Clunky and static, but it gave us useful still shots. Nice enough, but we really wanted something dynamic. Because we had the use of sliders on our TI-nSpire software, on Geogebra, and on the Desmos calculator, the solution we sought was closer than we suspected. REALIZATION & WHY IT WORKS: Last week, we discovered that we could use $g(x)=\sqrt \frac{\left | x \right |}{x}$ to create what we wanted. The argument of the root is 1 for $x<0$, making $g(x)=1$. For $x>0$, the root’s argument is -1, making $g(x)=i$, a non-real number. Our insight was that multiplying any function $y=f(x)$ by an appropriate version of g wouldn’t change the output of f if the input to g is positive, but would make the product ungraphable due to complex values if the input to g is negative. If I make a slider for parameter a, then $g_2(x)=\sqrt \frac{\left | a-x \right |}{a-x}$ will have output 1 for all $x. That means for any function $y=f(x)$ with real outputs only, $y=f(x)\cdot g_2(x)$ will have real outputs (and a real graph) for $x only. Aha! Using a slider and $g_2$ would allow me to control the appearance of my graph from left to right. NOTE: While it’s still developing, I’ve become a big fan of the free online Desmos calculator after a recent presentation at the Global Math Department (join our 45-60 minute online meetings every Tuesday at 9PM ET!). I use Desmos for all of the following graphs in this post, but obviously any graphing software with slider capabilities would do. EXAMPLE 1: Graph $y=(x+2)^3x^2(x-1)$, a 6th degree polynomial whose end behavior is up for $\pm \infty$, “wiggles” through the x-axis at -2, then bounces off the origin, and finally passes through the x-axis at 1. Click here to access the Desmos graph that created the image above. You can then manipulate the slider to watch the graph wiggle through, then bounce off, and finally pass through the x-axis. EXAMPLE 2: Graph $y=\frac{(x+1)^2}{(x+2)(x-1)^2}$, a 6th degree polynomial whose end behavior is up for $\pm \infty$, “wiggles” through the x-axis at -2, then bounces off the origin, and finally passes through the x-axis at 1. Click here to access the Desmos graph above and control the creation of the rational function’s graph using a slider. EXAMPLE 3: I believe students understand polar graphing better when they see curves like the limacon $r=2+3cos(\theta )$ moving between its maximum and minimum circles. Controlling the slider also allows users to see the values of $\theta$ at which the limacon crosses the pole. Here is the Desmos graph for the graph below. EXAMPLE 4: Object A leaves (2,3) and travels south at 0.29 units/second. Object B leaves (-2,1) traveling east at 0.45 units/second. The intersection of their paths is (2,1), but which object arrives there first? Here is the live version. OK, I know this is an overly simplistic example, but you’ll get the idea of how the controlling slider works on a parametrically-defined function. The$latex \sqrt{\frac{\left | a-x \right |}{a-x}}\$ term only needs to be on one of parametric equations. Another benefit of the slider approach is the ease with which users can identify the value of t (or time) when each particle reaches the point of intersection or their axes intercepts. Obviously those values could be algebraically determined in this problem, but that isn’t always true, and this graphical-numeric approach always gives an alternative to algebraic techniques when investigating parametric functions.
ASIDE 1–Notice the ease of the Desmos notation for parametric graphs. Enter [r,s] where r is the x-component of the parametric function and s is the y-component. To graph a point, leave r and s as constants. Easy.
EXAMPLE 5: When teaching calculus, I always ask my students to sketch graphs of the derivatives of functions given in graphical forms. I always create these graphs one part at a time. As an example, this graph shows $y=x^3+2x^2$ and allows you to get its derivative gradually using a slider.
ASIDE 2–It is also very easy to enter derivatives of functions in the Desmos calculator. Type “d/dx” before the function name or definition, and the derivative is accomplished. Desmos is not a CAS, so I’m sure the software is computing derivatives numerically. No matter. Derivatives are easy to define and use here.
I’m hoping you find this technology tip as useful as I do.
Exponential Derivatives and Statistics
This post gives a different way I developed years ago to determine the form of the derivative of exponential functions, $y=b^x$. At the end, I provide a copy of the document I use for this activity in my calculus classes just in case that’s helpful. But before showing that, I walk you through my set-up and solution of the problem of finding exponential derivatives.
Background:
I use this lesson after my students have explored the definition of the derivative and have computed the algebraic derivatives of polynomial and power functions. They also have access to TI-nSpire CAS calculators.
The definition of the derivative is pretty simple for polynomials, but unfortunately, the definition of the derivative is not so simple to resolve for exponential functions. I do not pretend to teach an analysis class, so I see my task as providing strong evidence–but not necessarily a watertight mathematical proof–for each derivative rule. This post definitely is not a proof, but its results have been pretty compelling for my students over the years.
Sketching Derivatives of Exponentials:
At this point, my students also have experience sketching graphs of derivatives from given graphs of functions. They know there are two basic graphical forms of exponential functions, and conclude that there must be two forms of their derivatives as suggested below.
When they sketch their first derivative of an exponential growth function, many begin to suspect that an exponential growth function might just be its own derivative. Likewise, the derivative of an exponential decay function might be the opposite of the parent function. The lack of scales on the graphs obviously keep these from being definitive conclusions, but the hypotheses are great first ideas. We clearly need to firm things up quite a bit.
Numerically Computing Exponential Derivatives:
Starting with $y=10^x$, the students used their CASs to find numerical derivatives at 5 different x-values. The x-values really don’t matter, and neither does the fact that there are five of them. The calculators quickly compute the slopes at the selected x-values.
Each point on $f(x)=10^x$ has a unique tangent line and therefore a unique derivative. From their sketches above, my students are soundly convinced that all ordered pairs $\left( x,f'(x) \right)$ form an exponential function. They’re just not sure precisely which one. To get more specific, graph the points and compute an exponential regression.
So, the derivatives of $f(x)=10^x$ are modeled by $f'(x)\approx 2.3026\cdot 10^x$. Notice that the base of the derivative function is the same as its parent exponential, but the coefficient is different. So the common student hypothesis is partially correct.
Now, repeat the process for several other exponential functions and be sure to include at least 1 or 2 exponential decay curves. I’ll show images from two more below, but ultimately will include data from all exponential curves mentioned in my Scribd document at the end of the post.
The following shows that $g(x)=5^x$ has derivative $g'(x)\approx 1.6094\cdot 5^x$. Notice that the base again remains the same with a different coefficient.
OK, the derivative of $h(x)=\left( \frac{1}{2} \right)^x$ causes a bit of a hiccup. Why should I make this too easy? <grin>
As all of its $h'(x)$ values are negative, the semi-log regression at the core of an exponential regression is impossible. But, I also teach my students regularly that If you don’t like the way a problem appears, CHANGE IT! Reflecting these data over the x-axis creates a standard exponential decay which can be regressed.
From this, they can conclude that $h'(x)\approx -0.69315\cdot \left( \frac{1}{2} \right)^x$.
So, every derivative of an exponential function appears to be another exponential function whose base is the same as its parent function with a unique coefficient. Obviously, the value of the coefficient depends on the base of the corresponding parent function. Therefore, each derivative’s coefficient is a function of the base of its parent function. The next two shots show the values of all of the coefficients and a plot of the (base,coefficient) ordered pairs.
OK, if you recognize the patterns of your families of functions, that data pattern ought to look familiar–a logarithmic function. Applying a logarithmic regression gives
For $y=a+b\cdot ln(x)$, $a\approx -0.0000067\approx 0$ and $b=1$, giving $coefficient(base) \approx ln(base)$.
Therefore, $\frac{d}{dx} \left( b^x \right) = ln(b)\cdot b^x$.
Again, this is not a formal mathematical proof, but the problem-solving approach typically keeps my students engaged until the end, and asking my students to discover the derivative rule for exponential functions typically results in very few future errors when computing exponential derivatives.
Feedback on the approach is welcome.
Classroom Handout:
Here’s a link to a Scribd document written for my students who use TI-nSpire CASs. There are a few additional questions at the end. Hopefully this post and the document make it easy enough for you to adapt this to the technology needs of your classroom. Enjoy.
An unexpected lesson from technology
This discovery happened a few years ago, but I’ve just started ‘blogging, so I guess it’s time to share this for the “first” time.
I forget whether my calculus class at the time was using the first version of the TI-Nspire CAS or if we were still on the TI-89, but I had planned a very brief introduction to the CAS syntax for computing symbolic derivatives, but my 5-minute introduction in the first week of introducing algebraic rules of derivatives ended up with my students discovering antiderivative rules simply because they had technology tools which allowed them to explore beyond where their teacher had intended them to go.
They had absolutely no problem computing algebraic derivatives of power functions, so the following example was used not to demonstrate the power of CAS, but to give easily confirmed outputs. I asked them for the derivative of $x^5$, and their CAS gave the top line of the image below.
(In case there are readers who are TI-Nspire CAS users who don’t know the shortcut for computing higher order derivatives, use the left arrow to place the cursor after the dx in the “denominator” of $\frac{d}{dx}$ and press the carot (^) key. Then type the integer of the derivative you want.)
I wanted my students to compute the 2nd and 3rd derivatives and confirm the power rule which they did with the following screen.
That was the extent of what I wanted at the time–to establish that a CAS could quickly and easily confirm algebraic results whether or not a “teacher” was present. Students could create as many practice problems as were appropriate for themselves and get their solutions confirmed immediately by a non-judgmental expert. Of course, one of my students began to explore in ways my “trained” mind had long ago learned not to do. In my earlier days of CAS, I had forgotten the unboundedness of mathematical exploration.
Shortly after my syntax 5-minutes had passed and I had confirmed everyone could handle it, a young man called me to his desk to show me the following.
He understood what a 1st or 2nd derivative was, but what in the world was a negative 1st derivative? Rather than answering, I posed it to the class who pondered a few moments before recognizing that “underivatives” (as they called them in that moment) of power functions added one to the current exponent before dividing by the new exponent. They had discovered and explained (at least algebraically) antiderivatives long before I had intended. Technology actually inspired and extended my students’ learning!
Then I asked them what the CAS would give if we asked it for a 0th derivative. It was another great technology-inspired discussion.
I really need to explore more about the connections between mathematics as a language and the parallel language of technology. | {"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": 157, "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": 0.8442742824554443, "perplexity": 651.2277230623163}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267155413.17/warc/CC-MAIN-20180918130631-20180918150631-00617.warc.gz"} |
https://www.physicsforums.com/threads/momentum-impulse-etc.272963/ | # Momentum, impulse, etc.
1. Nov 18, 2008
### eiktmywib
1. The problem statement, all variables and given/known data
My question is:
Now assume that the pitcher in Part D throws a 0.145 kg baseball parallel to the ground with a speed of 32 m/s in the +x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's velocity just after leaving the bat if the bat applies an impulse of -8.4 N*s to the baseball?
2. Relevant equations
The first part of the question is this:
Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the +x direction.
The right answer (that I got correct) is this:
The impulse on the ball caused by the bat will be in the negative x direction.
3. The attempt at a solution
Well I know that impulse is just the change in momentum...
So I did
mv=mv
8.4 N*s=(0.145)(v)
and I got 57.9 m/s... which was wrong
1. The problem statement, all variables and given/known data
My question is:
Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 kg that is traveling horizontally at 10.9 m/s. Olaf's mass is 73.5 kg.
If the ball hits Olaf and bounces off his chest horizontally at 7.30 m/s in the opposite direction, what is his speed vf after the collision?
2. Relevant equations
The first part of the question is:
If Olaf catches the ball, with what speed v_f do Olaf and the ball move afterward?
I got 5.90 cm/s, which is right.
3. The attempt at a solution
I found the initial momentum which is (-7.3 m/s * 100cm/1m)(0.400 kg) = -292 kgcm/s
And this was right
And then I tried momentum final = mv
-292 = (73.5kg)(v)
And it was wrong...
2. Nov 18, 2008
### jamesrc
For the first problem: Impulse is equal to the change in momentum, not the momentum...
For the second problem: I don't see you trying to set up the problem correctly. Initially, the momentum of the system is all in the ball (that momentum would be equal to the mass of the ball times the velocity of the ball). After the collision, both the ball and Olaf are moving, so the total momentum would be the sum of the ball's momentum (its mass times velocity) and Olaf's (his mass times velocity; his velocity is what you are solving for). | {"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": 0.9183516502380371, "perplexity": 797.1998647597893}, "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/1480698541142.66/warc/CC-MAIN-20161202170901-00175-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/61153/a-question-about-the-number-of-intersections-of-lines-in-r3/61178 | # A question about the number of intersections of lines in $R^{3}$
Suppose I have n lines in $R^{3}$ with the conditions that: no 3 lines in one plane, no 3 lines intersect at one point, for fixed 2 lines, no 3 lines intersect these 2 lines at the same time. what is the up bound of the number of intersections? The up bound $n^{\frac{3}{2}}$ is a simple corollary of Guth-Katz's paper or one can prove it directly by algebraic method. Is it possible to establish the up bound like $n^{\frac{4}{3}}$ or some better one?
The up bound will also be a up bound for Erdos's unit distance problem in $R^{2}$.
-
Maybe you could give a summary of or reference for the Guth-Katz paper and for the Erdos unit distance problem? – Gerry Myerson Apr 10 '11 at 0:14
The best summary of Guth-Katz paper I can think is the link in JSE's answer below, for unit distance problem, one can find reference in the reference of cs.tau.ac.il/~michas/pst5.pdf. – user13289 Apr 10 '11 at 4:14
If the lines are indexed by $1,\ldots,n$ and the set of lines intersecting line $i$ is called $A_i$, an upper bound is the maximum of [\sum_{i=1}^n|A_i|,] subject to $|A_i\cap A_j|\leqslant 2$ for all $i\neq j$. But it might be obvious that this bound is worse than $n^{3/2}$. – Thomas Kalinowski Apr 10 '11 at 7:51
George Purdy (U. Cincinnati) is an expert on this general topic. He is giving a seminar @NYU tomorrow on this. Maybe contact him? cs.nyu.edu/~raghavan/geometry/spring11/Purdy.pdf – Joseph O'Rourke Apr 11 '11 at 13:23
It looks like "no five lines in a quadric" but not exactly same. n lines in a (singly) ruled surface of degree $n^{\frac{1}{2}}$ is a situation appeared if one try to prove the up bound $n^{\frac{3}{2}}$, but still the full strength of that condition will not be used... – user13289 Apr 10 '11 at 15:26 | {"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": 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": 0.9464259743690491, "perplexity": 403.39525334456687}, "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/1418802775225.37/warc/CC-MAIN-20141217075255-00039-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/yukawa-potential.294757/ | # Yukawa Potential
1. Feb 23, 2009
### wofsy
I would appreciate an explanation of Yukawa potential
2. Feb 23, 2009
### Tom Mattson
Staff Emeritus
The Yukawa potential is the potential that arises from a massive scalar field. It is:
$$V(r)=-k\frac{exp(-\mu r)}{r}$$
where $k>0$ and $\mu$ is the mass of the mediating field.
Note that:
1.) It is attractive (that is, $F=-\frac{\partial V}{\partial r}$ is negative).
2.) It reduces to the Coulomb potential for $\mu =0$.
3. Feb 23, 2009
### Staff: Mentor
4. Feb 23, 2009
### Roosta
The Yukawa Potential can be roughly thought of as a generalization of an inverse-square force potential that takes into account a massive mediator or force. This would mean that instead of massless photons exchanging the force, as is the case with electromagnetism, some other particle with mass exchanges the force between two particles.
5. Feb 24, 2009
### RedX
http://en.wikipedia.org/wiki/Yukawa_potential
it says that the Fourier transformation of the Yukawa potential is the amplitude for two fermions to scatter. But the Fourier transform ignores 4-momentum and only has 3-momentum. The amplitude to scatter should depend on a 4-momentum squared, and not 3-momentum. So how is this reconciled?
6. Feb 24, 2009
### Brian_C
There is a problem in Jackson's E&M book which asks you to derive the charge distribution corresponding to this potential. I could never quite get it right.
7. Feb 25, 2009
### sp105
may i know the page no and the problem to be solved in Jackson Book of E and M
8. Feb 25, 2009
### Roosta
I'm not as familiar with relativistic QM as I would like to be, but it seems like the given formula actually does depend on the four momentum. The k in transform corresponds to the three spatial components of the four-momentum while the m corresponds to mass, which in turn depends on the time component of the four-momentum. Why the fourth component of the four-momentum is left out of the Fourier Transform, on the other hand, is unfamiliar to me.
It is known that, from the definition of V (in a static field) and the first of Maxwell's equations that:
$\nabla\cdot\vec{E}=-\nabla ^2V=\frac{\rho}{\epsilon _0}$
So that to find the charge density, one must simply take the negative Laplacian of the potential. This will work every where except for the origin, where you have to apply gauss's law and gauss's vector calculus equation to find the charge. I found:
$\rho (r)=4\pi g^2\epsilon _0 \delta (r)-g^2m^2\frac{e^{-mr}}{r}$
9. Feb 25, 2009
### clem
The simplified version of the Yukawa derivation takes place in the barycentric system where the energy component of the 4-momentum transfer vanishes. Then the 3D Fourier T can be made.
10. Feb 25, 2009
### RedX
Does barycentric system mean center of mass frame between the two fermions? Does this mean that the Yukawa potential is only valid in a center of mass frame, since you break Lorentz invariance by choosing a specific frame?
11. Mar 25, 2009
### sp105
May I come to know about the barycentric systems
I have listened the word for the first time and I am curious to know about it because you have mentioned that it reduces the 4 momentum to 3- momentum
12. Mar 25, 2009
### clem
The "barycentric system" is the Lorentz system in which the total momentum equals zero. It is usually loosely called (even by me) the center of mass (cm) system, even though the term "center of mass" has no clear meaning in relativity. The Yukawa potential is usually derived for two nucleons. Then the energy component of the 4-momentum transfer vanishes in the cm system because the two individual energies are equal. It is valid only in the cm system, where most calculations are made anyway. The simple Yukawa potential is used mainly in nonrelativitic calculations , because other effects become important at higher energies. The Yukawa potential by itself is now useful only for simple order of magnitude estimates because the full N-N interaction is more complicated. It does describe the long range part of the N-N potential.
13. Mar 26, 2009
### sp105
thanks a lot for such an eloborative reply
butit has increased my curicity
will you help me to tell what are the other factors responsible for N-N interactions
14. Mar 26, 2009
### clem
You really need to go to a book on strong interactions now.
15. May 13, 2009
### sp105
I know it very well
that I am not known to all these things thats why I sk you for recommending me a book
(a specific book)
because I can have book from a library or may have it on rent before purchasing it
Similar Discussions: Yukawa Potential | {"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": 0.9302760362625122, "perplexity": 771.3865989919914}, "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/1508187825812.89/warc/CC-MAIN-20171023073607-20171023093607-00722.warc.gz"} |
http://mathhelpforum.com/calculus/151141-basic-definition-matching.html | # Thread: basic definition matching.
1. ## basic definition matching.
Im having problems trying to match up "Linear Approximation and Differential basic terms". The answers i chose should be correct but its not. Can anyone figure out why?
Thanks guys
http://img714.imageshack.us/img714/5822/weirdx.jpg
2. Originally Posted by ASUSpro
Im having problems trying to match up "Linear Approximation and Differential basic terms". The answers i chose should be correct but its not. Can anyone figure out why?
Thanks guys
http://img714.imageshack.us/img714/5822/weirdx.jpg
what are the answers you chose? ... and why.
3. the answers i chose are as follows in the following. B, A, E is from textbook diagrams. F is the tangent line because it touches curve at one point. Not sure about D and C, a friend helped me.
4. Originally Posted by ASUSpro
the answers i chose are as follows in the following. B, A, E is from textbook diagrams. F is the tangent line because it touches curve at one point. Not sure about D and C, a friend helped me.
D looks like it's only another label for f(x) ... the linearization of f at x is the tangent line at f(x).
Everything else looks fine. | {"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": 0.8926976323127747, "perplexity": 1131.557769655732}, "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-2017-26/segments/1498128320264.42/warc/CC-MAIN-20170624152159-20170624172159-00301.warc.gz"} |
https://economics.stackexchange.com/questions/24805/comparative-statics-question-with-an-application | # Comparative statics question with an application
In the state of Mexas, two politicians (Mr. BO, or "Politician 1" and Mr. TC, or "Politician 2") are competing intensely for a senate seat. The two politicians spend on advertising to increase the number of supporters. A political consultant finds that the optimal advertising expenditure of Mr. BO, $$S_{1}$$, depends on the spending $$S_{2}$$ by Mr. TC and a "likability" parameter $$\alpha$$ that influences the popularity of Mr. BO in Mexas: $$\text{Equation 1: }S_{1} = f\left ( S_{2}, \alpha \right ),$$ where $$f:\mathbb{R}_{+}^{2}\rightarrow \mathbb{R}$$ is twice continuously differentiable, $$0<\frac{\partial f\left ( S_{2}, \alpha \right )}{\partial S}<1$$ and $$0<\frac{\partial f\left ( S_{2}, \alpha \right )}{\partial \alpha}<1$$ for all $$S_{2} \geq 0$$ and all $$\alpha \geq 0$$.
The optimal expenditure of politician 2 depends on the spending $$S_{1}$$ of politician 1 and a "redness" parameter $$\beta$$, which influences how many people would stick to Mr. TC: $$\text{Equation 2: }S_{2} = g\left ( S_{1}, \beta \right ),$$ where $$g:\mathbb{R}_{+}^{2}\rightarrow \mathbb{R}$$ is twice continuously differentiable, $$0<\frac{\partial g\left ( S_{1}, \alpha \right )}{\partial S}<1$$ and $$0<\frac{\partial g\left ( S_{1}, \alpha \right )}{\partial \alpha}<1$$ for all $$S_{1} \geq 0$$ and all $$\beta \geq 0$$.
The equilibrium values of $$S_{1}$$ and $$S_{2}$$ are given by the solution of the simultaneous equations (1) and (2). Suppose that there is a unique solution $$S_{1}^{*}>0$$ and $$S_{2}^{*}>0$$.
Does an increase in $$\alpha$$(holding $$\beta$$ constant) necessarily increase or necessarily decrease $$S_{1}^{*}$$? Explain.
My attempt: Here I have used Implicit Function Theorem to answer the question, as we are essentially looking for the comparative statics of $$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }$$.
Since from equation 2, $$S_{2}^{*} = g\left ( S_{1}^{*}, \beta \right )$$, I substituted $$S_{2}^{*}$$ in equation to obtain $$S_{1}^{*} = f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ).$$ Then since $$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F,$$ I can apply the Implicit Function Theorem (IFT) to derive $$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }$$.
So $$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\frac{\partial F}{\partial \alpha}}{\frac{\partial F}{\partial S_{1}}} = - \frac{\frac{\partial f\left ( \right )}{\partial \alpha}}{1 - \frac{\partial f \left ( \right )}{\partial g} \frac{\partial g \left ( \right )}{\partial S_{1}^{*}}} \frac{>}{<} 0.$$
Since we do not know the sign of the derivative $$\frac{\partial f \left ( \right )}{\partial g}$$, the effect of $$\alpha$$ on $$S^{*}_{1}$$ is ambiguous.
• Any thoughts to the question and my work? – OGC Oct 3 '18 at 8:31
You can obtain a definite answer for $$\text{sign}\left\{\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha }\right\}$$ given the assumptions.
From
$$S_{1}^{*} - f\left ( g\left ( S^{*}_{1}, \beta \right ), \alpha \right ) = 0 \equiv F$$
and the Implicit Function Theorem
$$\frac{\mathrm{d} S_{1}^{*} }{\mathrm{d} \alpha } = - \frac{\partial F/\partial \alpha}{\partial F/\partial S^*_{1}}$$
we have
$$\frac{\partial F}{\partial \alpha} = -\frac{\partial f}{\partial \alpha}$$
and
$$\frac{\partial F}{\partial S^*_{1}} = 1-\frac{\partial f}{\partial S_2}\cdot \frac{\partial g}{\partial S^*_{1}}$$
For these expressions we know not only the signs but also the magnitudes. The result follows.
• So the effect is negative then? I don't understand why you don't have $\frac{\partial f\left ( \right )}{\partial g\left ( \right )}$ but instead have $\frac{\partial f\left ( \right )}{\partial S_{2} \left ( \right )}$. – OGC Oct 3 '18 at 15:03
• @OGC It is the same thing, since $g$ has the exact same position in $f$ as $S_2$. Also, it appears you are forgetting to take into account one minus sign. The effect is positive. – Alecos Papadopoulos Oct 3 '18 at 15:07
• Thanks a lot. It's easy to see some of these silly mistakes once someone points them to you. – OGC Oct 3 '18 at 19:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 39, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9224722981452942, "perplexity": 433.92240131134577}, "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/1579251802249.87/warc/CC-MAIN-20200129194333-20200129223333-00519.warc.gz"} |
http://quics.umd.edu/research/publications?s=author&%3Bf%5Bag%5D=N&%3Bf%5Bauthor%5D=1760&f%5Bauthor%5D=1624 | # Publications
Export 9 results:
[ Author] Title Type Year
Filters: Author is Carl A. Miller [Clear All Filters]
M
, Optimal robust self-testing by binary nonlocal XOR games, in 8th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2013, vol. 22, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2013, pp. 254–262.
C. Miller, Evasiveness of Graph Properties and Topological Fixed-Point Theorems, Foundations and Trends in Theoretical Computer Science, vol. 7, pp. 337-415, 2013.
C. Miller, An Euler–Poincaré bound for equicharacteristic étale sheaves, Algebra & Number Theory, vol. 4, no. 1, pp. 21 - 45, 2010.
C. Miller, Exponential iterated integrals and the relative solvable completion of the fundamental group of a manifold, Topology, vol. 44, no. 2, pp. 351 - 373, 2005.
, Robust Protocols for Securely Expanding Randomness and Distributing Keys Using Untrusted Quantum Devices, Journal of the ACM, vol. 63, no. 4, pp. 33:1–33:63, 2016.
, Randomness in nonlocal games between mistrustful players, Quantum Information and Computation, vol. 17, no. 7&8, pp. 0595-0610, 2017.
, Keyring models: an approach to steerability, Journal of Mathematical Physics, vol. 59, p. 022103, 2018.
, Universal Security for Randomness Expansion from the Spot-Checking Protocol, SIAM Journal on Computing, vol. 46, no. 4, 2017. | {"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": 0.8267933130264282, "perplexity": 2324.9108962585224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000044.37/warc/CC-MAIN-20190626013357-20190626035357-00370.warc.gz"} |
http://ultrawaffle.livejournal.com/ | Or connect using:
Ultrafilter! Waffle! Below are the 10 most recent journal entries recorded in the "Ultrafilter! Waffle!" journal:
May 7th, 2011
02:34 am
FLMPotD
Let K/k be the splitting field of an irreducible quintic over k. Show that K cannot contain a root of any irreducible cubic over k.
Current Mood: tired
Tags:
March 31st, 2011
01:10 pm
FLMPotD
Compute \sum_{k=0}^n (-1)^k \binom{n}{k}/(k+1).
Current Mood: moody
Tags:
March 26th, 2011
02:17 pm
FLMPotD
Find a commutative ring R, a multiplicative subset S \subset R, and finitely generated R-modules M and N such that the natural map S^{-1}Hom(M,N) \to Hom(S^{-1}M,S^{-1}N) is not an isomorphism.
Current Mood: busy
Tags:
March 7th, 2011
10:04 pm
FLMPotD
Show that a finite p-group G can be recovered from the mod p cochains on BG, as an E_\infty-algebra.
(This implies that a finite nilpotent group can be recovered from its integral cochains...I think the same should be true for any finite group, but I don't quite know enough about representation theory and group cohomology over Z to be sure about a certain step of the argument, and maybe my intuitions from how things work for p-groups really only apply to nilpotent groups. Note that this is not true for infinite groups, as there are nontrivial groups with trivial cohomology, a fact which is crucial to the proof of the Kan-Thurston theorem.)
Current Mood: okay
Tags:
February 23rd, 2011
05:32 pm
FLMPotD
Show that if the homology of a spectrum is free, then its Atiyah-Hirzebruch spectral sequence for any (co)homology theory with torsion-free coefficients degenerates at E_2.
Current Mood: okay
Tags:
10:02 am
On cleverness in LARPs
While I've never written a LARP, one of the subtler points about writing them that I've noticed from playing is that it is difficult to make plots that require people to be clever in character. The basic difficulty is that if I'm presented with a challenge to which I (the player) see an obvious solution (or even just first step towards a solution), but my character sheet doesn't even mention this obvious idea, I will assume that for some reason it is not obvious to my character. If my character sheet says "you hope to figure out a way to do X" and there's an obvious way to do X, in character it doesn't make sense for me to be "figuring out a way" if the obvious way was obvious to my character. So if you're writing a character and part of achieving that character's goals involves coming up with a solution to a problem, you need to make sure the solution is difficult enough (or inaccessible with the knowledge that the character starts out with at the beginning of the game) and that you don't conspicuously leave out any obvious ideas the character should have for going about the solution.
Anyways, the moral of the story is that character sheets need to be very thorough, because an omission of an idea in a character sheet is just as significant as an inclusion.
Current Mood: thoughtful
February 17th, 2011
05:34 pm
Bah
I don't understand why people ever say things in math without accompanying them with an explanation of what's actually going on in them. It's just so...why???
(Specifically, this was prompted by thinking about the fact that a simplicial group is automatically a Kan complex. Like two years ago I read about this and they just gave some unexplained formula for how you can use the group structure to fill horns. Then today, as I was bored in a seminar, I figured out that it's nothing but a souped-up version of the Eckmann-Hilton argument saying that the composition of paths in a group is canonically homotopic to their pointwise product using the group structure. Why didn't they explain this in the book where I read it before??)
Current Mood: ranting
February 4th, 2011
03:37 pm
FLMPotD
Prove Picard's big theorem (in a neighborhood of an essential singularity, a holomorphic function's image misses at most one point of C) using only the fact that the holomorphic universal cover of C-{0,1} is the unit disk, elementary complex analysis, and covering space theory.
(In my complex analysis class years ago, we learned the proof of Picard's little theorem along these lines, which I found very beautiful, and then proceeded to do a very analytic and unenlightening proof of Picard's big theorem. I had always assumed that there must not exist a similarly nice proof, and then recently I tried to come up with one and discovered that there was one.)
Current Mood: sick
Tags:
February 3rd, 2011
12:14 am
FLMPotD
Show that the ring Q[sin,cos] (say as a subring of the ring of functions on R) is spanned by the functions sin nx and cos nx as n varies.
(This one's from . There's an unenlightening trivial solution using obscure trig identities, and then a more interesting conceptual solution.)
Current Mood: merp
Tags:
February 2nd, 2011
10:34 pm | {"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": 0.8316588997840881, "perplexity": 637.1044856789101}, "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-2015-35/segments/1440645359523.89/warc/CC-MAIN-20150827031559-00169-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2154461/how-to-construct-integer-polynomial-of-degree-n-which-takes-n-times-the-valu/2154484 | # How to construct integer polynomial of degree $n$ which takes $n$ times the value $1$ and $n$ times the value $-1$ (in integer points)
Question is like in title:
For positive integer $n$ I want to construct a polynomial $w(x)$ of degree $n$ with integer coefficients such that there exist $a_1<a_2<\cdots<a_n$ (integer numbers) satisfying $w(a_1)=w(a_2)=\cdots=w(a_n)=1$ and $b_1<b_2<\cdots<b_n$ (also integer numbers) satisfying $w(b_1)=w(b_2)=\cdots=w(b_n)=-1$.
I am not sure if such polynomial even exists for every $n$.
EDIT: $a_i, b_i$ are not predetermined, as someone mentioned in a comment.
This is not possible for $n>3$. In fact, it is not possible to find a degree-$n$ polynomial $w(x)$ which has $n$ different integer roots of $w(x)=1$ and even one integer root (say $b$) of $w(x)=-1$. This is because, if $a_1,...,a_n$ are the roots of $w(x)=1$, $w(x)-1=c(x-a_1)...(x-a_n)$, where $c$ is an integer. Thus $w(b)-1$ is $c$ times a product of $n$ different integers, and since at most two of these can be $\pm 1$, and all others have absolute value at least $2$, if $n>3$ then $w(b)-1$ can't equal $-2$ -- it must have absolute value at least $2^{n-2}$.
In fact what you asked for is not possible for $n=3$ either. For $w(b)-1=-2$ to have a solution we must have $(b-a_1)(b-a_2)(b-a_3)=-2$ so the three terms are $-1,+1,+2$. But if those are the three terms, then the order of them is determined, so there is at most one solution for $b.$ It is possible for $n=2$: take $w(x)=x^2-3x+1$. Then $w(x)=1$ has two solutions $x=0,3$ and $w(x)=-1$ has two solutions $x=1,2$.
$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Set}[1]{\left\{ #1 \right\}}Just to complement the excellent answer (+1), if n = 1 you have$$ w(x) = C (x - a_{1}) + 1, $$and$$w(b_{1}) = C (b_{1} - a_{1}) = -2,$$so there is a solution iff b_{1} - a_{1} \in \Set{\pm 1, \pm 2}. If n = 2 you have$$ w(x) = C (x - a_{1})(x - a_{2}) + 1, $$and need$$ C (b_{1} - a_{1})(b_{1} - a_{2}) = -2 = C (b_{2} - a_{1})(b_{2} - a_{2}), $$and there are solutions for instance for$$\tag1\label1 b_{1} = a_{1} + 1 = a_{2} - 2, $$so that a_{2} = a_{1} + 3 and$$\tag2\label2 b_{2} = a_{1} + 2 = a_{2} - 1, $$say a_{1} = 0, a_{2} = 3, b_{1} = 1, b_{2} = 2, and then C = 1. Actually, for the case n = 2 we have$$ a_{2} - a_{1} = \Size{a_{2} - a_{1}} \le \Size{b_{1} - a_{1}} + \Size{b_{1} - a_{2}} \le 3,$$so I think the general conditions for the existence of a solution are indeed$\eqref1$and$\eqref2\$. | {"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": 0.9503229260444641, "perplexity": 112.60496228741482}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986668994.39/warc/CC-MAIN-20191016135759-20191016163259-00026.warc.gz"} |
http://physics.stackexchange.com/questions/72926/in-nuclear-fusion-reaction-what-is-the-percentage-of-mass-converted-to-energy | # In nuclear fusion reaction, what is the percentage of mass converted to energy?
I read somewhere that it is about one percent of the mass, but I find this too high. Also I have done some calculations, for example, the Tsar Bomba was 50 MT bomb and weighed about 27 tons. Although I don't know how much exactly fusion fuel was used, I think it is safe to assume that a large percentage of this 27 tons was fusion fuel. So, if we want to get the same amount energy from mass-energy equation, we would need something with a mass of 2.2 kg. Which proves what I mean : if fusion reaction converted one percent of the fuel mass to energy, then we would need only 100*2.2 kg = 220 kg of fusion fuel to make the Tsar Bomba, which I find much much lower than the actual number.
Tell me what you think, please.
-
WolframAlpha is the perfect tool for such calculations:
1. Fraction of mass converted to energy, result is 0.0037681, less than 0.4%.
2. How much is mass defect of 50 MT of TNT, result is 2.3 kg.
As for Tsar Bomba the fusing material is of course only small part of the actual device.
-
I'm sorry, but how could you get the fraction of mass converted to energy that way ? – Abanob Ebrahim Aug 1 '13 at 15:56
@AbanobEbrahim: Click on the link. We take reaction D + T -> He-4 + n and subtract mass of reaction products from the mass of reactants. The result would be the mass defect (~17MeV/c^2). Dividing it by the mass of fusion products we receive the fraction of mass converted to energy. – user23660 Aug 1 '13 at 16:14
In nuclear fusion reaction, what is the percentage of mass converted to energy?
First a note.
A fusion bomb generates a lot of its energy by fission
Briefly the sequence is something like this:
1. Set of a small fission bomb.
This generates X-rays and neutrons.
2. The x-rays are used with a Teller Ulam device to compress a fusion part. This will contain some fuel such as lithium deuteride. (Not Tritium because it is rare and it has an annoying short half live of 12.3 years. And its decays product is He3 which likes to absorb neutrons).
3. More fusion fuel is generated by the neutrons.
Neutron + Li6 -> neutron + He + tritium.
Alternatively Li7 can be used (as unexpectedly discovered during Castle Bravo).
4. The compressed tritium and deuterium reacts, producing some energy and more neutrons.
5. The extra neutrons react with the remaining fission material (either in the initial device, or in the spark plug, or with the tamper which can be made from U238.
I read somewhere that it is about one percent of the mass, but I find this too high. Also I have done some calculations, for example, the Tsar Bomba was 50 MT bomb and weighed about 27 tons.
Well, you know the yield of the bomb (somewhat over 50 Megaton TNT).
Thus you can look up how much energy is liberated by 50 Megaton TNT
Then convert that number to mass.
Although I don't know how much exactly fusion fuel was used, I think it is safe to assume that a large percentage of this 27 tons was fusion fuel.
I am not that sure. The Tzar bomba was a 100MT design, which used a uranium tamper. Later it was downgraded to 50MT to avoid a lot of pollution. So in the initial design at least half the yield would not have been directly generated by fusion but by fission. Also U238 or lead (as alternative tamper) are very heavy, while the fusion fuel is likely lithium deuteride.
(Granted, the neutrons to release the energy from the tamper are mostly generated by the fusion reaction).
So, if we want to get the same amount energy from mass-energy equation, we would need something with a mass of 2.2 kg. Which proves what I mean : if fusion reaction converted one percent of the fuel mass to energy, then we would need only 100*2.2 kg = 220 kg of fusion fuel to make the Tsar Bomba, which I find much much lower than the actual number.
When fission fuel gets used it does not completely change to energy. Instead the atom is smashed apart and there will the several pieces. Typically this will be 2 or 3 neutrons and 2 smaller atoms (e.g. 92U to 50Tin, 42Molybdenum and neutrons).
It is just that the sum of the mass of these fragments is slightly smaller.
It is not the case that a while atom just disappears and somehow releases energy.
Part of that energy will be the new fragment, which usually move at a very high speed. And this will result in collisions, x-rays and, well, heat. Lots of it.
-
I realize this is two years old, but would still like to point this out:
When saying that 1% of the mass of fuel is converted to energy, it only means the percentage of the mass of what fuel actually undergoes fusion.
Thus, your 2.2kg of fuel converted to energy would represent 2200kg (edit: oops, 220kg) of fuel which actually underwent fusion, NOT the amount which was contained in the bomb. THAT number is going to be highly variable, dependent on the efficiency of the design and construction of the device. In this instance, it would appear that approximately 1% of the fuel (depending on how much was fuel and not bomb structure + fission trigger) reacted, and the other 98+% just got blown outward at high velocity.
-
Your first part of the answer is correct and I agree with it. But "it would appear that approximately 1% of the fuel reacted" is not correct. As you mentioned this question was two years ago, and I found that the fusion fuel efficiency is around 25-50% of the available fuel. So to get 2.2 kg of mass-energy equivalent, we would need 220 kg of fuel to undergo fusion, and this 220 kg would represent 25-50% of the fuel in the bomb so we would actually need to have somewhere between 440 and 880 kilograms of fuel there to end up with 2.2 kg of mass converted to energy. – Abanob Ebrahim Dec 10 '15 at 20:27 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8809757828712463, "perplexity": 760.871089405127}, "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/1466783393442.26/warc/CC-MAIN-20160624154953-00029-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://www.scientificlib.com/en/Physics/LX/AvogadrosLaw.html | - Art Gallery -
# .
Definition: Under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules.
Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) is a gas law named after Amedeo Avogadro who, in 1811,[1] hypothesized that two given samples of an ideal gas, of the same volume and at the same temperature and pressure, contain the same number of molecules. Thus, the number of molecules or atoms in a specific volume of gas is independent of their size or the molar mass of the gas.
As an example, equal volumes of molecular hydrogen and nitrogen contain the same number of molecules when they are at the same temperature and pressure, and observe ideal gas behavior. In practice, real gases show small deviations from the ideal behavior and the law holds only approximately, but is still a useful approximation for scientists.
Mathematical definition
Avogadro's law is stated mathematically as:
$$\frac{V}{n} = k$$
Where:
V is the volume of the gas.
n is the amount of substance of the gas.
k is a proportionality constant.
The most significant consequence of Avogadro's law is that the ideal gas constant has the same value for all gases. This means that:
$$\frac{p_1\cdot V_1}{T_1\cdot n_1}=\frac{p_2\cdot V_2}{T_2 \cdot n_2} =$$ constant
Where:
p is the pressure of the gas in the cell
T is the temperature in kelvin of the gas
Ideal gas law
A common rearrangement of this equation is by letting R be the proportionality constant, and rearranging as follows:
pV = nRT
This equation is known as the ideal gas law.
Molar volume
Taking STP to be 101.325 kPa and 273.15 K, we can find the volume of one mole of a gas:
$$V_{\rm m} = \frac{V}{n} = \frac{RT}{p} = \frac{(8.314 \mathrm{ J} \mathrm{ mol}^{-1} \mathrm{ K}^{-1})(273.15 \mathrm{ K})}{101.325 \mathrm{ kPa}} = 22.41 \mathrm{ dm}^3 \mathrm{ mol}^{-1}$$
For 100.000 kPa and 273.15 K, the molar volume of an ideal gas is 22.414 dm3mol-1.
Boyle's law
Charles's law
Combined gas law
Gay-Lussac's law
Ideal gas
Edrei Arabo
References
^ Avogadro, Amadeo (1810). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique 73: 58–76. English translation. | {"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": 0.876680314540863, "perplexity": 1321.531244835094}, "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/1679296945323.37/warc/CC-MAIN-20230325095252-20230325125252-00385.warc.gz"} |
http://math.stackexchange.com/questions/241318/showing-that-the-orthogonal-projection-in-a-hilbert-space-is-compact-iff-the-sub | # Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional
Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$.
I have to show that $T$ is compact iff $M$ is finite dimensional.
So if we assume that $M$ is finite dimensional then $\overline{T(B(0,1))}$ is a closed bounded set in a finite dim vector normed space and so it is compact. Which gives that $T$ is compact.
But I am unsure how to prove that if $T$ is compact then $M$ is finite dimensional?
Thanks for any help
-
Assume $T$ compact. As $T(M\cap B(0,1))=M\cap B(0,1)$, then $M\cap B(0,1)$ has a compact closure. Conclude by Riesz theorem (which is easier to prove in the context of Hilbert spaces).
-
If $M$ is infinite dimensional, then there exists $\{e_n\}_{n\in \mathbb{N}}\subset M$, which is an orthonormal set. Evidently $\{e_n\}_{n\in \mathbb{N}}\subset \overline{T(B(0,1))}$, but $\{e_n\}_{n\in \mathbb{N}}$ has no convergent subsequence, which contraditicts to the compactness of $T$.
-
My first attempt would have been what the answer by Davide and Richard did. But here's another approach.
Since $T$ is an orthogonal projection, it is a positive operator. The equation $T^2=T$ guarantees that all eigenvalues are either $1$ or $0$. As $T$ is compact, the multiplicity of $1$ as an eigenvalue is finite (because $0$ is the only possible accumulation point in the spectrum of a compact operator). So $\dim\, M=\text{Tr}\,(T)<\infty$.
(depending on context, the last inequality might not be obvious. Then we could just say that $T$ is a finite sum of rank-one projections, i.e. it is a finite-rank projection. So $M$, being the range of a finite-rank projection, is finite-dimensional)
- | {"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": 0.9895880222320557, "perplexity": 82.04378695135921}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701162648.4/warc/CC-MAIN-20160205193922-00270-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3065579/understanding-the-legendre-transform | # Understanding the Legendre transform
In physics, I've seen the Legendre transform motivated by "changing the variable $$x$$ of a function $$x \mapsto f(x)$$ to the variable $$u = \frac{df}{dx}$$."
I don't quite see what that means and why the Legendre transform is the answer to this heuristic. I'd understand it as follows:
Let $$f: \mathbb{R} \to\mathbb{R}$$ be strictly convex and differentiable. Then for every $$x_0 \in \mathbb{R}$$ the slope $$\frac{df(x_0)}{dx}$$ is unique. We want to find a function $$f^*: f'(\mathbb{R}) \to \mathbb{R}$$ such that $$f(x_0)=f^*(\frac{df(x_0)}{dx})$$ for every $$x_0 \in \mathbb{R}$$. In fact we can view the function $$f^*$$ as a function on a subset of the dual space $$\mathbb{R^*}$$ such that $$f^*(df(x_0))=f(x_0)$$.
However, this doesn't seem to capture the Legendre transform, for example by the above the Legendre transform of the exponential function should be the identity function. How can the physicists heuristic be made precise? | {"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": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9263536930084229, "perplexity": 84.45117454574775}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195529481.73/warc/CC-MAIN-20190723172209-20190723194209-00513.warc.gz"} |
http://www.wetsavannaanimals.net/wordpress/the-adjoint-representation-of-the-lorentz-group/ | # The Adjoint Representation of the Lorentz Group
This is my answer to the Physics Stack Exchange question “Is it possible to construct an adjoint representation for the Lorentz group?”. I was a little taken aback by this question: the adjoint representation is a very fundamental thing that all Lie groups have. Actually, within the adjoint representation we truly see the real meaning of the Jacobi identity.
Every Lie group has an adjoint representation. I’m not sure what definition you come at the adjoint representation from, but here’s the fundamental one which I’m sure you’ll see is always meaningful.
Think of a $C^1$ path $\sigma:[-1,1]\to\mathfrak{G}$ through the identity in a Lie group $\mathfrak{G}$ with $\sigma(0) =\mathrm{id}$ and with tangent $X$ there.
Now think of a general group member $\gamma \in\mathfrak{G}$ acting on that path so that $\sigma(t) \mapsto \gamma^{-1} \sigma(t) \gamma$. This too is naturally a path through the identity and has a transformed tangent $X^\prime$ there. This transformation is linear. We say the group “acts on its own Lie algebra $\mathfrak{g}$” in this way and write the transformation $X\mapsto\gamma^{-1} X \gamma$. This is just notation, but in a matrix Lie group it is also a literal matrix product. Perhaps, then, less confusingly we write $X\mapsto {\rm Ad}(\gamma) X$, where the linear transformation ${\rm Ad}(\gamma)$ wrought by $\gamma$ on the Lie algebra is now a member of $GL(\mathfrak{g})$, the group of invertible linear transformations of the Lie algebra $\mathfrak{g}$ thought of as a plain vector space. The association:
$$\rho:\mathfrak{G}\to GL(\mathfrak{g});\;\rho(\gamma)= {\rm Ad}(\gamma)$$
is a homomorphism as is readily shown. This is sometimes called the Adjoint Representation of the Lie group. Through the big Ad Adjoint representation $\mathfrak{G}$ is mapped to a new Lie group, this time always a matrix Lie group, a subgroup of $GL(\mathfrak{g})$.
Now, then, we can look at:
$${\rm ad}(X)\stackrel{def}{=}\left.\mathrm{d}_t {\rm Ad}(e^{t\,X})\right|_{t=0}$$
This too is a linear operator on $\mathfrak{g}$, although in general not an invertible one. Indeed, you can show without too much strife that:
$${\rm ad}(X) Y = [X,\,Y]$$
So in fact the big Ad adjoint representation induces a homomorphism of Lie algebras $\rho^\prime:\mathfrak{g} \to {\rm Lie}(\rho(\mathfrak{G}))$. This too is a homomorphism of linear spaces and moreover a homomorphism that respects Lie brackets. It is thus a Lie algebra homomorphism and it is also called the adjoint representation (of the Lie algebra). I like to quaintly call it little ad adjoint representation.
Now here for me is one of the most beautiful equations around:
$${\rm ad}([X, \,Y]) = [{\rm ad}(X),\,{\rm ad}(Y)] = {\rm ad}(X) . {\rm ad}(Y)-{\rm ad}(Y) . {\rm ad}(X)$$
This is a restatement of the fact that little ad respects Lie brackets. But it is also a form of the Jacobi identity in disguise. Wow! That’s the real meaning of the Jacobi identity: it’s there so the adjoint representation of a Lie group, clearly a very basic and fundamental thing, induces a homomorphism in the corresponding Lie algebras that respects Lie brackets. Everything’s exactly as we would expect and so, if you’re ever designing a Universe, that’s why you must remember to throw the Jacobi identity in! Write a note for yourself now so you don’t forget! Now my LaTeX skills aren’t up to drawing a commutative diagram from memory, so I hope you can see there is a pretty neat and simple one.
Just another couple of interesting facts. The kernel of Big Ad is the centre of the the Lie group. The kernel of little ad is then the centre of the Lie algebra. So if the Lie group is simple, i.e. contains no normal Lie subgroups, then there cannot be a continuous centre to annihilate by the homomorphism. Same is true if the group simply has no continous centre for reasons other than simplicity. So there are no bits of the Lie algebra that get wiped out. The original Lie group and the image of big Ad then have exactly the same Lie algebra. If further there is no discrete centre in a Lie group and if the group is connected, then the Lie group and the image of the adjoint representation are the same Lie group.
Okay. So now let’s specialise to the Lorentz group. There is no continuous centre, by inspection of the commutation relationships. Therefore the Lie algebra of the image of big Ad is exactly the same as the Lie algebra of the Lorentz group. Nor is there a discrete centre in the Lorentz group. $\ker(\rho) = \{\mathrm{id}\}$, so, by the homomorphism theorem, the image of $O(1,3)$ under the big Ad adjoint representation is the Lorentz group itself. | {"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": 0.9161041378974915, "perplexity": 199.65089596611952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583667907.49/warc/CC-MAIN-20190119115530-20190119141530-00604.warc.gz"} |
https://www.physicsforums.com/threads/a-few-questions-on-hookes-law.131934/ | # A few questions on Hooke's Law
1. Sep 13, 2006
### OVB
I think im getting some concepts confused, so i need help clarifying the answers to the following questions:
1. can you use a rubber band instead of a steel spring in a scale to measure weight?
- I said no, because it would break
2. Should the wires supporting a suspension bridge be elastic or inelastic?
- I said elastic so that it can return to its original position, but I dont know how to expand on this...
3. How is a steel wire "very elastic" if it does not stretch much with large forces?
- I am not sure at all about this one
And k is independent of gravity right? (i.e., if you have a force and a displacement on earth, would those be in the same proportion on the moon? I'm thinking F/x = k equals a direct proportion but I might be breaking smoe rules).
I spent a good hour before and that is the progress I have made.
Last edited: Sep 13, 2006
2. Sep 13, 2006
### OVB
does anyone know?
3. Sep 13, 2006
### OVB
I explained rather well what I knew and what I didn't. Am I being ignored for something I left out or does no one know?
4. Sep 13, 2006
### Staff: Mentor
1. can you use a rubber band instead of a steel spring in a scale to measure weight?
- I said no, because it would break
Well, no is the right answer, but the reasoning is not correct.
1. In order to measure weight, the deflection must be linearly proportional to the weight - that is the point of Hooke's law. Only this way can one obtain a uniformly graduated scale. Steel spring could also be overloaded into the inelastic region, in which case it would be permanently deformed.
A rubber band is non-linear. The cross-section changes as it stretches. One might try and experiment with a suitable rubber band. Load it up with gram weights e.g. 1g, 2g, 3g, . . . and see if the deflection is a constant multiple of the mass.
The spring constant is a property of the metal and independent of gravity. The weight of something is a measure of the gravitational force on the mass - W = mg.
2. If a metal is stressed into the inelastic range, there will be permanent deformation. Structures are designed such the materials in the components operate well below the yield strength, i.e. well within the elastic range. When a structure is loaded then unloaded, as happens when a vehicle crosses a bridge, then the structure must return to its original unloaded form. Bridges are designed to deflect under load, but not by much. Many (or most) structures tend to be very stiff - i.e. resist displacement.
3. If a metal is highly loaded, and still does not stretch (which I take to mean 'permanently deform'), then it has a 'large elastic' range.
When metals are deformed, they are worked beyond the elastic range (yield strength) and they are plastically deformed.
Last edited: Sep 13, 2006
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook
Similar Discussions: A few questions on Hooke's Law | {"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": 0.8134880065917969, "perplexity": 1064.7405366053686}, "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-2017-39/segments/1505818689779.81/warc/CC-MAIN-20170923213057-20170923233057-00041.warc.gz"} |
https://brilliant.org/problems/equal-but-opposite/ | # Equal But Opposite
Geometry Level 1
Suppose two vectors $\vec{u}$ and $\vec{v}$ have the same magnitude but have opposite directions. What is $\vec{u} + \vec{v}$?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "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": 0.8907725811004639, "perplexity": 1085.8245727958583}, "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/1585371660550.75/warc/CC-MAIN-20200406200320-20200406230820-00170.warc.gz"} |
https://physicsoverflow.org/35946/the-dual-behavior-of-quantum-fields-and-the-big-bang?show=36052 | # The dual behavior of quantum Fields and the big Bang
Originality
+ 0 - 0
Accuracy
+ 0 - 3
Score
-3.00
708 views
Referee this paper: arXiv:1604.01253 by Malik Matwi
Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)
Original Abstract; We modify the propagation for the quarks and gluons, with that we have finite results, without ultra violet divergence in perturbed interaction of the quarks and gluons, this makes it easily for the interaction renormalization, like the self energy. Then we search for a way to remove our modification, with fixing the Lagrange parameters. so we can ignore our modification. We relate the modification to interaction situation, this is, we need it only for interaction renormalization. we see for the free the modification is removed. then We try to give the modification terms modification physical aspects, for this we see the corresponding terms in the Lagrange. To do that we find the role of those terms in the Feynman diagrams, in self energies, quarks gluons vertex. We see we can relate the propagation modification to fields dual behavior, pairing particle with antiparticle appears as scalar particles with high mass. For the quarks we can interrupt these particles as pions.
requested Apr 22, 2016
summarized
paper authored Mar 14, 2016 to physics
edited Apr 23, 2016
## 2 Reviews
+ 4 like - 0 dislike
I'm very happy to see sparks of science in Syria, but I have to say this is a very confused paper, and the mistakes are elementary.
The author starts with a higher-derivative regularization of the propagator that takes the form of
$$\frac{1}{(k^2-i\epsilon)(1+a^2k^2)}.$$
I don't have an immediate gripe with this although it's well known such regulator is not guaranteed to respect renormalizablity and unitariry of QCD, as oppposed to dim-reg in which the two are concretely proven. But the author seems to be in spirit more concerned with the "physical" aspects, so I can let this go for the moment. However, later the author makes the hypothesis that the regulator is physical and makes no attempt to remove it, and interprets the second pole mass offered by $1+a^2k^2$ as a pion. This is very wrong, notwithstanding the completely sabotaged gauge invariance/renormalizablity/unitarity, now you have two poles in an allegedly-particle propagator at free field level, that normally just means you don't really have a particle interpretation of your field theory, and to have a second particle at free propagator level, you need a second fundamental field. I guess the author confuses himself partly because he only comes to realize the existence of such a pole after a loop calculation on page 12, while in fact the pole is there at the very beginning. Plus, if one considers chiral symmetry breaking, the degrees of freedom just run wild if a pion is really what the author claims it to be.
In addition, the author has a confused discussion on confinement. First he gets the definition of confinement wrong, confinement is a long distance phenomenon, while the author demonstrates a linear potential at short distance $r<a$, here (page10) he also confuses cutoff scale with energy scale and gives a wrong justisfication for the scale at which confinement happens, on top of that here he seems to be willing to freely tune $a \to 0$ while this is in contradiction with identifying $1/a$ as the physical mass of a pion. Also, the way he gets linear potential is basically expanding a Yukawa potential to $O(r)$, in such way he might as well claim massive $\phi^4$, or any interacting theory with a massive particle for that matter, to be confining. And even if he had done everything right, it's destined to fail trying to prove confinement within perturbative QCD, since perturbation series is probably divergent (at best asymptotic), using perturbation theory to probe confinement, which is a strongly interacting effect, is doomed both in principle and practice.
I didn't read the rest of the paper.
reviewed Apr 23, 2016 by (2,640 points)
edited Apr 24, 2016
malik matwi
for the low distance confinement r<a, if the confinement survive at long distance, the the quarks of the protons and the neutrons in the same nucleus will be confinement in whole nucleus, and this is wrong.
malik matwi, also the dual behavior explaines how the confinement quarks of different baryons interacte
to satisfy the symmetries I related that modification to dual fields behavior, as I think the nature is scalar and it is not charged. I said in the abstract when the length a could not be removed then the perturbation is broken, and that depends on the coupling constant behavior.
the more details in page 10 when r<a, the total energy of the quarks become negative so the quarks disappear(condensation in hadrons)
for the confinement, here is defined in linear potential sigma *r it likes the string force, it is impossible to escape from it, so r<a
later I tried to say the varies in the length a is geodesic expanding, so r/a is invariant when a expands.
malik matwi
at first I tried to remove x1=x2 from the propagation
https://www.docdroid.net/TYzB01L/the-time-stop-in-the-quantum-fields-fluctuation.pdf.html
the propagation modification is legal according to remove x1=x2
at first I tried to remove x1=x2 from the propagation
https://www.docdroid.net/TYzB01L/the-time-stop-in-the-quantum-fields-fluctuation.pdf.html
the propagation modification is legal according to remove x1=x2
the propagation modification in my paper is exactly same the Pauli–Villars regularization for the scalar field propagation
http://isites.harvard.edu/fs/docs/icb.topic792163.files/15-regschemes.pdf
and
http://paperity.org/p/58839061/ambiguities-in-pauli-villars-regularization
but the difference is in the meaning of M and 1/a
in Pauli–Villars regularization the cutoff scale M is a heavy particle mass.
while in my paper the energy scale 1/a is a mass of paired particle-antiparticle only at low energy for the quarks at high energy 1/a is ignored while the mass M is taken at any energy.
your classifying my paper as confused is injustice.
the pairing particle antiparticle is like to say(for electrons photons interaction) that the long wave photons could not interact with the high energy electrons as the possibility of the interaction with the low energy electrons.
that is, the long wave photons see the high energy electrons and positrons as paired electrons-positrons.
at first I tried to satisfy the chiral symmetry, in that place, I thought that the chiral symmetry breaking is due to vacuum classical polarization, and this problem is solved by dual behavior of fields, some of these notes are in the first paper http://iiste.org/Journals/index.php/APTA/article/view/26837
at first I tried to satisfy the chiral symmetry, in that place, I thought that the chiral symmetry breaking is due to vacuum classical polarization, and this problem is solved by dual behavior of fields, some of these notes are in the first paper http://iiste.org/Journals/index.php/APTA/article/view/26837
notice(page 37): we assume that the universe was created in every point in two dimensions space XY then the explosion in Z direction. That is by the quarks, in each point in XY at the quarks were created and then they expanded in each point XY to the length a0 then the explosion in Z direction, the result is the universe in the space XYZ.
This assumption appears to be strange, I used it to calculate
Vq/Vh=Hh/Hq=Sdq/Sdh (page 44)
We can make this assumption more legal, we assume, the quarks universal condensation in hadrons occurred in one direction let it Z direction, this means, to spend less energy for the condensation, the correspond force must effect only in one direction Z . We let X, Y, Z as proper distances.
@MK
+ 0 like - 0 dislike
This paper apparently claims to: modify perturbative QCD to make it UV finite, derive free and confined phases, and build from this a cosmology that doesn't need a dark sector... Normally I would just glance at such a paper, but this one is from Syria. I don't see many (any!) QFT papers from Syria, right or wrong, and I think it might contribute in some small way to the recovery of that country, to engage with a work so ambitious as this.
reviewed Apr 22, 2016 by (1,650 points) | {"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": 0.8885275721549988, "perplexity": 1072.756526278953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195528687.63/warc/CC-MAIN-20190723022935-20190723044935-00538.warc.gz"} |
http://tex.stackexchange.com/questions/38775/hiding-section-titles-when-the-section-is-empty?answertab=oldest | # Hiding section titles when the section is empty
I have created a macro like this:
``````\newcommand{\showsection}[2]{
\ifstrequal{#2}{}{}{\section{#1} #2}
}
``````
It should be used like this in the document:
``````\showsection{Books}{
Do you know any good books?
}
``````
If `#2` is empty, as in this case below, the section title to not be shown:
``````\showsection{Books}{
}
``````
Unfortunately, with some macros inside, which should sometimes appear blank, somehow something is getting through. Perhaps it is an extra space? No text is visible. I have put `%` after every line of the macros which I placed inside. How can I get the section titles to disappear in this case?
-
Please have a look at the documentation of `etoolbox`. There are also described the commands `\ifblank` and `\ifstrempty`. Another mehtod is using the pacakge `xparse`. – Marco Daniel Dec 19 '11 at 12:52
Please mention important packages in the text, not just as tags. It is always good to post a add a minimal working example (MWE) that illustrates your problem exactly and lets people test their solutions easily. – Martin Scharrer Dec 19 '11 at 12:55
You have an end-of-line character after the opening `{` which causes a space. It might be better to check if the argument is a single space instead of being empty. You can add one space to the argument yourself to make sure the test is true for a real empty content.
Also you should add `%` after any `{` or `}` at the end of a line inside the macro definition.
``````\newcommand{\showsection}[2]{% | {"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": 0.9527686238288879, "perplexity": 1036.6665961443302}, "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/1454701150206.8/warc/CC-MAIN-20160205193910-00020-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://leanprover-community.github.io/archive/stream/116395-maths/topic/infinitary.20logic.html | ## Stream: maths
### Topic: infinitary logic
#### Reid Barton (Oct 11 2018 at 19:51):
Is anyone aware of a formalization of infinitary logic (I think I am interested in what is called $L_{\infty,\infty}$)? Either in Lean or in another DTT-based system
Last updated: May 09 2021 at 09:11 UTC | {"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": 0.8548678755760193, "perplexity": 4598.844658099823}, "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/1620243988966.82/warc/CC-MAIN-20210509092814-20210509122814-00630.warc.gz"} |
http://physics.stackexchange.com/questions/35333/can-hydrogen-stay-frozen-in-vacuum | # can hydrogen stay frozen in vacuum?
I've look into the hydrogen state diagram, and it seems that it can be frozen under pressure.
Question: Does this mean that hydrogen cannot be kept frozen in a vacuum chamber?
- | {"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": 0.9012495875358582, "perplexity": 709.2278566739418}, "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-2015-48/segments/1448398468971.92/warc/CC-MAIN-20151124205428-00346-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/r19/section/5.17/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 5.17: Solve Real World Problems Involving Simple Interest
Difficulty Level: Basic Created by: CK-12
Estimated9 minsto complete
%
Progress
Practice Percent Equations
MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated9 minsto complete
%
Estimated9 minsto complete
%
MEMORY METER
This indicates how strong in your memory this concept is
Have you ever tried to figure out a problem involving interest? Take a look at this dilemma.
With a goal of boosting student attendance at football games, the student council has decided to invest a portion of their savings for middle school decorations. They figure that when the games are happening that they can decorate the middle school with balloons, banners and flyers.
“I think it will help make it a priority for students,” Jeremy said at the weekly student council meeting.
“It will be a lot of fun too. We could even host a pep rally to help get the kids charged up,” Candice suggested.
“We put 4000 in the bank in the sixth grade. Now that we are \begin{align*}8^{th}\end{align*} graders that money has been sitting in the bank for two years at a 4% interest rate,” Jeremy explained. Candice began working out the math in her head. If they did put4000 in the bank for two years and they had a 4% interest rate, then there definitely is more money in there now. She began to complete the calculations in her head.
Do you have an idea how to figure this out? This problem involves principal, interest rates and time. This Concept will teach you all about calculating simple interest. Pay close attention and you will see this problem again at the end of the Concept.
### Guidance
Money is a necessary part of everyday life, and as you get older, your relationship to money will change. In this lesson, we will explore some of the ways in which you will relate to money as you get older.
Saving money and making wise investments will be an important part of your financial planning. Part of making investing is earning interest. When you save money in the bank, the bank uses that money for its own investments. In return for using your money, the bank pays you a certain percent. This percent is your interest. Interest is the percent that a bank pays you for using keeping your money in their bank.
Banks compete with each other for your money because they want you to put your money in their bank. They try to give you the best “interest rate” that they can. This means that they will pay you a greater percentage than another bank to try to get your business. The greater the interest rate that they pay you; the more likely you are to invest your money with them in a savings account. The more money you save, the more they have to invest. They publish an interest rate \begin{align*}r\end{align*} which tells you what percent they will pay you per year \begin{align*}t\end{align*}. The principal, \begin{align*}p\end{align*} is the amount of money that you have put into the bank.
You can use this information in the formula \begin{align*}I = prt\end{align*} in order to calculate the interest that you will earn on your principal \begin{align*}p\end{align*}.
Take a few minutes and write this formula in your notebook.
Now let’s look at how we can use this formula to calculate interest.
You invest 5,000 in a bank for 2 years at a 4% interest rate. What is the interest you have earned after this time? We start by looking at the given information. Then use the formula to calculate interest. \begin{align*}p = 5000, r = .04, t = 2\end{align*}. Use the formula to calculate interest. \begin{align*}I &= prt\\ I &= 5000 \cdot .04 \cdot 2\\ I &= 400\end{align*} The bank will pay you400 in interest over two years at that rate.
Many investors may have specific goals—they want to earn a certain amount of interest on their investments. Because of this, they need to figure out the time that it takes to earn a certain amount of money. The formula \begin{align*}I = prt\end{align*} is an equation. We can use the Multiplication Property of Equations to solve for \begin{align*}t\end{align*} if we know \begin{align*}I, r\end{align*}, and \begin{align*}p\end{align*}.
Mrs. Duarte has $20,000 to invest. She wants to earn$10,000 in interest. She is considering a savings and loans bank that is offering her 5.6% interest per year. For how long will she have to leave her money in the bank in order to reach her goal of 10,000? Start by looking at the given information. \begin{align*}I = 10000, p = 20000, r = .056\end{align*} Solve for \begin{align*}t\end{align*}. Next, we substitute the given values into the formula and solve the equation. \begin{align*}I &= prt\\ 10000 &= 20000 \cdot .056 \cdot t\\ 10000 &= 1120t\\ \frac{10000}{1120} &= \frac{1120t}{1120}\\ 8.93 &= t\end{align*} She will have to leave her money in the bank for nearly 9 years. Exactly! We are using what we have learned about solving equations to figure out missing information regarding interest and banking.You can use the simple interest formula \begin{align*}I = prt\end{align*} to find any of the missing variables if you are given values of the others. We have used it to solve for \begin{align*}I\end{align*} and \begin{align*}t\end{align*}. Of course, once the bank pays you interest, your account balance grows. You start of with your principal \begin{align*}p\end{align*} and then you add your interest \begin{align*}I\end{align*}. Now let’s see how much a bank balance would be after a given time at a given interest rate. Jessica invests3,000 in a credit union at an interest rate of 3.9%. She leaves the money there for 5 years. What is her balance after that time?
To answer this question, we will need to do two things. First, we will need to figure out the amount of the interest. Then we can add this amount to the principal that Jessica first invested. This will give us the new balance.
First find the interest that she earned:
\begin{align*}p &= 3000, r = .039, t = 5\\ I &= prt\\ I &= 3000 \cdot .039 \cdot 5\\ I &= 585\end{align*}
She earned $585 in interest. Her principal was$3,000. How much does she have now?
\begin{align*}585 + 3000 = 3585\end{align*}
Solution: You would owe 8,113. Now let's go back to the dilemma from the beginning of the Concept. Now we need to figure out the interest and the final balance in the student council bank account. First, let’s find the amount of the interest. \begin{align*}I &= PRT\\ I &= (4000)(.04)(2)\\ I &= \320.00\end{align*} Next, we add this to the original amount invested. \begin{align*}\4000 + \320 = \4320.00\end{align*} This is the new balance in the student council account. ### Vocabulary Interest The amount of money paid or owed after a period of time. It is based on a percentage. Rate The percent charged or paid by a bank given a savings account or a loan amount. Principal The amount of the original loan or original deposit. ### Guided Practice Here is one for you to try on your own. A nurse put22,000 in the bank 15 years ago. She has earned \$21,450 in interest—nearly as much as her initial investment. What was the interest rate that the bank was paying her?
Solution
Using the simple interest formula \begin{align*}I = prt\end{align*}, we can calculate the interest rate \begin{align*}r\end{align*} if we are given the \begin{align*}I, p\end{align*} and \begin{align*}t\end{align*} values. As before, we will substitute the known values and then use inverse operations to find the missing value.
\begin{align*}I &= 21450, p = 22000, t = 15\\ I &= prt\\ 21450 &= 22000 \cdot r \cdot 15\\ 21450 &= 330000r\\ \frac{21450}{330000} &= \frac{330000r}{330000}\\ .065 &= r\end{align*}
Because we are looking for a percent-an interest rate, we have to change the decimal to a percent.
.065 = 6.5%
The bank was paying 6.5%.
### Practice
Directions: Use the simple interest formula \begin{align*}I = prt\end{align*} to solve for the Interest.
1. Find \begin{align*}I\end{align*} if \begin{align*}p = 62,300, r = .0525, t = 14\end{align*}.
2. Find \begin{align*}I\end{align*} if \begin{align*}p = 9800, r = .028, t = 9\end{align*}.
3. Find \begin{align*}I\end{align*} if \begin{align*}p = \600, r = .05, t=8\end{align*}
4. Find \begin{align*}I\end{align*} if \begin{align*}p = \2300, r = .06, t=12\end{align*}
5. Find \begin{align*}I\end{align*} if \begin{align*}p = \5500, r = .08, t=7\end{align*}
6. Find \begin{align*}I\end{align*} if \begin{align*}p = \400, r = .05\end{align*} and \begin{align*}t=5\end{align*}
7. Find \begin{align*}I\end{align*} if \begin{align*}p = \700, r = .03\end{align*} and \begin{align*}t=9\end{align*}
8. Find \begin{align*}I\end{align*} if \begin{align*}p = \500, r = .06\end{align*} and \begin{align*}t=12\end{align*}
9. Find \begin{align*}I\end{align*} if \begin{align*}p = \800, r = .09\end{align*} and \begin{align*}t=7\end{align*}
10. Find \begin{align*}I\end{align*} if \begin{align*}p = \950, r = .06\end{align*} and \begin{align*}t=4\end{align*}
Directions: Find the new interest and then find the new balance with the given information. There are two steps to solving these problems.
1. \begin{align*}p = 43000, r = .0365, t = 11\end{align*}
2. \begin{align*}p = 7000, r = .079, t = 4\end{align*}
3. \begin{align*}p = 8000, r = .06, t = 3\end{align*}
4. \begin{align*}p = 18000, r = .04, t = 5\end{align*}
5. \begin{align*}p = 25000, r = .05, t = 3\end{align*}
6. \begin{align*}p = 3000, r = .05, t = 7\end{align*}
7. \begin{align*}p = 12000, r = .04, t = 5\end{align*}
8. \begin{align*}p = 9000, r = .06, t = 10\end{align*}
9. \begin{align*}p = 7500, r = .03, t = 8\end{align*}
10. \begin{align*}p = 27500, r = .04, t = 6\end{align*}
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
### Vocabulary Language: English
Compound interest
Compound interest refers to interest earned on the total amount at the time it is compounded, including previously earned interest.
future value
In the context of earning interest, future value stands for the amount in the account at some future time $t$.
Interest
Interest is a percentage of lent or borrowed money. Interest is calculated and accrued regularly at a specified rate.
present value
In the context of earning interest, present value stands for the amount in the account at time 0.
Principal
The principal is the amount of the original loan or original deposit.
Rate
The rate is the percentage at which interest accrues.
Simple Interest
Simple interest is interest calculated on the original principal only. It is calculated by finding the product of the the principal, the rate, and the time.
Show Hide Details
Description
Difficulty Level:
Basic
Authors:
Tags:
Subjects: | {"extraction_info": {"found_math": true, "script_math_tex": 66, "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": 1, "texerror": 0, "math_score": 0.8274190425872803, "perplexity": 1654.651172531496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540915.89/warc/CC-MAIN-20161202170900-00033-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://cracku.in/jee-advanced-2015-paper-2-question-paper-solved?page=4 | Instructions
For the following questions answer them individually
Question 31
Question 32
Question 33
Question 34
Question 35
Question 36
# One mole of a monoatomic real gas satisfies the equation $$p(V - b) = RT$$ where b is a constant. The relationship of interatomic potential V(r) and interatomic distance r for the gas is given by
Instructions
In the following reactions
Question 37
Question 38
# The major compound Y is
Instructions
When 100 mL of 1.0 M HCl was mixed with 100 mL of 1.0 M NaOH in an insulated beaker at constant pressure, a temperature increase of $$5.7 ^{\circ}C$$ was measured for the beaker and its contents (Expt. 1). Because the enthalpy of neutralization of a strong acid with a strong base is a constant $$(-57.0 kJ mol^{-1})$$, this experiment could be used to measure the calorimeter constant. In a second experiment (Expt. 2), 100 mL of 2.0 M acetic acid $$(K_a = 2.0 \times 10^{-5})$$ was mixed with 100 mL of 1.0 M NaOH (under identical conditions to Expt. 1) where a temperaturerise of $$5.6 ^{\circ}C$$ was measured.
(Consider heat capacity of all solutions as $$4.2 J g^{-1} K^{-1}$$ and densityof all solutions as $$1.0 g mL^{-1})$$
Question 39
Question 40
OR | {"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": 0.9433103799819946, "perplexity": 2538.536318651011}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500094.26/warc/CC-MAIN-20230204044030-20230204074030-00142.warc.gz"} |
https://www.nature.com/articles/s41467-019-10340-8?error=cookies_not_supported&code=6306d4d8-b751-4836-b948-f7be3eb1dab3 | Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.
# Low-energy electrons transform the nimorazole molecule into a radiosensitiser
## Abstract
While matter is irradiated with highly-energetic particles, it may become chemically modified. Thereby, the reactions of free low-energy electrons (LEEs) formed as secondary particles play an important role. It is unknown to what degree and by which mechanism LEEs contribute to the action of electron-affinic radiosensitisers applied in radiotherapy of hypoxic tumours. Here we show that LEEs effectively cause the reduction of the radiosensitiser nimorazole via associative electron attachment with the cross-section exceeding most of known molecules. This supports the hypothesis that nimorazole is selectively cytotoxic to tumour cells due to reduction of the molecule as prerequisite for accumulation in the cell. In contrast, dissociative electron attachment, commonly believed to be the source of chemical activity of LEEs, represents only a minor reaction channel which is further suppressed upon hydration. Our results show that LEEs may strongly contribute to the radiosensitising effect of nimorazole via associative electron attachment.
## Introduction
Here we study the prototype compound, nimorazole (NIMO), of the class of nitroimidazolic (NI) radiosensitisers. For few decades, NIs have been investigated in biochemical and oncologic studies. Those compounds have been considered to mimic the well-known oxygen effect which is absent in hypoxic cells due to their anaerobic environment13,14. The chemical structure of NIMO is included in Fig. 1. The compound is characterized by its NI ring interconnected to a morpholine ring by a carbon chain. NIMO turned out to be the only successful compound to overcome tumour hypoxia with reasonable side effects for patients15 and therefore is used as standard chemical compound for pharyngeal and supra-glottic carcinoma in Danish radiotherapy centers16.
In our study we obtain that dissociation plays a minor role when an electron attaches to NIMO. Instead, the formation of the parent radical anion by associative electron attachment to NIMO is a very efficient process, in particular, when the molecule is hydrated. In this case, the only notable fragment anion NO2 becomes strongly quenched which is not related to a change in the probability of spontaneous electron emission (autodetachment). Our quantum chemical calculations for NIMO clustered with one or two water molecules fully support these experimental results. The calculations show that electron attachment to hydrated NIMO is stabilized by the presence of water. Ultimately, these results show that free radical anion formation by attachment of a low-energy electron could be the key process for radiosensitisation of hypoxic tumour cells by the electron-affinic NI radiosensitisers.
## Results
### Formation of the NIMO radical anion upon electron attachment
The present results indicate that free low-energy electrons are very effectively captured by NIMO. In contrast to halouracils, the dominant feature in electron attachment to NIMO is the very effective formation of the non-dissociated (metastable) parent anion, NIMO, without the cleavage of chemical bonds. This anion is observed within a very narrow main peak close to zero eV electron energy as shown in Fig. 1. We determined the absolute cross-section of this associative attachment (AA) process to be ~3 × 10−18 m2 (with an uncertainty of one order of magnitude, see “Methods” for the discussion on possible systematic errors). This cross-section is an extraordinary high value which exceeds the geometrical cross-section of the target molecule (about 0.5 × 10−18 m2).
Attachment of free electrons is accompanied with often significant energy release comprising the kinetic energy of the incoming electron and the electron affinity of the molecule, which was calculated to be 1.31 eV (at the M062x/6-311 + G(d,p) level of theory)17. The anion lifetime is typically short due to electron autodetachment or molecular dissociation5. The formation of (metastable) parent anions with high cross-sections on a microsecond detection time scale, as for the present case, is therefore restricted to only two types of molecular systems. The first enables effective redistribution of the excess energy over the vibrational degrees of freedom such as the sulphur hexafluoride molecule SF618 or fullerenes like C6019. These two compounds are characterised by the high symmetry of the molecule (e.g., in the former case there are six equal S–F bonds), which provides favourable conditions for the formation of a metastable parent anion. The second provides an effective sink for the excess energy via intramolecular bond breakage and rearrangement20. Though NIMO is of low symmetry, we suppose its appreciable number of 84 vibrational degrees of freedom provides an effective means for energy redistribution making the observation of an intense NIMO within our observation time window of few hundred microseconds possible.
### Dissociative electron attachment to NIMO
The only observable relevant DEA reaction is the formation of NO2. Figure 2 shows that the cross-section for this channel is about one order of magnitude lower than that for NIMO, occurring in the electron energy range between ~2–4 eV. Other observed fragment anions have cross-sections even two orders of magnitude lower than that for the parent anion and therefore they will not be discussed here. The electron energy dependence of their cross-sections is shown in the Supplementary Figs. 13. The formation of NO2 can formally be expressed as,
$${\mathrm{e}}^ - + \mathrm{NIMO} \leftrightarrow (\mathrm{NIMO}^{ - })^\# \to \left( {\mathrm{NIMO} - \mathrm{NO}_2} \right) + \mathrm{NO}_2^ -$$
(1)
with (NIMO)# representing the intermediate molecular anion. Reaction (1) represents the simple cleavage of the C–NO2 bond after initial formation of a repulsive shape resonance of σ*(C–NO2) character. In this case, the DEA reaction is expected as direct electronic dissociation along the repulsive C–N potential energy surface. We calculated the thermochemical threshold (free energy of reaction ∆G) for this process at the M062x/6–311+G(d,p) level of theory and obtained a value of +0.53 eV (release of the intact neutral fragment, as mentioned in reaction (1)). Since we can also observe a weak NO2 signal at electron energies close to 0 eV (see Supplementary Fig. 3), we investigated other dissociation pathways, which may lead to this anion yield. The only exothermic reaction found corresponds to the release of C3H3N2 + C6H11NO with ∆G = −0.26 eV. The formation of NO2 is driven by the appreciable adiabatic electron affinity (AEA) of NO2 (present value 2.40 eV). However, the weak NO2 ion yield at zero eV indicates a barrier on this reaction pathway. Therefore, autodetachment can effectively compete with the DEA channel of (NIMO)# at this energy. We further note that though our calculations for the (NIMO–NO2) fragment predict an appreciable AEA of +2.07 eV, and a DEA threshold of +0.85 eV for the release of NO2+ (NIMO–NO2), we do not observe this simple reaction channel leading to the reactive nitrogen species NO221.
### Solvation effects upon DEA to NIMO
It is important to note that the outcome of DEA reactions may be influenced by the presence of an environment. In order to elucidate this point, we performed electron attachment experiments with hydrated NIMO, i.e. NIMO(H2O)n clusters with <n> ≤ 14. We chose this approach since intense research in such finite cluster systems and in the condensed phase over the last years demonstrated that in bound molecules DEA can usually still be described on a molecular site, i.e., electron attachment proceeds to an individual molecule which is coupled to an environment. The latter affects both, the initial attachment process and the decomposition of the intermediate molecular anion22. The results for clustered NIMO indicate that DEA becomes suppressed in favour of AA. The ion yield ratio of NO2/(NIMO(H2O)n) (which can also be interpreted as the overall ratio DEA/AA) for different solvation conditions is shown in Fig. 3. A decrease by three orders of magnitude can be observed, indicating that DEA is reduced for solvated NIMO. This behaviour is within the general trend when going from the gas phase over microsolvation into solution23 and it is caused by parent anion stabilization via energy dissipation to the environment24. The detailed mechanism of the anion stabilization can be caused by (i) caging of dissociation products25,26 and (ii) ultrafast quenching of the transient anion27.
In view of the strong decrease of the ion yield ratio of NO2/(NIMO(H2O)n), one could also speculate about option (iii), a change in the autodetachment probability which leads to the decrease of the NO2 channel. This process will lead to a change of the position and the width of the resonance with the hydration28. To elucidate this question, we performed electron energy dependent ion yield measurements for the NO2 and the parent anion at different hydration conditions. The yields are shown in Fig. 4. As already discussed, in the isolated molecule the parent anion is formed at near zero energies and NO2 at around 3 eV. With increased hydration, the intensity of the low-energy peak in the spectrum of the parent anion decreases. This is caused by the fact that the attachment at low electron energies results into formation of NIMO(H2O)n anions instead of bare NIMO. The new feature is that when the molecule is hydrated with only few water molecules, the parent anion signal appears at around 3 eV. Therefore, the 3 eV resonance, which normally results in the dissociation, is now stabilized and the excess energy is presumably released by evaporation of water. Such evaporation then results in the formation of the parent anion. The 3 eV resonance shape weakly depends on the hydration degree: at lower hydration conditions (I, II) only the lower energy part of the resonance is stabilized, at higher hydration conditions (III, IV) the parent anion curve practically resembles that of the NO2 curve of the isolated molecule. Finally, at the highest hydration conditions (V), the 3 eV resonance practically disappears for both NO2 and the parent anion. Both resonances now result in the formation of larger NIMO(H2O)n cluster anions. The energy acquired by the attachment process is not enough to dissociate the molecule, nor to evaporate all the water monomers from the cluster.
On the basis of the aforementioned observation, we can clearly exclude option (iii) from the explanation of the NIMO stabilisation against DEA. If mechanism (iii) is operative, one would expect only the disappearance of the 3 eV resonance but no formation of the parent anions at the same energy. The data also demonstrates the important fact that the primary electron attachment target at low energies is still the NIMO molecule, despite of the possible electron attachment to the surrounding water network29. This effect can be ascribed to the large de-Broglie wavelength of the impinging low-energy electron, which is about 0.7 nm at the electron energy of 3 eV and to the positive electron affinity of NIMO.
Moreover, we have also calculated structures of anionic NIMO hydrated by one and two water molecules. The anions were optimized at M062x/6–31 + G(d,p) level of theory and basis set. The lowest energy conformers are shown in Fig. 5, while all the different binding motifs investigated are summarized in Supplementary Figs. 5 and 6. From all the structures found for the anionic NIMO hydrated by water, we can conclude that the anion is more and more stabilized with the increasing number of water molecules present. The AEAs of NIMO hydrated by one water molecule increases to 1.36–1.76 eV (see Supplementary Fig. 5), while for the NIMO hydrated by two water molecules increases to the range of 1.53–1.96 eV (Supplementary Fig. 6). Additionally, the vertical detachment energy (VDE) of the NIMO anion also increases with the increasing number of water molecules. While for the bare NIMO anion, the VDE is 1.68 eV, addition of one water molecule increases the VDE to 1.83–2.32 eV (see Supplementary Fig. 5), and the presence of two water molecules increases the VDE to 2.11–2.64 eV (Supplementary Fig. 6). Notably, for the most stable conformers of the hydrated NIMO anion in Fig. 5, the water molecules seem to cluster around the −NO2 group, however, this is not the case for neutral structures. It can be seen in Supplementary Fig. 5 that the preference for water binding to neutral NIMO are, e.g., the O atom of the morpholine ring or the N atom of the imidazole ring. Nevertheless, the relative energies of these neutrals within the error of the calculation can be assumed to be isoenergetic.
## Discussion
Together with the remarkably high electron attachment cross-section, the quenching of DEA in the solvated molecule is the second key outcome of our studies. The calculations for NIMO clustered with one or two water molecules indicate that the attachment of an electron to hydrated NIMO is stabilized by the presence of water, the attraction of water molecules to the −NO2 group and a fast dissipation of the excess energy due to the strong interactions between the O−H oscillators of water molecules30,31 and their evaporation, while the increased VDE makes the autodetachment also less probable. Though very simplified systems are studied here compared to the complex cellular environment, the time evolution of radiation damage and the peculiar features of electron attachment mentioned above may allow nonetheless some predictions on the action of NIMO as radiosensitiser in vivo. Due to the favourable adiabatic electron affinity of NIs, it was previously suggested that NIs become only active after the reduction13,32, where the rate of the reduction determines the uptake of the radiosensitiser by the cell. Therefore, free radical anion formation by low-energy electrons seems to be the key process for radiosensitisation of hypoxic tumour cells by NIs. This conclusion is further supported by the result that NIs have to be present at the instant of radiation for a radiosensitising effect, while when adding them at a later stage (after few ms) the effect vanished13.
After the reduction process of NIs in a cellular environment, it was suggested that the intact anion itself is not the cytotoxic species which attacks DNA. Instead, the NI radical anion may become protonated in hypoxic cells32,33, and then the neutralised radical compound could bind to DNA. In such case, chemical reaction schemes exist, which lead to strand breaks in DNA34,35. These schemes rely on the binding of neutral NIs at DNA sites attacked by hydroxyl radicals. The latter radicals are formed simultaneously by radiolysis processes. Therefore, NIs were suggested to mimic the oxygen effect in hypoxic tumours.
Finally, we note that hypoxic cells without a radiosensitiser usually require two to three times higher radiation dose for cell death compared to normally-oxygenated cells13. Electron-affinic radiosensitisers such as NIs may significantly lower the doses and consequently the side effects of the therapy36. Here, we demonstrate a fundamental mechanism of their accumulation in tumour cells, which may be used in further improvement of these important therapeutic agents.
## Methods
### Experimental design
The present study was carried out at two different crossed electron-molecular/cluster beam setups. The single-molecule data was collected at the Wippi apparatus in Innsbruck, a detailed description can be found in ref. 37. An oven serves as inlet for the NIMO sample. A capillary of 1 mm diameter is mounted onto it to guide the evaporated sample towards the interaction region. As ionisation source serves a hemispherical electron monochromator (HEM). It provides electrons with a narrow energy distribution (~100 meV) with Gaussian profile. The attachment processes take place in the region where molecular beam and electrons cross. Measurements at different electron energies are enabled by applying an appropriate acceleration potential in the HEM shortly before the interaction region. The negatively charged parent and fragment ions formed are subsequently extracted into a quadrupole mass analyser by a weak electrostatic field. The quadrupole has a nominal mass range of 2048 u and is utilised for mass selection. Thus, combining the HEM and the mass filter, the formation efficiency of selected fragments at varying energies can be studied. The ions are detected by a channel electron multiplier and counted by a preamplifier with analog-to-digital converter unit. The mass spectrometer is operated under high vacuum (~10−8 mbar background pressure).
For cluster experiments, the CLUster Beam (CLUB) apparatus in Prague was used, for a detailed review refer to ref. 38. In the present study, the configuration of the experiment was identical to that one described in ref. 25. For cluster production, helium or neon gas is humidified by a Pergo gas humidifier. A Nafion tubing gas line passes through a water bath and its membrane selectively permeate water vapour. The humidified gas is introduced into a heated oven filled with NIMO. At the opposite end a 90 μm conical nozzle is mounted. The mixture of humidified buffer gas and NIMO is co-expanded through the nozzle, which leads to the formation of NIMO(H2O)n clusters. The cluster beam is skimmed after a distance of ~2.5 cm and crossed by an electron beam in the interaction region ~1.5 m downstream. The electron energy can be varied by an accelerating potential. The created anions are extracted by a 2 μs long high-voltage pulse into a reflectron time-of-flight (RTOF) mass analyser with a mass resolution of ~5 × 103. A delay of 0.5 μs between electron pulse and ion extraction excludes any effects caused by those. With each extraction pulse, all anions are analysed, detected by a multichannel plate and recorded as mass spectrum.
### Materials
The NIMO used in both experiments was purchased from Toronto Research Chemicals (Canada) with a stated purity of ≥99%. For the doped water cluster study, type II pure water was prepared by reverse osmosis. Helium 4.6 and neon 5.0 served as buffer gases.
### Experimental procedures
NIMO appears as powder but the focus lies on interactions with single molecules or molecules in a microhydration environment, the sample reservoirs of both experiments are heated to evaporate the sample. For Wippi, the oven was heated to about 95 °C, for CLUB between 80 and 110 °C. The temperatures were set carefully avoiding thermal decomposition of the sample but having reasonable signal rates. At CLUB, the mean cluster sizes are controlled by the pressure of the buffer gas. Here, higher pressures lead to higher humidification conditions and thus larger cluster sizes.
Additionally, background spectra were taken at identical experimental conditions to exclude both background gas and instrumentally caused peaks. For Wippi this includes both measurements with empty oven and at random masses, for CLUB measurements at empty oven and during a blocked cluster beam. Background data was subtracted from the signal traces.
For both setups, an energy calibration is required. At Wippi, the well-known 0 eV resonance positions in the ion yields of the SF6/SF6 18 and Cl/CCl439 arising from s-wave electron attachment processes are used. Additionally, the energy resolution is determined based on the full-width at half maximum (FWHM) of this peak to be ~100 meV. At CLUB, electrons below 1.3 eV are strongly suppressed due to the design of the electron gun (see ref. 40 for details). Thus, the 4.4 eV resonance in the ion yield of O/CO241 is used for the calibration of the energy scale. The energy resolution amounts to ~0.7 eV. Cross-sections were determined with the data taken at Wippi by comparing the ion yields of NIMO and the well-known cross-sections of the 0.8 eV peak of Cl/CCl439 measured under the same conditions. Pressure calibrations caused by different sample introduction methods were implemented based on earlier experiments. Partial pressures were taken into account by normalizing the signal traces according to the related values. Only the order of magnitude can be derived due to systematic uncertainties. Two main influencing factors exist. First, the partial pressure determination arises uncertainties as the partial pressure cannot be measured directly but can only be evaluated by subtracting the pressures with and without presence of NIMO in the vacuum chamber. Additionally, the correction factor of the hot cathode used as pressure gauge for different gases must be estimated since it is not available (O(30%)). Second, resolution and transmission effects of the used quadrupole mass analyser result in varying signal heights (O(30%)).
### Data analysis for Figs. 1 and 2
Statistical significance and reproducibility were verified by repeating each measurement several times (Figs. 1 and 2). In case of Wippi data (Figs. 1 and 2), this includes data obtained at different days for various oven fillings, making up to about 200–300 set of measurements for the NIMO and nitrogen dioxide anion each, with gate times of 1 s/mass step and step size of 0.01 u. The figures in the main text show the mean of the according measurements already converted into cross-sections. Error bars refer to the statistically caused standard error of the mean which is calculated as $$s/\sqrt m$$ with s the standard deviation of the mean and m the number of averaged measurements. Here, the individual curves are normalised for error calculations to exclude systematic uncertainties caused by the quadrupole and pressure determination described before and hence only refer to the statistical variation of the shape of the resonance.
### Data analysis for Figs. 3 and 4
The ratios and mean number of water molecules <n > in the mixed clusters NIMO(H2O)n were determined from the mass spectra depicted in the Supplementary Fig. 4 (Figs. 3 and 4). The mass spectra were obtained by averaging the values of cumulative mass spectra obtained in three independent measurements for every hydration condition except of the mass spectrum for the isolated molecule taken only once. The isolated molecule was studied in detail using the Wippi apparatus introduced above. The individual cumulative spectra used for averaging were taken at different days and different spectrometer settings. To avoid systematic errors, we performed the set of measurements at the experimental setting preferring the detection of high m/z´s in the present TOF setup and the standard deviation of these measurements was used for estimation of error bars in Fig. 3. Every individual cumulative mass spectrum was obtained as a sum of 21, background subtracted, and electron current divided, mass spectra measured at electron energies from 0.6 to 5.6 eV with step size 0.25 eV. The dynamic range of the single measured mass spectrum was 2 × 106.
### Quantum chemical calculations
The thermodynamic threshold (free energy of reaction ∆G) for a DEA reaction, considering precursor molecule M and a release of a neutral fragment X (see e.g., reaction (1)), can be expressed by ∆G([M − X]) = DE(M−X) − EA(M − X), where DE(M − X) is the bond dissociation energy and EA(M − X) the electron affinity of the corresponding fragment. The threshold energy for the experimental observation of [M − X] in electron attachment experiments coincides with ∆G([M − X]) if the fragments are formed with no excess energy. Fragmentation reactions with kinetic energy release occur at electron energies above the thermodynamic threshold ∆G([M − X]). Quantum chemical calculations employing the density functional M062x42,43 were carried out to calculate free energies of reactions, dissociation energies (including the zero-point energy correction), and adiabatic electron affinities. For NIMO we used the lowest structure reported in ref. 17. All structures where optimized at the M062x/6–311 + G(d,p) level of theory and basis set with the Gaussian-09D01 program package44. Structures of NIMO hydrated by one and two water molecules were optimized at M062x/6–31 + G(d,p) level of theory and basis set. We determined AEAs and vertical electron affinities of the neutrals, and VDEs of the anions. Frequencies were calculated in all cases to confirm that the structures are local minima on the potential energy surface and not the transition states.
## Data availability
All data that led to the present findings are available upon request to the corresponding author. The source data underlying Fig. 14 and Supplementary Figs. 14 are provided as a Source Data file.
## References
1. 1.
Gokhberg, K., Kolorenč, P., Kuleff, A. I. & Cederbaum, L. S. Site- and energy-selective slow-electron production through intermolecular Coulombic decay. Nature 505, 661–663 (2014).
2. 2.
Ren, X. G., Al Maalouf, E. J., Dorn, A. & Denifl, S. Direct evidence of two interatomic relaxation mechanisms in argon dimers ionized by electron impact. Nat. Commun. 7, 11093 (2016).
3. 3.
Westphal, K. et al. Irreversible electron attachment - a key to DNA damage by solvated electrons in aqueous solution. Org. Biomol. Chem. 13, 10362–10369 (2015).
4. 4.
Ma, J., Wang, F., Denisov, S. A., Adhikary, A. & Mostafavi, M. Reactivity of prehydrated electrons toward nucleobases and nucleotides in aqueous solution. Sci. Adv. 3, e1701669 (2017).
5. 5.
Baccarelli, I., Bald, I., Gianturco, F. A., Illenberger, E. & Kopyra, J. Electron-induced damage of DNA and its components: Experiments and theoretical models. Phys. Rep. 508, 1–44 (2011).
6. 6.
Ptasinska, S., Denifl, S., Scheier, P., Illenberger, E. & Märk, T. D. Bond- and site-selective loss of H atoms from nucleobases by very-low-energy electrons (<3 eV). Angew. Chem. - Int Ed. 44, 6941–6943 (2005).
7. 7.
Michael, B. D. & O’Neill, P. Molecular biology. A sting in the tail of electron tracks. Science 287, 1603–1604 (2000).
8. 8.
Boudaïfffa, B., Cloutier, P., Hunting, D., Huels, M. A. & Sanche, L. Resonant formation of DNA strand breaks by low-energy (3 to 20 eV) electrons. Science 287, 1658–1660 (2000).
9. 9.
Ma, J. et al. Observation of dissociative quasi-free electron attachment to nucleoside via excited anion radical in solution. Nat. Commun. 10, 102 (2019).
10. 10.
Chomicz, L. et al. How to find out whether a 5-substituted uracil could be a potential DNA radiosensitizer. J. Phys. Chem. Lett. 4, 2853–2857 (2013).
11. 11.
Abdoul-Carime, H., Huels, M. A., Illenberger, E. & Sanche, L. Sensitizing DNA to secondary electron damage: resonant formation of oxidative radicals from 5-halouracils. J. Am. Chem. Soc. 123, 5354–5355 (2001).
12. 12.
Schürmann, R. et al. Resonant formation of strand breaks in sensitized oligonucleotides induced by low-energy electrons (0.5-9 eV). Angew. Chem. Int Ed. Engl. 56, 10952–10955 (2017).
13. 13.
Wardman, P. Chemical radiosensitizers for use in radiotherapy. Clin. Oncol. 19, 397–417 (2007).
14. 14.
Wang, H., Mu, X., He, H. & Zhang, X. D. Cancer radiosensitizers. Trends Pharm. Sci. 39, 24–48 (2018).
15. 15.
Baumann, M. et al. Radiation oncology in the era of precision medicine. Nat. Rev. Cancer 16, 234–249 (2016).
16. 16.
Henk, J. M., Bishop, K. & Shepherd, S. F. Treatment of head and neck cancer with CHART and nimorazole: phase II study. Radio. Oncol. 66, 65–70 (2003).
17. 17.
Feketeová, L. et al. Formation of radical anions of radiosensitizers and related model compounds via electrospray ionization. Int J. Mass Spectrom. 365-366, 56–63 (2014).
18. 18.
Klar, D., Ruf, M. W. & Hotop, H. Attachment of electrons to molecules at meV resolution. Aust. J. Phys. 45, 263–291 (1992).
19. 19.
Matejčik, Š. et al. Formation and decay of C60 - following free-electron capture by C60. J. Chem. Phys. 102, 2516–2521 (1995).
20. 20.
Sommerfeld, T. & Davis, M. C. Ring-opening attachment as an explanation for the long lifetime of the octafluorooxolane anion. J. Chem. Phys. 149, 084305 (2018).
21. 21.
Cerón-Carrasco, J. P., Requena, A., Zúñiga, J. & Jacquemin, D. Mutagenic effects induced by the attack of NO2 radical to the guanine-cytosine base pair. Front. Chem. 3, 13 (2015).
22. 22.
Neustetter, M., Aysina, J., da Silva, F. F. & Denifl, S. The effect of solvation on electron attachment to pure and hydrated pyrimidine clusters. Angew. Chem. Int Ed. 54, 9124–9126 (2015).
23. 23.
Bald, I., Langer, J., Tegeder, P. & Ingólfsson, O. From isolated molecules through clusters and condensates to the building blocks of life. A short tribute to Prof. Eugen Illenberger’s work in the field of negative ion chemistry. Int J. Mass Spectrom. 277, 4–25 (2008).
24. 24.
Poštulka, J., Slavíček, P., Fedor, J., Farník, M. & Kočišek, J. Energy transfer in microhydrated uracil, 5-fluorouracil, and 5-bromouracil. J. Phys. Chem. B 121, 8965–8974 (2017).
25. 25.
Kočišek, J., Pysanenko, A., Farník, M. & Fedor, J. Microhydration prevents fragmentation of uracil and thymine by low-energy electrons. J. Phys. Chem. Lett. 7, 3401–3405 (2016).
26. 26.
Kohanoff, J., McAllister, M., Tribello, G. A. & Gu, B. Interactions between low energy electrons and DNA: a perspective from first-principles simulations. J. Phys. -Condens Mat. 29, 383001 (2017).
27. 27.
Zawadzki, M., Ranković, M., Kočišek, J. & Fedor, J. Dissociative electron attachment and anion-induced dimerization in pyruvic acid. PCCP 20, 6838–6844 (2018).
28. 28.
Fabrikant, I. I. Electron attachment to molecules in a cluster environment: suppression and enhancement effects. Eur. Phys. J. D. 72, 96 (2018).
29. 29.
Barnett, R. N., Landman, U., Scharf, D. & Jortner, J. Surface and internal excess electron-states in molecular clusters. Acc. Chem. Res 22, 350–357 (1989).
30. 30.
Cowan, M. L. et al. Ultrafast memory loss and energy redistribution in the hydrogen bond network of liquid H2O. Nature 434, 199–202 (2005).
31. 31.
Zhang, Z., Piatkowski, L., Bakker, H. J. & Bonn, M. Ultrafast vibrational energy transfer at the water/air interface revealed by two-dimensional surface vibrational spectroscopy. Nat. Chem. 3, 888–893 (2011).
32. 32.
Edwards, D. I. Nitroimidazole drugs - Action and resistance mechanisms I. Mechanisms of action. J. Antimicrob. Chemother. 31, 9–20 (1993).
33. 33.
Wardman, P. Electron transfer and oxidative stress as key factors in the design of drugs selectively active in hypoxia. Curr. Med. Chem. 8, 739–761 (2001).
34. 34.
Wardman, P. The mechanism of radiosensitization by electron-affinic compounds. Radiat. Phys. Chem. 30, 423–432 (1987).
35. 35.
von Sonntag C. Free-Radical-Induced DNA Damage and Its Repair, A Chemical Perspective, 1 edn. (Springer-Verlag, Berlin Heidelberg, 2006).
36. 36.
Adams, G. E. et al. Electron-affinic sensitization. VII. A correlation between structures, one-electron reduction potentials, and efficiencies of nitroimidazoles as hypoxic cell radiosensitizers. Radiat. Res. 67, 9–20 (1976).
37. 37.
Denifl, S. et al. Free-electron attachment to coronene and corannulene in the gas phase. J. Chem. Phys. 123, 104308 (2005).
38. 38.
Kočišek, J., Lengyel, J. & Farnik, M. Ionization of large homogeneous and heterogeneous clusters generated in acetylene-Ar expansions: Cluster ion polymerization. J. Chem. Phys. 138, 124306 (2013).
39. 39.
Gallup, G. A., Aflatooni, K. & Burrow, P. D. Dissociative electron attachment near threshold, thermal attachment rates, and vertical attachment energies of chloroalkanes. J. Chem. Phys. 118, 2562–2574 (2003).
40. 40.
Kočišek, J., Grygoryeva, K., Lengyel, J., Farnik, M., & Fedor, J. Effect of cluster environment on the electron attachment to 2-nitrophenol. Eur. Phys. J. D. 70, 98 (2016).
41. 41.
Dressler, R. & Allan, M. Energy partitioning in the O/CO2 dissociative attachment. Chem. Phys. 92, 449–455 (1985).
42. 42.
Zhao, Y. & Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc. 120, 215–241 (2008).
43. 43.
Hehre, W. J., Radom, L., Schleyer, PvR, Pople, JA. Ab Initio Molecular Orbital Theory. (Wiley, Hoboken, NJ, 1986).
44. 44.
Frisch, M. J. et al. Gaussian 09, Revision D.01. (Gaussian Inc., Wallington CT, 2013).
## Acknowledgements
This work was supported by FWF, Vienna (P30332). R.M. and P.L.V. acknowledge the Portuguese National Funding Agency FCT-MCTES through PD/BD/114452/2016 and research grants UID/FIS/00068/2019 (CEFITEC) and PTDC/FIS-AQM/31281/2017. This work was also supported by Radiation Biology and Biophysics Doctoral Training Programme (RaBBiT, PD/00193/2010); UID/Multi/ 04378/2019 (UCIBIO). J.K. acknowledges the support by the Czech Science Foundation Grant 19–01159S. L.F. is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Université de Lyon for its financial support within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR). The crucial computing support from CCIN2P3 (France) is acknowledged gratefully. S.D. thanks Ass. Prof. Dr. Andreas Seppi from the Medical University Innsbruck for discussions.
## Author information
Authors
### Contributions
R.M. and J.K.: data aquisition experiment, data analysis; L.F.: quantum chemical calculations; R.M., J.K., L.F., J.F., M.F., P.L.-V., E.I. and S.D.: interpretation of data and manuscript preparation.
### Corresponding author
Correspondence to Stephan Denifl.
## Ethics declarations
### Competing interests
The authors declare no competing interests.
Journal peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Rights and permissions
Reprints and Permissions
Meißner, R., Kočišek, J., Feketeová, L. et al. Low-energy electrons transform the nimorazole molecule into a radiosensitiser. Nat Commun 10, 2388 (2019). https://doi.org/10.1038/s41467-019-10340-8
• Accepted:
• Published:
• ### The effect of solvation on electron capture revealed using anion two-dimensional photoelectron spectroscopy
• Aude Lietard
• , Golda Mensa-Bonsu
• & Jan R. R. Verlet
Nature Chemistry (2021) | {"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": 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": 0.8362702131271362, "perplexity": 4410.6381672634525}, "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-2021-39/segments/1631780056578.5/warc/CC-MAIN-20210918214805-20210919004805-00127.warc.gz"} |
http://www.aimath.org/WWN/rh/articles/html/111a/ | # Examples
See the [website] for many specific examples.
Ramanujan's tau-function defined implicitly by
also yields the simplest cusp form. The associated Fourier series satisfies
for all integers with which means that it is a cusp form of weight 12 for the full modular group.
The unique cusp forms of weights 16, 18, 20, 22, and 26 for the full modular group can be given explicitly in terms of (the Eisenstein series)
and
where is the sum of the th powers of the positive divisors of :
Then, gives the unique Hecke form of weight 16; gives the unique Hecke form of weight 18; is the Hecke form of weight 20; is the Hecke form of weight 22; and is the Hecke form of weight 26. The two Hecke forms of weight 24 are given by
where .
An example is the L-function associated to an elliptic curve where are integers. The associated L-function, called the Hasse-Weil L-function, is
where is the conductor of the curve. The coefficients are constructed easily from for prime ; in turn the are given by where is the number of solutions of when considered modulo . The work of Wiles and others proved that these L-functions are associated to modular forms of weight 2.
Back to the main index for The Riemann Hypothesis. | {"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": 0.9528982043266296, "perplexity": 485.9469889744708}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860110805.57/warc/CC-MAIN-20160428161510-00088-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physique.usherbrooke.ca/pages/en/biblio?page=3&s=year&o=asc&f%5Bauthor%5D=6583 | # Publications
Export 236 results:
Author Keyword Title Type [ Year]
Filters: Author is A.-M. S. Tremblay [Clear All Filters]
1991
, The influence of spin fluctuations on the temperature dependance of the magnetic susceptibility and nuclear relaxation in high Tc superconductors, in 10th Specialized Colloque AMPERE, NMR/NQR in High-Tc Superconductors, Zurich, Suisse, 1991.
, Magnetic-properties of the 2-dimensional Hubbard-model, Physical Review Letters, vol. 66, pp. 369–372, 1991. (182.55 KB)
, Noise and crossover exponent In the two-component random resistor network, Physical Review B, vol. 43, pp. 11546–11549, 1991. (204.64 KB)
, Noise and crossover exponent In the two-component random resistor network, Physical Review B, vol. 43, pp. 11546–11549, 1991. (204.64 KB)
, Random Mixtures With Orientational Order, and the Anisotropic Resistivity Tensor of High-t(c) Superconductors, Journal of Applied Physics, vol. 69, pp. 379–383, 1991. (691.17 KB)
1992
, Determinant Monte-carlo For the Hubbard-model With Arbitrarily Gauged Auxiliary Fields, International Journal of Modern Physics B, vol. 6, pp. 547–560, 1992.
, How Many Correlation Lengths For Multifractals, Physica A, vol. 183, pp. 398–410, 1992.
, How Many Correlation Lengths For Multifractals, Physica A, vol. 183, pp. 398–410, 1992.
, Noise and Crossover Exponent In Conductor-insulator Mixtures and Superconductor-conductor Mixtures, Physical Review B, vol. 45, pp. 755–767, 1992. (682.75 KB)
, Noise and Crossover Exponent In Conductor-insulator Mixtures and Superconductor-conductor Mixtures, Physical Review B, vol. 45, pp. 755–767, 1992. (682.75 KB)
, One-dimensional vibrations and disorder - the Zr1-xhfxs3 solid-solution, Physical Review B, vol. 46, pp. 5183–5193, 1992. (502.99 KB)
1993
, Flux-quantization In Rings For Hubbard (attractive and Repulsive) and T-j-like Hamiltonians - Comment, Physical Review B, vol. 47, pp. 15316–15318, 1993. (121.44 KB)
, Magnetic Neutron-scattering From 2-dimensional Lattice Electrons - the Case of La2-xsrxcuo4, Physical Review B, vol. 47, pp. 15217–15241, 1993. (1.13 MB)
, Neutron-scattering Measurements As A Test of Theories of High-temperature Superconductivity, Physical Review B, vol. 47, pp. 589–592, 1993. (193.34 KB)
, Scaling Behavior of Multifractal-moment Distributions Near Criticality, Journal De Physique I, vol. 3, pp. 323–330, 1993. (444.64 KB)
, Scaling Behavior of Multifractal-moment Distributions Near Criticality, Journal De Physique I, vol. 3, pp. 323–330, 1993. (444.64 KB)
, Symmetry and Nodes of the Superconducting Gap, Journal of Physics and Chemistry of Solids, vol. 54, pp. 1381–1384, 1993.
1995
, Destruction of Fermi liquid by spin fluctuations in two dimensions, Journal of Physics and Chemistry of Solids, vol. 56. pp. 1769-1771, 1995. (595.99 KB) | {"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": 0.9346160888671875, "perplexity": 4501.975213175348}, "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/1642320304859.70/warc/CC-MAIN-20220125160159-20220125190159-00639.warc.gz"} |
https://simple.m.wikipedia.org/wiki/Prime_counting_function | # Prime counting function
Function representing the number of primes less than or equal to a given number
In mathematics, the prime counting function is the function counting the number of prime numbers less than or equal to some real number x. It is written as ${\displaystyle \pi (x)}$,[1] but it is not related to the number π. Some key values of the function include ${\displaystyle \pi (1)=0}$, ${\displaystyle \pi (2)=1}$ and ${\displaystyle \pi (3)=2}$.
The 60 first values of π(n)
In general, if ${\displaystyle p_{n}}$stands for the n-th prime number, then ${\displaystyle \pi (p_{n})=n}$.[2]
## References
1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-07.
2. Weisstein, Eric W. "Prime Counting Function". mathworld.wolfram.com. Retrieved 2020-10-07. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.9836748242378235, "perplexity": 772.761272777463}, "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/1659882573630.12/warc/CC-MAIN-20220819070211-20220819100211-00180.warc.gz"} |
https://kb.osu.edu/dspace/handle/1811/13418 | # AB INITIO CALCULATIONS ON THE PHOTOLYSIS OF CHLORINE NITRATE ($CIONO_{2}$)
Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/13418
Files Size Format View
1996-FD-10.jpg 96.81Kb JPEG image
Title: AB INITIO CALCULATIONS ON THE PHOTOLYSIS OF CHLORINE NITRATE ($CIONO_{2}$) Creators: Chiu, Lue-Yung Chow; Lin, M. H.; Lai, S. T. Issue Date: 1996 Publisher: Ohio State University Abstract: Chlorine Nitratge ($CIONO_{2}$), which serves as a temporary reservoir in the $stratosphere^{1}$ for both $NO_{x}$ (i.e. $NO$ and $NO_{2}$) and $ClO_{x}$ (i.e. $Cl$ and $ClO$) species, may have three different UV photolysis pathways: (1) $CIONO_{2} \rightarrow Cl + NO_{3}, (2) ClONO_{2} \rightarrow ClO + NO_{2}$ and (3) $CIONO_{2} \rightarrow O + CIONO$. We have analyzed the first path way. Geometries and energies of the initial state $CIONO_{2} (I^{1}A)$, the final states $NO_{3} (^{2}A) + Cl (^{2}P)$ and the transition state $Cl\cdots ONO_{2}$ have been optimized by using $HF/6-31G^{*}$ basis set of the GAMESS program. The results obtained for the intial state agrees with that of Grana et $al.^{2}$ The energy of the transition state is 0.1058 h above the initial state and 0.0807 h above the final state. Description: a) Now at Department of Chemistry, Xiamen university, Xiamen 361005, China. 1. R. Zander, C. P. Rinsland, C. B, Farmer, L.R.Brown, and R. H. Norton, Geophys. Res. Leut. 13, 757 (1986). 2. A. M. Grana, T. J. Lee, and M. Head-Gordon, J. Phys. Chem. 99, 3493 (1995). Author Institution: Department of Chemistry, Howard University; Department of Chemistry, The Catholic University of America; Vitreous State Laboratory, The Catholic University of America URI: http://hdl.handle.net/1811/13418 Other Identifiers: 1996-FD-10 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8399166464805603, "perplexity": 3688.3551832197163}, "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/1438042988511.77/warc/CC-MAIN-20150728002308-00036-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/explanation-of-the-breaking-up-of-a-derivative.351509/ | # Explanation of the breaking up of a derivative
1. Nov 3, 2009
### jamesd2008
If you have dv/dt you say to yourself its the derivative of v with respect to t. But in an example of deriving the first kinematic equation for constant acceleration you go from a=dv/dt, to dv=a.dt and then you integrate this equation to give you the velocity. i.e v=u+1/2a(tsquared), using initial conditions. In this sense what is the dv part and what is the dt part? I hope you understand as finding it hard to put into words. Is dv and dt both derivatives?
Thanks
James
2. Nov 3, 2009
### mathman
dv/dt is the derivative. dv and dt and called differentials, and have meaning only in terms of carrying out integration.
3. Nov 3, 2009
### jamesd2008
Hi thanks for the reply could explain further, for example,
see attached word doc sorry can't type in here with the symbols so put it in a word document.
James
File size:
26 KB
Views:
50
4. Nov 3, 2009
### slider142
I do not understand where you believe the last equation you have in your document should come from.
5. Nov 3, 2009
### jamesd2008
I just mean that when you integrate the differential dt you get t so should the last equation not include a extra t term from the differential dt. As on the other side of the equation dx becomes x. Thanks for taking the time to look at it james
6. Nov 3, 2009
### slider142
At the current level, treating the derivative as if it were a fraction is simply an aid to manipulation and is not rigorously defined by what you have learned so far. In fact, it is simply an application of the fundamental theorem of calculus. That is, given f = dv/dt, the fundamental theorem tells us that $$\int f = v(t) + C$$ where C is an arbitrary constant of integration. As you can see, "multiplying both sides by dt" is unnecessary.
The proper presentation of dv and dt as separate entities called differentials will be covered in a course on differential equations, or differential geometry.
7. Nov 3, 2009
### jamesd2008
Thanks slide again for taking the time to look at this. Will let it go and just except thats the way it is. James
8. Nov 3, 2009
### slider142
dx is not the entity that becomes x. In the integral
$$\int dx$$
you are integrating the function between the integral sign and the dx symbol. This is the identity function 1, whose primitive with respect to x is x ( + C).
Similarly, the primitive of at with respect to t is at2/2. The dt at your level is only telling you which variable you are integrating over and is not an active participant in the integral.
9. Nov 3, 2009
### jamesd2008
ok so if there is no function to be integrated with respect to x then the identity function of 1 is used.
You have been a great help
Thanks
James
10. Nov 4, 2009
### HallsofIvy
Be careful with your terminology here. The identity function is f(x)= x. You mean the constant function, f(x)= 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": 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": 0.9688173532485962, "perplexity": 708.4953348412798}, "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-2018-13/segments/1521257647322.47/warc/CC-MAIN-20180320072255-20180320092255-00165.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/219042-splitting-field-x-3-5-a.html | # Thread: Splitting field for X^3-5
1. ## Splitting field for X^3-5
Hi,
Let $f=X^3 -5$, the $\Sigma_f = \mathbb{Q}[\alpha,\omega]$ with $\alpha$ real cube root of 5 and $\omega$ primitive root of 1.
$[\mathbb{Q}(\alpha):\mathbb{Q}]=3$ because $X^3-5$ is irreducible.
But what is $[\mathbb{Q}(\alpha,\omega):\mathbb{Q}(\alpha)]$?
$[\mathbb{Q}(\alpha,\omega):\mathbb{Q}(\alpha)] >1$ because $\omega \in \mathbb{C} \setminus \mathbb{R}$ and yet $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{R}$. Also the degree is $\leq 3$ because $X^3 -1$ has $\omega$ as a root. So the possibilities are 2 or 3.
If it's 2. Then because the min poly has to be monic and $\omega^2 + \omega = \sqrt{2}$
I think that we have to express $\sqrt{2}$ as a linear combination (with rational coefficients) of $\alpha$ and $\alpha^2$.
I can't see how do to this?
Thanks for any help!
2. ## Re: Splitting field for X^3-5
The roots of $X^3-1$ are $X_1=1, X_2=e^{\frac{2 \pi i}{3}}, X_3 = e^{\frac{4 \pi i}{3}}$ whereby $X_2$ and $X_3$ have minimal polynomial $T^2+T+1$
3. ## Re: Splitting field for X^3-5
Of course, I made a stupid mistake:
$\omega^2 + \omega = -1$!
thanks | {"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": 24, "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": 0.839455246925354, "perplexity": 287.20639539626615}, "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/1490218189198.71/warc/CC-MAIN-20170322212949-00244-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://programmer.ink/think/621a47705d985.html | # [machine learning] basic learning notes of DS 4: neural network and back propagation algorithm
Posted by garethdown on Sat, 26 Feb 2022 16:30:27 +0100
# Neural network and back propagation algorithm
## 4.1 cost function
in the previous section, we learned the basic structure of neural network and forward propagation algorithm. In the supporting operation, the weights of neural network have been given, but how can we train the appropriate weights ourselves? Therefore, we need to understand the cost function and back propagation algorithm.
we need to know some labeling methods. Suppose the neural network training samples have m m m, each containing a set of inputs x x x and a set of output signals y y y， L L L represents the number of neural network layers, S I S_I SI ＾ represents the number of neurons in each layer, S L S_L SL ﹤ indicates the number of processing units in the last layer.
the classification of neural networks can be divided into two categories: Class II Classification and multi class classification. Class II Classification: S L = 2 , y = 0 o r 1 S_L=2,y=0 or 1 SL=2,y=0or1； Multi category classification: S L = k , y i = 1 S_L=k,y_i=1 SL = k,yi = 1 indicates the score to the second i i Class i. ( k > 2 k>2 k>2)
we review that the cost function in logistic regression is
J ( θ ) = − 1 m [ ∑ i = 1 n y ( i ) l o g h θ ( x ( i ) ) + ( 1 − y ( i ) ) l o g ( 1 − h θ ( x ( i ) ) ) ] + λ 2 m ∑ j = 1 n θ j 2 J(\theta)=-\frac{1}{m}[\sum_{i=1}^ny^{(i)}logh_\theta(x^{(i)})+(1-y^{(i)})log(1-h_\theta(x^{(i)}))]+\frac{\lambda}{2m}\sum_{j=1}^n\theta_j^2 J(θ)=−m1[i=1∑ny(i)loghθ(x(i))+(1−y(i))log(1−hθ(x(i)))]+2mλj=1∑nθj2
but because there are multiple outputs in the neural network, the cost function is more complex
h Θ ( x ) ∈ R K ( h Θ ( x ) ) i = i t h o u t p u t J ( Θ ) = − 1 m [ ∑ i = 1 m ∑ k = 1 K y k ( i ) l o g ( h Θ ( x ( i ) ) ) k + ( 1 − y k ( i ) ) l o g ( 1 − ( h Θ ( x ( i ) ) ) k ) ] + λ 2 m ∑ l = 1 L − 1 ∑ i = 1 s l ∑ j = 1 s l + 1 ( Θ j i ( l ) ) 2 h_\Theta(x)\in \mathbb{R}^K\qquad (h_\Theta(x))_i=i^{th} output\\ J(\Theta)=-\frac{1}{m}[\sum_{i=1}^m\sum_{k=1}^Ky_k^{(i)}log(h_\Theta(x^{(i)}))_k+(1-y_k^{(i)})log(1-(h_\Theta(x^{(i)}))_k)]+\frac{\lambda}{2m}\sum_{l=1}^{L-1}\sum_{i=1}^{s_l}\sum_{j=1}^{s_{l+1}}(\Theta_{ji}^{(l)})^2 hΘ(x)∈RK(hΘ(x))i=ithoutputJ(Θ)=−m1[i=1∑mk=1∑Kyk(i)log(hΘ(x(i)))k+(1−yk(i))log(1−(hΘ(x(i)))k)]+2mλl=1∑L−1i=1∑slj=1∑sl+1(Θji(l))2
although the cost function is complex, the idea behind it is the same: we hope to know how far the prediction result of the algorithm is from the real situation by observing the cost function. Unlike logistic regression, neural networks have K K For K predictions, we select the one with the highest possibility through circulation and compare it with y y Compare the actual value in y. Similarly, we do not regularize the bias term, so i i i start with 1.
## 4.2 back propagation algorithm
### 4.2.1 algorithm overview
in the forward propagation algorithm, we calculate layer by layer from the input layer h Θ ( x ) h_\Theta(x) h Θ (x). Now let's calculate the partial derivative of the cost function ∂ ∂ Θ i j ( l ) J ( Θ ) \frac{\partial}{\partial \Theta_{ij}^{(l)}}J(\Theta) ∂ Θ ij(l)∂J( Θ)， We need to use the back-propagation algorithm, that is, first calculate the error of the last layer, and then calculate the error of each layer in reverse layer by layer until the penultimate layer. In my personal understanding, it is essentially the natural result of the chain derivation rule.
the example in the course is that there is only one training sample in the training set ( x ( 1 ) , y ( 1 ) ) (x^{(1)},y^{(1)}) (x(1),y(1)), the neural network structure is shown in the figure:
revisit the forward propagation algorithm:
we calculate the error from the last layer, which refers to the prediction of the activation unit( a k ( 4 ) a_k^{(4)} ak(4)) and actual value( y k y_k Error between yk)
( k = 1 : k k=1:k k=1:k), we use δ \delta δ Represents the error, which we can understand temporarily here δ ( l ) = ∂ ∂ z ( l ) J ( Θ ) \delta^{(l)}=\frac{\partial}{\partial z^{(l)}}J(\Theta) δ (l)=∂z(l)∂J( Θ)， be δ ( 4 ) = a ( 4 ) − y \delta^{(4)}=a^{(4)}-y δ (4)=a(4) − y, so we use this error value to calculate the error of the previous layer: δ ( 3 ) = ∂ ∂ z ( 3 ) J ( Θ ) = ∂ ∂ z ( 4 ) J ( Θ ) ⋅ ∂ z ( 4 ) ∂ a ( 3 ) ⋅ ∂ a ( 3 ) ∂ z ( 3 ) = ( Θ ( 3 ) ) T δ ( 4 ) . ∗ g ′ ( z ( 3 ) ) \delta^{(3)}=\frac{\partial}{\partial z^{(3)}}J(\Theta)=\frac{\partial}{\partial z^{(4)}}J(\Theta)\cdot\frac{\partial z^{(4)}}{\partial a^{(3)}}\cdot\frac{\partial a^{(3)}}{\partial z^{(3)}}=(\Theta^{(3)})^T\delta^{(4)}.*g'(z^{(3)}) δ (3)=∂z(3)∂J( Θ)= ∂z(4)∂J( Θ) ⋅∂a(3)∂z(4)⋅∂z(3)∂a(3)=( Θ (3))T δ (4). * g '(z(3)), where . ∗ .* . * is point multiplication, corresponding element multiplication. Here is the chain rule in the derivation of composite function. Among them, we should also understand a property of sigmoid function: g ′ ( z ) = g ( z ) ∗ ( 1 − g ( z ) ) g'(z)=g(z)*(1-g(z)) g′(z)=g(z)∗(1−g(z)).
we can also get the error of the second layer:
δ ( 2 ) = ∂ ∂ z ( 2 ) J ( Θ ) = ∂ ∂ z ( 3 ) J ( Θ ) ⋅ ∂ z ( 3 ) ∂ a ( 2 ) ⋅ ∂ a ( 2 ) ∂ z ( 2 ) = ( Θ ( 2 ) ) T δ ( 3 ) . ∗ g ′ ( z ( 2 ) ) \delta^{(2)}=\frac{\partial}{\partial z^{(2)}}J(\Theta)=\frac{\partial}{\partial z^{(3)}}J(\Theta)\cdot\frac{\partial z^{(3)}}{\partial a^{(2)}}\cdot\frac{\partial a^{(2)}}{\partial z^{(2)}}=(\Theta^{(2)})^T\delta^{(3)}.*g'(z^{(2)}) δ(2)=∂z(2)∂J(Θ)=∂z(3)∂J(Θ)⋅∂a(2)∂z(3)⋅∂z(2)∂a(2)=(Θ(2))Tδ(3).∗g′(z(2))
with the expression of all errors, we can calculate the partial derivative of the cost function, assuming λ = 0 \lambda=0 λ= 0, that is, when we do not do regularization: ∂ ∂ Θ i j ( l ) J ( Θ ) = ∂ ∂ z i ( l + 1 ) J ( Θ ) ⋅ ∂ z i ( l + 1 ) ∂ Θ i j ( l ) = a j ( l ) δ ( l + 1 ) \frac{\partial}{\partial \Theta_{ij}^{(l)}}J(\Theta)=\frac{\partial}{\partial z_i^{(l+1)}}J(\Theta)\cdot\frac{\partial z_i^{(l+1)}}{\partial \Theta_{ij}^{(l)}}=a_j^{(l)}\delta^{(l+1)} ∂Θij(l)∂J(Θ)=∂zi(l+1)∂J(Θ)⋅∂Θij(l)∂zi(l+1)=aj(l)δ(l+1)
In the above formula, the superscript and subscript mean: l l l represents the layer currently calculated; j j j represents the subscript of the active unit in the current computing layer; i i i represents the subscript of the error unit in the next layer.
when there are many training samples in the training set and regularization processing is required, we use Δ i j ( l ) \Delta_{ij}^{(l)} Δ ij(l) is the total error, which means the sum of the partial derivatives of each sample without regularization. We express the algorithm as:
we cycle each sample, use the forward propagation algorithm to calculate the activation unit of each layer, and then use the back propagation method to calculate the error sum of multiple training samples. We also need to consider regularization, and finally obtain the partial derivative of the cost function:
D i j ( l ) = 1 m ( Δ i j ( l ) + λ Θ i j ( l ) ) i f j ≠ 0 D i j ( l ) = 1 m Δ i j ( l ) i f j = 0 D_{ij}^{(l)}=\frac{1}{m}(\Delta_{ij}^{(l)}+\lambda\Theta_{ij}^{(l)})\qquad if\quad j\neq0\\ D_{ij}^{(l)}=\frac{1}{m}\Delta_{ij}^{(l)}\qquad if\quad j=0 Dij(l)=m1(Δij(l)+λΘij(l))ifj=0Dij(l)=m1Δij(l)ifj=0
## 4.3 summary
### 4.3.1 random initialization
any optimization algorithm requires some initial parameters. We initialized the parameters to 0 earlier. Such an initial method is not feasible for neural networks. If we set all initial parameters to 0, it will mean that all active units in our second layer will have the same value. Similarly, if all our initial parameters are the same non-zero number, the result is the same.
we usually have an initial parameter of ± ε ±\varepsilon ± ε Random value of.
def random_init(size):
return np.random.uniform(-0.12, 0.12, size)
compared with the first two learning algorithms, neural network will be more complex, so we may make some imperceptible errors in code implementation, which means that although the cost is decreasing, the final result may not be the optimal solution.
in order to avoid such problems, we adopt a numerical test method called gradient. Its essence is difference approximate differential. Select two very close points along the tangent direction in the cost function, and then calculate the average of the two points to estimate the gradient. For a particular θ \theta θ， We calculated that θ − ε \theta-\varepsilon θ − ε Chuhe θ + ε \theta+\varepsilon θ+ε Generation value of( ε \varepsilon ε Is a very small value, usually 0.001), and then the average of the two costs is used to estimate θ \theta θ The value of the agency.
f i ( θ ) ≈ J ( θ ( i + ) ) + J ( θ ( i − ) ) 2 ε f_i(\theta)≈\frac{J(\theta^{(i+)})+J(\theta^{(i-)})}{2\varepsilon} fi(θ)≈2εJ(θ(i+))+J(θ(i−))
if the difference between the two is within a reasonable range, then the implementation of our algorithm is correct.
### 4.3.3 summary
if the number of hidden layers is greater than 1, ensure that the number of units in each hidden layer is the same. Generally, the more units in the hidden layer, the better. What we really need to decide is the number of hidden layers and the number of units in each middle layer.
• Random initialization of parameters
• Using forward propagation algorithm to calculate all h θ ( x ) h_\theta(x) hθ(x)
• Write calculation cost function J J J's code
• All partial derivatives are calculated by back propagation algorithm
• An optimization algorithm is used to minimize the cost function
## 4.4 Python implementation of supporting operations
### 4.4.1 review of the previous section
in this section, we also use handwritten data sets to implement BPNN
previously, we imported the library, visualized and read, and the weight has been given to propagate forward. The code is as follows, and the specific steps are shown in the previous section.
import matplotlib.pyplot as plt
import numpy as np
import scipy.io as sio
import matplotlib
import scipy.optimize as opt
from sklearn.metrics import classification_report#This package is the evaluation report
y = data.get('y') # (5000,1)
y = y.reshape(y.shape[0]) # make it back to column vector
X = data.get('X') # (5000,400)
if transpose:
# for this dataset, you need a transpose to get the orientation right
X = np.array([im.reshape((20, 20)).T for im in X])
# and I flat the image again to preserve the vector presentation
X = np.array([im.reshape(400) for im in X])
return X, y
def expand_y(y):
res = []
for i in y:
y_array = np.zeros(10)
y_array[i-1] = 1
res.append(y_array)
return np.array(res)
def plot_100_image(X):
""" sample 100 image and show them
assume the image is square
X : (5000, 400)
"""
size = int(np.sqrt(X.shape[1]))
# sample 100 image, reshape, reorg it
sample_idx = np.random.choice(np.arange(X.shape[0]), 100) # 100*400
sample_images = X[sample_idx, :]
fig, ax_array = plt.subplots(nrows=10, ncols=10, sharey=True, sharex=True, figsize=(8, 8))
for r in range(10):
for c in range(10):
ax_array[r, c].matshow(sample_images[10 * r + c].reshape((size, size)),
cmap=matplotlib.cm.binary)
plt.xticks(np.array([]))
plt.yticks(np.array([]))
plot_100_image(X)
plt.show()
X = np.insert(X_raw, 0, np.ones)
y = expand_y(y_raw)
def deserialize(seq): # Disassembly parameters
return seq[:25*401].reshape(25, 401), seq[25*401: ].reshape(10, 26)
def sigmoid(z):
return 1 / (1 + np.exp(z))
def feed_forward(theta, X): # Forward propagation
t1, t2 = deserialize(theta)
m = X.shape[0]
a1 = X
z2 = a1 @ t1.T
a2 = np.insert(sigmoid(z2), 0, np.ones(m), axis=1)
z3 = a2 @ t2.T
h = sigmoid(z3)
return a1, z2, a2, z3, h
### 4.4.2 cost function
J ( Θ ) = − 1 m [ ∑ i = 1 m ∑ k = 1 K y k ( i ) l o g ( h Θ ( x ( i ) ) ) k + ( 1 − y k ( i ) ) l o g ( 1 − ( h Θ ( x ( i ) ) ) k ) ] + λ 2 m ∑ l = 1 L − 1 ∑ i = 1 s l ∑ j = 1 s l + 1 ( Θ j i ( l ) ) 2 J(\Theta)=-\frac{1}{m}[\sum_{i=1}^m\sum_{k=1}^Ky_k^{(i)}log(h_\Theta(x^{(i)}))_k+(1-y_k^{(i)})log(1-(h_\Theta(x^{(i)}))_k)]+\frac{\lambda}{2m}\sum_{l=1}^{L-1}\sum_{i=1}^{s_l}\sum_{j=1}^{s_{l+1}}(\Theta_{ji}^{(l)})^2 J(Θ)=−m1[i=1∑mk=1∑Kyk(i)log(hΘ(x(i)))k+(1−yk(i))log(1−(hΘ(x(i)))k)]+2mλl=1∑L−1i=1∑slj=1∑sl+1(Θji(l))2
def regularized_cost(theta, X, y): # Regularization cost function
m = X.shape[0]
t1, t2 = deserialize(theta)
_, _, _, _, h = feed_forward(theta, X)
pair_computation = -np.multiply(y, np.log(h)) - np.multiply((1-y), np.log(1-h))
cost = pair_computation.sum() / m
reg_t1 = (1 / (2*m)) * np.power(t1[:, 1:], 2).sum()
reg_t2 = (1 / (2*m)) * np.power(t2[:, 1:], 2).sum()
return cost + reg_t1 + reg_t2
### 4.4.3 back propagation algorithm
def serialize(a, b): # Flattening parameters
return np.concatenate((np.ravel(a), np.ravel(b)))
return np.multiply(sigmoid(z), 1 - sigmoid(z))
t1, t2 = deserialize(theta)
m = X.shape[0]
delta1 = np.zeros(t1.shape)
delta2 = np.zeros(t2.shape)
a1, z2, a2, z3, h = feed_forward(theta, X)
for i in range(m):
a1i = a1[i, :]
z2i = z2[i, :]
z2i = np.insert(z2i, 0, np.ones(1), axis=0)
a2i = a2[i, :]
hi = h[i, :]
yi = y[i, :]
d3i = hi - yi
d2i = np.multiply(t2.T @ d3i, sigmoid_gradient(z2i))
delta2 += (1/m) * np.matrix(d3i).T @ np.matrix(a2i) # Convert to matrix, (1,10) T@(1,26)->(10,26)
delta1 += (1/m) * np.matrix(d2i[1:]).T @ np.matrix(a1i) # (1,25).T@(1,401)->(25,401)
t1[:, 0] = 0
t2[:, 0] = 0
reg_term1 = (1 / m) * t1
reg_term2 = (1 / m) * t2
delta2 += reg_term2
delta1 += reg_term1
return serialize(delta1, delta2)
we use two norms to measure numerical and analytical gradients.
∥ n u m e r i c a l _ g r a d − a n a l y t i c _ g r a d ∥ 2 ∥ n u m e r i c a l _ g r a d ∥ 2 + ∥ a n a l y t i c _ g r a d ∥ 2 \frac{\Vert numerical\_grad - analytic\_grad\Vert_2}{\Vert numerical\_grad\Vert_2+\Vert analytic\_grad\Vert_2} ∥numerical_grad∥2+∥analytic_grad∥2∥numerical_grad−analytic_grad∥2
def expand_array(arr):
"""
input [1,2,3]
out
[[1,2,3],
[1,2,3],
[1,2,3]]
"""
return np.array(np.matrix(np.ones(arr.shape[0])).T @ np.matrix(arr))
return (regularized_cost(plus, X, y)-regularized_cost(minus, X, y)) / (2 * epsilon)
theta_matrix = expand_array(theta)
epsilon_matrix = np.identity(len(theta)) * epsilon
plus_matrix = theta_matrix + epsilon_matrix
minus_matrix = theta_matrix - epsilon_matrix
print("If your backpropagation implementation is correct,\n"
"the relative difference will be smaller than 10e-7(assume epsilon=10e-7).\n"
"Relative Difference:{}\n".format(diff))
def random_init(size):
return np.random.uniform(-0.12, 0.12, size)
init_theta = random_init(10285)
If your backpropagation implementation is correct,
the relative difference will be smaller than 10e-7(assume epsilon=10e-7).
Relative Difference:1.0262913293779404e-08
it should be noted that when the algorithm passes the gradient test, it must turn off the gradient test step before training the network, otherwise the training time is very long.
### 4.4.5 model training
def nn_training(X, y):
init_theta = random_init(10285)
res = opt.minimize(fun=regularized_cost,
x0=init_theta,
args=(X, y, 1),
method='TNC',
options={'maxiter': 400})
return res
res = nn_training(X, y)
final_theta = res.x
def show_accuracy(theta, X, y):
_, _, _, _, h = feed_forward(theta, X)
y_pred = np.argmax(h, axis=1) + 1
print(classification_report(y, y_pred))
### 4.4.6 show hidden layers
def plot_hidden_layer(theta):
final_theta1, _ = deserialize(theta)
hidden_layer = final_theta1[:, 1:]
fig, ax_array = plt.subplots(nrows=5, ncols=5, sharex=True, sharey=True, figsize=(5, 5))
for r in range(5):
for c in range(5):
ax_array[r, c].matshow(hidden_layer[5*r + c].reshape(20, 20),
cmap=matplotlib.cm.binary)
plt.xticks([])
plt.yticks([])
plot_hidden_layer(final_theta)
plt.show() | {"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": 0.9560785293579102, "perplexity": 3027.1696096619285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103877410.46/warc/CC-MAIN-20220630183616-20220630213616-00232.warc.gz"} |
https://windowsontheory.org/2012/06/12/balls-and-bins-on-graphs/ | “Balls and Bins?”, you ask, “Is there anything left to prove there?” Surprisingly, there are really natural questions that are open. Today I want to talk about one such question.
First a quick primer. Balls and Bins processes model randomized allocations processes, used in hashing or more general load balancing schemes. Suppose that I have ${m}$ balls (think items) to be thrown into ${n}$ bins (think hash buckets). I want a simple process that will keep the loads balanced, while allowing quick decentralized lookup.
The simplest randomized process one can think of is what’s called the one choice process: for each ball, we independently and uniformly-at-random pick one of the ${n}$ bins and place it there.
We measure the balance in terms of the additive gap: the difference between the maximum load and the average load. Many of us studied the ${m=n}$ case in a randomized algorithms class, and we know that the gap is ${\Theta(\frac{\ln n}{\ln \ln n})}$ except with negligible probability (for the rest of the post, I will skip this “except with negligible probability” qualifier). What happens when ${m}$ is much larger than ${n}$? It can be shown that for large enough ${m}$, the gap is ${\Theta(\sqrt{\frac{m\log n}{n}})}$ (see this paper for more precise behaviour of the gap).
Can we do better? Azar, Broder, Karlin and Upfal analyzed the following two choice process: balls come sequentially, and each ball picks two bins independently and uniformly at random (say, with replacement), and goes into the less loaded of the two bins (with ties broken arbitrarily). They showed that the gap of the two choice process when ${m=n}$ is significantly better: only ${\Theta(\ln\ln n)}$. What about the case when ${m}$ is much larger than ${n}$? Berenbrink et al. showed that the gap stays at ${\Theta(\ln \ln n)}$, for arbitrary ${m}$.
While the decrease from ${\Theta(\frac{\ln n}{\ln \ln n})}$ to ${\Theta(\ln \ln n)}$ is surprising enough, I find the latter distinction much more striking. When ${m}$ is a large polynomial in ${n}$, the gap for one choice scheme keeps going up, while that in two choice case remains put. For some quick intuition on why this happens, pause for a minute to think about the case ${n=2}$ (A hint is in the comments). Then read on.
So now that we understand one and two choice, let’s move on (or rather, dig in between). Yuval Peres, Udi Wieder and I analyzed the ${(1+\beta)}$-choice process, where we place a ball in a uniformly random bin with probability ${(1-\beta)}$, and in the lesser loaded to two random bins with probability ${\beta}$. The ${n=2}$ intuition would suggest that this slight bias towards balance would keep the gap from growing with ${m}$, and this is indeed what we show: for ${\beta}$ bounded away from ${1}$, the gap is ${\Theta(\log n/\beta)}$.
Analyzing the ${(1+\beta)}$-choice process helps us understand another natural process. Given a regular graph ${G}$ on ${[n]}$, one can define the balls and bins process on ${G}$ as follows: balls come sequentially, and each ball picks an edge of ${G}$ at random, and goes to the (bin corrseponding to the) lesser loaded endpoint. Thus when ${G}$ is made up of ${n}$ self loops, we get the one choice process. When ${G}$ is the complete graph (with self loops), we get the two choice process. Using the ${(1+\beta)}$-choice analysis, it can be shown that whenever ${G}$ is connected, the gap is independent of ${m}$. And when ${G}$ is an expander, the gap is ${\Theta(\log n)}$.
And that brings us to a simple open question: the case of ${G}$ being a cycle. To restate the question: ${n}$ bins on a cycle. Repeatedly pick two adjacent bins, and put a ball in the lesser loaded of the two. How large does the gap get?
We can show the gap is ${O(n\log n)}$ and ${\Omega(\log n)}$. Empirically neither of these bounds seems remotely tight. Can you improve them?
June 12, 2012 6:54 am
First, an apology for skipping a lot of other relevant literature.
The promised hint for the 2 bin case: Let X_1 and X_2 denote the loads of the two bins. Notice that the gap is |X_1-X_2|/2. How does the random variable (X_1-X_2) evolve in the one choice case? How about the two choice case?
June 12, 2012 3:36 pm
Perhaps you could emphasize that by “gap”, you mean it in the standard English sense of “difference”, as opposed to ratio (which is what usually “integrality gap” means, say).
June 12, 2012 3:42 pm
Thanks Arvind for the remark. i have added the qualificiation “additive” to clarify that.
Is the $\Omega(\log n)$ lower bound with respect to $m=n$? If so, I’m a bit surprised. If one always picks the bin with even index (according to some fixed enumeration of the bins along the cycle) instead of picking the lesser loaded one, one arrives at the single-choice model with $n$ balls and $n/2$ bins, which should upper-bound the two-choice cycle and whose gap I thought is also $O(\ln n / \ln\ln n)$ …
You are right. For $m=n$, pretty much any process would give you a maximum load of $O(log n/log log n$. The lower bound is for larger $m$. More precisely, if $m=0.25 n log n$, the average is $0.25 log n$. For one choice, there will then be a bin with load more than $0.75 log n$. In the cycle case, you get an edge which is picked $0.75 log n$ times, so that at least one of its endpoints has load at least $0.35 log n$, which is $\Omega(log n)$ more than the average. | {"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": 45, "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": 0.878722608089447, "perplexity": 489.265556102069}, "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-2019-09/segments/1550247508363.74/warc/CC-MAIN-20190221193026-20190221215026-00411.warc.gz"} |
http://www.biomedsearch.com/nih/Model-Migratory-B-Cell-Oscillations/23296998.html | Document Detail
A Model for Migratory B Cell Oscillations from Receptor Down-Regulation Induced by External Chemokine Fields. Jump to Full Text MedLine Citation: PMID: 23296998 Owner: NLM Status: Publisher Abstract/OtherAbstract: A long-standing paradigm in B cell immunology is that effective somatic hypermutation and affinity maturation require cycling between the dark zone and light zone of the germinal center. The cyclic re-entry hypothesis was first proposed based on considerations of the efficiency of affinity maturation using an ordinary differential equations model for B cell population dynamics. More recently, two-photon microscopy studies of B cell motility within lymph nodes in situ have revealed the complex migration patterns of B lymphocytes both in the preactivation follicle and post-activation germinal center. There is strong evidence that chemokines secreted by stromal cells and the regulation of cognate G-protein coupled receptors by these chemokines are necessary for the observed spatial cell distributions. For example, the distribution of B cells within the light and dark zones of the germinal center appears to be determined by the reciprocal interaction between the level of the CXCR4 and CXCR5 receptors and the spatial distribution of their respective chemokines CXCL12 and CXCL13. Computer simulations of individual-based models have been used to study the complex biophysical and mechanistic processes at the individual cell level, but such simulations can be challenging to parameterize and analyze. In contrast, ordinary differential equations are more tractable, but traditional compartment model formalizations ignore the spatial chemokine distribution that drives B cell redistribution. Motivated by the desire to understand the motility patterns observed in an individual-based simulation of B cell migration in the lymph node, we propose and analyze the dynamics of an ordinary differential equation model incorporating explicit chemokine spatial distributions. While there is experimental evidence that B cell migration patterns in the germinal center are driven by extrinsically regulated differentiation programs, the model shows, perhaps surprisingly, that feedback from receptor down-regulation induced by external chemokine fields can give rise to spontaneous interzonal and intrazonal oscillations in the absence of any extrinsic regulation. While the extent to which such simple feedback mechanisms contributes to B cell migration patterns in the germinal center is unknown, the model provides an alternative hypothesis for how complex B cell migration patterns might arise from very simple mechanisms. Authors: Cliburn Chan; Matthew Billard; Samuel A Ramirez; Harald Schmidl; Eric Monson; Thomas B Kepler Related Documents : 22951908 - Glutathione peroxidase 1 activity dictates the sensitivity of glioblastoma cells to oxi...22931348 - Effects of a cell-imprinted poly(dimethylsiloxane) surface on the cellular activities o...22992018 - Metabolism of inorganic arsenic in intestinal epithelial cell lines.23722198 - Induction of apoptosis by high-dose gold nanoparticles in nasopharyngeal carcinoma cells.17065808 - Activation of kinases upon volume changes: role in cellular homeostasis.3383578 - Quantitative effects of short- and long-term vasectomy on mouse spermatogenesis and spe... Publication Detail: Type: JOURNAL ARTICLE Date: 2013-1-8 Journal Detail: Title: Bulletin of mathematical biology Volume: - ISSN: 1522-9602 ISO Abbreviation: Bull. Math. Biol. Publication Date: 2013 Jan Date Detail: Created Date: 2013-1-8 Completed Date: - Revised Date: - Medline Journal Info: Nlm Unique ID: 0401404 Medline TA: Bull Math Biol Country: - Other Details: Languages: ENG Pagination: - Citation Subset: - Affiliation: Department of Biostatistics and Bioinformatics, Duke University Medical Center, Durham, NC, 27705, USA, [email protected]. Export Citation: APA/MLA Format Download EndNote Download BibTex MeSH Terms Descriptor/Qualifier:
From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine
Full Text Journal Information Journal ID (nlm-ta): Bull Math Biol Journal ID (iso-abbrev): Bull. Math. Biol ISSN: 0092-8240 ISSN: 1522-9602 Publisher: Springer-Verlag, New York Article Information Download PDF © The Author(s) 2012 Received Day: 14 Month: 5 Year: 2012 Accepted Day: 15 Month: 11 Year: 2012 Electronic publication date: Day: 8 Month: 1 Year: 2013 pmc-release publication date: Day: 8 Month: 1 Year: 2013 Print publication date: Month: 1 Year: 2013 Volume: 75 Issue: 1 First Page: 185 Last Page: 205 PubMed Id: 23296998 ID: 3547247 Publisher Id: 9799 DOI: 10.1007/s11538-012-9799-9
A Model for Migratory B Cell Oscillations from Receptor Down-Regulation Induced by External Chemokine Fields Cliburn ChanAff1 Address: [email protected] Matthew BillardAff2 Samuel A. RamirezAff3 Harald SchmidlAff1 Eric MonsonAff4 Thomas B. KeplerAff5 Department of Biostatistics and Bioinformatics, Duke University Medical Center, Durham, NC 27705 USA Thurston Arthritis Research Center, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 USA Program in Computational Biology and Bioinformatics, Duke University, Durham, NC 27710 USA Duke University Visualization Technology Group, Duke University, Durham, NC 27708 USA Department of Microbiology, Boston University School of Medicine, 72E Concord St, Boston, MA 02118 USA
Introduction
The evolution of high-affinity specific antibodies by long-lived B cells is driven by a process known as affinity maturation that occurs in the germinal center of lymph nodes. In this process, the germinal center (GC) is partitioned into a dark zone (DZ), consisting largely of rapidly dividing B cells known as centroblasts, and a light zone (LZ), consisting largely of B cells known as centrocytes interacting with follicular dendritic cells (FDC). It is believed that somatic hypermutation which introduces random changes in the antibody nucleotide sequence occurs within centroblasts in the DZ, while centrocytes in the LZ interact with and compete for immune complexes bound to FDC (Allen et al. 2007a; Shlomchik and Weisel 2012). A long-held hypothesis of cyclic re-entry is that the periodic migration of B cells from the DZ to the LZ and vice versa is critical for the efficiency of affinity maturation (Kepler and Perelson 1993; Kepler et al. 1993; Meyer-Hermann et al. 2001). FDCs present antigen bound on Fc receptor-captured antibodies on their cell surface, and centrocytes compete for binding to these antigens. Centrocytes with high affinity B cell receptors are more likely to successfully bind antigen and receive survival signals, while centrocytes with low affinity receptors fail to bind and undergo apoptosis. Successful centrocytes may then reenter the DZ for proliferation and another iteration of selection, or exit the germinal center as memory B cells or long-lived plasma cells.
How B cells migrate in the lymph node is hence critical for understanding the generation of high affinity long-lived memory and plasma cells that are the basis of humoral immunity. Naive B cells are believed to be attracted to the preactivation follicle primarily by the chemokine CXCL13, although lipid ligands that bind to the EBI2 receptor and CCR7:CCL19/CCL21 receptor-ligand interactions also modulate the B cell spatial distribution in the follicle (Gatto et al. 2011). In the post-activation germinal center, the migration of B cells between the DZ and LZ is driven by the chemokines CXCL12 found mainly in the LZ and CXCL13 in the DZ. These chemokines are recognized by the G-protein coupled receptors (GPCR) CXCR4 and CXCR5, with CXCR4 binding to CXCL12 and CXCR5 binding to CXCL13 (Allen et al. 2004, 2007a).
Pioneering work by Sally Zigmond has described receptor internalization (and resulting loss of sensitivity to a chemokine gradient) as an important aspect of GPCR-mediated chemotaxis (Zigmond 1981; Zigmond et al. 1982). Estimated receptor levels are in the 104 range (10,000–50,000) of receptors per cell (Zigmond 1981). Upon ligand binding, GPCRs signal to G-protein, become phosphorylated by GPCR kinase (effectively desensitizing the receptor by dissociating G-protein subunits), and internalize via one of two major pathways. One internalization pathway is fast and involves clathrin-mediated endocytosis. The other is slower and uses a lipid-raft/caveolae pathway. The clathrin pathway involves the recruitment of arrestin to the receptor, which can act as a scaffold for further signaling events. Receptors internalized in either pathway can potentially be recycled, or degraded. How chemokine receptors respond to the local chemokine field over time is hence likely to be a major regulatory mechanism for the migration behavior of B cells. Indeed, the literature describes alterations in chemokine receptor expression balance as the fundamental basis for directional migration within the lymph node and germinal center (Allen et al. 2004; Hardtke et al. 2005; Reif et al. 2002).
With the advent of two-photon microscopy, we can now observe individual B cell dynamics in situ within a developing germinal center (Allen et al. 2007b; Schwickert et al. 2007; Hauser et al. 2007b; Victora et al. 2010). However, two-photon microscopy is restricted to the visualization of relatively small regions and short time-spans. Computational modeling is therefore a valuable adjunct for inference beyond these short time and space scales, providing mechanistic insight into long range/long duration phenomena such as the relationship between B cell migration patterns and the efficacy of somatic hypermutation (Kleinstein 2002; Meyer-Hermann et al. 2009; Figge and Meyer-Hermann 2011). As traditional ordinary differential equation (ODE) models ignore the spatial inhomogeneity of the chemokine fields, computational simulations of individual-based models (IBM) may be more appropriate vehicles for understanding how emergent behavior arises from the interactions of single B cells with their environment and other cells (Figge 2005; Bogle and Dunbar 2009; Germain et al. 2011; Beltman et al. 2011).
The detailed biology of chemotaxis is complex, and existing models of chemotaxis are in general either mechanistic or phenomenological (Palsson and Othmer 2000; Hauser et al. 2007a; Figge et al. 2008). We use phenomenological models in this paper as our interest is in the feedback between receptors and an external chemokine field, and not so much in the detailed mechanism of chemotaxis. Phenomenological models are typically based on some variation of a persistent random walk biased in the direction of the chemokine gradient (Weiner 2002). To bridge between deterministic and stochastic motility models in continuous time, we use the classical Langevin process stochastic differential equation formalism for persistent random walks to model chemotaxis and reduce to a deterministic version by removing the Wiener noise component where appropriate. We also explicitly incorporate GPCR desensitization by an external chemokine field in the Langevin process model.
Chemokine receptors are regulated on multiple levels, and receptor dynamics can be complex (Lauffenburger and Linderman 1996). As an illustrative example (Beyer and Meyer-Hermann 2008) present a detailed formalism (comprising 6 differential equations with 13 free parameters) to model the dynamics of a single receptor type interacting with its chemokine. Implementing a model with that degree of complexity would focus attention on detail and detract from our intention to show that very simple mechanisms suffice to induce complex migratory behavior. We therefore chose to derive a new phenomenological model for the receptor incorporating just ligand binding, constitutive and binding-induced down-regulation, and de novo synthesis. The use of singular perturbation analysis leads to the formulation of a single equation to model the dynamics of each chemokine receptor.
Based on the considerations above, we propose a simple ODE model of individual B cells coupled to static chemokine fields. We used the model to investigate the range of dynamical behaviors exhibited in the presence of static chemokine field distributions representing the DZ and LZ of the germinal center. Our hypothesis was that study of a phenomenological model integrating spatial chemokine distributions, receptor regulation, and chemotaxis could provide a template for understanding the broad-stroke dynamics of B cells in the germinal center. This would complement the use of IBM simulations to fill in the fine details and reveal unexpected emergent behavior resulting from individual cell interactions. For our model, we chose to include just three components—a spatial distribution of chemokines in one dimension, a model for the regulation of chemokine receptors, and a chemotactic model for cell locomotion.
This manuscript describes the application of the spatially-driven ODE model to explore the migratory response of B cells to chemokines in the germinal center. We show that chemokine-induced receptor down-regulation and receptor-mediated chemotaxis in the presence of a simple fixed spatial distribution of the relevant chemokines is sufficient to induce complex migratory patterns, including intrazonal and interzonal oscillations.
Model Definition and Analysis
Static Chemokine Fields
The chemokine-driven ODE model is a deterministic nonlinear dynamical system in one spatial dimension, in which chemotaxis of a single cell is modulated by the levels of two chemokine receptors that are reciprocally regulated by the static 1D spatial distribution of their cognate chemokines.
Spatial Distribution of Chemokines
Chemokine fields are set up by the expression of chemokines by stromal cells in the germinal center with dynamics determined by diffusion, absorption and degradation, but in the steady state over short periods of time, we make the assumption that the chemokine field is static. We further simplify by assuming that each chemokine field has a Gaussian distribution, and only consider the dynamics along the axis that runs through both the follicle or germinal center centroids. Chemokines are modeled as functions of the cell displacement x—even though the field is static, cells with different displacements respond to the local chemokine field. This representation of chemokine fields as a function of cell displacement is flexible—it is possible to set up arbitrarily complex chemokine fields in this system if necessary to model in vivo measurements, for example, by using mixtures of Gaussians to represent multimodal fields.
Germinal Center Model
The chemokine concentration is given as a function of the cell displacement x. For the examples in the paper, we use Gaussian distributions f1 and f2 to represent the CXCL12 and CXCL13 chemokine distributions respectively, i.e.,
[Formula ID: Equ1]
[Formula ID: Equ2]
where ci and wi determine the height and width of the distribution, and k=(k1+k2)/2 is the half-distance between the dark and light zones in μm. The chemokine concentrations and gradients for CXCL12 and CXCL13 are shown in the first two panels in the top row of Fig. 1.
Toy Model for Receptor Regulation and Chemotaxis
We begin with a toy model for receptor regulation and chemotaxis to illustrate the basic requirements for chemokine-driven oscillations. We assume that the chemokine receptors are synthesized at a rate π and degraded at a rate δ. To couple the receptor dynamics to the chemokine field, we assume that receptors are also down-regulated at a rate proportional to the product of the receptor and the local cognate chemokine concentration. In other words, the chemokine drives the down-regulation of its receptor.
[Formula ID: Equ3]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dr_{i=1,2}}{dt} = \pi_{i} - r_{i} f_{i}(x) - \delta_{i} r_{i}$$\end{document}]
For chemotaxis, we assume that the cell velocity depends on the product of the receptor and the local cognate chemokine gradient, with a drag coefficient γ.
[Formula ID: Equ4]
[Formula ID: Equ5]
Analysis of Toy Model
In this section, we first analyze the toy model components individually to gain insight into the origin of specific dynamical behaviors, then integrate the components and explore the resulting system dynamics.
Regulation of Receptor Density
From Eq. (3), it follows that the steady state receptor density rSS at any given position x is given by
[Formula ID: Equ6]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_i^{\text{SS}} = \frac{\pi_i}{\delta_i+f_i(x)}$$\end{document}]
In the rightmost upper panel of Fig. 1, we plot the steady state solution for the receptor density at a particular position. It is clear that the effect of binding-induced down-regulation is to decrease the receptor density the greatest where the chemokine concentration is highest. Where the level of cognate chemokine is low, synthesis of new receptor outpaces down-regulation, and the saturating density of receptor is achieved.
Stability of Cell Velocity
Next, we examine the dynamics of the velocity v with respect to relative changes in receptor concentration using a reduced undamped model (γ=0)
[Formula ID: Equ7]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dv}{dt} = s f_1'(x) + f_2'(x)$$\end{document}]
where we rescale so that s=r1/r2. We also assume that the chemokine fields f1(x) and f2(x) are standard normal distributions with centers set 1.5 units from the origin.
In the lower panels of Fig. 1, we plot the rate of change of the velocity as the position of a B cell is varied. There are three different sets of steady state solutions for v possible. When s, the ratio of the densities of the two chemokine receptors, is small (so that r2 dominates), there is a single steady state at the mean of f2(x). As s increases, a new steady state is created at the mean of f1(x) by a saddle-node bifurcation, and the system is bistable. As s continues to rise, the steady state at f2(x) vanishes in a reverse saddle node bifurcation, and the system becomes monostable again. This implies that under these conditions, the DZ, LZ or both can be equilibria for a B cell, depending only on the ratio of CXCR4 and CXCR5 expressed.
Coupling Receptor and Velocity Dynamics Results in Spontaneous Oscillations
Referring to the bottom panels of Fig. 1, we see how oscillations can arise from coupling of the receptor and velocity dynamics in the presence of opposing chemokine fields. Suppose a cell starts with a low density of CXCR4 and high CXCR5 at the CXCL12 peak. Under appropriate conditions, the only stable equilibrium is at the CXCL13 peak and the cell moves to the right (bottom left). When it reaches the CXCL13 peak, the chemokine drives the down-regulation of the CXCR5 receptor and CXCR4 is up-regulated. Now we are in the situation illustrated by the bottom right panel; the stable equilibrium at the CXCL13 peak vanishes, and the cell is forced to return to the CXCL12 peak, setting up a system where oscillations result.
Bifurcation Analysis
To better understand the conditions where the system exhibits oscillatory behavior, we can systematically study the dynamics under changes of parameters using software for continuation of equilibria (Dhooge et al. 2003; Clewley et al. 2007). Continuation software “follows” the equilibrium solution as some parameter is changed, and also checks for the occurrence of specific bifurcations at each parameter value. This allows us to identify parameter regions where interesting or desired system behavior is found, and provides insight into the parameter values where there is a qualitative change of behavior. Figure 2 shows the bifurcation plots from different parameter regions for the toy model.
Biologically Motivated Phenomenological Model for Receptor Regulation and Chemotaxis
The toy model described above shows that a combination of receptor adaptation (modeled as down-regulation) and receptor-mediated chemotaxis can give rise to autonomous oscillations in the presence of a suitable static chemokine field. By design, the model abstracts away all other biological considerations. In this section, we describe simple biologically-motivated models that accommodate standard mass-action kinetics for receptor dynamics and chemotaxis with saturable chemokine receptor signals, and show that these models preserve similar autonomous oscillations.
Model for GPCR Regulation
As discussed in a recent review by Bennett et al. (2011), the regulation of chemokine receptor levels is highly complex with multiple different processes that can affect GPCR levels and activity. However, the mechanism of migration is thought to be independent of transcription and regulated primarily by receptor trafficking dynamics in response to agonist binding (Schaeuble et al. 2012). Agonist-dependent desensitization in response to agonist binding results in GPCR endocytosis and degradation as discussed in the Introduction. Some of the internalized receptor may be recycled rather than degraded, and the path taken depends on both cell type and the duration of ligand engagement. While probably too slow a process to directly influence lymphocyte migration, new receptor synthesis is also essential for long-term maintenance of surface chemokine receptor levels. These processes of new receptor synthesis and agonist-induced internalization, recycling and degradation that together determine the dynamics of chemokine receptor expression are illustrated in Fig. 3. The mass-action kinetics corresponding to Fig. 3 are given by
[Formula ID: Equ8]
[Formula ID: Equ9]
[Formula ID: Equ10]
10
where the first-order degradation term τ for U is necessary to ensure that the unbound receptor remains finite in the absence of ligand. To simplify the model, we neglect the contribution of receptor recycling on the available intracellular pool. That is, we assume that πμB, and hence that I is constant. With these assumptions, we can derive the following model for the dynamics of the CXCR4 and CXCR5 receptors in the presence of their cognate chemokine ligands (full derivation given in Appendix A)
[Formula ID: Equ11]
11 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{dr_{i=1,2}}{dt} = \pi_{i} - \frac{r_{i} \tau_{i} \kappa_{i} f_{i}(x)}{1 + \kappa_{i} f_{i}(x)} - \delta_{i} r_{i}$$\end{document}]
where κ is a rescaled equilibrium association constant for GPCR:lignad interactions, and δ+τ is the removal rate for bound GPCR that incorporates the first-order degradation of unbound receptor U.
Chemotactic Model for Cell Locomotion
The model for B cell chemotaxis assumes that the cell velocity is governed by a chemotactic process biased by saturable chemokine receptor signals generated by receptor ligand interactions that depend on both ligand concentration and density. The velocity has a drag coefficient γ, and a tuning factor for the degree of responsiveness to the underlying chemokine field given by χ. The model equations (derived in Appendix B) are
[Formula ID: Equ12]
12
[Formula ID: Equ13]
13
We have set the effective equilibrium association constant ϵ in the chemotactic model to be distinct from the value κ in the receptor regulation model to allow for differential coupling of bound receptor to signal transduction pathways involved in the two processes.
Bifurcation Analysis of Phenomenological Model
The full model is reproduced below for convenience.
[Formula ID: Equ14]
14
[Formula ID: Equ15]
15
[Formula ID: Equ16]
16
[Formula ID: Equ17]
17
As with the toy model, the bifurcation analysis suggests that spontaneous oscillations only occur for a rather restricted set of parameter combinations. For example, the leftmost panel of Fig. 4 shows that spontaneous oscillations only occur when the distance separating the simulated DZ and LZ fall within a narrow range.
Diversity of Dynamical Behaviors in 1D
A surprisingly rich variety of periodic behavior can be found in the 1D ODE model system. The range of behaviors include direct passage to a steady state equilibrium, damped oscillations to steady state and a variety of oscillatory behaviors with periods ranging from minutes to many hours. These oscillations may be from DZ to LZ, within a single zone or have components of both small intrazonal and large inter-zonal circulations. Oscillations can be highly asymmetrical, with a disproportionate amount of time spent in a single zone. Oscillations may even be apparently chaotic. Representative examples of the numerical simulations of displacement over time illustrating the diversity of oscillations are shown in Fig. 5.
Individual-Based Model Simulations of Receptor Dynamics and Chemotaxis
Finally, we implemented the phenomenological model in a 3D IBM simulation of immune cells (Kepler and Chan 2007; Mitha et al. 2008), extending the chemotactic model to incorporate stochastic deviations. We show that very similar dynamical behavior is observed in the 3D simulation as in the simpler ODE models.
Stochastic Model for Chemotaxis
For the IBM, we rewrite the phenomenological model for chemotaxis as a stochastic differential equation in Ito form, giving rise to the following Langevin process equation
[Formula ID: Equ18]
18
[Formula ID: Equ19]
19
where dW is the differential Wiener process.
There are three main differences between the phenomenological model and IBM simulation—the IBM is in 3D while the phenomenological model is 1D; cells in the IBM have a stochastic chemotactic motility model rather than a deterministic one (i.e., σ≠0); and there are extrinsic forces in the IBM when cells collide with each other or environmental boundaries. While we have closed form solutions for the spatial distribution of chemokines in the two models shown here, the simulation system uses a spatially discretized numerical approximation in order to generalize to arbitrary (and potentially evolving) chemokine distributions. Numerically, the differences between the IBM simulation and 1D phenomenological models are the use of a three-dimensional grid to store chemokine concentrations and gradients (5 μm per side voxels with trilinear interpolation between voxel centroids) as compared with values given by the closed forms f1 and f2 in the ODE model. In addition, cells in the IBM can have more complex behaviors such as division, death, and activation and the possibility of collision-induced forces when multiple cells are simulated. In the IBM simulation, cells are also constrained to be within a specified volume.
Dynamical Behavior in Individual-Based Simulation
Figure 6 shows three snapshots of the 3D IBM simulation. Oscillatory behavior is preserved in the presence of external forces and stochasticity, although of course, the periodicity is no longer synchronized between cells.
In the absence of stochasticity (Fig. 7 middle panel), the 3D simulation model behavior corresponds very closely to that of the 1D phenomenological model (Fig. 7 top panel). With stochasticity (Fig. 7 bottom panel), the 3D simulation behavior begins to diverge. However, the dynamical analysis of the minimal model remains informative for the 3D simulation behavior.
Discussion
We have described a simple mathematical model of chemotaxis-driven B cell migration in the germinal center. The model incorporates a static chemokine field, chemokine-induced receptor modulation, and chemotaxis driven by the interaction of the chemokine receptor with the local chemokine concentration and gradient. The model is specified using coupled first-order differential equations, lending itself to detailed analysis using techniques from nonlinear dynamics. Using this basic setup, we investigated the dynamics of B cell migration under a simple chemokine field comprising of two Gaussian distributions representing CXCL12 in the light zone and CXCL13 in the dark zone of the germinal center.
In this simple germinal center model, we show that spontaneous oscillations between the light and dark zone can arise, and the periodicity can be tuned so that the residence times in the dark and light zones is consistent with experimental observations. An interesting prediction of the model is that for a fixed width of the chemokine fields, oscillations only occur for a narrow range of separations between the dark and light zone. When the light and dark zones are too close or far apart, no oscillations are observed. Oscillations can also be elicited in an alternative model where one receptor is fixed, and only one receptor is regulated by the chemokine field (supplementary Fig. A.1), but then the allowed range of separations is even narrower. This suggests that reciprocal regulation of both CXCR4 and CXCR5 receptors gives more robust oscillatory behavior than regulation of a single receptor.
While the simple mechanism of chemokine-driven receptor down-regulation is sufficient for inducing autonomous oscillations of some complexity, the extent to which such a mechanism contributes to B cell cycling in the germinal center is unknown. In fact, there is substantial evidence that B cell cycling in the germinal center is largely driven by extrinsic influences (e.g., B cell:FDC or B cell:T follicular helper cell interactions) that trigger differentiation programs regulating the expression of chemokine receptors. However, our model shows that surprisingly complex migratory patterns can emerge from very simple mechanisms, a recurring theme in the study of nonlinear dynamical systems. We believe that this provides a useful alternative perspective on the causal mechanisms of complex immune cell migration patterns, such as those observed in the germinal center.
This work was originally motivated by the desire to simplify IBM simulations of B cell behavior in order to gain insight into observed motility patterns and to facilitate parameter calibration. The 1D phenomenological model described in Eqs. (14)–(17) differ from the single cells in the 3D IBM simulations in the restriction to one dimension, the absence of a stochastic component, and the absence of collisions with other cells and the environment boundaries. However, we show that the phenomenological model effectively predicts the large-scale behavior of the IBM simulation when parameters are matched. Dynamical behaviors of interest can be rapidly identified in the phenomenological model configuration using bifurcation analysis and numerical simulations, and then studied in the more realistic 3D stochastic context with the IBM simulation using the same parameter values as the 1D phenomenological model. This is much more efficient than the brute-force search over parameter space otherwise necessary for IBM simulations, since such models are analytically intractable and highly demanding of computational resources. A caveat is that the extent to which such ODE-based model simplifications can replicate the dynamics of richer IBM that incorporate phenomena such as cell-cell interactions is not known. We conjecture that ODE models with mean-field approximations of cell-cell interactions will still be useful for providing insight into the parameters of these more challenging simulations and their calibration, and plan to investigate such approximations.
In conclusion, the ODE models for B cell motility described offer potential for a thorough analysis of the surprising complexity engendered by simple environment/cell interactions, and highlight the importance of considering the chemokine environment in understanding migration patterns of B cells. In addition, the ODE models provide flexibility to perform rapid prototyping of B cell migration dynamics, and may serve as a tractable bridge to more detailed IBM simulations.
Appendix A: Model for GPCR Regulation
Let r be the density of unbound GPCR, and r the density of GPCR bound by its ligand, whose concentration is l. The dynamical equations for this system are
[Formula ID: Equ20]
20
[Formula ID: Equ21]
21
where π is the rate of production of GPCR where, δ is the removal rate for GPCR in the unbound state, and δ+τ is the removal rate for the bound GPCR.
[Formula ID: Equa]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\kappa}{\varepsilon}$$\end{document}]
is the forward rate constant for ligand binding,
[Formula ID: Equb]
[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\varepsilon}$$\end{document}]
is the rate constant for dissociation. We use these expressions to facilitate taking the limit where the binding and dissociation reactions are much faster than the cellular processes. We reduce the complexity of the dynamical system by taking some of its subprocesses as occurring on a much faster time-scale than others. These fast subprocesses are treated as if in equilibrium with the more slowly varying components, thus eliminating dynamic degrees of freedom. The mathematics used is singular perturbation theory (Jones 1995). In the present case, the justification for making the approximation comes from the measured rates for the subprocesses. The binding and dissociation of CXCL12 and CXCR4 has been characterized using surface plasmon resonance, giving kon=4.20×105 M s1, koff=8.24×10−3 s1, and KD=3.47×10−8 M, showing that the reactions equilibrate with a characteristic time of about 44 seconds for CXCL12 concentration equal to the reactions’ Kd (35 nM) (Vega et al. 2011). In contrast, the characteristic time for CXCR4 internalization upon binding by CXCL12 was found to be between 450 and 600 seconds (estimated from Signoret et al. 1997, Fig. 8A).
Define the total GPCR density rTr+r , whose rate equation is
[Formula ID: Equ22]
22 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{d{r_T}}}{dt} = \pi - \delta{r_T} - \tau{r^*}$$\end{document}]
Equation (21) can now be written
[Formula ID: Equ23]
23 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{d{r^*}}}{dt} = - ( {\delta + \tau} ){r^*} + \frac {\kappa}{\varepsilon} \bigl( {{r_T} - {r^*}} \bigr)l - \frac {1}{\varepsilon}{r^*}$$\end{document}]
We now perform the singular perturbation analysis, taking the limit ε→0. We first construct the “outer solution” by expanding the state variables as
[Formula ID: Equ24]
24 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned}[c] {r_T} ( {t,\varepsilon} ) &\equiv{r_{T,0}} + \varepsilon {r_{T,1}} + O \bigl( {{\varepsilon^2}} \bigr) \hfill \\ {r^*} ( {t,\varepsilon} ) &\equiv r_0^* + \varepsilon r_1^* + O \bigl( {{\varepsilon^2}} \bigr) \end{aligned}\end{document}]
and then matching coefficients of ε in Eqs. (22) and (24). To lowest order, we have
[Formula ID: Equ25]
25 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned}[c] \frac{{d{r_{T,0}}}}{dt} &= \pi - \delta{r_{T,0}} - \tau r_0^* \\ 0 &= \kappa \bigl( {{r_{T,0}} - r_0^*} \bigr)l - r_0^* \end{aligned}\end{document}]
the second of Eqs. (25) has solution
[Formula ID: Equ26]
26 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0^* = \frac{{\kappa{r_{T,0}}l}}{{1 + \kappa l}}$$\end{document}]
so that the first of Eqs. (25) becomes
[Formula ID: Equ27]
27 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{d{r_{T,0}}}}{dt} = \pi - \biggl( {\delta + \frac{{\tau\kappa l}}{{1 + \kappa l}}} \biggr){r_{T,0}}$$\end{document}]
To impose initial conditions, we must compute the “inner solution” obtained by rescaling time as
[Formula ID: Equ28]
28 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T = \frac{t}{\varepsilon}$$\end{document}]
and letting
[Formula ID: Equ29]
29 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R_T} ( {T,\varepsilon} ) = {r_T} \bigl( {t(T, \varepsilon ),\varepsilon} \bigr)$$\end{document}]
etc.
Now, matching coefficients of ε in the resulting differential equations gives
[Formula ID: Equ30]
30 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned}[c] \frac{{d{R_{T,0}}}}{dt} &= 0 \\ \frac{{dR_0^*}}{dt} &= \kappa \bigl( {{R_{T,0}} - R_0^*} \bigr)l - R_0^* \end{aligned}\end{document}]
Now the initial value problem in the inner solution has solution
[Formula ID: Equ31]
31 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R_{T,0}} ( T ) = {r_T} ( 0 )$$\end{document}]
and
[Formula ID: Equ32]
32 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0^* ( T ) = \biggl( {{r^*} ( 0 ) - \frac{{\kappa l{r_T}(0)}}{{1 + \kappa l}}} \biggr){e^{ - ( {1 + \kappa l} )T}} + \frac{{\kappa l{r_T}(0)}}{{1 + \kappa l}}$$\end{document}]
Matching the inner and outer solutions requires setting
[Formula ID: Equ33]
33 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r_{T,0}} ( 0 ) = \lim _{T \to\infty} {R_{T,0}} ( T ) = {r_{T,0}} ( 0 )$$\end{document}]
and
[Formula ID: Equ34]
34 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0^* ( 0 ) = \lim_{T \to\infty} R_0^* ( T ) = \frac{{\kappa l{r_T}(0)}}{{1 + \kappa l}}$$\end{document}]
Equation (11) from the text is recovered by dropping subscripts and substituting the chemokine fields for the ligand concentration,
[Formula ID: Equ35]
35
Initial parameter values for the spatial measurements and receptor dynamics used for bifurcation analysis and numerical simulations were derived from the literature (Lin and Butcher 2008; Hauser et al. 2007a; Allen et al. 2004; Victora et al. 2010; Zigmond 1981; Hoffman et al. 1996; Ricart et al. 2011), or estimated when no experimental data was available.
Appendix B: Model for Cellular Locomotion
The model for cellular locomotion starts with a third-order Langevin process in Ito form:
[Formula ID: Equ36]
36
where X(t)∈ℝ3 is the cell’s position at time; V(t)∈ℝ3 and P(t)∈ℝ3 are the velocity and polarization, respectively. We use upper-case letters to remind us that these variables are stochastic. The effective drag coefficient is γ, and the polarization decorrelation rate is ζ. Φ is the external force exerted on the cell, and Γ is the signal due to an external orientation field. dW is a Wiener process generating fluctuations in the polarization, and σ controls the size of those fluctuations.
We proceed from here by providing a model for Γ in the case where the orientation field is due to an inhomogeneous chemokine distribution. We suppose that the binding of chemokine receptors on the cell’s surface generates a local signal, and the global orientation signal is the vector average of these signals over the cell’s surface.
[Formula ID: Equ37]
37 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma\propto\int_{S}{{d^{2}}y \,{r^{*}} ( y )\widehat {n} ( y )}$$\end{document}]
where, as in Appendix A, r is the density of bound receptor, now considered a function of position y on the cell surface S, [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widehat {n} ( y )$\end{document}] is the unit vector normal to the cell surface at surface point y. We use the same singular perturbation method to get
[Formula ID: Equ38]
38 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r^{*}} ( y )\propto\frac{f ( x+y )r}{1+\kappa f ( x+y )}=\frac{f ( x )r}{1+\kappa f ( x )}+r{y^{T}} \nabla\frac{f ( x )}{1+\kappa f ( x )}+O \bigl( {{\Vert y \Vert }^{2}} \bigr)$$\end{document}]
where x is the position of the center of the cell, κ is the equilibrium association constant, and f(x) is the concentration of chemokine at x. The expression to the right of the equals sign results from a Taylor expansion.
Substituting Eq. (38) into Eq. (37) gives
[Formula ID: Equ39]
39 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma=\chi\frac{r\nabla f ( x )}{{{ [ 1+\kappa f ( x ) ]}^{2}}}$$\end{document}]
where the constant χ is the chemotactic coefficient.
Finally, we let the coefficients ζ,χ, and σ become very large while their ratios remain constant. We further assume that there are no external forces. In this limit, we get the system of equations
[Formula ID: Equ40]
40 [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\begin{aligned}[c] X &=V\,dt \\ dV &=-\gamma V\,dt+\chi\frac{r\nabla f ( x )}{{{ [ 1+\kappa f ( x ) ]}^{2}}}\,dt+\sqrt{\gamma}\sigma \,dW \end{aligned}\end{document}]
where the parameters have been rescaled to give the form displayed.
Values for the motility parameters used in the simulation were calculated by fitting to data from 3D trajectory data of individual lymphocytes from 2-photon data (Kepler and Chan 2007).
Acknowledgements
This work was supported by NIH/NIAID research contract HHSN272201000053C (TB Kepler, PI).
Open Access | {"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": 0.8791230320930481, "perplexity": 4199.061779888511}, "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/1438042988962.66/warc/CC-MAIN-20150728002308-00046-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.esrf.eu/UsersAndScience/Publications/Highlights/2004/HRRS/HRRS10 | Characterising the effect of pressure on the propagation of elastic waves in condensed matter is singularly important in many fundamental and applied research fields. An important case is related to the elasticity of hcp iron, the main constituent of the Earth's core. While the elastic anisotropy of Earth's inner core is now well established, its origin is still poorly understood. Theoretical predictions of the elastic moduli of iron yield very different results that are in disagreement with the scarce experimental results. Possible causes are: (i) the difficulty to perform reliable calculations at high pressures and temperatures, and (ii) the impossibility to carry out experiments on single crystal hcp iron.
Here we present the experimental determination of the five independent elastic moduli of hcp cobalt under hydrostatic compression to 39 GPa. Experimental and theoretical evidences suggest that hcp cobalt is a suitable analogue for hcp iron, with the advantage of being available as a single crystal. Thus, the knowledge of the elasticity of cobalt can be utilised to address the elastic anisotropy of hcp iron.
The experiments were carried out on the Inelastic X-ray Scattering (IXS) beamline II (ID28) utilising a focussed beam and an overall energy resolution of 3 meV. High quality single crystals (45 to 85 µm diameter, 20 µm thickness) were loaded in diamond anvil cells (DAC), using helium as pressure transmitting medium. The sound velocity of five independent acoustic phonon branches was determined as a function of pressure, permitting the derivation of the five independent elastic moduli. The results obtained compare well with ambient pressure ultrasonic measurements and, despite some discrepancies, an overall agreement with ab initio calculations [2] can be observed.
Determination of the full elastic tensor and its pressure evolution allows the mapping of the sound velocities in all along arbitrary directions in the crystal . The variation of the longitudinal acoustic sound velocity, VL{}, where is the angle from the c-axis in the meridian (a-c) plane is, in the case of iron, related to the observed seismic-wave anisotropy in Earth's inner core. Figure 16 shows VL{}, derived from IXS mesurements on cobalt, calculated at 0 K for cobalt and for iron [2], and derived from radial X-ray diffraction (RXRD) measurements on iron [1]. The results are compared at the same compression ratio 0/ = 0.86 (P~36 GPa for Co and P~39 GPa for Fe). We note a substantial agreement between the IXS results and the calculations, even if in the calculations the magnitude of the anisotropy is underestimated. This comparison validates the theoretical results and suggests a sigmoidal shape for VL{} in hcp iron as well, as indeed the calculations show.
Fig. 16: VL{}, from c- to a-axis. For a clear comparison, the velocities are normalised to one for = 90°. Solid line: IXS measurements for hcp Co; dotted line: calculations for hcp Co [2]; dashed line: calculations for hcp Fe [2]; dash-dotted line: RXRD measurements for hcp Fe [1].
The above experiments on single-crystalline Co were complemented by IXS measurements on a textured polycrystalline Fe-sample at 22 and 112 GPa. The aggregate longitudinal acoustic (LA) phonon dispersions, determined at the two pressures for two different orientations of the diamond anvil cell (at 50 and 90 respect to the compression axis of the cell), are reported in Figure 17, together with the differences in the derived sound velocities. While at 22 GPa the two dispersions almost overlap, leading to velocities equal within the error bars, at 112 GPa the difference becomes significant. Considering the known texture, with the c-axis of the crystallites preferentially oriented along the main compression axis of the cell, we can conclude that at 112 GPa the sound propagates faster by 4 to 5% at 50 from the c-axis than at 90 (i.e. in the basal plane). The measured anisotropy on a textured polycrystalline sample at 112 GPa is of the same order of magnitude as the anisotropy of seismic waves in the Earth's inner core (3-4%).
Fig. 17: LA phonon dispersions of hcp Fe at room temperature and at pressures of 22 GPa (lower curve) and 112 GPa (upper curve) for the two orientations of the DAC; full (open) circles: sound propagation at 90° (50°) to the DAC loading axis. Where not visible the errors are within the Symbols. The inset reports the relative difference in the sound speeds for the two orientations at the two pressures investigated.
Our results support the hypothesis of a sigmoidal shape of the longitudinal acoustic sound velocity in hcp iron at high pressure and ambient temperature, with a significant higher speed along the c-axis. A moderate alignment of the c-axis of the iron crystallites along the Earth's rotation axis, in an otherwise randomly oriented medium, can alone qualitatively explain compressional travel time anomalies observed in the Earth's inner core.
References
[1] H.K. Mao et al., Nature 396, 741 (1998); correction, Nature 399, 280 (1999).
[2] G. Steinle-Neumann et al.,Phys. Rev. B 60, 791 (1999).
Principal Publications and Authors
D. Antonangeli (a), M. Krisch (a), G. Fiquet (b), D.L. Farber (c), C.M. Aracne (c), J. Badro (b,c), F. Occelli (c), H. Requardt (a), Phys. Rev. Lett., 93, 215505 (2004); D. Antonangeli (a), F. Occelli (a), H. Requardt (a), J. Badro (b), G. Fiquet (b), M. Krisch (a), Earth Planet. Sci. Lett., 225, 243 (2004).
(a) ESRF
(b) Laboratorie de Minéralogie-Cristallographie de Paris, Institut de Physique du Globe de Paris (France)
(c) Lawrence Livermore National Laboratory (USA) | {"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": 0.9253209233283997, "perplexity": 2423.4238729516333}, "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-2018-13/segments/1521257647758.97/warc/CC-MAIN-20180322014514-20180322034514-00318.warc.gz"} |
https://www.physicsforums.com/threads/right-inverse.282106/ | # Right Inverse?
1. Dec 30, 2008
### jgens
Could someone please explain what is implied if a function has a right inverse? Thanks.
2. Dec 30, 2008
### NoMoreExams
3. Dec 30, 2008
### Tac-Tics
Let f be a function.
If r is the right inverse of f, then for all x, f(r(x)) = x. That is, the composition of f and r, f * r, is the identity function.
If l is a left inverse of f, then for all x, l(f(x)) = x. Again, this means l * f is the identity function.
If a function g is both a left and a right inverse, it is called a full inverse (or just simple, THE inverse). The full inverse of of f is usually designated f-1.
Some examples:
The squaring function, f(x) = x^2, is not one-to-one, and so it has no full inverse. However, it does have a partial inverse (a left inverse) which is the square root function. We know this because sqrt(x^2) = x. We can show it is not a full inverse by demonstrating that for some x, (sqrt(x))^2 /= x, and we can let x be any negative number. (Note in the complex numbers, sqrt is in fact a full inverse).
4. Dec 30, 2008
### jgens
NoMoreExams: Thanks, I had not read that article. That clears a lot of things up.
Tac-Tics: Thanks for the example.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook | {"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": 0.9580061435699463, "perplexity": 876.3204256841166}, "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-2018-39/segments/1537267156724.6/warc/CC-MAIN-20180921013907-20180921034307-00101.warc.gz"} |
https://www.physicsforums.com/threads/difference-equation-my-answer-different-from-books.884178/ | # Difference equation my answer different from books
1. Sep 4, 2016
### fahraynk
1. The problem statement, all variables and given/known data
When drugs are used to treat a medical condition, doctors often recommend starting with a higher dose on the first day than on subsequent days. In this problem, we consider a simple model to understand why. Assume that the human body is a tank of blood and that drugs instantly dissolve in the blood when ingested. Further assume that the drug vanishes from the body at a rate that is proportional to drug concentration.
Let X[n] represent the amount of drug taken on day n, and let y[n] represent the total amount of drug in the blood on day n, just after the dos x[n] has dissolved int eh blood, so that :
y[n]=x[n]+$\alpha$[n-1].
Assume that no drug is in the blood before day 0, and that one unit of drug is taken each day, starting with day 0. Determine an expression for the amount of drug in the blood immediately after the dose on day n has dissolved.
2. Relevant equations
3. The attempt at a solution
I plug $y=P^n$ into the difference equation to get $y_h=K\alpha^n$ and I plug $y=C$ into the equation for the particular solution and get for an answer :
$y = K\alpha^n - \frac{x(n)}{\alpha-1}$
The book says this:
Solve by iteration :
$n : y[n]$
$0 : 1$
$1:1+\alpha$
$2:1+\alpha+\alpha^2$
$3:1+\alpha+\alpha^2+\alpha^3$
$...$ : $...$
$n : \frac{1-\alpha^{n+1}}{1-\alpha}$
Books answer is : $\frac{1-\alpha^{n+1}}{1-\alpha}$
I think the book is using a power series representation or something like that. Is my answer wrong for x(n)=1 ? If so what am I doing wrong ? I rather not use a power series if I can answer the difference equation using methods I already know like undetermined coefficients.
2. Sep 4, 2016
### Krylov
Do you mean: $y[n] = x[n] + \alpha y[n-1]$ where $\alpha \in (0,1)$ determines the rate at which the drug disappears from the blood?
It is calculating the first few terms of the solution, recognizing a geometric series and then using the standard formula for the partial sums of such a series.
3. Sep 4, 2016
### vela
Staff Emeritus
I'm not sure why you're writing the solution in terms of x(n) since you've assumed x(n) is a constant to find the particular solution. Just write the constant.
You didn't finish solving the problem. You must still determine the arbitrary constant K.
Have something to add?
Draft saved Draft deleted
Similar Discussions: Difference equation my answer different from books | {"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": 0.8463610410690308, "perplexity": 587.5009257911015}, "config": {"markdown_headings": true, "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-2018-05/segments/1516084888135.38/warc/CC-MAIN-20180119204427-20180119224427-00205.warc.gz"} |
https://chemistry.stackexchange.com/questions/138882/mechanism-for-the-addition-of-hydrogen-iodide-to-3-3-dimethylbut-1-yne/138890 | # Mechanism for the addition of hydrogen iodide to 3,3‐dimethylbut‐1‐yne
The following reaction mechanism was given as a solution to a solved problem in my textbook1 for the addition of hydrogen iodide to 3,3‐dimethylbut‐1‐yne:
It can be seen that that 2,2-diiodo-3,3-dimethylbutane (a geminal dihalide) is the product according to this mechanism. I arrived at a different product 2,3-diiodo-2,3-dimethylbutane (a vicinal dihalide) by using the following mechanism:
The difference is due to the methanide shift I did on the second step, converting 3,3-dimethylbut-1-en-2-ylium cation to 2,3‐dimethylbut‐3‐en‐2‐ylium cation. I consider this to be a reasonable carbocation rearrangement because after the methanide shift, the 2,3‐dimethylbut‐3‐en‐2‐ylium cation is tertiary as well as in conjugation with the double bond. Moreover, the initial carbocation has a positive charge on a more electronegative $$\mathrm{sp^2}$$ hybridized carbon atom. Due to this, I believe, the methanide shift produces a more stable carbocation. But, why did the author proceed without making this rearrangement?
When I discussed this in chat, there was a consensus that the product of this reaction must be a geminal dihalide as obtained by the author. The only way, I could think of, by which I can obtain the geminal dihalide even after the rearrangement discussed earlier is to do another rearrangement during the addition of second molar equivalent of hydrogen iodide as given below:
The problem with this mechanism is the carbon bearing the postitive charge after the methanide shift also has an iodine atom attached to it. Earlier, I learnt that chlorine atom is the only halogen for which the positive mesomeric effect is stronger than the negative inductive effect thereby stabilizing the positive charge. But here, due to the presence of iodine, I think the carbocation is not stable after rearrangement.
Even after neglecting this fact, there seems to be a major difference betweeen the author's mechanism and the modified mechanism, which I've emphasized using a carbon-12 labelled reactant as given below:
It can be seen that even though we obtain geminal dihalides through either of the mechanism, the products formed aren't exactly the same. One has the iodine attoms attached to the normal carbon whereas in the other they are attached to the carbon-12 atom.
In short, what happens when hydrogen iodide is added to 3,3‐dimethylbut‐1‐yne?
### Reference
1. Solomons, et al. Organic Chemistry for JEE (Main & Advanced). Edited by MS Chouhan, Third Edition; Wiley India Private Limited. ISBN 978-81-265-6065-3
There seems to be an issue with both mechanisms on the front that an SN1 reaction would not take place since the carbocation formed is a vinylic carbocation which is highly unstable.
The actual reaction follows a termolecular mechanism where the rate of reaction is given to be:
$$\text{Rate}=[\ce{HX}]^2[\text{alkyne}]$$
Now, according to Advanced Organic Chemistry by Francis A. Carey, the reaction mechanism would be as follows:$$^1$$
Step $$1$$:
A concerted termolecular reaction...
This involves an acid/base reaction, protonation of the alkyne developing positive charge on the more substituted carbon. The π electrons act pairs as a Lewis base.
The other part is attack of the nucleophilic bromide ion on the more electrophilic carbocation creates the alkenyl bromide.
Step $$2$$:
In the presence of excess reagent, a second protonation occurs to generate the more stable carbocation.$$^2$$
Step $$3$$:
Attack of the nucleophilic bromide ion on the electrophilic carbocation creates the geminal dibromide.
[$$1$$]: The reaction mechanism stated above uses $$\ce{HBr}$$ and not $$\ce{HI}$$ instead. However this reaction takes place for $$\ce{HX}$$.
[$$2$$]: The same reaction using $$\ce{HI}$$ would have a comparatively lower yield of the geminal product since geminal diiodides are unstable due to the steric hinderance posed by the large size of the iodine atoms. | {"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": 13, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8672093152999878, "perplexity": 2710.5709083815395}, "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-2022-05/segments/1642320300934.87/warc/CC-MAIN-20220118152809-20220118182809-00023.warc.gz"} |
https://math.stackexchange.com/questions/4250525/counting-in-base-13/4250531 | # Counting in base 13
So, I was counting in base $$8$$:
$$1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,21,22,23,24,25,26...$$
Then I tried counting in base $$13$$ and got confused:
$$1,2,3,4,5,6,7,8,9,10,11,12$$ (confused here, maybe:) $$1(01),1(02),1(03),....1(12),2(00).$$
Is there a nice or more standard way list the counting numbers in a base above $$10$$?
• It is perhaps more common to use letters, e.g., a=10, b=11, c=12. For instance the hexadecimal (16) base has 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f as “digits”. Sep 14 '21 at 21:26
• $10_{13}$ would be $13_{10}$. You need new symbols. You could write, e.g., $a_{13}, b_{13}, c_{13}$ for $10,11,12$ in base $10$.
– lulu
Sep 14 '21 at 21:26
• Base $16$ (used in computing contexts) takes "ABCDEF" as extra digits. Sep 14 '21 at 21:27
• Indeed, in the computer world, where base $16$ is common, we use $$A=9+1,B=9+2,C=9+3,\\D=9+4,E=9+5,F=9+5$$ But in theory, you could use any symbols, as long as everybody with whom you are conversing in a base agree. Sep 14 '21 at 21:42
• For new digits, either use new symbols or (for large number bases) use decimal numbers separated by a separator. For example, $n=12\cdot13^2+12\cdot13+12$ would be $ccc_{13}$ or $12:12:12_{13}$ (or $(12,12,12)_{13}$). Sep 14 '21 at 21:43
In order to work in base $$13$$, you need three extra symbols, which represent $$10$$, $$11$$, and $$12$$. Suppose that these symbols are $$A$$, $$B$$, and $$C$$ respectively. Then you have:$$1,2,3,4,5,6,7,8,9,A,B,C,10,11,12,13,14,15,16,17,18,19,1A,1B,1C,20,\ldots$$
The standard way (at least as it's used conventionally in base 16 in computing contexts) is to start using letters. So you get $$0,1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C,\\ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C,\\ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B, 2C, \\ \vdots$$ Some prefer to use lower case instead, and that's fine too. And of course you're free to invent your own symbols if you'd like, or steal symbols from somewhere other than the Latin alphabet. Just be sure to inform your readers which symbol means what.
The standard way to count this (which is used in hexadecimal: base-$$16$$) is to use the letters of the alphabet like so: $$1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E ...$$ So in base-$$13$$ it would be: \begin{align} &1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C \\ &10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C \\ &20... \end{align} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 19, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9224733710289001, "perplexity": 370.20196592282093}, "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/1642320303512.46/warc/CC-MAIN-20220121162107-20220121192107-00546.warc.gz"} |
http://tex.stackexchange.com/questions/4223/what-parsers-for-latex-mathematics-exist-outside-of-the-tex-engines/4231 | # What parsers for (La)TeX mathematics exist outside of the TeX engines?
Inspired by the author's motivation for asking Is there a BNF grammar of tex language.
Are there any well done libraries that can parse some subset of TeX mathematics independently of the TeX engine? Important points to consider in answers:
• How large of a subset of TeX mathematical notation is supported?
• Is the parser portable? Does it have any dependencies?
• Is the parser closely tied to a particular backend or could it easily be used to support multiple output formats. In other words, how easily could it be integrated into a new system that had to support support output to PDF, HTML, PNG, etc.
For example, I know of the following parsers but not much about their applicability outside the use cases for which they were designed (Matplotlib graphics and math rendering in web browsers):
-
I've been looking into this too, so I'll share some observations that fall rather short of a proper answer, which would really involve looking at a whole lot of source code and asking the right questions about it.
Parsers generating HTML+Math ML
1. Nick Drakos & Ross Moore's Latex2html converter, written in Perl, which I think was the first converter to map equations to Math ML. In 1998, Ross Moore outlined his goals for Latex2html, tied to the now defunct, closed-source WebEq mathematics rendering software, and Webtex, which was an alternative syntax for mathematics designed for use in web pages. From the WebEq documentation: WebTeX always translates unambiguously into MathML, while LaTeX does not.
2. itex2mml, in C by Paul Gartside & others, also based on Webtex, but with support for some Latex not supported in Webtex.
3. tex4ht, written in C by Eitan Gurari and other eminent figures. It avoids having to parse Latex source by running latex with modified macros that insert specials into the DVI output, and parses the DVI output instead.
4. John McFarlan's Pandoc, as mentioned by Aditya, written in Haskell. Note that Pandoc supports generation of HTML, both with and without Math ML.
5. MathJax allows generation of Math ML besides the usual boxes plus image fonts output. It has an impressive degree of support for Latex, including limited support for user macros.
Parsers generating XML
Jason Blevins has a list of tools that convert Latex documents to XML-based formats, and that handle equations reasonably. Romeo Anghelache's Hermes, which is part of a full Latex parser that generates XML with semantic markup, is worth singling out: like tex4ht, it works by running the Tex engine with macros to put specials in the DVI output, which it then parses; it supports a wider set of semantic markup.
Fragments of Latex or DVI
With the exception of the systems referencing Webtex, there doesn't seem to be much interest in clearly codifying subsets of Latex to be parsed, I guess because these are regarded as moving targets. Instead, lists of commands supported, like that I mentioned for Mathjax, seems to be the way things are done.
With DVI-based converters, the issue of parsing Latex goes away, replaced by the relatively trivial issue of parsing marked-up DVI and the trickier issue of identifying the semantically significant macros and constructing markup-issuing replacements that do not improperly interfere. I haven't looked at how this is done for equational layout. It would be a useful exercise to see how a converter from Tex formulae to those of It's worth noting that the representation of expressions is essentially a superset of that used by Heckmann & Wilhelm (1997) would work.
Syntax highlighting
A completely different kind of parsing is involved in syntax highlighting, where the idea is to help the author see the significance of the parts of the formulae. I don't know of any syntax highlighters that do an interesting job here: Auctex only raisers/lowers super&subscripts, but i haven't really looked.
Reference
Heckmann & Wilhelm, 1997, A Functional Description of TeX's Formula Layout.
-
I know a little about itex2MML. Firstly, it's list of supported commands is very well documented (via the link that you have). Secondly, it's not written in Ruby; it's a C library and the current maintainer (Jacques Distler) is most interested in the Ruby extensions. I've successfully compiled Perl, PHP, and Python extensions. Thirdly, its output is naturally MathML but it can be coaxed into producing SVG or PNG. – Loop Space Oct 19 '10 at 10:28
@Andrew – Ruby: oops, fixed; apart from Pandoc and tex4ht, I've not looked at all at the implementation of these, so what I say should be taken with a pinch of salt. About documentation: what I had written was obviously was unclear; what I meant was that Webtex was well-specified, and the two systems, itex2mml and latex2html, that were based on it, were clear as a result. I hope this is now clear in my answer. – Charles Stewart Oct 19 '10 at 11:54
Similar to MathJax there is the more recent KaTex
-
which is really really good. – percusse Jul 10 '15 at 17:06
@Jesse Why is this not an answer? I find it perfectly fine. – Gonzalo Medina Jul 10 '15 at 18:05
This seems to require node.js on the server? Does this mean one has to ask their ISP to install node.js software on the server for the pages to work? it was not clear. With mathjax, one can put the javascript code in own folder, or load it directly from the mathjax servers.... – Nasser Jul 10 '15 at 21:50
...but a major problem with all these tools (mathjax, KaTex) and other like them that work at the HTML level, is that one loses a major item in the process. Which is the PDF files. Using Latex->tex4ht->HTML instead , one still have the original Latex source and hence can make PDF files as well as HTML from the same source. – Nasser Jul 10 '15 at 21:52
@GonzaloMedina -- I was expecting more information be provided. Current expression serves as a comment to me due to quality and length concerns. – Jesse Jul 11 '15 at 1:39
Pandoc uses the Haskell Text.TeXMath.Parser library to parse inline and display math. This is not complete. It only parses most common inline math expressions and does not support amsmath display environments.
• I don't know if there is an official documentation of what subset is supported. The source code will give some idea about that.
• It is as portable as Haskell. So, it should work on most popular OS.
• Pandoc is specifically designed to support multiple output formats. IIRC, the output can be translated to MathML or to images using mimetex, gladtex, etc.
-
It's worth noting that the representation of expressions is essentially a superset of that used by Heckmann & Wilhelm, 1997, A Functional Description of TeX's Formula Layout. – Charles Stewart Oct 19 '10 at 8:05 | {"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": 0.8926419019699097, "perplexity": 2337.8825790603073}, "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/1464049276759.73/warc/CC-MAIN-20160524002116-00062-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://conceptmap.cfapps.io/wikipage?lang=en&name=Compact_Muon_Solenoid | # Compact Muon Solenoid
The Compact Muon Solenoid (CMS) experiment is one of two large general-purpose particle physics detectors built on the Large Hadron Collider (LHC) at CERN in Switzerland and France. The goal of CMS experiment is to investigate a wide range of physics, including the search for the Higgs boson, extra dimensions, and particles that could make up dark matter.
LHC experiments A Toroidal LHC Apparatus Compact Muon Solenoid LHC-beauty A Large Ion Collider Experiment Total Cross Section, Elastic Scattering and Diffraction Dissociation LHC-forward Monopole and Exotics Detector At the LHC ForwArd Search ExpeRiment Linear accelerators for protons (Linac 2) and Lead (Linac 3) Proton Synchrotron Booster Proton Synchrotron Super Proton Synchrotron
View of the CMS endcap through the barrel sections. The ladder to the lower right gives an impression of scale.
CMS is 21 metres long, 15 m in diameter, and weighs about 14,000 tonnes.[1] Over 4,000 people, representing 206 scientific institutes and 47 countries, form the CMS collaboration who built and now operate the detector.[2] It is located in a cavern at Cessy in France, just across the border from Geneva. In July 2012, along with ATLAS, CMS tentatively discovered the Higgs boson.[3][4][5] By March 2013 its existence was confirmed.[6]
## Background
Recent collider experiments such as the now-dismantled Large Electron-Positron Collider and the newly renovated Large Hadron Collider (LHC) at CERN, as well as the (as of October 2011) recently closed Tevatron at Fermilab have provided remarkable insights into, and precision tests of, the Standard Model of Particle Physics. A principal achievement of these experiments (specifically of the LHC) is the discovery of a particle consistent with the Standard Model Higgs boson, the particle resulting from the Higgs mechanism, which provides an explanation for the masses of elementary particles.[7]
However, there are still many questions that future collider experiments hope to answer. These include uncertainties in the mathematical behaviour of the Standard Model at high energies, tests of proposed theories of dark matter (including supersymmetry), and the reasons for the imbalance of matter and antimatter observed in the Universe.
## Physics goals
Panorama of CMS detector, 100m below the ground.
The main goals of the experiment are:
• to explore physics at the TeV scale
• to further study the properties of the Higgs boson, already discovered by CMS and ATLAS
• to look for evidence of physics beyond the standard model, such as supersymmetry, or extra dimensions
• to study aspects of heavy ion collisions.
The ATLAS experiment, at the other side of the LHC ring is designed with similar goals in mind, and the two experiments are designed to complement each other both to extend reach and to provide corroboration of findings. CMS and ATLAS uses different technical solutions and design of its detector magnet system to achieve the goals.
## Detector summary
CMS is designed as a general-purpose detector, capable of studying many aspects of proton collisions at 0.9–13 TeV, the center-of-mass energy of the LHC particle accelerator.
The CMS detector is built around a huge solenoid magnet. This takes the form of a cylindrical coil of superconducting cable that generates a magnetic field of 4 tesla, about 100 000 times that of the Earth. The magnetic field is confined by a steel 'yoke' that forms the bulk of the detector's weight of 12 500 t. An unusual feature of the CMS detector is that instead of being built in-situ underground, like the other giant detectors of the LHC experiments, it was constructed on the surface, before being lowered underground in 15 sections and reassembled.
It contains subsystems which are designed to measure the energy and momentum of photons, electrons, muons, and other products of the collisions. The innermost layer is a silicon-based tracker. Surrounding it is a scintillating crystal electromagnetic calorimeter, which is itself surrounded with a sampling calorimeter for hadrons. The tracker and the calorimetry are compact enough to fit inside the CMS Solenoid which generates a powerful magnetic field of 3.8 T. Outside the magnet are the large muon detectors, which are inside the return yoke of the magnet.
A cutaway diagram of the CMS detector
## CMS by layers
For full technical details about the CMS detector, please see the Technical Design Report.[8]
### The interaction point
This is the point in the centre of the detector at which proton-proton collisions occur between the two counter-rotating beams of the LHC. At each end of the detector magnets focus the beams into the interaction point. At collision each beam has a radius of 17 μm and the crossing angle between the beams is 285 μrad.
At full design luminosity each of the two LHC beams will contain 2,808 bunches of 1.15×1011 protons. The interval between crossings is 25 ns, although the number of collisions per second is only 31.6 million due to gaps in the beam as injector magnets are activated and deactivated.
At full luminosity each collision will produce an average of 20 proton-proton interactions. The collisions occur at a centre of mass energy of 8 TeV. But, it is worth noting that for studies of physics at the electroweak scale, the scattering events are initiated by a single quark or gluon from each proton, and so the actual energy involved in each collision will be lower as the total centre of mass energy is shared by these quarks and gluons (determined by the parton distribution functions).
The first test which ran in September 2008 was expected to operate at a lower collision energy of 10 TeV but this was prevented by the 19 September 2008 shutdown. When at this target level, the LHC will have a significantly reduced luminosity, due to both fewer proton bunches in each beam and fewer protons per bunch. The reduced bunch frequency does allow the crossing angle to be reduced to zero however, as bunches are far enough spaced to prevent secondary collisions in the experimental beampipe.
### Layer 1 – The tracker
Momentum of particles is crucial in helping us to build up a picture of events at the heart of the collision. One method to calculate the momentum of a particle is to track its path through a magnetic field; the more curved the path, the less momentum the particle had. The CMS tracker records the paths taken by charged particles by finding their positions at a number of key points.
The tracker can reconstruct the paths of high-energy muons, electrons and hadrons (particles made up of quarks) as well as see tracks coming from the decay of very short-lived particles such as beauty or “b quarks” that will be used to study the differences between matter and antimatter.
The tracker needs to record particle paths accurately yet be lightweight so as to disturb the particle as little as possible. It does this by taking position measurements so accurate that tracks can be reliably reconstructed using just a few measurement points. Each measurement is accurate to 10 µm, a fraction of the width of a human hair. It is also the inner most layer of the detector and so receives the highest volume of particles: the construction materials were therefore carefully chosen to resist radiation.[9]
The CMS tracker is made entirely of silicon: the pixels, at the very core of the detector and dealing with the highest intensity of particles, and the silicon microstrip detectors that surround it. As particles travel through the tracker the pixels and microstrips produce tiny electric signals that are amplified and detected. The tracker employs sensors covering an area the size of a tennis court, with 75 million separate electronic read-out channels: in the pixel detector there are some 6,000 connections per square centimetre.
The CMS silicon tracker consists of 14 layers in the central region and 15 layers in the endcaps. The innermost four layers (up to 16 cm radius) consist of 100 × 150 μm pixels, 124 million in total. The pixel detector was upgraded as a part of the CMS phase-1 upgrade in 2017, which added an additional layer to both the barrel and endcap, and shifted the innermost layer 1.5 cm closer to the beamline. [10]
The next four layers (up to 55 cm radius) consist of 10 cm × 180 μm silicon strips, followed by the remaining six layers of 25 cm × 180 μm strips, out to a radius of 1.1 m. There are 9.6 million strip channels in total.
During full luminosity collisions the occupancy of the pixel layers per event is expected to be 0.1%, and 1–2% in the strip layers. The expected HL-LHC upgrade will increase the number of interactions to the point where over-occupancy would significantly reduce trackfinding effectiveness. An upgrade is planned to increase the performance and the radiation tolerance of the tracker.
This part of the detector is the world's largest silicon detector. It has 205 m2 of silicon sensors (approximately the area of a tennis court) in 9.3 million microstrip sensors comprising 76 million channels.[11]
### Layer 2 – The Electromagnetic Calorimeter
The Electromagnetic Calorimeter (ECAL) is designed to measure with high accuracy the energies of electrons and photons.
The ECAL is constructed from crystals of lead tungstate, PbWO4. This is an extremely dense but optically clear material, ideal for stopping high energy particles. Lead tungstate crystal is made primarily of metal and is heavier than stainless steel, but with a touch of oxygen in this crystalline form it is highly transparent and scintillates when electrons and photons pass through it. This means it produces light in proportion to the particle's energy. These high-density crystals produce light in fast, short, well-defined photon bursts that allow for a precise, fast and fairly compact detector. It has a radiation length of χ0 = 0.89 cm, and has a rapid light yield, with 80% of light yield within one crossing time (25 ns). This is balanced however by a relatively low light yield of 30 photons per MeV of incident energy. The crystals used have a front size of 22 mm × 22 mm and a depth of 230 mm. They are set in a matrix of carbon fibre to keep them optically isolated, and backed by silicon avalanche photodiodes for readout.
The ECAL, made up of a barrel section and two "endcaps", forms a layer between the tracker and the HCAL. The cylindrical "barrel" consists of 61,200 crystals formed into 36 "supermodules", each weighing around three tonnes and containing 1,700 crystals. The flat ECAL endcaps seal off the barrel at either end and are made up of almost 15,000 further crystals.
For extra spatial precision, the ECAL also contains preshower detectors that sit in front of the endcaps. These allow CMS to distinguish between single high-energy photons (often signs of exciting physics) and the less interesting close pairs of low-energy photons.
At the endcaps the ECAL inner surface is covered by the preshower subdetector, consisting of two layers of lead interleaved with two layers of silicon strip detectors. Its purpose is to aid in pion-photon discrimination.
### Layer 3 – The Hadronic Calorimeter
The Hadron Calorimeter (HCAL) measures the energy of hadrons, particles made of quarks and gluons (for example protons, neutrons, pions and kaons). Additionally it provides indirect measurement of the presence of non-interacting, uncharged particles such as neutrinos.
The HCAL consists of layers of dense material (brass or steel) interleaved with tiles of plastic scintillators, read out via wavelength-shifting fibres by hybrid photodiodes. This combination was determined to allow the maximum amount of absorbing material inside of the magnet coil.
The high pseudorapidity region ${\displaystyle \scriptstyle (3.0\;<\;|\eta |\;<\;5.0)}$ is instrumented by the Hadronic Forward (HF) detector. Located 11 m either side of the interaction point, this uses a slightly different technology of steel absorbers and quartz fibres for readout, designed to allow better separation of particles in the congested forward region. The HF is also used to measure the relative online luminosity system in CMS.
About half of the brass used in the endcaps of the HCAL used to be Russian artillery shells.[12]
### Layer 4 – The magnet
The CMS magnet is the central device around which the experiment is built, with a 4 Tesla magnetic field that is 100,000 times stronger than the Earth's. CMS has a large solenoid magnet. This allows the charge/mass ratio of particles to be determined from the curved track that they follow in the magnetic field. It is 13 m long and 6 m in diameter, and its refrigerated superconducting niobium-titanium coils were originally intended to produce a 4 T magnetic field. The operating field was scaled down to 3.8 T instead of the full design strength in order to maximize longevity.[13]
The inductance of the magnet is 14 Η and the nominal current for 4 T is 19,500 A, giving a total stored energy of 2.66 GJ, equivalent to about half-a-tonne of TNT. There are dump circuits to safely dissipate this energy should the magnet quench. The circuit resistance (essentially just the cables from the power converter to the cryostat) has a value of 0.1 mΩ which leads to a circuit time constant of nearly 39 hours. This is the longest time constant of any circuit at CERN. The operating current for 3.8 T is 18,160 A, giving a stored energy of 2.3 GJ.
The job of the big magnet is to bend the paths of particles emerging from high-energy collisions in the LHC. The more momentum a particle has the less its path is curved by the magnetic field, so tracing its path gives a measure of momentum. CMS began with the aim of having the strongest magnet possible because a higher strength field bends paths more and, combined with high-precision position measurements in the tracker and muon detectors, this allows accurate measurement of the momentum of even high-energy particles.
The tracker and calorimeter detectors (ECAL and HCAL) fit snugly inside the magnet coil whilst the muon detectors are interleaved with a 12-sided iron structure that surrounds the magnet coils and contains and guides the field. Made up of three layers this “return yoke” reaches out 14 metres in diameter and also acts as a filter, allowing through only muons and weakly interacting particles such as neutrinos. The enormous magnet also provides most of the experiment's structural support, and must be very strong itself to withstand the forces of its own magnetic field.
### Layer 5 – The muon detectors and return yoke
As the name “Compact Muon Solenoid” suggests, detecting muons is one of CMS's most important tasks. Muons are charged particles that are just like electrons and positrons, but are 200 times more massive. We expect them to be produced in the decay of a number of potential new particles; for instance, one of the clearest "signatures" of the Higgs Boson is its decay into four muons.
Because muons can penetrate several metres of iron without interacting, unlike most particles they are not stopped by any of CMS's calorimeters. Therefore, chambers to detect muons are placed at the very edge of the experiment where they are the only particles likely to register a signal.
To identify muons and measure their momenta, CMS uses three types of detector: drift tubes (DT), cathode strip chambers (CSC) and resistive plate chambers (RPC). The DTs are used for precise trajectory measurements in the central barrel region, while the CSCs are used in the end caps. The RPCs provide a fast signal when a muon passes through the muon detector, and are installed in both the barrel and the end caps.
The drift tube (DT) system measures muon positions in the barrel part of the detector. Each 4-cm-wide tube contains a stretched wire within a gas volume. When a muon or any charged particle passes through the volume it knocks electrons off the atoms of the gas. These follow the electric field ending up at the positively charged wire. By registering where along the wire electrons hit (in the diagram, the wires are going into the page) as well as by calculating the muon's original distance away from the wire (shown here as horizontal distance and calculated by multiplying the speed of an electron in the tube by the time taken) DTs give two coordinates for the muon's position. Each DT chamber, on average 2 m x 2.5 m in size, consists of 12 aluminium layers, arranged in three groups of four, each with up to 60 tubes: the middle group measures the coordinate along the direction parallel to the beam and the two outside groups measure the perpendicular coordinate.
Cathode strip chambers (CSC) are used in the endcap disks where the magnetic field is uneven and particle rates are high. CSCs consist of arrays of positively charged “anode” wires crossed with negatively charged copper “cathode” strips within a gas volume. When muons pass through, they knock electrons off the gas atoms, which flock to the anode wires creating an avalanche of electrons. Positive ions move away from the wire and towards the copper cathode, also inducing a charge pulse in the strips, at right angles to the wire direction. Because the strips and the wires are perpendicular, we get two position coordinates for each passing particle. In addition to providing precise space and time information, the closely spaced wires make the CSCs fast detectors suitable for triggering. Each CSC module contains six layers making it able to accurately identify muons and match their tracks to those in the tracker.
Resistive plate chambers (RPC) are fast gaseous detectors that provide a muon trigger system parallel with those of the DTs and CSCs. RPCs consist of two parallel plates, a positively charged anode and a negatively charged cathode, both made of a very high resistivity plastic material and separated by a gas volume. When a muon passes through the chamber, electrons are knocked out of gas atoms. These electrons in turn hit other atoms causing an avalanche of electrons. The electrodes are transparent to the signal (the electrons), which are instead picked up by external metallic strips after a small but precise time delay. The pattern of hit strips gives a quick measure of the muon momentum, which is then used by the trigger to make immediate decisions about whether the data are worth keeping. RPCs combine a good spatial resolution with a time resolution of just one nanosecond (one billionth of a second).
## Collecting and collating the data
### Pattern recognition
New particles discovered in CMS will be typically unstable and rapidly transform into a cascade of lighter, more stable and better understood particles. Particles travelling through CMS leave behind characteristic patterns, or ‘signatures’, in the different layers, allowing them to be identified. The presence (or not) of any new particles can then be inferred.
### Trigger system
To have a good chance of producing a rare particle, such as a Higgs boson, a very large number of collisions is required. Most collision events in the detector are "soft" and do not produce interesting effects. The amount of raw data from each crossing is approximately 1 megabyte, which at the 40 MHz crossing rate would result in 40 terabytes of data a second, an amount that the experiment cannot hope to store, let alone process properly. The full trigger system reduces the rate of interesting events down to a manageable 1,000 per second.
To accomplish this, a series of "trigger" stages are employed. All the data from each crossing is held in buffers within the detector while a small amount of key information is used to perform a fast, approximate calculation to identify features of interest such as high energy jets, muons or missing energy. This "Level 1" calculation is completed in around 1 µs, and event rate is reduced by a factor of about 1,000 down to 50 kHz. All these calculations are done on fast, custom hardware using reprogrammable field-programmable gate arrays (FPGA).
If an event is passed by the Level 1 trigger all the data still buffered in the detector is sent over fibre-optic links to the "High Level" trigger, which is software (mainly written in C++) running on ordinary computer servers. The lower event rate in the High Level trigger allows time for much more detailed analysis of the event to be done than in the Level 1 trigger. The High Level trigger reduces the event rate by a further factor of 100 down to 1,000 events per second. These are then stored on tape for future analysis.
### Data analysis
Data that has passed the triggering stages and been stored on tape is duplicated using the Grid to additional sites around the world for easier access and redundancy. Physicists are then able to use the Grid to access and run their analyses on the data.
There are a huge range of analyses performed at CMS, including:
• Performing precision measurements of Standard Model particles, which allows both for furthering the knowledge of these particles and also for the collaboration to calibrate the detector and measure the performance of various components.
• Searching for events with large amounts of missing transverse energy, which implies the presence of particles that have passed through the detector without leaving a signature. In the Standard Model only neutrinos would traverse the detector without being detected but a wide range of Beyond the Standard Model theories contain new particles that would also result in missing transverse energy.
• Studying the kinematics of pairs of particles produced by the decay of a parent, such as the Z boson decaying to a pair of electrons or the Higgs boson decaying to a pair of tau leptons or photons, to determine various properties and mass of the parent.
• Looking at jets of particles to study the way the partons (quarks and gluons) in the collided protons have interacted, or to search for evidence of new physics that manifests in hadronic final states.
• Searching for high particle multiplicity final states (predicted by many new physics theories) is an important strategy because common Standard Model particle decays very rarely contain a large number of particles, and those processes that do are well understood.
## Milestones
1998 Construction of surface buildings for CMS begins. 2000 LEP shut down, construction of cavern begins. 2004 Cavern completed. 10 September 2008 First beam in CMS. 23 November 2009 First collisions in CMS. 30 March 2010 First 7 TeV proton-proton collisions in CMS. 7 November 2010 First lead ion collisions in CMS.[14] 5 April 2012 First 8 TeV proton-proton collisions in CMS.[15] 29 April 2012 Announcement of the 2011 discovery of the first new particle generated here, the excited neutral Xi-b baryon. 4 July 2012 Spokesperson Joe Incandela (UC Santa Barbara) announced evidence for a particle at about 125 GeV at a seminar and webcast. This is "consistent with the Higgs boson". Further updates in the following years confirmed that the newly discovered particle is the Higgs boson.[16] 16 February 2013 End of the LHC 'Run 1' (2009–2013).[17] 3 June 2015 Beginning of the LHC 'Run 2' with an increased collision energy of 13 TeV.[18] 28 August 2018 Observation of the Higgs Boson decaying to a bottom quark pair.[19] 3 December 2018 Planned end of the LHC 'Run 2'.[20] 3 March 2021 Planned end of CERN Long Shutdown 2 and planned start of LHC 'Run 3'.[21]
## Etymology
The term Compact Muon Solenoid comes from the relatively compact size of the detector, the fact that it detects muons, and the use of solenoids in the detector.[22] "CMS" is also a reference to the center-of-mass system, an important concept in particle physics.
## Notes
1. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2014-10-18. Retrieved 2014-10-18.CS1 maint: archived copy as title (link)
2. ^ "CMS Collaboration - CMS Experiment". cms.cern. Retrieved 28 January 2020.
3. ^ Biever, C. (6 July 2012). "It's a boson! But we need to know if it's the Higgs". New Scientist. Retrieved 2013-01-09. 'As a layman, I would say, I think we have it,' said Rolf-Dieter Heuer, director general of CERN at Wednesday's seminar announcing the results of the search for the Higgs boson. But when pressed by journalists afterwards on what exactly 'it' was, things got more complicated. 'We have discovered a boson – now we have to find out what boson it is'
Q: 'If we don't know the new particle is a Higgs, what do we know about it?' We know it is some kind of boson, says Vivek Sharma of CMS [...]
Q: 'are the CERN scientists just being too cautious? What would be enough evidence to call it a Higgs boson?' As there could be many different kinds of Higgs bosons, there's no straight answer.
[emphasis in original]
4. ^ Siegfried, T. (20 July 2012). "Higgs Hysteria". Science News. Retrieved 2012-12-09. In terms usually reserved for athletic achievements, news reports described the finding as a monumental milestone in the history of science.
5. ^ Del Rosso, A. (19 November 2012). "Higgs: The beginning of the exploration". CERN Bulletin. Retrieved 2013-01-09. Even in the most specialized circles, the new particle discovered in July is not yet being called the “Higgs boson". Physicists still hesitate to call it that before they have determined that its properties fit with those the Higgs theory predicts the Higgs boson has.
6. ^ O'Luanaigh, C. (14 March 2013). "New results indicate that new particle is a Higgs boson". CERN. Retrieved 2013-10-09.
7. ^ "The Higgs Boson". CERN: Accelerating Science. CERN. Retrieved 11 June 2015.
8. ^ http://cds.cern.ch/record/922757
9. ^ "Tracker detector - CMS Experiment". cms.web.cern.ch. Retrieved 20 December 2017.
10. ^ Weber, Hannsjorg (2016). "The phase-1 upgrade of the CMS pixel detector". 2016 IEEE Nuclear Science Symposium, Medical Imaging Conference and Room-Temperature Semiconductor Detector Workshop (NSS/MIC/RTSD). pp. 1–4. doi:10.1109/NSSMIC.2016.8069719. ISBN 978-1-5090-1642-6.
11. ^ CMS installs the world's largest silicon detector, CERN Courier, Feb 15, 2008
12. ^ "Using Russian navy shells - CMS Experiment". cms.web.cern.ch. Retrieved 20 December 2017.
13. ^ Precise mapping of the magnetic field in the CMS barrel yoke using cosmic rays
14. ^ "First lead-ion collisions in the LHC". CERN. 2010. Retrieved 2014-03-14.
15. ^ "New world record - first pp collisions at 8 TeV". CERN. 2012. Retrieved 2014-03-14.
16. ^ "ATLAS and CMS experiments shed light on Higgs properties". CERN. 2015. Retrieved 2018-09-13. ...the decay of the Higgs boson to tau particles is now observed with more than 5 sigma significance...
17. ^ "LHC report: Run 1 - the final flurry". CERN. 2013. Retrieved 2014-03-14.
18. ^ "LHC experiments back in business at record energy". CERN. 2015. Retrieved 2018-09-13.
19. ^ "LHC Schedule 2018" (PDF). CERN. 2018. Retrieved 2018-09-13.
20. ^ "Long-sought decay of Higgs boson observed". CERN. 2018. Retrieved 2018-09-13.
21. ^ "MASTER SCHEDULE OF THE LONG SHUTDOWN 2 (2019-2020)" (PDF). CERN. 2018. Retrieved 2018-09-13.
22. ^ Aczel, Ammir D. "Present at the Creation: Discovering the Higgs Boson". Random House, 2012 | {"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": 0.8465695977210999, "perplexity": 1764.8505214942136}, "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/1603107907213.64/warc/CC-MAIN-20201030033658-20201030063658-00096.warc.gz"} |
https://proceedings.neurips.cc/paper/2019/file/0d441de75945e5acbc865406fc9a2559-Reviews.html | NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID: 3579 Efficient Pure Exploration in Adaptive Round model
### Reviewer 1
------------------------------------------ After author feedback: Thanks to the authors for their response. This is a paper with strong theoretical results which are well supported by the experiments. ------------------------------------------- Original review: Summary: The object of study is the round complexity of (epsilon, delta)-PAC best-k-arms identification. In that particular instance of the best-k-arms bandit problem, an algorithm successively samples arms, with the goal of finally returning a set of k arms, which with probability bigger than 1-delta verifies that the mean or the j^th arm of the set is within epsilon of the j^th highest mean. The focus is on algorithms which have a small number of rounds at which they make decisions: at each such round, the algorithm queries a number of samples, and no adaptive decision is made between rounds. An algorithm is good if the total number of samples it requires is small (sample complexity) and if the number of rounds is small (round complexity). New algorithms and complexity bounds are provided for two settings: either the algorithm is free to choose the number of rounds, or that number is set in advance. An adaptation to exact top-k arm identification (epsilon=0) is also presented. Strengths: - Strong theoretical guarantees: the algorithms have both close to optimal sample complexities and round complexities. These algorithms are the first ones with such round complexities which do not require additional information on the problem. - Convincing experimental evaluation. The new algorithms improve over existing ones by order of magnitude with respect to the round complexity, while having sample complexity of comparable order to the best ones. The results on the exact top-k problem are particularly striking. - Clear presentation of the algorithms, in particular on page 4. Weaknesses: - the description and comparison of the different existing top-k settings could be clearer. For example, some papers require the means of all the k arms to be above the k^th highest mean, while this paper compares each of the k arms to different reference arms. This is mentioned in the related work section at the end of the paper, but the clarity would be enhanced if it was explained in the introduction. - The references to existing results could be more precise. For example the lower bound mentioned line 168 could be cited more precisely. Instead of referring to papers [16] and [14] without more precisions, the authors could write for example [16, Theorem X]. Lines 100-102, a notion of "near-optimality" coming from a previous paper is also referred to in a vague way. Remarks: - I think that the notation log^*(a) is defined, but not log^*_b(a). It took me a moment to realize that this was the logarithm in base b. - Instead of calling the problems 1, 2 and 3, it would be easier for the reader to have them be called by names. Writing "exact-top-k" is not much longer than "problem 3" and is much more descriptive.
### Reviewer 2
- The existing algorithms need \log^\star(n) rounds. Given the slow growth of $\log^\star(n)$, trying to further reduce this number does not seem that interesting in practice. This number is smaller than 5 for $n \le 2^{65536}$. - While there might exist differences in the setting, the authors of [4] claim that minimal round complexity scales at least as $\Omega( \log^\star(n))$, so that the authors should clarify how their setting differs from [4], as otherwise their results would simply prove that [4] contains a mistake. - As a minor remark, it could be good for clarity to explicity define the $\log^\star(n)$ function for an arbitrary base (other than 2). - Also, the authors mention that the algorithm proposed in [4] requires the knowledge of a number $\Delta$, such that $\Delta_k > \Delta$. Therefore, a straightforward approch to extend it to the present setting (where $\Delta$ is unknown) would be to apply the algorithm of [4] for \Delta = 1, then check that $\Delta_k > \Delta$ by sampling. It is noted that one can always estimate $\Delta_k$ by sampling all arms enough time (and this takes one "round") If the test fails, do the same for $\Delta = \gamma$, $\Delta = \gamma^2$ (where $\gamma < 1$) and so on and so forth until the test succeeds. Would such an approach fail and if not why ?
### Reviewer 3
This paper focused mainly on the PAC top-k-subset selection problem in an adaptive round model, which means the authors are not only interested in the sample complexity of the proposed algorithm, but also try to achieve minimal number of rounds used. The motivation of this setting has been stated and the main text is globally well written. The algorithms are elimination-based that try to kill as much arms as they can in each round to reduce the round complexity. It appears that the query complexity in the fixed-confidence setting stays optimal and shows a good empirical performance as well. I have a major technique concern, however, regarding the proof of Lemma 1. In equation 7, you are using a union bound over all iterations r. But the Hoeffding bound only holds for iteration r if i^* has been updated at this iteration. What if it is not updated at this iteration? \hat{\theta_i^*} >= \theta_i^* - \epsilon/8 holds for iteration r only when it holds for iteration r-1, but that cannot be true from the very beginning. Maybe I am missing something, but I really need more clarification on this part, which seems to be a crucial lemma in this paper. Otherwise, regarding the experiments, have you ever thought of comparing \deltaE to some other fixed confidence BAI algorithms and \deltaER to some fixed budget BAI algorithms. Of course, it is not comparable straightforwardly, but have you thought of proposing some naive adaptation of those algorithms to your setting? Minor: - I would suggest the authors to make the motivation more clear as you only stated the use of muliple testing for several applications but didn't really explain why we need to do so in the real world - The Exponential-Gap-Elimination alogorithm could have been detailed in the paper (or appendix), otherwise the paper is not self-contained Update after author feedback: I am convinced by the author's feedback regarding the proof, and thus increased the score. In summary, this paper has some strong theoretical results that worth being published in NeurIPS. | {"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": 0.865349531173706, "perplexity": 497.9817090349762}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039503725.80/warc/CC-MAIN-20210421004512-20210421034512-00548.warc.gz"} |
http://science.sciencemag.org/content/early/2018/07/03/science.aat7932?rss=1 | Report
# Unusual high thermal conductivity in boron arsenide bulk crystals
See allHide authors and affiliations
Science 05 Jul 2018:
eaat7932
DOI: 10.1126/science.aat7932
## Abstract
Conventional theory predicts that ultrahigh lattice thermal conductivity can only occur in crystals composed of strongly-bonded light elements, and that it is limited by anharmonic three-phonon processes. We report experimental evidence that is a departure from these long-held criteria. We measured a local room-temperature thermal conductivity exceeding 1000 W m−1 K−1 and an average bulk value reaching 900 W m−1 K−1 in bulk boron arsenide (BAs) crystals, where boron and arsenic are light and heavy elements, respectively. The high values are consistent with a proposal for phonon band engineering and can only be explained with higher order phonon processes. These findings yield new insight into the physics of heat conduction in solids and show BAs to be the first known semiconductor with ultrahigh thermal conductivity.
View Full Text | {"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": 0.8182563781738281, "perplexity": 4445.163735800019}, "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-2018-39/segments/1537267160400.74/warc/CC-MAIN-20180924110050-20180924130450-00229.warc.gz"} |
http://mathhelpforum.com/calculus/113538-word-problem.html | # Math Help - Word Problem
1. ## Word Problem
A tank in the shape of a right circular cone is being filled with water at a rate of 5 ft^3 per minute, but water is also flowing out at a rate of 1 ft^3 per minute. The tank is 60 feet deep and 40 feet across the top.
A) How quickly is the depth of the water changing when the water is 42 ft deep?
B) If the tank is empty, how long will it take for the tank to be 1/2 full?
C) How deep is the water when the tank is 1/2 full? How quickly is the depth of the water changing when the tank is 1/2 full?
I’m really not sure if these are right
A) dv/dt=5-1=4, h=60, r=20
find dh/dt when h=42
h/r=60/20, r=h/3, dr/dt=1dh/3dt
60/20=42/r r=14ft.
V=pr^2h/3 dv/dt=p/3[2rhdr/dt+r^2dh/dt]
4=p/3[2(14)(42)(1dh/3dt)+196dh/dt]
4=p/3[392dh/dt+196dh/dt] = p/3[588dh/dt] = dh/dt=1/49p ft/min.
B) V=pr^2h/3 p(20)^2(60)/3 = 8000p(1/2) = 4000p
dv/dt=4ft/min if 1 min for 4ft, 16 hours 40 min for 4000ft
C) find h when V=4000
3v/pr^2=h h=3(4000)/p(20)^2 = 12000/400p = 30pft^3
find dh/dt when h=30
4=p/3[2(20)(30)(1dh/3dt)+20^2dh/dt]
4=p/3[800dh/dt]
dh/dt=1/66.67p ft/min.
2. A and B look correct.
For C, using the formula
$V=\frac{1}{3}\pi\left(20\cdot\frac{h}{60}\right)^2 h=\frac{1}{27}\pi h^3,\;\;\;\;\;\mbox{(Corrected)}$
I derived
\begin{aligned}
\frac{1}{27}\pi h^3&=4000\\
h^3&=\frac{4000\cdot 27}{\pi}\\
h&=30\sqrt[3]{\frac{4}{\pi}}.
\end{aligned}
3. Originally Posted by Scott H
A and B look correct.
For C, using the formula
$V=\frac{1}{3}\pi\left(20\cdot\frac{h}{60}\right)^2 h=\frac{1}{27}\pi r^3,$
I derived
\begin{aligned}
\frac{1}{27}\pi h^3&=4000\\
h^3&=\frac{4000\cdot 27}{\pi}\\
h&=30\sqrt[3]{\frac{4}{\pi}}.
\end{aligned}
wow thanks! I didn't know about that formula. Is there one I could use for the second part of C? My answer for that looks completely wrong.
4. I corrected $\frac{1}{27}\pi r^3$ to $\frac{1}{27}\pi h^3$.
To find the rate of change of the depth at $h=30\sqrt[3]{\frac{4}{\pi}}$, we may use the Chain Rule:
$\frac{dV}{dt}=\frac{d}{dt}\left(\frac{1}{27}\pi h^3\right)=\frac{1}{9}\pi h^2\frac{dh}{dt}.$
5. It's the same cone-volume formula that you used successfully in part A, but Scott has made things easier by eliminating r because it's just h/3. You could have done the same in part A (you pointed out the ratio) but you used the product rule instead. In part C you need the neater version, which Scott has given you as
$V = \frac{1}{27} \pi h^3$
which is of course correct, but beware his typo where he states it with r in place of h.
Now differentiate V with respect to h, and use the chain rule to deduce dh/dt. You need to restore the pi which you dropped from your value for the half-full volume during part B, and consequently you also want to remove pi from Scott's corresponding value of h.
Just in case a picture helps organise the chain rule...
Spoiler:
... where
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
Edit: sorry Scott, thought you weren't online ... however, perfectstorm does want to note the rogue pi.
__________________________________________
Don't integrate - balloontegrate!
Balloon Calculus: Gallery
Balloon Calculus Drawing with LaTeX and Asymptote!
6. I think I got it now thanks a lot Scott, and Tom your picture helped! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "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": 0.9139032959938049, "perplexity": 1307.876027678184}, "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-2015-35/segments/1440645167576.40/warc/CC-MAIN-20150827031247-00178-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=3637449&postcount=7 | Thread: 0 divided by 0 View Single Post
P: 1,188 Actually, in complex analysis, it's common to say that the function 1/z maps 0 to infinity when infinity is considered as the point at infinity on the Riemann sphere. So, in that sense, you could say 1/0 = infinity (when it's done naively by calculus students, this is wrong because they don't have a mapping in mind). You could restrict this to get 1/x when x is a real variable. The trick here is that you have to identify negative infinity with positive infinity. This isn't to say that 0 has an inverse. It is just that it is now included in the domain of the function 1/x and the point at infinity is added to the range. But, still, 0/0 wouldn't have a good interpretation because that would correspond to the function 0/x, which is zero everywhere. I guess you could send 0 to 0, so that the function is continuous. So, you could define 0/0 to be zero. But it would be very confusing and bad notation that wouldn't accomplish anything, since there's no need to describe the constant function equal to 0 by such a convoluted means. And again, you would need to be careful to point out that it's a mapping, rather than taking an inverse, but that's a moot point. Better not to discuss it at all than to cause all this confusion. So, yes, 0/0 is undefined. | {"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": 0.937658429145813, "perplexity": 163.06090504925308}, "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/1405997858581.26/warc/CC-MAIN-20140722025738-00019-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://grittyjello.blogspot.com/ | ## Monday, August 17, 2015
### MAT133 Practice Problems: Financial Math, Linear Algebra, and Lagrange Multipliers
It's like I blinked and suddenly eight months have passed since my last update. That's enough time to gestate an entire human baby. Sadly(??), that's not today's topic.
I recently created some practice problems with solutions for the University of Toronto course MAT133, Calculus and Linear Algebra for Commerce. Download the practice problems here (PDF). It isn't much, but should help anyone looking for extra practice in financial math, matrices, and the method of Lagrange multipliers.
I'll update this post if I ever generate more MAT133-related practice material. The Fall/Winter MAT133 website hosts huge amounts of excellent material, including past tests.
That's all for today. See you in probably fewer than eight months!
## Sunday, December 28, 2014
### Driving A Car Over a Circular Hill
Here's a textbook physics question about circular motion:
A car drives over a hill with a circular vertical profile. At what speed will the car lift off?
Finding the minimum liftoff speed at the top of the hill is easy enough. Here's the free body diagram:
Figure 1
Two forces act on the car: weight, $$mg$$, pulls downward, and a normal force, $$N$$, pushes upward. If the car stays on the hill, then $$N$$ must be less than $$mg$$, since the net force $$F_\text{net} = mg - N$$ must point downward to hold the car on a circular path around the hill's curve. As usual, the centripetal force required for the car's circular motion is $$F_\text{c} = mv^2/r$$, where $$v$$ is the car's speed and $$r$$ is the hill's radius of curvature. This centripetal force must be provided entirely by $$F_\text{net}$$:
$\frac{mv^2}{r} = mg - N. \label{eq1}$As soon as the car leaves the ground, the normal force will have nothing to act on, so it will vanish. As a result, setting $$N = 0$$ gives the threshold speed at which the car lifts off:
\begin{align} \frac{mv^2}{r} &= mg - 0 \nonumber \\ v &= \sqrt{gr} \end{align}
The car will lift off if its speed at the top of the hill exceeds $$\sqrt{gr}$$.
That was easy because we limited our analysis to the top of the hill. But the question didn't ask specifically about just the top of the hill. If the car begins its climb at some initial speed, would it still lift off right at the crest of the hill? Or would it fly out of control immediately? Or maybe it would lift off somewhere on the other side of the hill?
Depending on your hobbies, you might not have any intuition about how cars behave on circular hills with red and blue lights strobing in the bullet-cracked rear-view, your buddy firing back with his good arm while shrieking about churches in his native tongue. So let's analyze a more general question:
A car coasts up a circular hill with initial speed $$v_0$$. Where on the hill does the car lift off, and for what initial speed?
Figure 2. Notes from a speed bump designer's last day on the job.
The car's position on the hill is specified by its angle from vertical, $$\theta$$. The angle subtended by the hill from vertical is $$\theta_0$$. The hill's height is $$h_0 = r - r\cos\theta_0$$.
#### Some Restrictions
• $$\theta_0 > 0$$, otherwise there would be no hill.
• Let's stick to the regime $$\theta_0 < \pi/2$$, otherwise the "hill" would really be a terrain bubble with walls steeper than vertical.
• We'll require $$\theta \leq \theta_0$$. If not, then the car is beyond the circular extent of the hill, which doesn't make sense.
### One Way to Proceed: Energy Conservation
The car rolls up the hill without applying power to its wheels, so its mechanical energy (kinetic + gravitational potential) is conserved. (Another plausible model would be to hit the gas to keep the car moving at constant speed, but that turns out not to be as interesting.) Initially, the car has total energy $$mv_0^2/2$$. After climbing through a height $$h$$, the car will slow to speed $$v < v_0$$, having gained gravitational potential energy $$mgh$$. Energy conservation gives the car's speed in terms of height:
\begin{align} \frac{1}{2}mv_0^2 + 0 &= \frac{1}{2}mv^2 + mgh \nonumber \\ v^2 &= v_0^2 - 2gh \label{asdfjkl} \end{align}
#### Getting Over the Hill
Before trying to find the liftoff speed, let's find the minimum initial speed required to get over the hill at all. Equating initial kinetic energy with potential energy at the crest of the hill gives $$mv_0^2/2 = mgh_0$$, or $$v_0 = \sqrt{2gh_0}$$. Substituting $$h_0 = r - r\cos \theta_0$$ gives $$v_0 = \sqrt{2gr(1 - \cos \theta_0)}$$ as the threshold initial speed for cresting the hill. If the car starts up the hill slower than this, it will roll to a stop partway up and roll back down the same side.
It seems reasonable that if the car can't even get over the hill, it wouldn't have enough speed to fly off the road. We'll look at this again toward the end.
### The Liftoff Criterion
The car will stay on the hill if the radial component of its weight, $$mg \cos \theta$$, minus the opposing normal force, $$N$$, is strong enough to provide the centripetal force required for circular motion. This gives something very close to Eq. ($$\ref{eq1}$$):
\begin{align} m\frac{v^2}{r} &= mg\cos \theta - N. \end{align}
As before, setting $$N = 0$$ gives the speed above which the car will lift off:
\begin{align} v^2 &= gr\cos \theta. \nonumber \end{align}
The car will lift off if its speed exceeds this threshold, corresponding to the inequality
\begin{align} v^2 > gr\cos \theta. \nonumber \end{align}
Substituting $$v^2 = v_0^2 - 2gh$$ from Eq. ($$\ref{asdfjkl}$$) gives
\begin{align} v_0^2 - 2gh &> gr \cos \theta. \label{someeq} \end{align}
Finally, inserting $$h = r\cos \theta - r\cos \theta_0$$ gives
\begin{align} v_0^2 - 2gr(\cos \theta - \cos \theta_0) &> gr \cos \theta \nonumber \\ v_0^2 + 2gr\cos \theta_0 &> 3gr \cos \theta \nonumber \\ \frac{v_0^2}{3gr} + \frac{2}{3} \cos \theta_0 &> \cos \theta \label{coastsol} \end{align}
The car will leave the ground whenever this liftoff criterion is met. Note that everything on the left-hand side is constant.
#### A Closer Look at the Left Side of Equation ($$\ref{coastsol}$$)
Let's start with the cosine term on the left side of the liftoff criterion. Since $$\cos \theta_0$$ is bounded above by 1, $$0 < \cos \theta_0 < 1$$, so $$0 < \frac{2}{3} \cos \theta_0 < \frac{2}{3}.$$ Nothing surprising here. For any sizable hill, this term will be somewhat less than $$\frac{2}{3}$$.
The second term on the left side of Eq. ($$\ref{coastsol}$$), $$v_0^2/(3gr)$$, is never less than zero, and can be made arbitrarily large by choosing a high enough $$v_0$$. If $$v_0$$ is zero, the car obviously shouldn't lift off. Sure enough, setting $$v_0 = 0$$ in Eq. ($$\ref{coastsol}$$) makes the liftoff criterion $$\frac{2}{3}\cos\theta_0 > \cos\theta$$. In other words, $$\cos\theta_0$$ must be so large that a mere $$2/3$$ of it is enough to exceed $$\cos\theta$$. Obviously, $$\theta = \theta_0$$ won't work, since the inequality goes the wrong way: $$\frac{2}{3}\cos\theta < \cos\theta$$ for $$0 < \theta < \pi/2$$. We'll definitely need the strict inequality $$\theta_0 < \theta$$ to make the car to lift off when $$v_0 = 0$$. But we've already noted the restriction that $$\theta_0 ≥ \theta$$ for any hill, so the liftoff criterion is never satisfied for $$v_0 = 0$$.
Reassuringly, then, Eq. ($$\ref{coastsol}$$) makes sense in the zero-speed limit: given zero initial speed, the car will never lift off (unless $$\theta$$ and $$\theta_0$$ both exceed $$90°$$, and the car just falls off). Furthermore, choosing a large enough $$v_0$$ will ensure that the inequality ($$\ref{coastsol}$$) is satisfied for any value of $$\theta$$, meaning that if the car begins with a high enough speed, it will leave the hill immediately.
### The Minimum Initial Speed Required for Liftoff
Given $$g$$, $$r$$, and $$\theta_0$$, we can choose a particular initial speed $$v_0$$ such that $$v_0^2/(3gr) = \frac{1}{3}\cos \theta_0$$. The special $$v_0$$ value that makes this true is
$v_0 = \sqrt{gr\cos\theta_0}. \label{liftoffspeed}$
At this special speed, the liftoff criterion, Eq. ($$\ref{coastsol}$$), becomes
\begin{align} \frac{1}{3}\cos \theta_0 + \frac{2}{3}\cos \theta_0 = \cos \theta_0 &> \cos \theta \nonumber \\ \implies \theta_0 &< \theta \end{align}
As noted above, this inequality is never true, since it implies that the car's position has exceeded the maximum angular extent of the hill. Thus, the car will remain grounded if its initial speed is lower than $$\sqrt{gr\cos\theta_0}$$.
If the car starts up the hill at a speed greater than $$\sqrt{gr\cos\theta_0}$$, then $$v_0^2/(3gr) > \frac{1}{3}\cos \theta_0$$, and the liftoff criterion becomes $$A \cos \theta_0 > \cos \theta$$, where $$A > 1$$. Even if $$\cos \theta_0$$ is smaller than $$\cos \theta$$, we can still satisfy $$A \cos \theta_0 > \cos \theta$$ with a big enough $$A$$. In particular, whenever $$A$$ is a tiny bit bigger than 1, then $$\theta_0 = \theta$$ is always enough to satisfy the liftoff criterion. But $$\theta_0 = \theta$$ describes the car's initial position. Thus, if $$v_0 > \sqrt{gr\cos\theta_0}$$, then the car flies off the surface of the hill right away!
In conclusion, there is no way to send a car coasting up a hill so that it lifts off at the top of the hill, or anywhere else, for that matter. The car will either stay on the hill for its entire motion, or it will lift off immediately (and may crash back onto the hill a short time later).
Incidentally, recall that the minimum speed needed to get over the hill is $$v_0 = \sqrt{2gr(1-\cos\theta_0)}$$. This will be lower than the minimum lift-off speed $$\sqrt{gr\cos\theta_0}$$ if
\begin{align} 2 - 2\cos\theta_0 &< \cos\theta_0 \nonumber \\ 2 &< 3\cos\theta_0 \nonumber \\ \theta_0 &< \cos^{-1}\left(\frac{2}{3}\right) \approx 48.2° \nonumber \\ \end{align}
So, for a small to sizable hill with half-angle $$\theta_0 < 48.2°$$, the car can roll over smoothly to the other side if it starts with speed $$\sqrt{2gr(1-\cos\theta_0)} < v_0 < \sqrt{gr\cos\theta_0}$$. For a steeper hill with $$\theta_0 > 48.2°$$, the inequality reverses to $$\sqrt{2gr(1-\cos\theta_0)} > \sqrt{gr\cos\theta_0}$$, meaning the liftoff speed is less than the speed required to get over the hill. On such a steep hill, either the car stops and rolls back down, or it starts with enough speed that it lifts off immediately; there's no way to roll it to the other side without catching air.
### Some Actual Numbers
For a concrete example, let's choose $$\theta_0 = \pi/4 = 45°$$, so $$\cos\theta_0 = 1/\sqrt{2}$$. Let's give the car an initial speed below the liftoff speed given by Eq. ($$\ref{liftoffspeed}$$) of $$v_0^2 = gr\cos\theta_0$$. Let's go with $$v_0^2 = \frac{1}{2}gr\cos\theta_0 = gr/(2\sqrt{2})$$. The car will lift off as soon as $$\theta$$ satisfies
\begin{align} \cos \theta &< \frac{2}{3} \cos \left( \frac{\pi}{4}\right) + \frac{gr/(2\sqrt{2})}{3gr} \nonumber \\ &= \frac{2}{3\sqrt{2}} + \frac{1}{6\sqrt{2}} \nonumber \\ &= \frac{4}{6\sqrt{2}} + \frac{1}{6\sqrt{2}} = \frac{5}{6\sqrt{2}}\nonumber \\ \theta &> \cos^{-1}\left(\frac{5}{6\sqrt{2}}\right) = 53.9° \end{align}
The car would lift off if $$\theta$$ could exceed 53.9°, but it can't, since the hill's half-angle is only 45°.
Now, what if the initial speed slightly exceeds the lift-off threshold? Let's set $$v_0^2 = 1.1 gr\cos\theta_0$$ $$= 1.1 gr/\sqrt{2}$$. The liftoff criterion in this case is
\begin{align} \cos \theta &< \frac{2}{3} \cos \left(\frac{\pi}{4}\right) + \frac{1.1gr/\sqrt{2}}{3gr} \nonumber \\ &= \frac{2}{3\sqrt{2}} + \frac{1.1}{3\sqrt{2}} \nonumber \\ \theta &> 43.1° \end{align}
Given this initial speed, the car will take off whenever it's more than 43.1° from vertical. This first occurs at the car's initial position, where $$\theta = 45°$$, so the car lifts off immediately.
## Monday, November 17, 2014
### A Vector Sum Trap in a Related-Rates Problem
A student of mine recently showed me a math problem that's easy to get wrong, as I did, through hasty vector addition. Here's the problem (adapted from this PDF):
Two radar stations, A and B, are tracking a ship generally north of both stations. Station B is located 6 km east of station A. At a certain instant, the ship is 5 km from A and also 5 km from B. At the same instant, station A reads that the distance between station A and the ship is increasing at the rate of 28 km/h. Station B reads that the distance between station B and the ship is increasing at 4 km/h. How fast and in what direction is the ship moving?
### Solution 1: Related Rates
If you've studied related rates before (who hasn't!), you'll look for an equation involving distances, differentiate that equation with respect to time, and solve for the unknown rate.
Let's use Newton's notation for time derivatives: $$\dot a = \frac{da}{dt}$$. This problem gives you two rates: $$\dot a = v_a = 28 \text{ km/h}$$ and $$\dot b = v_b = 4 \text{ km/h}$$. You need to find the ship's speed and direction, i.e., its velocity vector $$v$$, which you can express as the vector sum of horizontal and vertical components: $$v_x + v_y$$. This is useful here because $$v_x$$ and $$v_y$$ are easily related to $$x$$ and $$h$$, as follows.
The large triangle's top corner stays anchored to the ship as it moves; the ship will "drag" the top corner while $$h$$ remains vertical. So if the ship has horizontal speed $$v_x$$, the right triangle base $$x$$ must change at exactly $$v_x$$ to move the top corner along with the ship, which means $$v_x = \dot x$$. Similarly, $$v_y = \dot h$$. So we just need to find $$\dot x$$ and $$\dot h$$ to solve this problem.
The right triangle with base $$x$$ gives us $$h^2 + x^2 = a^2$$, or $$h^2 = a^2 - x^2$$. The other right triangle with base $$(6 - x)$$ gives $$h^2 + (6 - x)^2 = b^2$$. Combining these equations gives
$b^2 = (6 - x)^2 + a^2 - x^2 \label{asdf}$in which $$x$$ is the only unknown. Good! The rate of change of $$x$$ will give us the ship's horizontal speed. Differentiate both sides of ($$\ref{asdf}$$) with respect to time, using the chain rule on $$a(t), b(t),$$ and $$x(t)$$:
\begin{align} 2b \dot b &= 2(6 - x)(-\dot x) + 2a \dot a - 2x \dot x \nonumber \\ b \dot b &= (x - 6) \dot x + a \dot a - x \dot x \nonumber \\ (6\text{ km}) \dot x &= a \dot a - b \dot b = (5\text{ km})(28\text{ km/h}) - (5\text{ km})(4\text{ km/h}) \nonumber \\ \dot x &= 20 \text{ km/h} \nonumber \end{align}And from $$h^2 = a^2 - x^2$$,
\begin{align} h \dot h &= (4\text{ km})\dot h = a \dot a - x \dot x = (5\text{ km})(28\text{ km/h}) - (3\text{ km})(20\text{ km/h}) \nonumber \\ \dot h &= 20 \text{ km/h} \nonumber \end{align} The ship's velocity is $$\mathbf{(v_x, v_y) = (20, 20)}$$ km/h, or $$\mathbf{20\sqrt{2} \approx 28.8}$$ km/h northeast.
To check that answer and withdraw from the bleakness of the real world for a little while longer, let's find the sum of the given velocities $$v_a$$ and $$v_b$$ by resolving them into their horizontal and vertical components and adding those:
Note that we don't actually need the value of $$\theta$$, since it only ever appears as $$\cos \theta = 3/5$$ or $$\sin \theta = 4/5$$ (these values are readable straight from the first diagram). Adding up components gives:
\begin{align} v_x &= v_{a}\cos \theta - v_{b} \cos \theta \nonumber \\ &= \frac{3}{5}(28 - 4) \nonumber \\ &= \mathbf{14.4 \text{ km/h}} \nonumber \\ v_y &= v_{a}\sin \theta + v_{b} \sin \theta \nonumber \\ &= \frac{4}{5}(28 + 4) \nonumber \\ &= \mathbf{25.6 \text{ km/h}} \nonumber \end{align}...Wait, what the hell!? $$(v_x, v_y) = (14.4, 25.6)$$ km/h translates to a speed of 29.4 km/h in a direction 60.6° north of east, shown in red below. This sum clearly disagrees with the related-rates answer of 28.8 km northeast, shown in blue.
How could this happen? Adding two vectors is dead simple; no way did you screw that up. Did you flip a sign somewhere in the related-rates analysis? Maybe $$\dot x$$ doesn't actually equal $$v_x$$ or something? What is going on!?
### What Is Going On
It's true that the unknown velocity vector $$v$$ has components $$v_a = 28 \text{ km/h}$$ and $$v_b = 4 \text{ km/h}$$. The problem is that these components are taken along non-orthogonal (not right-angled) lines $$a$$ and $$b$$, so they can't simply add back up to $$v$$! Some of $$v_a$$ points in the direction of $$v_b$$, and some of $$v_b$$ points along $$v_a$$. Here's a surefire way to convince yourself that these measured velocities shouldn't add up to the ship's velocity: picture multiple radar stations, say 10 of them, distributed along the line AB. It wouldn't make sense to add all of their velocity measurements.
Here's a geometric way to realize why the vector sum $$v_a + v_b \ne v$$. The a-component of the unknown velocity $$v$$ is $$v_a$$. This defines a whole family of vectors whose component along the line defined by $$a$$ is $$v_a = 28 \text{ km/h}$$. Here's that family of vectors in green:
Similarly, there is a family of vectors whose component along the line defined by $$b$$ is $$v_b = 4 km/h$$:
Only one vector belongs to both families. That vector is the correct velocity of the ship, which we easily found using related rates to be $$20\sqrt{2} \approx 28.8 \text{ km/h}$$ 45 degrees north of east:
You could of course solve this problem using vector geometry: the unknown vector $$v$$ forms some angle $$\alpha$$ with the positive x-axis, so the angle between $$v$$ and $$v_a$$ is $$\theta - \alpha$$, etc. However, it's quicker to use related rates... As long as you don't add up vector components that point along non-orthogonal directions.
## Tuesday, April 22, 2014
### Bayes' Theorem, Part 2
As implied in Part One, this article series is supposed to be an easy introduction to Bayes' Theorem for non-experts (by a non-expert), not a thinly veiled job application directed at government agencies that don't officially exist.
### Review: Testing Positive for a Rare Disease Doesn't Mean You're Sick
The previous article in this series illustrated a surprising fact about disease screening: if the disease you're testing for is sufficiently rare, then a positive diagnosis is probably wrong. This seemingly WTF outcome is an instance of the false positive paradox. It arises when the event of interest (in this case, being diseased) is so statistically rare that true positives are drowned out by a background of false positives.
Bayes' Theorem allows us to analyze this paradox, as shown below. But first, we need to define true and false positives and negatives.
### False Positives and False Negatives
No classification test is perfect. Any real-world diagnostic test will sometimes mistakenly report disease in a healthy person. This type of error is defined as a false positive. If you test for the disease in a large number of people who are known to be healthy, a certain percentage of the test results will be false positives. This percentage is called the false positive rate of the test. It's the probability of getting a positive result if you test a healthy person.
The other type of classification error is the false negative -- for example, a clean bill of health mistakenly issued to someone who's actually sick. If you run your test on a large number of people known to be sick, the test will fail to detect disease in some percentage of them. This percentage is known as the test's false negative rate.
The lower the false positive rate and false negative rate, the better the test. Both rates are independent of population size and disease prevalence.
But now we get to the root of the false positive paradox: if the disease is rare enough, then the vast majority of people you test will be healthy. This unavoidable testing of crowds of healthy people represents plenty of opportunities to get false positives. These false positives drown out the relatively faint true positive signal coming from the few sick people in the population. And if the test obtains each true positive at the cost of many false positives, any given positive result is probably a false one. It's intuitive by this point that the error rate of a screening process depends not only on the accuracy of the test itself, but also on the rarity of what you're screening for.
For a more rigorous understanding, we need to derive Bayes' Theorem. To do that, we need some basic probability theory.
### Probability Basics
The probability that some event $$A$$ will occur is written $$P(A)$$. All probabilities are limited to values between 0 ("impossible") and 1 ("guaranteed"). If we let $$H$$ stand for the event that a fair coin lands heads up, then $$P(H) = 0.5$$, or 50%. If $$X$$ stands for rolling a "20" on a 20-sided die, then $$P(X) = 1/20$$, or 5%.
If two events $$A$$ and $$B$$ cannot occur at the same time, they are said to be mutually exclusive, and the probability that either $$A$$ or $$B$$ occurs is just $$P(A) + P(B)$$. Rolling a 19 and rolling a 20 on a 20-sided die are mutually exclusive events, so the probability of rolling 19 or 20 is $$1/20 + 1/20 = 1/10$$.
The opposite of an event, or its complement, is denoted with a tilde ($$\text{~}$$) before the letter. The probability that $$A$$ will not occur is written $$P(\text{~}A)$$. If $$A$$ has only two possible values, such as heads/tails, sick/healthy, or guilty/innocent, then $$A$$ is called a binary event and $$P(A) + P(\text{~}A) = 1$$, which just says that either $$A$$ happens or it doesn't. Heads and tails, sickness and health, and guilt and innocence are all mutually exclusive binary events.
### Conditional Probability
So far, we've considered probabilities of single events occurring in theoretical isolation: a single coin flip, a single die roll. Now, consider the probability of an event $$A$$ given that some other event $$B$$ has occurred. This new probability is read as "the probability of A given B" or "the probability of A conditional on B." Because this new probability quantifies the occurrence of A under the condition that B has definitely occurred, it is known as a conditional probability. Standard notation for conditional probability is:
$$$P(A|B)$$$
The vertical bar stands for the word "given." $$P(A|B)$$ means "the probability of $$A$$ given $$B$$."
It's really important to recognize right away that $$P(A|B)$$ is not the same as $$P(B|A)$$. To see why, dream up two related real-world events and think about their conditional probabilities:
• probability that a road is wet given that it's raining: $$P(\text{wet road} ~ | ~ \text{raining})$$
• probability that it's raining given that the road is wet: $$P(\text{raining} ~ | ~ \text{wet road})$$
The road will certainly get wet if it rains, but many things besides rain could result in a wet road (use your imagination). Therefore,
$$$P(\text{wet road} ~ | ~ \text{raining}) > P(\text{raining} ~ | ~ \text{wet road}).$$$
### Bayes' Theorem Converts between P(A|B) and P(B|A)
Okay, so $$P(A|B)$$ does not equal $$P(B|A)$$, but how are they related? If we know one quantity, how do we get the other? This section title blatantly gave away the answer.
To derive Bayes' Theorem, consider events $$A$$ and $$B$$ that have nonzero probabilities $$P(A)$$ and $$P(B)$$. Let's say that $$B$$ has just occurred. What is the probability that $$A$$ occurs given the occurrence of $$B$$? In symbols, what is $$P(A|B)$$?
Well, since $$A$$ occurs after $$B$$, it will certainly be true that both $$A$$ and $$B$$ will have occurred. The occurrence of both $$A$$ and $$B$$ is itself an event; let's call it $$AB$$, with probability $$P(AB)$$. Now, note that $$P(B)$$ will always be greater or equal to $$P(AB)$$, because the "$$A$$" in "$$AB$$" represents an added criterion for event completion. The chance of both $$A$$ and $$B$$ occurring has to be lower than the chance of just $$B$$ occurring (unless, of course, $$A$$ is guaranteed to occur).
The value of $$P(AB)$$ itself isn't as interesting as the ratio of $$P(AB)$$ to $$P(B)$$, and here's why. This ratio compares the probability of both A and B to the probability of B just by itself. The ratio gives the proportion of possible occurrences of $$AB$$ relative to the possible occurrences of $$B$$. You should be able to convince yourself that this ratio is none other than the conditional probability $$P(A|B)$$:
$$$P(A|B) = \frac{P(AB)}{P(B)}.$$$
Rearranging gives
$$$P(AB) = P(A|B)P(B). \label{whatstheuse}$$$
Similarly,
\begin{align} P(B|A) &= \frac{P(BA)}{P(A)} \\
P(BA) &= P(B|A)P(A). \end{align}
Since the order of $$A$$ and $$B$$ doesn't affect the probability of both occurring, we have $$P(AB) = P(BA)$$, so
$$$P(A|B)P(B) = P(B|A)P(A).$$$
$$$P(A|B) = \frac{P(B|A)P(A)}{P(B)} \label{existentialangst}$$$
There we have it: to convert from $$P(B|A)$$ to $$P(A|B)$$, multiply $$P(B|A)$$ by the ratio $$P(A)/P(B)$$. Let's see what this looks like in the disease example.
### Back to the Disease Example
Let the symbols $$+$$ and $$-$$ stand for a positive and negative diagnosis, and let $$D$$ stand for the event that disease is present. Since the test must return either $$+$$ or $$-$$ every time we test someone, $$P(+) + P(-) = 1$$. And since disease must either be present or absent, $$P(D) + P(\text{~}D) = 1$$.
Now, to determine precisely how much a positive diagnosis should worry us, we care about the probability of disease given a positive diagnosis. This is just $$P(D|+)$$. By Bayes' Theorem (Equation $$\ref{existentialangst}$$), we need to calculate
$$$P(D|+) = \frac{P(+|D)P(D)}{P(+)}$$$
Consider each term on the right side:
$$P(+|D)$$ is just the probability of getting a positive diagnosis given the presence of disease, i.e., the probability that the test works as advertised as a disease detector. This is the definition of the true positive rate, AKA the sensitivity, a very commonly quoted test metric.
$$P(D)$$ is the probability of disease in a person randomly selected from our population. In other words, $$P(D)$$ is the disease prevalence (e.g., 15 per 10,000 people).
What about the denominator, $$P(+)$$? It's the probability of getting a positive diagnosis in a randomly selected person. A positive diagnosis can be either 1. a true positive, or 2. a false positive.
1. A true positive is defined by the occurrence of both $$D$$ and $$+$$. Equation $$\ref{whatstheuse}$$ says that the probability of both $$D$$ and $$+$$ is $$P(+|D)P(D)$$. $$P(+|D)$$ is the true positive rate, and $$P(D)$$ is the disease prevalence.
2. A false positive is defined by the occurrence of both $$\text{~}D$$ and $$+$$. The probability of this is $$P(+|\text{~}D)P(\text{~}D)$$. $$P(+|\text{~}D)$$ is the false positive rate (after which the paradox is named), and $$P(\text{~}D) = 1 - P(D)$$.
Since true and false positives are mutually exclusive events, their probabilities add up to give the probability of a positive outcome = either a true positive or a false positive. Thus,
$$$P(+) = P(+|D)P(D) + P(+|\text{~}D)P(\text{~}D).$$$
Bayes' Theorem for the disease-screening example now looks like this:
$$$P(D|+) = \frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|\text{~}D)P(\text{~}D)} \label{thehorror}$$$
### Plugging in Example Numbers
Part One of this series gave concrete numbers for a hypothetical outbreak of dancing plague. Let's insert those numbers into our newly minted Equation $$\ref{thehorror}$$ to calculate the value of a positive diagnosis.
• Dancing plague was assumed to affect one in 1000 people, so $$P(D) = 1/1000$$. Since each person either has or does not have disease, $$P(D) + P(\text{~}D) = 1$$.
• Test sensitivity was 99%. Sensitivity is synonymous with the true positive rate, so this tells us that $$P(+|D) = 0.99$$. And since the test must return either $$+$$ or $$-$$ when disease is present, $$P(-|D) = 1 - 0.99 = 0.01$$.
• Test specificity was 95%. Specificity is synonymous with the true negative rate, so $$P(-|\text{~}D) = 0.95$$. Then $$P(+|\text{~}D) = 1 - 0.95 = 0.05$$.
Inserting these numbers into Equation $$\ref{thehorror}$$ gives
\begin{align} P(D|+) &= \frac{0.99 \cdot \frac{1}{1000}}{0.99 \cdot \frac{1}{1000} + 0.05 \cdot (1 - \frac{1}{1000})} \label{whyareyouevenwritingthis} \\ &= \frac{\frac{0.99}{1000}}{\frac{0.99}{1000} + 0.05 \cdot \frac{999}{1000}} \\ &= 0.01943 \\ &\simeq 1.9 \% \end{align}
As expected, this is the same answer we got in Part One through a less rigorous approach.
### Final Remarks
In this excessively long article, we derived Bayes' Theorem and used it to confirm our earlier reasoning in Part One that a positive diagnosis of dancing plague has only a 2% chance of being correct. This low number is an example of the false positive paradox, and Equation $$\ref{whyareyouevenwritingthis}$$ reveals its origin.
The form of Equation $$\ref{whyareyouevenwritingthis}$$ is [something] divided by [something + other thing], or $$\frac{t}{t + f}$$. If $$f$$ is small compared to $$t$$, then $$\frac{t}{t+f} \simeq \frac{t}{t} = 1$$, which means that the probability of disease given a positive test result is close to 100%. But if $$f$$ becomes much larger than $$t$$, then $$\frac{t}{t+f}$$ becomes much less than 1. Looking at Equations $$\ref{thehorror}$$ and $$\ref{whyareyouevenwritingthis}$$, you can see that $$t$$ matches up with the term $$P(+|D)P(D)$$, the probability of getting a true positive, and $$f$$ matches up with $$P(+|\text{~}D)P(\text{~}D)$$, the probability of getting a false positive. In our example, $$t \simeq 0.001$$ and $$f \simeq 0.05$$. Thus, the chance of getting a false positive is 50 times higher than the chance of getting a true positive. That's why someone who tests positive probably has nothing to worry about, other than the social stigma of getting tested in the first place.
### Cliffhanger Ending
In my next post, I'll explain the Bayesian methodology I used in the course of my involvement with the series To Catch A Killer. Essentially, the above analysis can be adapted to homicide investigations by replacing rare-disease prevalence with the homicide rate for a specific time, place, and demographic, and by treating the presence or absence of forensic evidence as positive or negative diagnostic test outcomes.
## Monday, January 27, 2014
### Bayes' Theorem, Part 1: Not Just a Mnemonic for Apostrophe Placement
If you're intimately familiar with Bayes' Theorem or profoundly bored of it, you may still find value in this post by taking a shot every time you read the words "theorem" and "disease."
I first encountered Bayes' Theorem in a high school conversation about email spam filters. I didn't retain much about either the theorem or spam filters, but promptly added the term "Bayes' Theorem" to my mental list of Things That Sound Vaguely Technical And Also Possibly Sinister. (That list includes the names of every military and/or aerospace contractor that ever existed. If you think of any exceptions, send them my way.)
Years afterward, Bayes' Theorem started cropping up in my medical biophysics studies and after-hours discussions about airport and border security. More recently, I used Bayes' Theorem to weigh forensic evidence in the upcoming documentary series To Catch a Killer. The theorem seems to appear everywhere and makes you sound smart, but just what is it?
Basically, Bayes' Theorem tells you how to update your beliefs using new information. That's the best plain-English definition I can think of. More generally, Bayes' Theorem tells you how to manipulate conditional probabilities, saving you from fallacious logic along the lines of "most Pabst drinkers are hipsters, so most hipsters drink Pabst." (It may be true that most Pabst drinkers are not hipsters, but that's not the point of this fallacy. The lesson for me is that I come up with poor examples.)
Bayes' Theorem follows directly from basic probability principles, but proper derivations tend to look like field notes by Will Hunting on how to outperform pompous Harvard grad students at impressing Minnie Driver. Accordingly, this post shall include zero equations, which is great, since I figured out how to embed equations in my last post. Instead, I'll try to show the importance of Bayes' Theorem by posing the following brain teaser to you, dear reader.
### Brain Teaser: You Tested Positive for a Rare Disease; Do You Really Have It?
Imagine that a disease afflicts 0.1% of the general population, or 1 in 1000 people. A particular diagnostic test returns either "positive" or "negative" to indicate the presence or absence of the disease. Let's say you know that this test is 99% sensitive. That's a compact way of saying that out of 100 people who truly do have the disease, 99 of them will correctly test positive, whereas 1 will erroneously test negative, even though they actually have the disease. Let's also say you know that the test is 95% specific. That means that out of 100 disease-free people, 95 will correctly test negative, but 5 of these healthy people will erroneously be told that they have the disease.
Suppose you run this test on yourself, and sweet buttery Jesus, it says you're positive. This deeply distresses you, as it should if the disease in question were, say, dancing plague. As psychosomatic head-bobbing sets in, you ask yourself the following question: given the positive test result, what are the chances that I actually have dancing plague?
Take another look at those goddamn numbers. The test is 99% sensitive and 95% specific. Should you embrace your groovy fate and invest in a bell-bottomed suit and unnervingly realistic John Travolta mask? Is all hope lost? Is the jig up?!
Think it over and decide on your final answer before reading on. At the very least, don't bother with precise numbers, but decide whether you think the chance of actually having dancing plague is more or less than 50%, given your positive diagnosis.
If you haven't seen this kind of question before, the chance that your answer exceeds 50% exceeds 50%. It turns out that even though you tested positive, the chance that you have the disease is only about 2%! Choreographed celebrations are in order.
### Explanation
You don't actually need to know anything about Bayes' Theorem to correctly answer the above question, though you might end up stepping through the theorem without knowing it. Here's one way to proceed.
Pick a sample of 1000 people from the general population. On average, only 1 of these people will actually have the disease. The vast majority, 999 out of 1000, will be healthy. Our initial sample thus consists of 999 healthy people and 1 sick person. Now, test them all.
Our test is 99% sensitive. That means that when the one diseased guy in our sample gets tested, he'll be correctly identified as sick 99 times out of 100. Very rarely, 1 time in 100, the test will mess up and give him a negative result.
The specificity of 95% means that most healthy people will test negative, as they should. 95% of the initial 999 healthy people, or 949.05 of them on average, will correctly be told that they're disease-free. However, the remaining 49.95 healthy people will erroneously receive positive test results, even though they're fine.
Therefore, by testing each of our starting 1000 people, we'd find an average of 0.99 correct positive diagnoses and 49.95 incorrect positive diagnoses, giving 50.94 positive diagnoses in total. Rounding off the numbers, it's obvious that about 51 people in our initial 1000 will be freaked out by positive test results. However, only one of these people will actually have the disease.
If you test positive, you could be any one of those 51 people, so try not to panic: the chance that you're the one person who actually has dancing plague is 1/51, or 1.96%.
### Final Remarks
What was that about Bayes' Theorem helping to update your beliefs? "Belief" refers to one possible way to interpret what it means for a random outcome to have some numerically determined chance of occurring. In the above disease example, it's sensible to think of the chance that someone is ill as a measure of how firmly you believe that they're ill.
If you randomly chose one person from the general population and didn't test them, you'd be pretty skeptical that they're ill, since the disease is so rare. The chance that you picked someone with the disease is 1/1000. Running the test then gives you new information -- specifically, the test outcome. That outcome is sometimes wrong, but you can still use the new information to update your prior belief that the person has the disease. If the person tests positive, your belief just jumped from a prior value of 1/1000 to a "posterior" value of 1/51, a 20-fold increase.
### Cliffhanger Ending
In a future post, we'll derive Bayes' Theorem and show how it applies to this and other problems. Until next time!
EDIT: Part 2 is here.
## Tuesday, January 7, 2014
### Typesetting Equations in Web Pages with MathJax
One blog feature that should have influenced my choice of blogging platform is the ability to typeset equations. In this imperfect world, I didn't think that far ahead, so I chose Blogger out of familiarity with Google's ways, and for the convenience of having created an account in 2011 that's lain dormant until recently. Fortunately, the equation-writing solution that I test in this post should work for just about any web document out there.
I'm sporadically active on the Astronomy Picture of the Day (APOD) forums, where discussion occasionally spirals into the astrophysical realm and waxes technical. That's where I first encountered the need to typeset equations online. I resorted to a quick and dirty solution:
1. Decide whether laziness trumps steps 2-4.
2. Use a LaTeX-to-image converter such as LaTeXiT to generate equation images.
3. Upload images to university webspace.
4. Embed images inline with text.
LaTeX is a mathematical markup language that's as common and useful in research circles as actual latex is in meatspace. There's a lot of general information out there about LaTeX, so here, I'll just focus on getting it to work in Blogger posts. At the risk of overstretching the bizarre dating/relationship theme that seems to pervade everything I write, introducing LaTeX on the second date post demonstrates the unrealistic optimism that it'll actually come in handy so the user won't have to. Okay, fun's over: it's pronounced LAH-tek or LAY-tek, probably in reference to the last name of LaTeX's creator, Leslie Lamport. (But how is that pronounced...?)
A cursory web search identified MathJax as the most magic-like way to type posts in LaTeX and see equations across all major browsers. I'd heard of MathML before, but since I'm so used to LaTeX already, MathJax looked way too easy to pass up, and it is. All the end user needs to do is to include the MathJax JavaScript library in their HTML via the following snippet [per official instructions]:
<script type="text/javascript"
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
(That's the secure snippet, which imports the script over HTTPS. An HTTP variant is provided through the above official instructions link for those who'd rather flirt with men in the middle.)
After embedding the above script in your HTML, everything should be hunky dory. Hack up some LaTeX in your HTML view; the default math delimiters for inline and paragraph equations are $$...$$ and $...$. So, this string:
$\nabla \times \mathbf E = - \frac{\partial \mathbf B}{\partial t}$
gives you this:
$\nabla \times \mathbf E = - \frac{\partial \mathbf B}{\partial t}$
Huge thanks go out to the MathJax team, and to you, dear reader, for reading all the way to the end of this healthcare.gov of a post. Until next time!
tl;dr: including a MathJax script in your web page lets you type LaTeX directly in HTML.
## Thursday, January 2, 2014
### Hello, world!
Thought I'd try out this newfangled "web log" fad that's been sweeping the BBSes! Wow, I hope my second sentence isn't that sarcastic.
My primary aim for this blog is to consolidate and share ideas that don't seem to fit elsewhere (i.e., anywhere), and to keep a publicly accessible electronic record of selected long-term projects. In other words, I hope that using this blog as an open lab notebook will force myself to maintain guilt-fueled progress on stuff I write about by fabricating a pervasive sense of accountability to a merciless multitude of silent and/or imaginary readers. Updating frequently enough should also give me some much-needed practice at putting together word series that look good and then doing this again multiple times per minute with different words.
Coming soon: geographic profiling of serial homicide! Statistical crime analysis! A personalized, curse-laden introduction to the Google Maps API! Blood flow in arteries! Sundry language quirks! Comments from spammers, probably! Less sensationalism.
Hoping that signing off like this ends up looking perfectly natural and savvy,
Peter | {"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": 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": 9, "x-ck12": 0, "texerror": 0, "math_score": 0.9838539958000183, "perplexity": 773.4893719566909}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262600.90/warc/CC-MAIN-20190527125825-20190527151825-00066.warc.gz"} |
https://www.physicsforums.com/threads/what-is-eulers-identity-really-saying.603246/ | # What is Euler's identity really saying?
1. May 4, 2012
So it is true that ei∏+1=0. But what does this mean? Why are all these numbers linked?
2. May 4, 2012
### DonAntonio
They are linked precisely by that equation, and since the equality $e^{i\theta}:=\cos\theta+i\sin\theta\,\,,\,\,\theta\in\mathbb{R}\,\,$ follows at once say from the definition
of the complex exponential function as power series (or as limit of a sequence), the above identity is really trivial.
DonAntonio
3. May 4, 2012
### HallsofIvy
Staff Emeritus
Look at the MacLaurin series for those functions:
$$e^x= 1+ x+ x^2/2!+ x^3/3!+ \cdot\cdot\cdot+ x^n/n!$$
$$cos(x)= 1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot+ (-1)^nx^{2n}/(2n)!$$
$$sin(x)= x- x^3/3!+ x^5/5!- x^7/7!+ \cdot\cdot\cdot+ (-1)^nx^{2n+1}/(2n)!$$
If you replace x with the imaginary number ix (x is still real) that becomes
$$e^{ix}= 1+ ix+ (ix)^2/2!+ (ix)^3/3!+ \cdot\cdot\cdot+ (ix)^n/n!$$
$$e^{ix}= 1+ ix+ i^2x^2/2!+ i^3x^3/3!+ \cdot\cdot\cdot+ i^nx^n/n!$$
But it is easy to see that, since $i^2= -1$, $(i)^3= (i)^2(i)= -i$, $(i)^4= (i^3)(i)= -i(i)= -(-1)= 1$ so then it starts all over: $i^5= (i^5)i= i$, etc. That is, all even powers of i are 1 if the power is 0 mod 4 and -1 if it is 2 mod 4. All odd powers are i if the power is 1 mod 4 and -i if it is 3 mod 4.
$$e^{ix}= 1+ ix- x^2/2!- ix^3/3!+ \cdot\cdot\cdot$$
Separating into real and imaginary parts,
$$e^{ix}= (1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot)+ i(x- x^3/3!+ x^5/5!+ \cdot\cdot\cdot)$$
$$e^{ix}= cos(x)+ i sin(x)$$
Now, take $= \pi$ so that $cos(x)= cos(\pi)= -1$ and $sin(x)= sin(\pi)= 0$ and that becomes
$$e^{i\pi}= -1$$
or
$$e^{i\pi}+ 1= 0$$
I hope that is what you are looking for. Otherwise, what you are asking is uncomfortably close to "number mysticism".
Last edited: May 4, 2012
4. May 5, 2012 | {"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": 0.9676054120063782, "perplexity": 503.87429191183463}, "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/1502886104631.25/warc/CC-MAIN-20170818082911-20170818102911-00121.warc.gz"} |
http://crypto.stackexchange.com/questions/2412/finding-the-lfsr-and-connection-polynomial-for-binary-sequence | # Finding the LFSR and connection polynomial for binary sequence.
I have written a C implementation of the Berlekamp-Massey algorithm to work on finite fields of size any prime. It works on most input, except for the following binary GF(2) sequence:
$0110010101101$ producing LFSR $\langle{}7, 1 + x^3 + x^4 + x^6\rangle{}$
i.e. coefficients $c_1 = 0, c_2 = 0, c_3 = 1, c_4 = 1, c_5 = 0, c_6 = 1, c_7 = 0$
However, when using the recurrence relation
$$s_j = (c_1s_{j-1} + c_2s_{j-2} + \cdots + c_Ls_{j-L}) \mbox{ for } j \geq L.$$ to check the result, I get back:
0110010001111, which is obviously not right.
Using the Berlekamp-Massey Algorithm calculator they say the (I believe) characteristic polynomial should be $x^7 + x^4 + x^3 + x^1$. Which, according to my paper working, the reciprocal should indeed be $1 + x^3 + x^4 + x^6$.
What am I doing wrong? Where is my understanding lacking?
-
This is probably just a difference in notation than any failure in understanding or implementation. Some people define recurrence relations with subscripts in reverse order than others; the original description given by Berlekamp in his 1968 book Algebraic Coding Theory began counting from $1$ instead of $0$ etc. Observe that $$x^7 + x^4 + x^3 + x^1 = x^7(1 + x^{-3} + x^{-4} + x^{-6})$$ in comparison to your $1 + x^3 + x^4 + x^6$ which you say is the correct reciprocal of what the web site's answer should be. So I would say that the web site seems to be following Berlekamp's original description and giving you an answer that is "off-by-one". | {"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": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.8505645990371704, "perplexity": 490.08344508047327}, "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-2013-48/segments/1386163051509/warc/CC-MAIN-20131204131731-00022-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://cms.math.ca/cmb/kw/generalized%20numerical%20range | location: Publications → journals
Search results
Search: All articles in the CMB digital archive with keyword generalized numerical range
Expand all Collapse all Results 1 - 2 of 2
1. CMB 2008 (vol 51 pp. 86)
Nakazato, Hiroshi; Bebiano, Natália; Providência, Jo\ ao da
The Numerical Range of 2-Dimensional Krein Space Operators The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained. Keywords:numerical range, generalized numerical range, indefinite inner product spaceCategories:15A60, 15A63, 15A45
2. CMB 2000 (vol 43 pp. 448)
Li, Chi-Kwong; Zaharia, Alexandru
Nonconvexity of the Generalized Numerical Range Associated with the Principal Character Suppose $m$ and $n$ are integers such that $1 \le m \le n$. For a subgroup $H$ of the symmetric group $S_m$ of degree $m$, consider the {\it generalized matrix function} on $m\times m$ matrices $B = (b_{ij})$ defined by $d^H(B) = \sum_{\sigma \in H} \prod_{j=1}^m b_{j\sigma(j)}$ and the {\it generalized numerical range} of an $n\times n$ complex matrix $A$ associated with $d^H$ defined by $$\wmp(A) = \{d^H (X^*AX): X \text{ is } n \times m \text{ such that } X^*X = I_m\}.$$ It is known that $\wmp(A)$ is convex if $m = 1$ or if $m = n = 2$. We show that there exist normal matrices $A$ for which $\wmp(A)$ is not convex if $3 \le m \le n$. Moreover, for $m = 2 < n$, we prove that a normal matrix $A$ with eigenvalues lying on a straight line has convex $\wmp(A)$ if and only if $\nu A$ is Hermitian for some nonzero $\nu \in \IC$. These results extend those of Hu, Hurley and Tam, who studied the special case when $2 \le m \le 3 \le n$ and $H = S_m$. Keywords:convexity, generalized numerical range, matricesCategory:15A60 | {"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": 0.9907617568969727, "perplexity": 601.8911659376912}, "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-2014-41/segments/1410657134514.93/warc/CC-MAIN-20140914011214-00299-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/89761/restriction-of-irreducible-representations | # Restriction of irreducible representations
Given a locally compact group $G$ with a compact subgroup $K$.
Assume we are given two irreducible, infinite dimensional, admissible representations $\pi$ and $\pi'$ of $G$.
What are examples, where $Res_K \pi$ and $Res_K \pi'$ are not isomorphic $K$ representation except for a finite dimensional representation?
Admissible in this context means that $Res_K \pi$ decomposes with finite multiplicity.
I would be mostly interested in examples of reductive groups over local fields. Also what will happen, if we replace irreducible by indecomposable?
If $G$ is connected, let $K$ be its maximal compact subgroup. If $G$ is totally disconnected, let $K$ be open.
Are there examples, where nuclear representations with linearly independant characters become isomorphic after restriction to $K$?
-
What does that mean, "isomorphic up to a finite dimensional representation". Does it mean that there is a morphism from one representation to the other with finite dimensional kernel and cockerel? That can't be right, because that there would be many trivial answers to your question, like $\pi$ any infinite dimensional representation (irreducible admissible) and $\pi'$ the trivial representation. – Joël Feb 28 '12 at 13:56
pm, Joël, perhaps "up to a finite dimensional representation" means the multiplicities of the $K$-types are different for only finitely many $K$-types. Of course, there are going to be trivial counter-examples when the representations are not infinite-dimensional. – B R Feb 28 '12 at 15:09
Yes, I am interested in infinite dimensional representations only. – Marc Palm Feb 28 '12 at 15:12
If Archimedean local fields are ok, then the simplest example probably occurs with $G=GL(2, \mathbb R)$ and $K=SO(2, \mathbb R).$ The irreducible representations of $K$ are in bijection with the integers. One can construct representations of $GL(2, \mathbb R)$ such that the set of $K$-types is the set of odd integers, the set of even integers, the set of odd integers with absolute value at least (or at most) $2n+1,$ or the set of even integers with absolute value at least (or at most) $2n.$ In particular, one can construct two representations such that the restrictions to $K$ differ by deleting a single representation of $K.$
In a bit more detail, take $s=(s_1, s_2)$ a pair of complex numbers and $\omega=(\omega_1, \omega_2)$ where for $i=1,2,$ $\omega_i$ is either trivial or the sign character. Consider smooth functions $GL(2, \mathbb R) \to \mathbb C$ such that $$f\left( \begin{pmatrix} a& b \\ 0& d \end{pmatrix} g \right) = |a|^{s_1} \omega_1(a) |d|^{s_2} \omega_2(d) f(g).$$ The $K$-type corresponding to an integer $n$ will appear in this representation if and only if $(-1)^n = \omega_1(-1)\omega_2(-1).$ (And all have multiplicity one.) You get reducibility when $s_1-s_2$ is an integer $m$ with this parity. If $m$ is positive there is a subrepresentation which lives on the $K$-types $\ge m$ in absolute value. If $m$ is nonpositive, there is a subrepresentation which lives on the $K$-types $\le -m$ in absolute value.
For obvious reasons, the reference I know best is the book Goldfeld and I wrote. We discuss reducibility of $(\mathfrak{g}, K)$-modules on p. 267. On p. 332, we give a fairly elementary description of each invariant subspace looks like in the representation of $GL(2, \mathbb R)$ on smooth functions. I see another reference in the comments as well.
Adding: in the $p$-adic case a similar phenomenon occurs and may be easier to see. Take $F$ a nonarchimedean local field with ring of integers $\mathfrak{o}.$ Take $G=GL(2, F)$ and $K=GL(2, \mathfrak{o}).$ Consider the representation $Ind_B^G \chi$ parabolically induced from a character of the Borel subgroup $B$: $$\chi\begin{pmatrix} a& b \\ 0& d \end{pmatrix} = |a|^{s_1}|d|^{s_2}.$$ As a $K$-module $Ind_B^G \chi$ is isomorphic to $Ind_{B\cap K}^K 1$ for all $s_1, s_2.$ (Here "$1$" means the one dimensional trivial representation of $B \cap K$.) As a $G$-module $Ind_B^G \chi$ is irreducible for most values of $s_1, s_2,$ but if $s_1=s_2$ then the subspace of $Ind_{B\cap K}^K 1$ corresponding to the trivial representation of $K$ is actually a $G$-invariant subspace spanned by the function $g \mapsto |\det g|^{s_1}.$
-
This example is okay: Can you please write down the representation in terms of induced representation, so that I can check this myself. I guess via restriction by steps, this is remains true for restriction to $O(2)$. This example you mention, does it stem from the fact $GL(2) = GL(2)^+ \rtimes \{ \pm 1 \}$? – Marc Palm Feb 28 '12 at 17:03
You can find this series of examples worked out completely in Howe and Tan's book on non-abelian harmonic analysis. – Victor Protsak Feb 28 '12 at 19:22
pm, it works for $SL(2,\mathbb R)$. It stems from the fact that the maximal torus of $SL(2,\mathbb R)$ is $\mathbb R^\times\simeq \mathbb R_{>0}\times\lbrace\pm 1\rbrace$. – B R Feb 29 '12 at 0:09 | {"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": 0.8915992975234985, "perplexity": 142.65069737268612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275645.9/warc/CC-MAIN-20160524002115-00168-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://socratic.org/questions/triangle-a-has-an-area-of-8-and-two-sides-of-lengths-8-and-7-triangle-b-is-simil | Geometry
Topics
# Triangle A has an area of 8 and two sides of lengths 8 and 7 . Triangle B is similar to triangle A and has a side with a length of 16 . What are the maximum and minimum possible areas of triangle B?
Maximum Area $= 361.28 \text{ }$square units
Minimum Area $= 9.29514 \text{ }$square units
#### Explanation:
I computed all possible triangles and there are 2 possible triangles for A and 6 possible triangles for B. Then I computed the area for each triangle to determine the maximum and minimum areas.
For first triangle A:
sides $a = 8$ , $b = 7$ , $c = 2.3809 \text{ }$,angle $C = {16.6015}^{\circ}$
For first triangle B:
sides $a ' = 16$, $\text{ } b ' = 14$, $\text{ } c ' = 4.76182$,angle $C = {16.6015}^{\circ}$,
Area$= 32$
sides $a ' ' = \frac{128}{7}$,$\text{ } b ' ' = 16$, $\text{ } c ' ' = 5.44208$,angle $C = {16.6015}^{\circ}$,
Area$= 41.7959$
sides $a ' ' ' = 53.7609$,$\text{ } b ' ' ' = 47.0408$,$\text{ } c ' ' ' = 16$,angle $C = {16.6015}^{\circ}$,
Area$= 361.28 \text{ }$Maximum Area
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For second triangle A:
sides $a = 8$ , $b = 7$ , $c = 14.8436 \text{ }$,angle $C = {163.398}^{\circ}$
For first triangle B:
sides $a ' = 16$, $\text{ } b ' = 14$, $\text{ } c ' = 29.6871$,angle $C = {163.398}^{\circ}$,
Area$= 32$
sides $a ' ' = \frac{128}{7}$,$\text{ } b ' ' = 16$, $\text{ } c ' ' = 33.9281$,
angle $C = {163.398}^{\circ}$,
Area$= 41.798$
sides $a ' ' ' = 8.62327$,$\text{ } b ' ' ' = 7.54536$,$\text{ } c ' ' ' = 16$,
angle $C = {163.398}^{\circ}$,
Area$= 9.29514 \text{ }$Minimum Area
God bless....I hope the explanation is useful.
##### Impact of this question
531 views around the world
You can reuse this answer | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 40, "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": 0.9134165644645691, "perplexity": 3217.5613678550585}, "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-2022-27/segments/1656104248623.69/warc/CC-MAIN-20220703164826-20220703194826-00498.warc.gz"} |
https://math.stackexchange.com/questions/549292/lower-bound-on-maximal-value | # Lower bound on maximal value
A bag contains $n$ items with different values. The total value is $1$. I am allowed to pick one item, and pick the item with the maximal value. Obviously, in the worst case I will get a value of $1 \over n$.
Now, suppose the bag is shaken so that some items break, and their value decreases. Specifically, there are $n$ known constants, $p_1,...,p_n \in (0,1]$, such that the value of item $i$ is now $p_i v_i$ (where $v_i$ is the original value). What is now the value I can get in the worst case (as a function of the $p_i$'s)?
Formally, I look for a lower bound on the solution of the following maximization problem:
$$\max_{i=1..n} p_i v_i$$ $$s.t. \sum_{i=1..n} v_i = 1$$
The lower bound is a function of the parameters $p_1,...,p_n$.
EXAMPLE: Suppose $n=2$, $p_1=1$, $p_2=0.5$. My value is the maximum between $v_1$ and $0.5 v_2 = 0.5(1-v_1)$. The worst case is when $v_1 = {1 \over 3}$. In this case the two expressions are equal, and my value is ${1 \over 3}$.
SOME BOUNDS:
• If only a single item has a value (i.e. $v_i=1$ for some $i$), then in the worst case my value will be $p_i$. So in general, the lower bound is at most $\min_{i} {p_i}$.
• If all items have the same value (i.e. $v_i={1 \over n}$ for all $i$), then in the worst case my value will be $\max_{i} {p_i \over n}$. So in general, the lower bound is at most ${\max_{i} p_i} \over {n}$
These two bounds are not tight: in the example I gave above, these bounds are both $1 \over 2$, while the actual lower bound is $1 \over 3$. Can you find a tight lower bound?
The worst case distribution of $v_i$ is when the values of the items equalise as you noticed, which is when each $v_i$ is inversely proportional to $p_i$, i.e. when $$v_i = \dfrac{\frac1{p_i}}{\sum \frac1{p_i}}$$
Clearly for this distribution $p_i v_i = \dfrac{1}{\sum \frac1{p_i}}$ for all $i$ and you have the desired lower bound.
In the specific example you mention, this gives $\dfrac{1}{\sum \frac1{p_i}} = \frac{1}{1+2} = \frac13$. | {"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": 0.9896422624588013, "perplexity": 76.44376621951041}, "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/1585370495413.19/warc/CC-MAIN-20200329171027-20200329201027-00537.warc.gz"} |
http://www.openwatcom.org/index.php/Search_Paths | # Search Paths
## Introduction
This page concerns search paths in wgml and gendev. As such, it will explore several topics:
• the documented search paths;
• the actual search paths;
• how the search paths are used; and
• what this implies for this project.
## Documented Search Paths
This is what the WGML Reference has to say:
Section 9.55 INCLUDE (wgml) and Section 15.6.2 INCLUDE (gendev):
```When working on a PC/DOS system, the DOS environment symbol
GMLINC may be set with an include file list. This symbol is
defined in the same way as a library definition list (see
"Defining a Library List" on page 297), and provides a list
of alternate directories for file inclusion. If an included
file is not defined in the current directory, the directories
specified by the include path list are searched for the file.
the DOS environment symbol PATH are searched.
```
This defines the intended use of the environment variable GMLINC: it is intended to tell wgml where to look when an :INCLUDE tag is encountered. The search path is:
• the current directory
• the directories given in GMLINC
• the directories given in PATH
Note that "include path list" here refers to the contents of GMLINC.
The README file produceable from the WGML 3.33 Update has this note relating to this path in section "4.4 New GML Features":
```The device library is now searched before the DOS PATH
when trying to open and include file.
```
So, the include path list would now be:
• the current directory
• the directories given in GMLINC
• the directories given in GMLLIB
• the directories given in PATH
Section 14 Running WATCOM Script/GML:
```The option file "default" is located and loaded before other
options are processed. The search path for the default option
file is the current disk location, the device library path,
followed by the document include path.
```
The "default" option file is an option file named "default.opt" which is always included if found. The search path is:
• the current directory
• the device library path
• the document include path
This is not clear: if the "document include path" includes the entire path used with the :INCLUDE tag, then both the current directory and the directories provided by GMLLIB are searched twice; if it only includes GMLINC, then the PATH directory list is ignored.
Section 14.3.6 DEVice:
```When working on a PC/DOS system, the DOS environment symbol
GMLLIB is used to locate the device information (see
"Libraries with IBM PC/DOS" on page 297). If the device
see"INCLUDE" on page 102).
```
This, then, is the search path for binary device libraries:
• the directories given in GMLLIB
• the document include path
Here, the "document include path" would make more sense if it were applied to the entire path used with the :INCLUDE tag. Or, at least, it would if this did not entail searching the directories in GMLLIB a second time!
Section 14.3.8 FILE:
```When working on a PC/DOS system, the DOS environment symbol
GMLLIB is used to locate the command file if it is not in the
current directory (see "Libraries with IBM PC/DOS" on page
297). If it is still not found, the document include path is
searched (see "INCLUDE" on page 102).
```
This, then, is the search path for non-default option files:
• the current directory
• the directories given in GMLLIB
• the document include path
Here, the "document include path" would make more sense if it were applied to the entire path used with the :INCLUDE tag. Or, at least, it would if this did not entail searching the directories in GMLLIB a second time!
So, how many paths are there, per the documentation? I suggest that there are three (source files, option files, and binary device libraries), but what they are depends on how "the document include path" and "the device library include path" are interpreted.
If "the document include path" is taken to refer solely to the contents of GMLINC and "the device library include path" to the contents of GMLLIB, then we have:
• for source files (.gml, .pcd, .fon, plus any extension specified by the user):
```the current directory
the directories given in GMLINC
the directories given in GMLLIB
the directories given in PATH
```
• for option files, including "default.opt":
```the current directory
the directories given in GMLLIB
the directories given in GMLINC
```
• for binary device libraries:
```the directories given in GMLLIB
the current directory
the directories given in GMLINC
```
If "the document include path" is taken to all the directories searched for document specification files and "the device library include path" to all the directories searched for binary device libraries, then we have:
• for source files (.gml, .pcd, .fon, plus any extension specified by the user):
```the current directory
the directories given in GMLINC
the directories given in GMLLIB
the directories given in PATH
```
• for the "default.opt" file:
```the current directory
the directories given in GMLLIB ("device library path")
the current directory ("device library path")
the directories given in GMLINC ("device library path")
the directories given in GMLLIB ("device library path")
the directories given in PATH ("device library path")
the current directory ("document include path")
the directories given in GMLINC ("document include path")
the directories given in GMLLIB ("document include path")
the directories given in PATH ("document include path")
```
• for option files:
```the current directory
the directories given in GMLLIB
the current directory ("document include path")
the directories given in GMLINC ("document include path")
the directories given in GMLLIB ("document include path")
the directories given in PATH ("document include path")
```
• for binary device libraries:
```the directories given in GMLLIB
the current directory ("document include path")
the directories given in GMLINC ("document include path")
the directories given in GMLLIB ("document include path")
the directories given in PATH ("document include path")
```
If we remove the duplicates from the second set, then we have:
• for source files (.gml, .pcd, .fon, plus any extension specified by the user):
```the current directory
the directories given in GMLINC
the directories given in GMLLIB
the directories given in PATH
```
• for option files, including "default.opt":
```the current directory
the directories given in GMLLIB
the directories given in GMLINC
the directories given in PATH
```
• for binary device libraries:
```the directories given in GMLLIB
the current directory
the directories given in GMLINC
the directories given in PATH
```
Interestingly, the filenames given on the command lines to gendev and wgml were not mentioned in any section found in the WGML Reference by searching on "path". Direct investigation finds this information:
Section 14 Running WATCOM Script/GML:
```The "file-name" specifies the file containing the source text
and GML tags for the document. If the file type part of the
file name (see "Files" on page 281) is not specified, WATCOM
Script/GML searches for source files with the alternate file
extension followed by the file type of GML. When a file type
is specified, WATCOM Script/GML searches for source files
with that file type.
```
Section 16 Running WATCOM GENDEV:
```The "file-name" specifies the file containing the device,
font and/or driver definitions. If the file type part of the
file name (see "Files" on page 281) is not specified, WATCOM
GENDEV searches for source files with the default file type
for device and driver definitions. The font definition file
type is the default alternate extension.
```
Exactly where these searches are conducted is not specified. The most likely search path used is the source file path, which is the one used with the :INCLUDE tag.
So, the situation is unclear, and testing is called for, especially since, even with the aid of the README, this is for version 3.3, and we are using version 4.0 of wgml and version 4.1 of gendev.
## Actual Search Paths
It might be wondered why it is important that our gendev and wgml duplicate the search paths used by gendev 4.1 and wgml 4.0. The theoretical reason is quite simple: if different files with the same name exist in directories which are listed in different environment variables, then the order of search will determine which of those files is used. Since our wgml is intended to produce the same output as wgml 4.0, it would be helpful if it used the same input files. Whether this applies to the Open Watcom document build system is not, at present, known.
### Search Targets
The various documents available suggest that wgml 4.0 searches for eleven different types of files:
• the source file given on the command line (wgml/source)
• an option file given on the command line (wgml/opt)
• a layout file given on the command line (wgml/lay)
• a source file named in an :IMBED block (wgml/imbed)
• a source file named in an :INCLUDE block (wgml/include)
• a source file named in an .AP block (wgml/.ap)
• a source file named in an .IM block (wgml/.im)
• the default option file (wgml/default)
• a device library file (wgml/cop)
• a binary file named in a :BINCLUDE block (wgml/binary)
• a graphic file named in a :GRAPHIC block (wgml/graphic)
Several objections might be raised to this list:
• The tag :IMBED is documented as being the same as :INCLUDE except for being allowed only in the pre-:GDOC section of the document specification (which turns out to be incorrect for wgml 4.0: :IMBED works just fine in the body of the document).
• .IM is short for "IMBED" (the odd spelling is used in place of "EMBED" because the digraph ".EM" was already taken)
• Two blocks naming source files (:MAILMERGE and :VALUESET, which are documented as being equivalents) are ignored
As to the last point, :MAILMERGE and :VALUESET are related to using wgml to produce correspondence, which is not our goal. Indeed, even if we were to release our version (some future version), it would probably not include the ability to produce correspondence; many other programs do that.
gendev 4.1 searches for two different types of files:
• the source file given on the command line (gendev/source)
• a source file named in an :INCLUDE block (gendev/include)
This may be a bit surprising, since the message list for gendev includes this message:
```CL--003 Missing or invalid command filename
```
but using "FILE" as an option produces the error:
```CL--001: Invalid option in command line
```
and no gendev command line option for a command file is documented in the WGML Reference. The implication is that gendev 4.1 does not use user-supplied option files.
To investigate whether or not gendev 4.1 searches for a file named "default.opt", I first invoked gendev with "( incl", which is documented to cause it to list each file as it is included. The file used had no :INCLUDE statements, but was itself listed when "incl" was given, and not listed when it was not. I then placed the command line into a "default.opt" file in the same directory that gendev 4.1 was invoked in. The results clearly showed that the file was not found, which implies that it is not searched for.
It may also be surprising that gendev 4.1 does not search for binary device libraries or files, since it creates them and creates or rewrites wgmlst.cop (the directory file) and so must surely search for that. However, if gendev 4.1 is invoked in a directory which is not listed in GMLLIB, then this warning is produced:
```SN--082: Current disk location and library path do not match
```
and the library is produced anyway.
So the situation is not that gendev 4.1 does not search for library files; the situation is that gendev 4.1 only looks in the directory it is invoked in -- and that it expects that directory to be listed in GMLLIB. Well, it does if GMLLIB is defined: if GMLLIB does not exist, no error message appears.
### Individual Location Tests
Now that the situations to test have been identified, let us start with two tests:
1. whether or not, for the current directory and each environment variable separately, the file type is found by the program only when it is in the location indicated; and
2. whether or not, for environment variable path lists written left-to-right in a locale which generally writes left-to-right, the directories are searched left-to-right (LtoR) or right-to-left (RtoL).
``` curdir GMLLIB GMLINC PATH
wgml/source yes LtoR LtoR LtoR
wgml/opt yes LtoR LtoR LtoR
wgml/lay yes LtoR LtoR LtoR
wgml/imbed yes LtoR LtoR LtoR
wgml/include yes LtoR LtoR LtoR
wgml/.ap yes LtoR LtoR LtoR
wgml/.im yes LtoR LtoR LtoR
wgml/default yes LtoR LtoR LtoR
wgml/cop no LtoR LtoR LtoR
wgml/binary yes LtoR LtoR LtoR
wgml/graphic yes LtoR LtoR LtoR
gendev/source yes no LtoR no
gendev/include yes no LtoR no
```
The layout file was found not only when named with the LAYOUT option, but also when named with the .AP and .IM control words and the :IMBED and :INCLUDE tags placed before :GDOC.
The SCRIPT control words, of course, only work when SCRIPT or WSCRIPT is given on the command line or in an option file.
The result shown for wgml/cop using the current directory only appears to be correct: not defining GMLLIB or defining it to point at a directory not containing the device library which includes the specified device (whether that directory contains a valid library or not) produces this error message:
```IO--008: For the device (or font) 'test':
The information file for this name cannot be found.
If the device/font has been defined, the problem may
be that the DOS SET symbol GMLLIB has not been
correctly set to point to the device library.
```
which is consistent with this statement in the WGML Reference:
```To locate the library, WATCOM Script/GML and WATCOM GENDEV
must have a list of library directories. This list is defined
with the DOS SET command.
```
This applies not only to the directory file but to the device, driver and font files as well. When GMLLIB is set to "." or ".\", then the device is found.
The device is also found when GMLLIB does not exist but GMLINC contains the directory of a library the directory file of which contains an entry for the defined name of the device. So the current directory is, in fact, skipped, and not checked automatically. In ow\docs\mif\onebook.mif, the value assigned to GMLINC for the help file build (as opposed to PS) starts with ".;", thus ensuring that the current directory is always searched.
The tag :BINCLUDE was tested with a text binary file and worked as described in the WGML Reference. The last line was included whether it ended with a newline character or not.
The tag :GRAPHIC was tested using the PS device and an EPS graphic file created from a .BMP file. It took a while to get a file that GhostView was happy with:
1. the :BINCLUDE tag had to be commented out; apparently, the file it included had lines so long that they made the PS invalid;
2. the file EZAMBLE.PS had to be (copied to the test directory and) prepended to the output file; otherwise, GhostScript regarded the file as an EPS file.
The second point, however, only applies to the PS device from the WGML 3.33 Update. When GMLLIB was pointed at ow\docs\gml\syslib, wgml produced a file which GhostView not only recognized as a PS file but which it could convert to a PDF file which Acrobat could display.
The results shown for gendev/source and gendev/include for GMLLIB and PATH are surprising: they suggest that neither GMLLIB nor PATH is searched by gendev for these files.
### Location Pair Testing
There are a total of six pairs to test; for easy viewing, the results will be spread over two tables. In this tables, "N/A" will be entered when either (or both) member of the pair is not used. When both pairs are used, then the member of the pair which is searched first will be entered.
``` curdir/GMLLIB curdir/GMLINC curdir/PATH
wgml/source curdir curdir curdir
wgml/opt curdir curdir curdir
wgml/lay curdir curdir curdir
wgml/imbed curdir curdir curdir
wgml/include curdir curdir curdir
wgml/.ap curdir curdir curdir
wgml/.im curdir curdir curdir
wgml/default curdir curdir curdir
wgml/cop N/A N/A N/A
wgml/binary curdir curdir curdir
wgml/graphic curdir curdir curdir
gendev/source N/A curdir N/A
gendev/include N/A curdir N/A
```
``` GMLLIB/GMLINC GMLLIB/PATH GMLINC/PATH
wgml/source GMLINC GMLLIB GMLINC
wgml/opt GMLLIB GMLLIB GMLINC
wgml/lay GMLINC GMLLIB GMLINC
wgml/imbed GMLINC GMLLIB GMLINC
wgml/include GMLINC GMLLIB GMLINC
wgml/.ap GMLINC GMLLIB GMLINC
wgml/.im GMLINC GMLLIB GMLINC
wgml/default GMLLIB GMLLIB GMLINC
wgml/cop GMLLIB GMLLIB GMLINC
wgml/binary GMLINC GMLLIB GMLINC
wgml/graphic GMLINC GMLLIB GMLINC
gendev/source N/A N/A N/A
gendev/include N/A N/A N/A
```
### Search Paths Observed
This section lists the search paths which are actually used, based on the above tests.
For wgml/source, wgml/lay, wgml/ap, wgml/im, wgml/imbed, wgml/include, wgml/binary, and wgml/graphic:
1. the current directory
2. any directories listed in the GMLINC environment variable
3. any directories listed in the GMLLIB environment variable
4. any directories listed in the PATH environment variable
For wgml/opt and wgml/default:
1. the current directory
2. any directories listed in the GMLLIB environment variable
3. any directories listed in the GMLINC environment variable
4. any directories listed in the PATH environment variable
For wgml/cop:
1. any directories listed in the GMLLIB environment variable
2. any directories listed in the GMLINC environment variable
3. any directories listed in the PATH environment variable
For gendev/source and gendev/include:
1. the current directory
2. any directories listed in the GMLINC environment variable
## Using Search Paths
Now that the search paths have been identified, how are they used? This turns out to be a bit more complicated than might be thought.
Note: The DOS version of wgml 4.0, at least, imposes a limit on the length of the value of the environment variables. Relative paths can produce a useable setup where absolute paths produce "File Not Found" errors. Of course, using relative paths restricts the directories in which wgml 4.0 can be used to those with respect to which the relative paths make sense.
### Finding One File
This is the most common use: the search path is followed until a file whose name matches the filename given is found. This will normally be the first match, and it will hide any other matches in locations further down the path.
Both gendev and wgml allow the use of multiple extensions. The WGML Reference discusses this topic in several locations.
Section 14.1.3 IBM PC/DOS Specifics:
```The following default file types are used by WATCOM Script/GML:
File Type Usage
GML document source files
LAY layout files created with the :save tag
OPT command files
VAL value files specified by the VALUESET command line
option
```
Section 16 Running WATCOM GENDEV:
```File Type Definition (IBM PC/DOS)
PCD default file type for the device and driver definition.
FON default file type for the font definition.
COP default file type for the created member name.
```
Since the mail-merge/correspondence capabilities of wgml are not being recreated (as mentioned above) and the .LAY extension is for files which wgml creates (all layout files in the Open Watcom document build system use extension .GML), these are the default extensions we need to consider:
```File Type Usage
GML document source files
OPT command files
PCD device and driver definition source files
FON font definition source files
COP binary device definition files
```
Using this list, it is possible to make the other statements in the WGML Reference clearer.
The first source file sought is the one given on the command line:
Section 14 Running WATCOM Script/GML:
```The "file-name" specifies the file containing the source text
and GML tags for the document. If the file type part of the
file name (see "Files" on page 281) is not specified, WATCOM
Script/GML searches for source files with the alternate file
extension followed by the file type of GML. When a file type
is specified, WATCOM Script/GML searches for source files
with that file type.
```
which appears to me to be saying that wgml looks for:
• the specified extension
or
• the alternate extension
• the extension .GML
Note that the first option is presumed to apply in all cases below, although it is only stated explicitly here.
Section 16 Running WATCOM GENDEV
```The "file-name" specifies the file containing the device, font
and/or driver definitions. If the file type part of the file
name (see "Files" on page 281) is not specified, WATCOM GENDEV
searches for source files with the default file type for device
and driver definitions. The font definition file type is the
default alternate extension.
```
which appears to me to be saying that gendev looks for:
• the specified extension
or
• the extension .PCD
• the alternate extension (which defaults to .FON)
Although, as shown above, there are at least four ways to include a source file, the only discussions in the WGML Reference are for files named by the :INCLUDE tag:
Section 9.55 INCLUDE:
```If the specified file does not have a file type, the default
document file type is used. For example, if the main document
file is manual.doc, doc is the default document file type. If
command line is used. If the file is still not found, the file
type GML is used.
```
which appears to me to probably be saying that wgml looks for:
• the specified extension
or
• the extension of the main source file
• the alternate extension
• the extension .GML
Section 15.6.2 INCLUDE:
```The value of the required attribute file is used as the name
of the file to include. The content of the included file is
processed by WATCOM GENDEV as if the data was in the original
file. This tag provides the means whereby a definition may be
specified using a collection of separate files. More than one
definition may be included into one file for processing by
WATCOM GENDEV.
```
which says nothing to the point, but the situation presumably is that gendev looks for:
• the specified extension
or
• the extension .PCD
• the alternate extension (which defaults to .FON)
and may or may not look for the extension of the main source file before using .PCD.
The attribute MEMBER_NAME occurs in the device, driver, and font definitions. This text is the first encountered, and is for the font definition. The other two, in Sections 15.9.1.2 & 15.8.1.2, are identical except that "driver" or "device" is used where "font" appears here.
Section 15.8.1.2 MEMBER_NAME Attribute:
```The member_name attribute specifies the member name of the
font definition. The value of the member name attribute must
be a valid file name. The member name must be unique among
the member names of the font, driver and device definitions.
When the GENDEV program processes the font block, it places
the font definition in a file with the specified member name
as the file name. If the file extension part of the file name
is not specified, the GENDEV program will supply a default
extension. Refer to "Running WATCOM GENDEV" on page 277 for
```
which appears to me to be saying that gendev will use:
• the extension given, if one is given
or
• the extension .COP
Nothing was found specifying the extension used by wgml in searching for these files. It would, however, be reasonable to suppose that wgml does this:
• the extension given, if one is given
or
• the extension .COP
It is, of course, possible that wgml, if given a member name which has an extension other than .COP which it cannot find as such, will replace the given extension with .COP and try again.
The command-line option ALTEXTENSION, it appears, works in a straightforward and obvious manner:
Section 14.3.1 ALTEXTension:
```When a GML source file is specified on the WGML command line,
or as an include file, the file type can be omitted. If a
source file with the default file type cannot be found, WATCOM
Script/GML will search for a file with the file type supplied
by the alternate extension option.
```
Section 16.1.1 ALTEXTension:
```When a GENDEV source file is specified on the GENDEV command
line, or as an include file, the file type may be omitted. A
default file type will be supplied by WATCOM GENDEV. If the
source file cannot be found with the default file type, the
alternate extension option supplies a second file type to
find with the source file.
```
Since I am using terms that usually describe methods of traversing trees, and since their application here may not be entirely clear, I will start by explaining how I am using them.
Suppose you have a file, testdoc.doc, which includes a line "#INCLUDE file='chapter1'". According to the material quoted above, wgml should look for these files:
• chapter1.doc
• chapter1.gml
Now suppose you have two different source file directories, docs1 and docs2, organized so that "..\" will access them from the directory in which wgml is running. If you set GMLINC to "..\docs1;..\docs2" and you place the file chapter1.doc in the directory docs2 and the file chapter1.gml in the directory docs1, will wgml include chapter1.doc or chapter1.gml? If it includes chapter1.doc, that is, if it searches the entire path using ".doc" before trying ".gml", then that is what I am calling "depth-first". If if includes chapter1.gml, that is, if it searches each directory on the path for each extension before moving on to the next directory, then that is what I am calling "breadth-first".
Initially, I tried to do this with a single table: this failed because the situation is too complicated to be summarized so simply. So now I will discuss various sets of search targets with similar search patterns in separate sections.
#### wgml Option Files
This section discusses two search targets:
• wgml/default
• wgml/opt
For these targets, wgml only uses one extension in any given instance.
For wgml/default, the file sought, "default.opt", is searched for automatically. There appears to be no way to specify a different name, and so the file name is almost certainly hard-coded into wgml.
For wgml/opt, the definitive tests showed that:
• if the extension is specified, then that is the only extension used
• if the extension is not specified, then .OPT is the only extension used
In particular, for this command line
```wgml testdoc.doc ( device test file testopt altext txt
```
wgml searches for "testopt.opt" only. Neither the extension of the source file given on the command line nor any alternate extension given on the command line is used with the search target wgml/opt.
Aditional testing confirms that, even with this command line
```wgml testdoc.doc ( device test file testopt.txt
```
if "file diffopt" appears in testopt.txt, then only diffopt.opt will be searched for. Prior usage of other extensions has no effect on the current search.
#### wgml Source Files
This section discusses these search targets:
• wgml/source
• wgml/lay
• wgml/imbed
• wgml/include
• wgml/.ap
• wgml/.im
• wgml/binary
• wgml/graphic
Starting with wgml/source, the search pattern derived from the documentation is:
• the specified extension
or
• the alternate extension
• the extension .GML
Given that the three files "testdoc.doc", "testdoc.txt", and "testdoc.gml" exist in the search path, then the command line
```wgml testdoc.doc ( device test altext txt
```
causes the file "testdoc.doc" to always be processed. If "testdoc.doc" does not exist in the search path, then wgml reports that "testdoc.doc" cannot be found and does not process any "testdoc" file (it does, however invoke the device functions in the :INIT block with "start" as the value of attribute place). This confirms that, if an extension is given, wgml searches for the file using only that extension.
With both file "testdoc.txt" and file "testdoc.gml" in the search path, then, when the command line
```wgml testdoc ( device test altext txt
```
is invoked, "testdoc.gml" is processed when it occurs first in the search path and "testdoc.txt" is processed when it occurs first in the search path.
If both files are in the same directory (and so neither appears first in the search path), then "testdoc.gml" is processed.
So, the actual search pattern is:
• the specified extension (only)
or
• the extension .GML is used first
• the alternate extension is used if .GML did not work
• each directory is checked for both extensions before the next directory is checked
which is not what the documentation described.
For the various included source files, the search pattern derived from the documentation (for :INCLUDE) is:
• the specified extension
or
• the extension of the main source file
• the alternate extension
• the extension .GML
The tests will be described in terms of the LAYOUT command line option, but each test was also performed for .AP, .IM, :IMBED, and :INCLUDE with both layout files and source files and :BINCLUDE and :GRAPHIC with binary or graphic files.
Given that the file "testdoc.gml" exists and that three files "testlay.doc", "testlay.txt", and "testlay.gml" exist in the search path, then the command line
```wgml testdoc ( device test altext txt layout testlay.doc
```
causes the file "testlay.doc" to always be processed. If "testlay.doc" does not exist in the search path, then wgml reports that "testlay.doc" cannot be found and does not process the "testdoc.gml" file it found (it does, however invoke the device functions in the :INIT block with "start" as the value of attribute place). This confirms that, if an extension is given with a layout file, wgml searches for the file using only that extension.
In some cases, when the document is found but an included source file is not, then the output of the functions in the :INIT block with "document" as the value of attribute place appear in the output file, but none of the document itself. In other cases the document is partially processed.
When file "testdoc.doc" and the files "testlay.doc", "testlay.txt", and "testlay.gml" are all in the search path, then the command line
```wgml testdoc.doc ( device test altext txt layout testlay
```
causes whichever of "testlay.doc", "testlay.gml", or "testlay.txt" occurs first in the search path to be processed. If "testlay.gml" and "testlay.txt" are in the same directory, but "testlay.doc" is not in that directory or in any directory ahead of it in the search path, then "testlay.txt" is processed.
When file "testdoc.gml" and the files "testlay.txt" and "testlay.gml" are all in the search path, then the command line
```wgml testdoc ( device test altext txt layout testlay
```
causes whichever of "testlay.gml" or "testlay.txt" occurs first in the command line to be processed. If "testlay.gml" and "testlay.txt" are in the same directory, then "testlay.gml" is processed.
So, the actual situation for the included files is:
• the specified extension (only)
or
• the extension of the main source file, if one is given
• the alternate extension is used first
• the extension .GML if the alternate extension did not work
• each directory is checked for all three extensions before the next directory is checked
or (if the main source file has no extension)
• the extension .GML is used first
• the alternate extension is used if .GML did not work
• each directory is checked for both extensions before the next directory is checked
which is far more complicated than the documentation suggests.
#### gendev Source Files
This section discusses these search targets:
• gendev/source
• gendev/include
For gendev/source, we saw above that the manual is fairly clear:
• the specified extension
or
• the extension .PCD
• the alternate extension (which defaults to .FON)
If the path contains a file "genall.doc", and this command line is executed:
```gendev genall.doc
```
then "genall.doc" is found and processed. If "genall.doc" does not exist in the search path, but "genall.pcd" and "genall.fon" both do, gendev reports that it cannot be found and generates no files. This confirms that, if an extension is provided, only that extension is used.
If files "genall.pcd" and "genall.fon" exist in the search path and this command line is executed:
```gendev genall
```
then it will process whichever file is found first in the path. If they are placed in the same directory and nowhere else in the path, then "genall.pcd" is processed.
If "genall.pcd", "genall.fon", and "genall.txt" exist in the search path and this command line is executed:
```genall genall ( altext txt
```
then it will process "genall.pcd" or "genall.txt", depending on which it encounters first, or "genall.pcd" if both are in the same directory and neither any earlier in the path, but, when forced to use "genall.fon" because that is all that is available, gendev states that it cannot find "genall.txt".
So, for gendev/source, the actual search pattern is:
• the specified extension (only)
or
• the extension ".PCD" is used first
• the alternate extension is used if ".PCD" did not work
• if no alternate extension was given on the command line, then ".FON" is used if ".PCD" did not work
• each directory is checked for both extensions before the next directory is checked
which is the documented behavior.
For gendev/include, there is no search pattern given in the manual; the discussion above simply re-uses the pattern for gendev/source.
If files "testinc.pcd" and "testinc.fon" exist and are used with #INCLUDE in a source file, then they are included. If they do not exist, then gendev reports that the file requested cannot be found: it does not include "testinc.fon" if it is told to look for "testinc.pcd" or "testinc.pcd" if it is told to look for "testinc.fon".
If the #INCLUDE statement merely shows "testinc", then "testinc.pcd" is included whether "genall.pcd" or "genall.fon" or, for that matter, "genall.txt" when ALTEXTENSION is used with value "txt", is processed. A file named "testinc.fon" or "testinc.txt" is used, if available, when no "testinc.pcd" file exists.
Specifying "genall.txt" on the command line has a very interesting effect: included files ending in ".PCD" are not found, only those ending in ".TXT" or ".FON".
So, for gendev/include, the search pattern is:
• the specified extension (only)
or
• the extension of the source file on the command line is used first
• the extension ".PCD" is used if no extension was used with the source file on the command line
• the alternate extension is used if ".PCD" did not work
• if no alternate extension was given on the command line, then ".FON" is used if the first extension tried did not work
• each directory is checked for both extensions before the next directory is checked
which is somewhat unexpected.
#### wgml Binary Device Library Files
wgml is given a defined name, not a filename, for the target device, so the first step is to find the device library whose wmglst.cop file has an entry for that defined name, as discussed in Finding Device Libraries. The member name can then be extracted for use in locating the binary file encoding the :DEVICE block for the target device.
This member name may or may not include an extension; if it does not, then ".COP" will be used as the extension by gendev when it creates the file and so must be used by wgml when it searches for it, presumably using the search path determined by actual test and summarized in Search Paths Observed. It is also not unreasonable to test whether or not, if the member name includes an extension but a file with that name but not that extension is found, a search using ".COP" is done before giving up.
The basic tool in testing this was a pair of binary libraries, "testlib1" and "testlib2", set up so that "testlib1" contained a device file "test.doc" while "testlib2" contained a device file "test.pcd", both of which had "test" as their defined name. As might be expected, whichever of "testlib1" and "testlib2" appeared first in the search path was used.
When any binary file, in whichever directory was listed first, was made inaccessable (by renaming, for easy reversibility), the same error (IO-008) occurred as is shown in Individual Location Tests in the context of searching the current directory for the binary device library files. If both "test.doc" and "test.cop" are in "testlib1" and "test.doc" is made inaccessible, then the error occurs: "test.cop" is not used.
The implication is clear: wgml only actually searches for the first directory file (wgmlst.cop) in the search path containing an entry for the defined name it is given. It looks for the member name only in the same directory, and only for the extension given (if one is given) or the extension ".COP", but not both.
### Finding Device Libraries
The search path used to convert a defined name to a member name is used, not to return the first wgmlst.cop file it finds, but rather to return the first wgmlst.cop file with an entry for the defined name is found. Once the defined name is found in a wgmlst.cop file, the search terminates and any additional wgmlst.cop files containing the defined name are never found.
Further, as shown in wgml Binary Device Library Files, a file with the member name (with .COP appended if it has no extension) must exist in that same directory, or wgml will report that it could not be found.
## Specified Paths
The bulk of this page deals with simple filenames: filenames which may have an extension but which are given with no path information at all. This section deals with filenames that include path information. As such, it does not apply to binary device library files, since those files are sought by defined name and not (directly) by filename.
There are two types of path information that may be prepended to the filename:
• an absolute path; and
• a relative path.
The WGML Reference does not address this issue, nor does the README file produceable from the WGML 3.33 Update. This leaves testing as the only source of information.
These are the file types discussed here whose filenames can include path information:
• the source file given on the command line (wgml/source)
• an option file given on the command line (wgml/opt)
• a layout file given on the command line (wgml/lay)
• a source file named in an :IMBED block (wgml/imbed)
• a source file named in an :INCLUDE block (wgml/include)
• a source file named in an .AP block (wgml/.ap)
• a source file named in an .IM block (wgml/.im)
• the default option file (wgml/default)
• a binary file named in a :BINCLUDE block (wgml/binary)
• a graphic file named in a :GRAPHIC block (wgml/graphic)
• the source file given on the command line (gendev/source)
• a source file named in an :INCLUDE block (gendev/include)
and they can be grouped into three categories:
• wgml Option Files
• wgml/default
• wgml/opt
• wgml Source Files
• wgml/source
• wgml/lay
• wgml/imbed
• wgml/include
• wgml/.ap
• wgml/.im
• wgml/binary
• wgml/graphic
• gendev Source Files
• gendev/source
• gendev/include
Preliminary results for relative paths are suggestive:
• If the filename includes path information, that path information is automatically prepended to simple filenames included in the file. It does this even if the filename has its own prepended information: that is, the original path replaces the given path.
• There is no third attempt, for wgml Source Files, to find the file using ".gml" if the filename has a different extension and the alternate extension either does not exist or is not ".gml".
It appears that gendev searches the normal paths if the file is not found: the filename reported as "not being found" includes path information from the PATH environment variable; wgml reports the filename as entered when a different path is provided and it cannot be found, and opens it if it in fact exists there. This suggests that, at least for relative paths, the search order is "enhanced" to include the original file's path information first, and then the given path information. No attempt to pursue this further was made.
Preliminary results for absolute paths are also interesting:
• The absolute path is prepended to simple filenames included in the file. If the filename has a path, whether relative or absolute, that is used instead.
• There is no indication that either the default extension or the third extension ".gml" are used for wgml Source Files if the filename includes an extension. If it does not include an extension, then the alternate extension, if one exists, is used.
## Design and Coding Implications
It is, of course, hardly possible at present to be certain about the meaning of the information determined on this page. This is a preliminary attempt to set down some general ideas.
It is not possible to both implement the search procedures as documented and to replicate the behavior of wgml 4.0 and gendev 4.1, for these are not consistent with each other. It might be helpful to build a model of how wgml was originally used. This model is, of course, speculative.
If the non-PC sections of the WGML Reference and such documents as script-tso.txt are considered, then it becomes apparent that, originally, wgml and gendev were intended to be used on IBM VM/CMS and DEC VAX/VMS computers. In the IBM VM/CMS environment, GMLLIB was not an environment variable, but the name of something called a "MACLIB", or, alternately, was created with the "MACLIB" command. The same applies to the DEC VAX/VMS environment, except that the command used was "LIBRARY". The net effect was that there could only be one device library, and its name was GMLLIB.
When wgml and gendev were ported to the PC, GMLLIB became an environment variable giving the dirctory of the device library. This immediately created the possibility of having more than one device library, since an environment variable can list several directories.
The environment variable GMLINC is only mentioned in the context of PCs: apparently, it did not exist in wgml or gendev as used in the IBM VM/CMS or DEC VAX/VMS environment. The environment variable PATH, of course, is very common on PC systems, at least the non-Linux systems which are currently fully supported by Open Watcom.
Now, it occurs to me that the use of GMLLIB and GMLINC by gendev can be taken as indicative of their originally intended use: GMLLIB was intended to point to device libraries (only), and GMLINC was intended to point to source code (only). Clearly the search patterns shown to be used by wgml no longer reflect this division: both are checked (the order varies), and then PATH is checked as well. I suggest that this reflects a different understanding of GMLLIB and GMLINC: that, in fact, GMLLIB is intended for the centralized system-wide device library which ordinary users cannot alter, and GMLINC was intended for local files, whether source files or local device libraries.
That GMLLIB is searched first for device libraries not only preserves the authority of the central authority, but, more to the point, ensures that the local device libraries will have to use different defined names, thus preventing unpleasant surprises where a local library redefines a device with the same defined name as a defined name used in the centralized library, while preserving the ability of local users to create and use modified devices if they feel the need to.
This eventually led to GMLLIB being searched (after GMLINC) for source files, and for PATH being searched in, presumably, a last, desperate effort to find the desired device library or file before giving up and disappointing the user.
The search paths found to be used by wgml for option files and for source files are the same as those documented, so there really is no reason to implement anything else in these cases. The search path found to be used by gendev for source files is not the same as the documented path, but can be regarded as a shortened form of it. Since gendev, unlike wgml, is used only occasionally, either search path would do.
The search procedure for binary device libraries is so different from the normal search procedure that it has to be implemented separately. Since it preserves the idea of a system-provided library, it should be implemented to work the way it works in wgml 4.0.
The issues of breadth versus depth and use of extensions turned out to be so easy and straightforward to implement that that is what was done. Our gendev and wgml should behave exactly as gendev 4.1 and wgml 4.0 do when searching for filenames that do not include any path information.
The issue of filenames which do contain path information is currently handled quite simply: it is treated as an error. If such filenames turn out to require support, additional testing will be needed to ensure that the behavior of gendev 4.1 and wgml 4.0 is completely understood. Based on what has already been done, this should work:
1. Create a global variable def_path.
2. Initialize def_path for both option and document specification files in wgml and source files in gendev if a relative path is used.
3. If def_path is not empty, and the filename sought does not have an absolute path, search the def_path first and any relative path given as part of the filename.
4. In terms of the existing code, this can be done by creating directory_list object(s) containing the path(s) to be used, either instead of (wgml) or before (gendev) the normal search pattern.
5. If absolute paths are always followed, then identifying filenames incorporating them and searching only that path for only that filename separately from the rest of the code might be an easy simplification. | {"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": 0.8012315630912781, "perplexity": 3197.840480166268}, "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-2014-49/segments/1416400379414.61/warc/CC-MAIN-20141119123259-00138-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/99326/set-multicolumn-width-to-default-table-width | # Set multicolumn width to default table width
I have a two column table with a single multicolumn spanning both of them. The text on that multicolumn is rather long so I need to set its width so it wraps around. I can set the width with \multicolumn{2}{p{whatever}}{text} but what should I set "whatever" to? What is the actual width of a table? Is there a variable holding that value?
I first thought to use \textwidth but that's not it. Building the document on draft mode shows that it's going out a tiny bit. The following table should show my problem. In draft mode will display a black box on its side.
\begin{tabular}{l l}
\toprule
\multicolumn{2}{p{\textwidth}}{\lipsum[1]} \\
\bottomrule
\end{tabular}
I do have more rows, I'm just keeping the example simple. I'm using the memoir document class. The actual text inside the multicolumn is using the seqsplit package.
EDIT: I should have more specific with the question. The only row that would make the table go over the limit is the one that I'm also trying to limit. And by limit I mean, so I don't get an overfull warning.
So my question can be better rephrased as how to find the available space for a table so it is not overfull?
The space available isp{\dimexpr\textwidth-2\tabcolsep\relax} due to the column padding LaTeX applies, assuming you have \noindent\begin{tabular} otherwise the table itself will be offset by \parindent.
However if the span is wider than the columns it spans, TeX puts all the extra width into the last column. If that is a problem, edit your example to be a complete (small) document that shows the problem.
• Wouldn't it be better to use \linewidth instead of \textwidth in case the table is used in some other environment such as \itemize where the available horizontal space is reduced. – Peter Grill Feb 21 '13 at 21:52
• Thank you, this solves my problem. The other rows should be quite small, this is the only one that should have to span multiple rows. – carandraug Feb 22 '13 at 0:01
Knowledge of the column entries (especially the widest elements) allows you to calculate the width of the \multicolumn accurately:
\documentclass{article}
\usepackage{booktabs,calc,lipsum}% http://ctan.org/pkg/{booktabs,calc,lipsum}
\begin{document}
\begin{tabular}{l l}
\toprule
Some text & Some more text \\
Short & Very super super long \\
Longer than others & Shrt \\
\multicolumn
{2}
{p{\widthof{Longer than others}+\widthof{Very super super long}+2\tabcolsep}}
{\lipsum[1]} \\
\bottomrule
\end{tabular}
\end{document}
calc provides \widthof{<stuff>} that returns the natural width of <stuff>. Each column is padded on both sides by \tabcolsep.
• That's not working for me. I should have been more explicit on the question, but the only row that will take the whole horizontal space of the table, is also the one I'm also trying to limit. – carandraug Feb 21 '13 at 23:56
The total width of the table is not available until the whole table has been typeset, unless all columns are specified with an explicit width (p columns). TeX chooses the column width only when the whole table has been loaded into memory, and so this makes quite difficult to solve your problem, because a text width has to be specified in order to typeset a paragraph.
For this particular problem, where all the columns must be spanned, there's an automatic solution: first typeset the table ignoring the paragraph spanning the columns to get the width, then retypeset it.
The first argument in \multipar is necessary, because the total number of columns is not available inside a tabular environment.
\documentclass{article}
\usepackage{booktabs,lipsum}
\usepackage{environ}
\newsavebox{\funnytabularbox}
\newlength{\funnytabularwd}
\makeatletter
\NewEnviron{funnytabular}[1]
{\begin{lrbox}{\funnytabularbox}
\let\multipar\@gobbletwo
\begin{tabular}{#1}\BODY\end{tabular}
\end{lrbox}%
\setlength{\funnytabularwd}{\wd\funnytabularbox}%
\let\multipar\@multipar
\begin{tabular}{#1}\BODY\end{tabular}}
\newcommand\@multipar[2]{\multicolumn{#1}{p{\funnytabularwd}}{#2}}
\makeatother
\begin{document}
\begin{funnytabular}{l l}
\toprule
Some text & Some more text \\
\midrule
Short & Very super super long \\
\midrule
Longer than others & Shrt \\
\midrule
\multipar{2}{\lipsum*[1]} \\
\bottomrule
\end{funnytabular}
\end{document}
• thank you for solution. But my problem is actually simpler. My apologies I should have made it more explicit on the question. See my edit and I found my solution with David's answer. – carandraug Feb 22 '13 at 0:14
I used the following, which may help your case (I needed the multicolumn width for \multirow):
\usepackage{tabularx}
\usepackage{multirow}
...
\begin{tabularx}{\textwidth}{|*{9}{>{\centering\arraybackslash}X|}}
\hline
start
& \multicolumn{7}{X}{\multirow{3}{\dimexpr(\hsize+2\tabcolsep)*7-2\tabcolsep\relax}{%
Hello, this can be a long paragraph.}
& \\
& & \\
& & end\\\hline
\end{tabularx}
The magic is in knowing the values for \hsize and \tabcolsep, with the calculation being done in the \dimexpr command from e-Tex. | {"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": 0.8692383766174316, "perplexity": 1144.8026845447669}, "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-2020-40/segments/1600400201826.20/warc/CC-MAIN-20200921143722-20200921173722-00539.warc.gz"} |
https://www.physicsforums.com/threads/simple-question-about-gravity-on-earth.301253/ | # Simple question about Gravity on Earth
1. Mar 20, 2009
### Ghost803
We keep getting drilled with the info in class that all objects would fall to the earth at the same speed. And it does not make sense to me
So the equation is (Ge * m1 * m2)/r2 = F
And F = m1 * a. There fore m1 * a = (Ge* m1*m2)/r2
This can be simplified and the m1 eliminated by dividing it from both sides, so that a = Ge *m2(mass of earth)/r2.
So my question is, how does this work with masses larger than the earth? Because according to this equation the only thing determining acceleration is the mass of the earth and the distance and the the Gravitational Constante Ge.
Do you just use the mass of the larger object to calculate acceleration and disregard the smaller object's mass?
2. Mar 20, 2009
### mgb_phys
Because you also have a force on the Earth, the Earth is accelerating up to meet the falling object but normally the effect for a tennis ball is rather small so you can ignore it.
For two stars/planets falling into each other you have to take it into account.
3. Mar 20, 2009
### DaveC426913
i.e the simpler form of the equation is only used for objects whose mass when compared to Earth's is insignificant.
4. Mar 20, 2009
### Integral
Staff Emeritus
Not quite.
Note that there are no approximations made in deriving the equation in the OP. It holds for all masses, but there is one key restriction built into it. You must use your distance from the center of the earth, that is the r in the denominator. If you use the radius of the earth you get the common value of 9.81 m/s2. So this value is only meaningful at or near the earths surface.
5. Mar 20, 2009
### pallidin
OK. here's one for you...
I''m sitting here on my chair veiwing PF.
I now raise with one hand a 1/lb weight 2-feet above me.
The earth does not move 2-feet backwards, of course, BUT, does it move at all?
6. Mar 20, 2009
### DaveC426913
Yes. It moves an amount equivalent to the Earth's mass divided your arm's mass times two feet.
7. Mar 20, 2009
### DaveC426913
Right. Forgot about the distance thing.
8. Mar 20, 2009
### Ghost803
Aah, I think I understand it now. I forgot to take into account the acceleration of the earth.
Which would be F= m2 * a.
m2 *a = (Ge *m1 *m2)/ r2
cancel out the m2s and you get a = Ge *m1/r2
So now my question is..
Does this mean that if any object how ever large were to come close to the earth, that it would have the same acceleration as us. And that any difference in speed for the two colliding, would solely be caused by the earth also accelerating at a great speed, which it usually does not because our masses are soo insignificant?
9. Mar 20, 2009
### DaveC426913
What?
10. Mar 20, 2009
### Ghost803
I meant. Say an object like Mars or something were to come close to the earth's surface. Would it accelerate towards earth, at 9.8m/s2, just like objects on earth with really small mass?
And would any difference in time it would take for impact by the large object, compared to how long an small object, like us would take to hit the earth be caused by the earth also accelerating. Which the earth would not usually do if small objects fall down, because their mass is so small compared to the mass of the earth.
11. Mar 20, 2009
### DaveC426913
Wellll... the part of Mars that is near Earth would feel the tug of 9.8m/s^2, yes. That there are a lot of confounding factors - mostly to do with tides (differences in pull at different points) and Roche limits - that make the question mostly academic.
Yes, a large object would contribute its gravitational pull to the equation. (Note that you're now talking about a system whose mass is Earth PLUS Mars, so it makes sense that the attraction is stronger).
Similar Discussions: Simple question about Gravity on Earth | {"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": 0.8675259947776794, "perplexity": 920.7870263334337}, "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-2017-43/segments/1508187824820.28/warc/CC-MAIN-20171021152723-20171021172723-00116.warc.gz"} |
http://tex.stackexchange.com/questions/30421/framed-minipage-of-dependent-dimensions | # Framed minipage of dependent dimensions
I am trying to prepare handouts of a ppt (I know). I would like a slide to be shown on the left half of a page and comments regarding it in a framed box of the same size as the slide on the right half. By
\begin{minipage}{.5\textwidth}
\includegraphics[page=4,scale=.3]{my_ppt}
\end{minipage}
I can get the slide in, but how do I create a minipage to the right of it that has the same height as the previous minipage?
Is minipage the environment of choice here? And is the framed package the best way to get a frame around it?
-
It doesn't make much sense putting a \includgraphics into a minipage. Also I would use width and/or height not scale to determine the size. You can put the image into a savebox and measure its size. Which class and layout do you like to use for this document? – Martin Scharrer Oct 4 '11 at 11:56
\documentclass[a4paper]{report}, right using the minipage here is not that useful, I just thought it might be possible crating an empty copy of it to the right somehow. How do I do this with the savebox? – D.Roepo Oct 4 '11 at 12:01
An environment form can be obtained as follows
\newsavebox\pptbox
\newenvironment{pptcomment}[2][]
{\sbox\pptbox{\includegraphics[#1]{#2}}%
\dimen0=\ht\pptbox
\parbox{.5\textwidth}{\centering\usebox\pptbox}%
\begin{lrbox}{\pptbox}\begin{minipage}[c][\dimexpr\dimen0-2\fboxsep\relax][c]
{\dimexpr.5\textwidth-2\fboxsep\relax}}
{\end{minipage}\end{lrbox}\fbox{\box\pptbox}}
and then used as
\begin{pptcomment}[page=4,scale=.3]{my_ppt}
The comment text that will be vertically centered
with respect to the image on its left
\end{pptcomment}
In other words, you pass the new environment the same parameters you'd pass to \includegraphics. However, setting scale is not recommended.
How does this work?
We allocate a new bin \pptbox for storing objects; the first use of it is for measuring the height of the image (\ht\pptbox, which is stored in the temporary dimension register \dimen0). Then we typeset the image in a \parbox so that it will be vertically centered with respect to other objects. Next we open lrbox, which is the "environment form of \mbox, storing the contents again in the \pptbox bin. This box consists of a minipage vertically centered ([c]), as high as \dimen0 (i.e., the height of the image) minus twice the clearance between a box and the frame (\fboxsep); the contents of this minipage will be centered with respect to the total height; finally the minipage will be as wide as half the textwidth. The contents of the environment is read in and then the bin is wrapped up (\end{lrbox}) and used inside \fbox.
It's best to use pptcomment inside a center environment, which can also be inserted directly in the environment:
\newsavebox\pptbox
\newenvironment{pptcomment}[2][]
{\begin{center}
\sbox\pptbox{\includegraphics[#1]{#2}}%
\dimen0=\ht\pptbox
\parbox{.5\textwidth}{\centering\usebox\pptbox}%
\begin{lrbox}{\pptbox}\begin{minipage}[c][\dimexpr\dimen0-2\fboxsep\relax][c]
{\dimexpr.5\textwidth-2\fboxsep\relax}}
{\end{minipage}\end{lrbox}\fbox{\box\pptbox}
\end{center}}
-
Thanks a lot. This is really nice! Could you for a novice as me comment a bit on what exactly is going on in your environment definition? Also is the slide indented, which, when making it .5\textwidth wide, results in the comment hanging over into the right side margin. Is there a work around? – D.Roepo Oct 4 '11 at 17:36
@user7091 Such an environment should be set inside a center environment. I'll add some explanations later. – egreg Oct 4 '11 at 17:41
It doesn't make much sense putting a \includgraphics into a minipage. Also I would use width and/or height not scale to determine the size. You can put the image into a savebox and measure its size.
Something along the lines of:
\documentclass{article}
\usepackage{graphicx}
\newsavebox\mysavebox
\begin{document}
\sbox\mysavebox{\includegraphics[width=.49\textwidth]{image}}
\usebox\mysavebox\hfill
\fbox{\begin{minipage}[b][\ht\mysavebox][t]{.49\textwidth}%
\end{minipage}}
\end{document}
This however makes the framed minipage have a depth of \fboxsep which you can correct using e.g. \raisebox{\fboxsep}{..}. Using the adjustbox package would simplify these combinations:
\documentclass{article}
\newsavebox\mysavebox
\begin{document}
\sbox\mysavebox{\includegraphics[width=.49\textwidth]{image}}
\usebox\mysavebox\hfill | {"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": 0.9352478981018066, "perplexity": 1496.951235966071}, "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/1438042988922.24/warc/CC-MAIN-20150728002308-00012-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://nrich.maths.org/422/solution | ### Golden Thoughts
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
### Pent
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
### Pentakite
ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.
# Darts and Kites
##### Stage: 4 Challenge Level:
This solution is from Arun Iyer of SIA High School and Junior College.
Part 1
First I will show that $POR$ is a straight line. For this I would like to state the perpendicular bisector theorem.
PERPENDICULAR BISECTOR THEOREM: Every point equidistant from the two ends of a line segment lies on the perpendicular bisector of the line segment.
Now consider the line segment $QS$.
$OQ=OS=1$ therefore by the perpendicular bisector theorem, $O$ must lie on the perpendicular bisector of $QS$.
$PQ=PS$ (as the sides of the rhombus are equal), therefore by the perpendicular bisector theorem, $P$ must lie on the perpendicular bisector of $QS$.
$RQ=RS$ (as the sides of the rhombus are equal), therefore by the perpendicular bisector theorem, $R$ must lie on the perpendicular bisector of $QS$.
Now the perpendicular bisector of a line segment is unique and hence $P$, $O$, $R$ must lie on the same perpendicular bisector and hence $POR$ is a straight line.
Part 2
Now I will get all the angles of the rhombus.
$\angle QPS$ = $72^{\circ}$ (given), $\angle QRS = \angle QPS = 72^{\circ}$ as they are opposite angles of a rhombus. The diagonal of the rhombus bisects the angles of a rhombus and therefore $\angle QPO = \angle SPO = \angle QRO = \angle SRO = 36^{\circ}$.
Triangles $OQR$ and $OSR$ are isosceles triangles, therefore $\angle OSR = \angle OQR = 36^{\circ}$.
Using the fact that sum of angles of a triangle is 180 degrees, we can see that $\angle SOR$ and $\angle QOR$ are equal to 108 degrees. Since we have proved that $POR$ is a straight line in Part 1, we can determine $\angle QOP$ and $\angle SOP$ to be 72 degrees.
Again using the fact that sum of angles of a triangle is 180 degrees, we can see that $\angle OQP$ and $\angle OSP$ are equal to 72 degrees.
Part 3
Let the side of the rhombus be $x$.
Consider triangle $OPS$ in which $PO=PS=x$ (since $\angle POS = \angle PSO$).
Applying the cosine rule $$\cos OPS = [PS^2 + PO^2 - OS^2]/[2\times PS \times PO]$$ therefore $$\cos 36 =[2x^2-1]/[2x^2]\quad (1).$$
Splitting the isosceles triangle $ORS$ into two right angled triangles gives $$\cos 36 = x/2 \quad (2).$$
From (1) and (2),
\eqalign{ [2x^2 - 1]/[2x^2] &= x/2\cr x^3 - 2x^2 + 1 &= 0 \cr (x - 1)(x^2 - x - 1) &= 0.}
Now $x\neq 1$ because triangle POS is not equilateral, therefore $x^2 - x - 1 = 0$ and hence
$x = [1 + \sqrt 5]/2$ or $x = [1 - \sqrt 5]/2$.
Clearly $x\neq [1 - \sqrt 5]/2$ because the side length cannot be negative, therefore $x = [1 + \sqrt 5]/2$, the Golden Ratio. | {"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": 0.9390331506729126, "perplexity": 258.12540365204813}, "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-2016-40/segments/1474738662166.99/warc/CC-MAIN-20160924173742-00011-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/251506/is-c-c-infty-dense-in-x-0-alpha?answertab=active | # Is $C_c^{\infty}$ dense in $X_0^{\alpha}$?
While reading papers on fractional Laplacian, I always meet space $X_0^{\alpha}(\mathcal{C}_{\Omega})$ which is defined as following: $$X_0^{\alpha}(\mathcal{C}_{\Omega})=\{z\in L^2(\mathcal{C}_{\Omega}): z=0 \text{ on }\partial_L\mathcal{C}_{\Omega}, \int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla z(x,y)|^2\,dxdy<\infty\},$$ Where $\Omega$ is bounded domain in $\mathbb{R}^n$, $\mathcal{C}_{\Omega}=\{(x,y):x\in \Omega,y\in \mathbb{R}_+\}\subset \mathbb{R}^{n+1}_+$, and $\partial_L\mathcal{C}_{\Omega}$ is the lateral boundary of $\mathcal{C}_{\Omega}$. And we equip $X_0^{\alpha}$ with norm $$\Vert z\Vert_{X_0^{\alpha}}^2=\int_{\mathcal{C}_{\Omega}}y^{1-\alpha}|\nabla z(x,y)|^2\,dxdy.$$
Then my question is: Is $C_c^{\infty}(\overline{\mathbb{R}^{n+1}_+})$ dense in $X_0^{\alpha}(\mathcal{C}_{\Omega})$, and how to prove it? I don't even know whether we have $\Vert \eta_{\varepsilon}*u-u\Vert_{X_0^{\alpha}}\to 0$, where $\eta_{\varepsilon}$ is the standard mollifier.
-
Well, technically it's those functions $z \in C^\infty(\mathbb{R}^{n+1}_+)$ whose support are contained inside the region bounded by $\partial_L C_\Omega$, to use your notation.
Let $u \in X_0^\alpha(C_\Omega)$, and $\varphi$ be your favorite bump function supported on a ball of radius 1 in $\mathbb{R}^{n+1}$, say, and make your space the whole cylinder $$C_\Omega' = \{ (x,y): x \in \Omega, y \in \mathbb{R} \}$$ and extend $u$ by even reflection (for $y < 0$, let $u(x,y) = u(x,-y)$). Let $d(X) = d(x,\partial \Omega)$ where $X=(x,y)$. Define $$u(X,\epsilon) = \begin{cases}u(X) & \text{if} \,\,d>2\epsilon \\ 0 &\text{otherwise}\end{cases}$$ and then subsequently $$u_\epsilon(X) = (u(X,\epsilon) *\frac{1}{\epsilon^n}\varphi(\frac{X}{\epsilon}))$$ From here, it's fairly simple to see that $u_\epsilon \in C^\infty_c(C_\Omega')$, and it remains only to show that $$\int_{C_\Omega'} y^{1-\alpha} |\nabla u - \nabla u_\epsilon|^2 dx dy \rightarrow 0$$ as $\epsilon \rightarrow 0$.
Notice that $y^{1-\alpha} dx dy$ is a nice bounded positive measure for $0 < \alpha < 2$ (which is always the case when you consider the extension for the fractional Laplacian), hence continuous functions of compact support are dense in $L^2(y^{1-\alpha} dx dy)$. Choose $n+1$ such nice functions $G=(g_1,\ldots,g_{n+1})$, such that $$\int_{C_\Omega'} |\nabla u - G|^2 y^{1-\alpha} dx dy < \frac{\eta}{6}$$ Define $G_\epsilon(X)$ the same way we defined $u_\epsilon$. Now $$|\nabla u - \nabla u_\epsilon| \leq |\nabla u - G| + |G - G_\epsilon| + |G_\epsilon - \nabla u_\epsilon|$$ It is clear (by simply using Fubini and doing the convolution first) that $$\int y^{1-\alpha} |G_\epsilon - \nabla u_\epsilon|^2 dx dy < \frac{\eta}{6}$$ so we are left only to consider the term in $|G_\epsilon - G|$. However, $G$ is uniformly continuous, hence for $\epsilon$ sufficiently small we have $|G_\epsilon - G|^2 < \frac{\eta}{C}$, except possibly when $d(X) < 3\epsilon$. Since the support of $G$ is bounded, pick $C$ so that $$\int y^{1-\alpha} |G_\epsilon - G|^2 dx dy < \frac{\eta}{6}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9974685311317444, "perplexity": 67.70239028389007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860114285.77/warc/CC-MAIN-20160428161514-00125-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://discuss.codechef.com/questions/125508/please-help-me-approach-this-question | ×
0 Can someone explain me how to approach this problem? I am not able to understand the question properly! https://www.codechef.com/problems/REN2013G asked 14 Apr '18, 12:23 1●1 accept rate: 0% 15.2k●1●18●59
0 We have a source, which is the castle, and a destination, which is the village, and also some watch towers. We are trying to get from the source to the destination by travelling in straight lines from one location to the next (minimum ammunition used) and we want to find the minimum ammunition used overall from the source to the destination. To solve this, think of the castle, village, and watch towers as nodes in a graph. From the castle we can directly (straight line) go to each watch tower and the village. From each watch tower we can go to all the other watch towers and the village. Of course, these connections or edges have a weight (or ammunition usage), which is the distance squared (as specified in the problem). So to solve the problem we simply build a graph with all of the connections and nodes and find the shortest path (Dijkstra's algorithm) from the source to the destination. The time complexity should be around O(n^2) because when building a graph there is a connection between every pair of nodes. answered 15 Apr '18, 02:20 4★c0deb0t 41●4 accept rate: 0% Thank You so much @codebot (15 Apr '18, 21:14) No problem! (16 Apr '18, 00:32) c0deb0t4★
toggle preview community wiki:
Preview
By Email:
Markdown Basics
• *italic* or _italic_
• **bold** or __bold__
• image?
• numbered list: 1. Foo 2. Bar
• to add a line break simply add two spaces to where you would like the new line to be.
• basic HTML tags are also supported
• mathemetical formulas in Latex between \$ symbol
Question tags:
×2,700
×1,056
×74
question asked: 14 Apr '18, 12:23
question was seen: 201 times
last updated: 16 Apr '18, 00:32 | {"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": 0.8276616930961609, "perplexity": 2207.0965901135964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583745010.63/warc/CC-MAIN-20190121005305-20190121031305-00291.warc.gz"} |
https://www.physicsforums.com/threads/some-kinematics-questions-with-cars-and-satallites.222837/ | # Some Kinematics questions with cars and satallites
1. Mar 18, 2008
### neonjr
Having difficulty with the followingCalculating this collision, as the standard equation does not apply, i don't think.
Question 1:
A 2000kg car travelling at 24 m/s EAST (x), collides with a 3600kg truck travelling at 10m/s SOUTH(y). What is there velocity immediately after impact.
Question 2:
An Earth Satellite travels in a circular orbit of radius 4 times the Earths radius. Calculate its acceleration in m/s^2
Question 3:
A satellite circles the Earth at an average altitude of 760km, with a period of 100 min. Calculate the mass of the Earth, if G = 6.67x10^-11 Nm^2/kg^2, and the radius of the Earth is 6.38x10^6m.
Can you advise me the correct equation to use and directions, i'm lost. Please and thank you.
Josh
2. Mar 19, 2008
### tiny-tim
Welcome to PF!
Hi Josh! Welcome to PF!
By "standard equation", I take it you mean conservation of momentum in one dimension?
It's ok - you just have two equations instead of one - conservation of momentum North, and conservation of momentum East!
(btw, I suppose you're expected to use conservation of energy as well - but in practice, it wouldn't be conserved, because the vehicles would be deformed in the collision. )
As to Questions 2 and 3: do you know a formula for the aceleration of something in a circle?
Similar Discussions: Some Kinematics questions with cars and satallites | {"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": 0.8803949356079102, "perplexity": 1365.5954390598956}, "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/1490218190234.0/warc/CC-MAIN-20170322212950-00482-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://gateforum.org/1519/which-one-of-the-following-options-is-the-closest-in-meaning-to-the-word-given-below-nadir | Which one of the following options is the closest in meaning to the word given below? Nadir
A Highest B Lowest C Medium D Integration
B
Lowest | {"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": 0.866813600063324, "perplexity": 1460.5793173401657}, "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-2017-04/segments/1484560279489.14/warc/CC-MAIN-20170116095119-00253-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/153472 | ## Metastable supergravity vacua with F and D supersymmetry breaking
We study the conditions under which a generic supergravity model involving chiral and vector multiplets can admit viable metastable vacua with spontaneously broken supersymmetry and realistic cosmological constant. To do so, we impose that on the vacuum the scalar potential and all its first derivatives vanish, and derive a necessary condition for the matrix of its second derivatives to be positive definite. We study then the constraints set by the combination of the flatness condition needed for the tuning of the cosmological constant and the stability condition that is necessary to avoid unstable modes. We find that the existence of such a viable vacuum implies a condition involving the curvature tensor for the scalar geometry and the charge and mass matrices for the vector fields. Moreover, for given curvature, charges and masses satisfying this constraint, the vector of F and D auxiliary fields defining the Goldstino direction is constrained to lie within a certain domain. The effect of vector multiplets relative to chiral multiplets is maximal when the masses of the vector fields are comparable to the gravitino mass. When the masses are instead much larger or much smaller than the gravitino mass, the effect becomes small and translates into a correction to the effective curvature. We finally apply our results to some simple classes of examples, to illustrate their relevance.
Published in:
Journal of High Energy Physics, 8, 42
Year:
2007
ISSN:
1126-6708
Keywords:
Laboratories: | {"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": 0.9497054815292358, "perplexity": 293.1328233986235}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590127.2/warc/CC-MAIN-20180718095959-20180718115959-00457.warc.gz"} |
https://tex.stackexchange.com/questions/277792/vertical-align-text-next-to-image-in-rhead-without-influencing-the-positioning-o | Vertical align text next to image in rhead without influencing the positioning of lhead
I try to vertical align a text next to an image in a fancyhdr environment. This is my LaTeX code:
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{graphicx}
Baz lorem \\
ipsum dolor}
\rhead{This should be vertically aligned (middle) \raisebox{-.5\height}{\includegraphics[scale=0.25]{wappen}}}
\pagestyle{fancy}
\fancypagestyle{titlestyle}
{
\fancyhf{}
\fancyfoot[C]{\Large \today}
}
\begin{document}
Test
\end{document}
It somehow works (the text is vertically aligned), but the multi-lined text on the left is influenced: Its baseline is also shifted upwards.
How can I vertically align the text on the right without modifying the baseline of the left header?
You can do that easily with the \Centerstack command, from the stackengine package.
Here is how, with one of my own images since you didn't provide one. Also, centring text w. r. t. the image is better if you lower of 0.4\height, to take into account the height of the line of text (the exact value depend on the height of the image, of course):
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{stackengine}
\setstackEOL{\\}
\Centerstack[l]{Foo bar \\
Baz lorem \\
ipsum dolor}}
\rhead{This should be vertically aligned (middle) \raisebox{-.4\height}
{\includegraphics[scale=0.16]{Hedgehog-in-the-Fog}}}
\pagestyle{fancy}
\fancypagestyle{titlestyle}
{
\fancyhf{}
\fancyfoot[C]{\Large \today}
}
\begin{document}
Test
\end{document}
Put everything in the left field of the header:
\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{lipsum}
\fancyhf{}
\begin{tabular}{@{}l@{}}
Foo bar \\
Baz lorem \\
ipsum dolor
\end{tabular}\hfill
\begin{tabular}{@{}l@{}}
\mbox{}\\
This should be vertically aligned (middle) \\
\mbox{}
\begin{tabular}{@{}c@{}}
\includegraphics[height=40pt]{duck}
\end{tabular}}
\pagestyle{fancy}
\fancypagestyle{titlestyle}{%
\fancyhf{}% | {"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": 0.9770129919052124, "perplexity": 4388.4073594132415}, "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-2020-34/segments/1596439737039.58/warc/CC-MAIN-20200806210649-20200807000649-00598.warc.gz"} |
https://www.physicsforums.com/threads/age-of-the-observable-universe.825790/ | # Age of the observable Universe?
Tags:
1. Aug 1, 2015
### G.D.
Forgive my ignorance?.. If we can see 13.8-ish billion light years away how can the universe be the same age? Matter cannot travel at the speed of light, so how are we as far away (in light years) as the universe is old?
2. Aug 1, 2015
### marcus
We routinely see stuff that is now between 45 and 46 billion LY away. But we see it as it was around year 380,000 of the expansion, not as it is now.
Relativity allows distances to grow at several times the speed of light---that is not subject to the same rules as ordinary motion, stuff moving thru its surrounding space.
3. Aug 1, 2015
### Bandersnatch
Hi G.D., welcome to PF!
Matter did not have to go to the edge of the observable universe before emitting its light. What we see now at the edges of observability (as marcus said, at about 46-ish billion light-years), started already some distance away from the point where we're at now. It emitted its light 13.8 billion years ago when it was about a thousand times closer than it is now (~44 million light years) and kept on moving away as its light kept on moving through the expanding space towards us. By the time the light reached us, after 13.8 billion years of travel, the object that was the source managed to recede to 46 billion light years.
The key point here is that at the time of emission of the light we observe, the stuff in the universe was already spatially separated.
4. Aug 1, 2015
### G.D.
Hello and thank you both. So let me see if I've got this. The universe is 13.8 ish billion yrs old, but in that amount of time the universe has expanded to the point where our furthest viewable objects are 46 billionish light years away. Matter can't travel faster than the speed of light, but the space inbetween galaxies can expand faster. So if something is 46 billionish light years away that doesn't mean it's that old.
5. Aug 1, 2015
### Bandersnatch
Just to make sure it is clear - while we can say that the farthest object we see is NOW at 46 billion ly because we can calculate how it will have receded, we still only see the light that it emitted 13.8 billion years ago, when it was much closer.
Feel free to ask if you'll have more questions. Although I have to say that this topic is the mainstay of the cosmology section of the forums, so perhaps you'll find your questions already answered in another thread.
6. Aug 1, 2015
### marcus
Hi G.D. would you like to know how to calculate that 46 billion LY for yourself? It is a simple integral from early time up to present, so if you have taken (even very beginning level would do) calculus, you might not be put off by the integral and enjoy using it to get the distance to the farthest matter we can see.
there is a website called numberempire.com that does integrals for you online. it's easy to use and free. You just go there, paste or type in the function you want to integrate over some range----and put the limits (start and finish of the desired range) of integration in and press calculate.
Basically the integral is telling you how far a flash of light can travel in 13.8 billion years when it is helped by expansion. You add up all the little cdt steps the light takes multiplied by how much each step gets expanded between the time it takes the step and the present. dt is a bit of time, cdt is the original length of the step, and then there is the expansion factor S(t) and the integral adds all these little S(t)dt steps up. We use units where c=1 so we don't have to include the speed of light explicitly.
If you don't want to bother with the integral yourself, it's fine, there's also an online calculator called "Lightcone" that does all that stuff for you. I keep the link in my PF signature to have it handy.
7. Aug 1, 2015
### marcus
http://www.numberempire.com/definiteintegralcalculator.php
What you put in for "function to be integrated" (I'll explain why later) is 1.3*(sinh(1.5*t))^(-2/3)
and because the variable is t you change x to t in the variable box
and for the limits you put in 0.00001 and 0.8
If you want the answer directly in billions of LY you can instead put in 17.3*1.3*(sinh(1.5*t))^(-2/3)
8. Aug 1, 2015
### marcus
It may be a little confusing because I'm using a time unit which is 17.3 billion years, to make the formulas simple. On that scale the present is 0.8.
(the universe is still young, only 80% of one of its time units)
and on that scale the universe expands as sinh(1.5t)2/3
So that between time t and the present 0.8 distances get enlarged by the ratio of the two sizes:
sinh(1.5*0.8)2/3/sinh(1.5t)2/3=1.3/sinh(1.5t)2/3. It happens that the present value of that size function is 1.3 so can we put that in for the numerator.
The integral adds each dt step taken around time t, scaled up by the appropriate ratio 1.3/sinh(1.5t)2/3
Last edited: Aug 1, 2015
9. Aug 1, 2015
### Chalnoth
What we see are the light rays that have been traveling for about 13.8 billion years.
10. Aug 1, 2015
### Chronos
While it is true matter cannot travel that fast there is nothing to prevent empty space from expanding that fast and it carries along the matter embedded in it like a surfboard.
11. Aug 2, 2015
### Stephanus
How can we determine that those things are 45 billion ly? The red shift?Doppler?
And from what I learnt in SR Forum.
Supposed V = 0.99c, so red shift is $k = \sqrt{\frac{1+V}{1-V}} = 14$ Even if its V is 0.99....99c still if we multiply it by the age of the universe it can't be farther than 13.5 billion ly.
So how can we know that this particular thing is farther than 13 billions ly?
12. Aug 2, 2015
### Stephanus
Yes, matter can't travel faster than the speed of light, it's the space that is expanded.
But how we measure that the object is 46 billion lys?
13. Aug 2, 2015
### Stephanus
Ahh, that's the answer. Sorry, just haven't read the incoming threads. Is it the combination of doppler shift factor and hubble law?
14. Aug 2, 2015
### Chronos
Expansion of the universe is related to cosmological redshift which is an entirely different beast from doppler shift.
15. Aug 2, 2015
### timmdeeg
We can measure the light emitted by the object then. From this combined with the knowledge of how the universe expands we can calculate the distance then, about 46 million lys, and the distance now, 46 billion lys; assuming that the universe expanded by a factor of 1000.
Last edited: Aug 2, 2015
16. Aug 2, 2015
### Chalnoth
Special relativity doesn't work in an expanding universe, as special relativity assumes flat space-time, and the expansion is curvature in space-time.
The redshift that we see for objects far away in the universe has nothing to do with the Doppler shift. Instead, it's due to the expansion itself. When the universe doubles in size, photon wavelengths are also doubled. The CMB photons have been redshifted by a factor of approximately 1090, meaning that the universe has expanded 1090 times in each direction since the CMB was emitted.
To get the distance to the CMB, we use what is known as the first acoustic peak. In the early universe plasma, there were pressure waves. These pressure waves sort of "bunch up" at a distance called the sound horizon: the distance that the waves could have traveled since our universe began. This creates the first acoustic peak, and its distance is approximately a separation of one degree on the sky.
So if we know how old the universe was when the CMB was emitted, and can model how far the sound waves in that early plasma could have traveled, then we can use the one degree separation to estimate how far away those peaks were, and the answer is approximately 46 million light years. As the universe has expanded by a factor of about 1090 since then, the current distance to the matter that emitted that light is around 50 billion light years*.
*The different numbers in this thread come from different observations. The numbers I'm quoting here are from a combination of data including the WMAP 9-year data. I use WMAP instead of the better Planck data because the website is better-designed and it's easier to look up the numbers.
17. Aug 5, 2015
### Gaz
Well that can't be true I think its safe to assume that the galaxies would have been moving around independently somewhat to ?
But I guess no one can really know how much ?
18. Aug 5, 2015
### marcus
Good point! Things do have their individual motions in the local space surrounding them and that does contribute a doppler bit on top of the main distance expansion redshift.
In many situations the cosmological redshift is the main thing and the individual random motions are (as you suggest) very hard to determine---they tend to be small (a few hundred km/s) and get neglected.
But there are some nice situations where one can estimate the small contribution they make.
For example if you can resolve a spiral galaxy seen edge on. Besides the cosmological redshift due to how much the distance to the galaxy has expanded---you have the stars on one side coming towards us and the stars on the other side going away, due to rotation. That gives a doppler effect over and above the cosmological redshift so you can measure the speed of rotation.
Also clusters of galaxies have the individual random motions of the galaxies within the cluster. So there is doppler (from the RADIAL component of those motions) which is over and above the redshift due to the change in overall distance to the cluster. One can get a very rough idea, at least.
Or so I'm told : ^)
not an expert in that kind of thing.
Last edited: Aug 5, 2015
19. Aug 5, 2015
### Chalnoth
I was a little bit short in that explanation, you're right. Especially within very massive clusters, the local motions can be pretty large, but not large compared to the cosmological redshift for far-away galaxies. The largest are, if I recall correctly, about 3,000 km/s, which translates to a redshift of $z = 0.01$. Most galaxies are going to have much smaller local motions (our own motion relative to the CMB is about 600km/s). Obviously for galaxies at a redshift of 1-2 or higher, this isn't going to be an issue (for the most part). Most cosmologists just ignore the local motions of galaxies entirely, and that approximation works extremely well.
As for how we know this, when astronomers are doing surveys of large numbers of galaxies, they measure the position of the galaxies by direction and redshift (since the redshift is directly measurable, while the distance is not). Galaxy clusters on such maps look as if they have been elongated along the line of sight: instead of a nearly spherical blob, clusters are long blobs oriented along the line of sight (when redshift is converted to the appropriate units to approximate distance). This fact can be used to estimate the velocities of the galaxies within the cluster.
Similar Discussions: Age of the observable Universe? | {"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": 0.8369948863983154, "perplexity": 691.3254894356285}, "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/1502886104636.62/warc/CC-MAIN-20170818121545-20170818141545-00281.warc.gz"} |
http://www.mathworks.com/help/stats/prob.multinomialdistribution-class.html?nocookie=true | # prob.MultinomialDistribution class
Package: prob
Superclasses: prob.ParametricTruncatableDistribution
Multinomial probability distribution object
## Description
`prob.MultinomialDistribution` is an object consisting of parameters and a model description for a multinomial probability distribution. Create a probability distribution object with specified parameters using `makedist`.
## Construction
`pd = makedist('Multinomial')` creates a multinomial probability distribution object using the default parameter values.
`pd = makedist('Multinomial','Probabilities',probabilities)` creates a multinomial distribution object using the specified parameter value.
collapse all
### `probabilities` — outcome probabilities`[0.500 0.500]` (default) | vector of scalar values in the range `[0,1]`
Outcome probabilities, specified as a vector of scalar values in the range `[0,1]`. Each vector element is the probability that a multinomial trial has a particular corresponding outcome. The values in `probabilities` must sum to 1.
Data Types: `single` | `double`
## Properties
collapse all
### `probabilities` — outcome probabilitiesvector of scalar values in the range `[0,1]`
Outcome probabilities for the multinomial distribution, stored as a vector of scalar values in the range `[0,1]`. The values in `probabilities` must sum to 1.
Data Types: `single` | `double`
### `DistributionName` — Probability distribution nameprobability distribution name string
Probability distribution name, stored as a valid probability distribution name string. This property is read-only.
Data Types: `char`
### `IsTruncated` — Logical flag for truncated distribution`0` | `1`
Logical flag for truncated distribution, stored as a logical value. If `IsTruncated` equals `0`, the distribution is not truncated. If `IsTruncated` equals `1`, the distribution is truncated. This property is read-only.
Data Types: `logical`
### `NumParameters` — Number of parameterspositive integer value
Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.
Data Types: `single` | `double`
### `ParameterDescription` — Distribution parameter descriptionscell array of strings
Distribution parameter descriptions, stored as a cell array of strings. Each cell contains a short description of one distribution parameter. This property is read-only.
Data Types: `char`
### `ParameterNames` — Distribution parameter namescell array of strings
Distribution parameter names, stored as a cell array of strings. This property is read-only.
Data Types: `char`
### `ParameterValues` — Distribution parameter valuesvector of scalar values
Distribution parameter values, stored as a vector. This property is read-only.
Data Types: `single` | `double`
### `Truncation` — Truncation intervalvector of scalar values
Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.
Data Types: `single` | `double`
## Methods
### Inherited Methods
cdf Cumulative distribution function of probability distribution object icdf Inverse cumulative distribution function of probability distribution object iqr Interquartile range of probability distribution object median Median of probability distribution object pdf Probability density function of probability distribution object random Generate random numbers from probability distribution object truncate Truncate probability distribution object
mean Mean of probability distribution object std Standard deviation of probability distribution object var Variance of probability distribution object
## Definitions
### Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution. While the binomial distribution gives the probability of the number of "successes" in n independent trials of a two-outcome process, the multinomial distribution gives the probability of each combination of outcomes in n independent trials of a k-outcome process. The probability of each outcome in any one trial is given by the fixed probabilities p1, ..., pk.
The multinomial distribution uses the following parameters.
ParameterDescriptionSupport
`probabilities`Outcome probabilities$0\le probabilities\left(i\right)\le 1\text{\hspace{0.17em}};\text{\hspace{0.17em}}\sum _{all\left(i\right)}probabilities\left(i\right)=1$
The probability density function (pdf) is
$f\left(x|n,p\right)=\frac{n!}{{x}_{1}!\dots {x}_{k}!}{p}_{1}{}^{{x}_{1}}\cdots {p}_{k}{}^{{x}_{k}}\text{ };\text{ }\sum _{1}^{k}{x}_{i}=n\text{\hspace{0.17em}},\text{\hspace{0.17em}}\sum _{1}^{k}{p}_{i}=1\text{\hspace{0.17em}},$
where x = (x1,...,xk) gives the number of each k outcome in n trials of a process with fixed probabilities p = (p1,...,pk) of individual outcomes in any one trial.
## Examples
collapse all
### Create a Multinomial Distribution Object Using Default Parameters
Create a multinomial distribution object using the default parameter values.
`pd = makedist('Multinomial')`
```pd = MultinomialDistribution Probabilities: 0.5000 0.5000```
### Create a Multinomial Distribution Object Using Specified Parameters
Create a multinomial distribution object for a distribution with three possible outcomes. Outcome 1 has a probability of 1/2, outcome 2 has a probability of 1/3, and outcome 3 has a probability of 1/6.
`pd = makedist('Multinomial','probabilities',[1/2 1/3 1/6])`
```pd = MultinomialDistribution Probabilities: 0.5000 0.3333 0.1667```
Generate a random outcome from the distribution.
```rng('default'); % for reproducibility r = random(pd)```
```r = 2 ```
The result of this trial is outcome 2. By default, the number of trials in each experiment, n, equals 1.
Generate random outcomes from the distribution when the number of trials in each experiment, n, equals 1, and the experiment is repeated ten times.
```rng('default'); % for reproducibility r = random(pd,10,1)```
```r = 2 3 1 3 2 1 1 2 3 3```
Each element in the array is the outcome of an individual experiment that contains one trial.
Generate random outcomes from the distribution when the number of trials in each experiment, n, equals 5, and the experiment is repeated ten times.
```rng('default'); % for reproducibility r = random(pd,10,5)```
```r = 2 1 2 2 1 3 3 1 1 1 1 3 3 1 2 3 1 3 1 2 2 2 2 1 1 1 1 2 2 1 1 1 2 2 1 2 3 1 1 2 3 2 2 3 2 3 3 1 1 2```
Each element in the resulting matrix is the outcome of one trial. The columns correspond to the five trials in each experiment, and the rows correspond to the ten experiments. For example, in the first experiment (corresponding to the first row), 2 of the 5 trials resulted in outcome 1, and 3 of the 5 trials resulted in outcome 2. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 2, "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": 0.9518137574195862, "perplexity": 1679.5591857905147}, "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/1438042988840.31/warc/CC-MAIN-20150728002308-00044-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/hi-i-need-some-help.379769/ | # Hi i need some help
1. Feb 19, 2010
### the pro
1. The problem statement, all variables and given/known data
the parametric equations of the motion are x=(t)^2m and y=(6t-1.5t^2)m i need to find the
velocity at the time moment of t=2s
2. Relevant equations
3. The attempt at a solution
so this are the four solutions v=o,v=2m/s,v=4m/s,6m/s .
2. Feb 19, 2010
### korican04
Do you know how to take a derivative?
dx/dt = 2t m/s
dy/dt= 6-3t m/s
At t=2
dx/dt=4 m/s
dy/dt=0 m/s
v= 4 m/s in the x direction.
3. Feb 19, 2010
### the pro
No i dont now how if you could post it how to do also the derivate then it would be ok and thanks for this post
4. Feb 21, 2010
### Astronuc
Staff Emeritus
See if this helps - http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html
One has two position parameters x(t) and y(t).
The velocity, or rather speed, in each direction is just the first derivative with respect to time,
vx(t) = dx(t)/dt, and vy(t) = dy(t)/dt.
http://hyperphysics.phy-astr.gsu.edu/hbase/deriv.html
http://hyperphysics.phy-astr.gsu.edu/hbase/deriv.html#c3
Then since velocity is a vectors,
v(t) = vx(t) i + vy(t) j, where i and j are just the unit vectors in x and y directions.
The magnitude of v(t) is given by the square root of the sum of the squares of the speeds in both direction, i.e.,
|v(t)| = sqrt (vx2(t) + vy2(t)) | {"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": 0.9632109999656677, "perplexity": 2122.4738176441847}, "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-2018-13/segments/1521257645413.2/warc/CC-MAIN-20180318013134-20180318033134-00031.warc.gz"} |
http://voer.edu.vn/c/photon-energies-and-the-electromagnetic-spectrum/0e60bfc6/15d6c4d0 | Giáo trình
# College Physics
Science and Technology
## Photon Energies and the Electromagnetic Spectrum
Tác giả: OpenStaxCollege
A photon is a quantum of EM radiation. Its energy is given by $E=\text{hf}$ and is related to the frequency $f$ and wavelength $\lambda$ of the radiation by
$E=\text{hf}=\frac{\text{hc}}{\lambda }\text{(energy of a photon),}$
where $E$ is the energy of a single photon and $c$ is the speed of light. When working with small systems, energy in eV is often useful. Note that Planck’s constant in these units is
$h=\text{4}\text{.}\text{14}×{\text{10}}^{\text{–15}}\phantom{\rule{0.25em}{0ex}}\text{eV}\cdot \text{s}.$
Since many wavelengths are stated in nanometers (nm), it is also useful to know that
$\text{hc}=\text{1240 eV}\cdot \text{nm}.$
These will make many calculations a little easier.
All EM radiation is composed of photons. [link] shows various divisions of the EM spectrum plotted against wavelength, frequency, and photon energy. Previously in this book, photon characteristics were alluded to in the discussion of some of the characteristics of UV, x rays, and $\gamma$ rays, the first of which start with frequencies just above violet in the visible spectrum. It was noted that these types of EM radiation have characteristics much different than visible light. We can now see that such properties arise because photon energy is larger at high frequencies.
Rotational energies of molecules ${\text{10}}^{-5}$ eV Vibrational energies of molecules 0.1 eV Energy between outer electron shells in atoms 1 eV Binding energy of a weakly bound molecule 1 eV Energy of red light 2 eV Binding energy of a tightly bound molecule 10 eV Energy to ionize atom or molecule 10 to 1000 eV
Photons act as individual quanta and interact with individual electrons, atoms, molecules, and so on. The energy a photon carries is, thus, crucial to the effects it has. [link] lists representative submicroscopic energies in eV. When we compare photon energies from the EM spectrum in [link] with energies in the table, we can see how effects vary with the type of EM radiation.
Gamma rays, a form of nuclear and cosmic EM radiation, can have the highest frequencies and, hence, the highest photon energies in the EM spectrum. For example, a $\gamma$-ray photon with $f{\text{= 10}}^{\text{21}}\phantom{\rule{0.25em}{0ex}}\text{Hz}$ has an energy $E=\text{hf}=6.63×{\text{10}}^{\text{–13}}\phantom{\rule{0.25em}{0ex}}\text{J}=4\text{.}\text{14 MeV.}$ This is sufficient energy to ionize thousands of atoms and molecules, since only 10 to 1000 eV are needed per ionization. In fact, $\gamma$ rays are one type of ionizing radiation, as are x rays and UV, because they produce ionization in materials that absorb them. Because so much ionization can be produced, a single $\gamma$-ray photon can cause significant damage to biological tissue, killing cells or damaging their ability to properly reproduce. When cell reproduction is disrupted, the result can be cancer, one of the known effects of exposure to ionizing radiation. Since cancer cells are rapidly reproducing, they are exceptionally sensitive to the disruption produced by ionizing radiation. This means that ionizing radiation has positive uses in cancer treatment as well as risks in producing cancer.
High photon energy also enables $\gamma$ rays to penetrate materials, since a collision with a single atom or molecule is unlikely to absorb all the $\gamma$ ray’s energy. This can make $\gamma$ rays useful as a probe, and they are sometimes used in medical imaging. x rays, as you can see in [link], overlap with the low-frequency end of the $\gamma$ ray range. Since x rays have energies of keV and up, individual x-ray photons also can produce large amounts of ionization. At lower photon energies, x rays are not as penetrating as $\gamma$ rays and are slightly less hazardous. X rays are ideal for medical imaging, their most common use, and a fact that was recognized immediately upon their discovery in 1895 by the German physicist W. C. Roentgen (1845–1923). (See [link].) Within one year of their discovery, x rays (for a time called Roentgen rays) were used for medical diagnostics. Roentgen received the 1901 Nobel Prize for the discovery of x rays.
While $\gamma$ rays originate in nuclear decay, x rays are produced by the process shown in [link]. Electrons ejected by thermal agitation from a hot filament in a vacuum tube are accelerated through a high voltage, gaining kinetic energy from the electrical potential energy. When they strike the anode, the electrons convert their kinetic energy to a variety of forms, including thermal energy. But since an accelerated charge radiates EM waves, and since the electrons act individually, photons are also produced. Some of these x-ray photons obtain the kinetic energy of the electron. The accelerated electrons originate at the cathode, so such a tube is called a cathode ray tube (CRT), and various versions of them are found in older TV and computer screens as well as in x-ray machines.
X-ray Photon Energy and X-ray Tube Voltage
Find the maximum energy in eV of an x-ray photon produced by electrons accelerated through a potential difference of 50.0 kV in a CRT like the one in [link].
Strategy
Electrons can give all of their kinetic energy to a single photon when they strike the anode of a CRT. (This is something like the photoelectric effect in reverse.) The kinetic energy of the electron comes from electrical potential energy. Thus we can simply equate the maximum photon energy to the electrical potential energy—that is, $\text{hf}=\text{qV.}$ (We do not have to calculate each step from beginning to end if we know that all of the starting energy $\text{qV}$ is converted to the final form $\text{hf.}$)
Solution
The maximum photon energy is $\text{hf}=\text{qV}$, where $\text{q}$ is the charge of the electron and $\text{V}$ is the accelerating voltage. Thus,
$\text{hf}=\left(1\text{.}\text{60}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\right)\left(\text{50.0}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{V}\right)\text{.}$
From the definition of the electron volt, we know $\text{1 eV}=\text{1}\text{.}\text{60}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{J}$, where $\text{1 J}=1 C\cdot V.$ Gathering factors and converting energy to eV yields
$\text{hf}=\left(\text{50.0}×{\text{10}}^{3}\right)\left(1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\cdot \text{V}\right)\left(\frac{\text{1 eV}}{1.60×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{C}\cdot \text{V}}\right)=\left(\text{50.0}×{\text{10}}^{3}\right)\left(\text{1 eV}\right)=\text{50.0 keV.}$
Discussion
This example produces a result that can be applied to many similar situations. If you accelerate a single elementary charge, like that of an electron, through a potential given in volts, then its energy in eV has the same numerical value. Thus a 50.0-kV potential generates 50.0 keV electrons, which in turn can produce photons with a maximum energy of 50 keV. Similarly, a 100-kV potential in an x-ray tube can generate up to 100-keV x-ray photons. Many x-ray tubes have adjustable voltages so that various energy x rays with differing energies, and therefore differing abilities to penetrate, can be generated.
[link] shows the spectrum of x rays obtained from an x-ray tube. There are two distinct features to the spectrum. First, the smooth distribution results from electrons being decelerated in the anode material. A curve like this is obtained by detecting many photons, and it is apparent that the maximum energy is unlikely. This decelerating process produces radiation that is called bremsstrahlung (German for braking radiation). The second feature is the existence of sharp peaks in the spectrum; these are called characteristic x rays, since they are characteristic of the anode material. Characteristic x rays come from atomic excitations unique to a given type of anode material. They are akin to lines in atomic spectra, implying the energy levels of atoms are quantized. Phenomena such as discrete atomic spectra and characteristic x rays are explored further in Atomic Physics.
Ultraviolet radiation (approximately 4 eV to 300 eV) overlaps with the low end of the energy range of x rays, but UV is typically lower in energy. UV comes from the de-excitation of atoms that may be part of a hot solid or gas. These atoms can be given energy that they later release as UV by numerous processes, including electric discharge, nuclear explosion, thermal agitation, and exposure to x rays. A UV photon has sufficient energy to ionize atoms and molecules, which makes its effects different from those of visible light. UV thus has some of the same biological effects as $\gamma$ rays and x rays. For example, it can cause skin cancer and is used as a sterilizer. The major difference is that several UV photons are required to disrupt cell reproduction or kill a bacterium, whereas single $\gamma$-ray and X-ray photons can do the same damage. But since UV does have the energy to alter molecules, it can do what visible light cannot. One of the beneficial aspects of UV is that it triggers the production of vitamin D in the skin, whereas visible light has insufficient energy per photon to alter the molecules that trigger this production. Infantile jaundice is treated by exposing the baby to UV (with eye protection), called phototherapy, the beneficial effects of which are thought to be related to its ability to help prevent the buildup of potentially toxic bilirubin in the blood.
Photon Energy and Effects for UV
Short-wavelength UV is sometimes called vacuum UV, because it is strongly absorbed by air and must be studied in a vacuum. Calculate the photon energy in eV for 100-nm vacuum UV, and estimate the number of molecules it could ionize or break apart.
Strategy
Using the equation $E=\text{hf}$ and appropriate constants, we can find the photon energy and compare it with energy information in [link].
Solution
The energy of a photon is given by
$E=\text{hf}=\frac{\text{hc}}{\lambda }.$
Using $\text{hc}=\text{1240 eV}\cdot \text{nm,}$ we find that
$E=\frac{\text{hc}}{\lambda }=\frac{\text{1240 eV}\cdot \text{nm}}{\text{100 nm}}=\text{12}\text{.}\text{4 eV}.$
Discussion
According to [link], this photon energy might be able to ionize an atom or molecule, and it is about what is needed to break up a tightly bound molecule, since they are bound by approximately 10 eV. This photon energy could destroy about a dozen weakly bound molecules. Because of its high photon energy, UV disrupts atoms and molecules it interacts with. One good consequence is that all but the longest-wavelength UV is strongly absorbed and is easily blocked by sunglasses. In fact, most of the Sun’s UV is absorbed by a thin layer of ozone in the upper atmosphere, protecting sensitive organisms on Earth. Damage to our ozone layer by the addition of such chemicals as CFC’s has reduced this protection for us.
# Visible Light
The range of photon energies for visible light from red to violet is 1.63 to 3.26 eV, respectively (left for this chapter’s Problems and Exercises to verify). These energies are on the order of those between outer electron shells in atoms and molecules. This means that these photons can be absorbed by atoms and molecules. A single photon can actually stimulate the retina, for example, by altering a receptor molecule that then triggers a nerve impulse. Photons can be absorbed or emitted only by atoms and molecules that have precisely the correct quantized energy step to do so. For example, if a red photon of frequency $f$ encounters a molecule that has an energy step, $\Delta E,$ equal to $\text{hf},$ then the photon can be absorbed. Violet flowers absorb red and reflect violet; this implies there is no energy step between levels in the receptor molecule equal to the violet photon’s energy, but there is an energy step for the red.
There are some noticeable differences in the characteristics of light between the two ends of the visible spectrum that are due to photon energies. Red light has insufficient photon energy to expose most black-and-white film, and it is thus used to illuminate darkrooms where such film is developed. Since violet light has a higher photon energy, dyes that absorb violet tend to fade more quickly than those that do not. (See [link].) Take a look at some faded color posters in a storefront some time, and you will notice that the blues and violets are the last to fade. This is because other dyes, such as red and green dyes, absorb blue and violet photons, the higher energies of which break up their weakly bound molecules. (Complex molecules such as those in dyes and DNA tend to be weakly bound.) Blue and violet dyes reflect those colors and, therefore, do not absorb these more energetic photons, thus suffering less molecular damage.
Transparent materials, such as some glasses, do not absorb any visible light, because there is no energy step in the atoms or molecules that could absorb the light. Since individual photons interact with individual atoms, it is nearly impossible to have two photons absorbed simultaneously to reach a large energy step. Because of its lower photon energy, visible light can sometimes pass through many kilometers of a substance, while higher frequencies like UV, x ray, and $\gamma$ rays are absorbed, because they have sufficient photon energy to ionize the material.
How Many Photons per Second Does a Typical Light Bulb Produce?
Assuming that 10.0% of a 100-W light bulb’s energy output is in the visible range (typical for incandescent bulbs) with an average wavelength of 580 nm, calculate the number of visible photons emitted per second.
Strategy
Power is energy per unit time, and so if we can find the energy per photon, we can determine the number of photons per second. This will best be done in joules, since power is given in watts, which are joules per second.
Solution
The power in visible light production is 10.0% of 100 W, or 10.0 J/s. The energy of the average visible photon is found by substituting the given average wavelength into the formula
$E=\frac{\text{hc}}{\lambda }.$
This produces
$E=\frac{\left(6\text{.}\text{63}×{\text{10}}^{\text{–34}}\phantom{\rule{0.25em}{0ex}}\text{J}\cdot \text{s}\right)\left(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)}{\phantom{\rule{0.25em}{0ex}}\text{580}×{\text{10}}^{\text{–9}}\phantom{\rule{0.25em}{0ex}}\text{m}}=\text{3.43}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{J}.$
The number of visible photons per second is thus
$\text{photon/s}=\frac{\text{10.0 J/s}}{3\text{.}\text{43}×{\text{10}}^{\text{–19}}\phantom{\rule{0.25em}{0ex}}\text{J/photon}}=\text{2.92}×{\text{10}}^{\text{19}}\phantom{\rule{0.25em}{0ex}}\text{photon/s}.$
Discussion
This incredible number of photons per second is verification that individual photons are insignificant in ordinary human experience. It is also a verification of the correspondence principle—on the macroscopic scale, quantization becomes essentially continuous or classical. Finally, there are so many photons emitted by a 100-W lightbulb that it can be seen by the unaided eye many kilometers away.
# Lower-Energy Photons
Infrared radiation (IR) has even lower photon energies than visible light and cannot significantly alter atoms and molecules. IR can be absorbed and emitted by atoms and molecules, particularly between closely spaced states. IR is extremely strongly absorbed by water, for example, because water molecules have many states separated by energies on the order of ${\text{10}}^{\text{–5}}\phantom{\rule{0.25em}{0ex}}\text{eV}$ to ${\text{10}}^{\text{–2}}\phantom{\rule{0.25em}{0ex}}\text{eV,}$ well within the IR and microwave energy ranges. This is why in the IR range, skin is almost jet black, with an emissivity near 1—there are many states in water molecules in the skin that can absorb a large range of IR photon energies. Not all molecules have this property. Air, for example, is nearly transparent to many IR frequencies.
Microwaves are the highest frequencies that can be produced by electronic circuits, although they are also produced naturally. Thus microwaves are similar to IR but do not extend to as high frequencies. There are states in water and other molecules that have the same frequency and energy as microwaves, typically about ${\text{10}}^{\text{–5}}\phantom{\rule{0.25em}{0ex}}\text{eV.}$ This is one reason why food absorbs microwaves more strongly than many other materials, making microwave ovens an efficient way of putting energy directly into food.
Photon energies for both IR and microwaves are so low that huge numbers of photons are involved in any significant energy transfer by IR or microwaves (such as warming yourself with a heat lamp or cooking pizza in the microwave). Visible light, IR, microwaves, and all lower frequencies cannot produce ionization with single photons and do not ordinarily have the hazards of higher frequencies. When visible, IR, or microwave radiation is hazardous, such as the inducement of cataracts by microwaves, the hazard is due to huge numbers of photons acting together (not to an accumulation of photons, such as sterilization by weak UV). The negative effects of visible, IR, or microwave radiation can be thermal effects, which could be produced by any heat source. But one difference is that at very high intensity, strong electric and magnetic fields can be produced by photons acting together. Such electromagnetic fields (EMF) can actually ionize materials.
It is virtually impossible to detect individual photons having frequencies below microwave frequencies, because of their low photon energy. But the photons are there. A continuous EM wave can be modeled as photons. At low frequencies, EM waves are generally treated as time- and position-varying electric and magnetic fields with no discernible quantization. This is another example of the correspondence principle in situations involving huge numbers of photons.
# Section Summary
• Photon energy is responsible for many characteristics of EM radiation, being particularly noticeable at high frequencies.
• Photons have both wave and particle characteristics.
# Conceptual Questions
Why are UV, x rays, and $\gamma$ rays called ionizing radiation?
How can treating food with ionizing radiation help keep it from spoiling? UV is not very penetrating. What else could be used?
Some television tubes are CRTs. They use an approximately 30-kV accelerating potential to send electrons to the screen, where the electrons stimulate phosphors to emit the light that forms the pictures we watch. Would you expect x rays also to be created?
Tanning salons use “safe” UV with a longer wavelength than some of the UV in sunlight. This “safe” UV has enough photon energy to trigger the tanning mechanism. Is it likely to be able to cause cell damage and induce cancer with prolonged exposure?
Your pupils dilate when visible light intensity is reduced. Does wearing sunglasses that lack UV blockers increase or decrease the UV hazard to your eyes? Explain.
One could feel heat transfer in the form of infrared radiation from a large nuclear bomb detonated in the atmosphere 75 km from you. However, none of the profusely emitted x rays or $\gamma$ rays reaches you. Explain.
Can a single microwave photon cause cell damage? Explain.
In an x-ray tube, the maximum photon energy is given by $\text{hf}=\text{qV}.$ Would it be technically more correct to say $\text{hf}=\text{qV}+\text{BE,}$ where BE is the binding energy of electrons in the target anode? Why isn’t the energy stated the latter way?
# Problems & Exercises
What is the energy in joules and eV of a photon in a radio wave from an AM station that has a 1530-kHz broadcast frequency?
$6.34×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{eV}$, $1.01×{\text{10}}^{-27}\phantom{\rule{0.25em}{0ex}}\text{J}$
(a) Find the energy in joules and eV of photons in radio waves from an FM station that has a 90.0-MHz broadcast frequency. (b) What does this imply about the number of photons per second that the radio station must broadcast?
Calculate the frequency in hertz of a 1.00-MeV $\gamma$-ray photon.
$2\text{.}\text{42}×{\text{10}}^{\text{20}}\phantom{\rule{0.25em}{0ex}}\text{Hz}$
(a) What is the wavelength of a 1.00-eV photon? (b) Find its frequency in hertz. (c) Identify the type of EM radiation.
Do the unit conversions necessary to show that $\text{hc}=\text{1240 eV}\cdot \text{nm,}$ as stated in the text.
$\begin{array}{lll}\text{hc}& =& \left(\text{6.62607}×{\text{10}}^{-\text{34}}\phantom{\rule{0.25em}{0ex}}J\cdot s\right)\left(\text{2.99792}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)\left(\frac{{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}\text{nm}}{1 m}\right)\left(\frac{\text{1.00000 eV}}{\text{1.60218}×{\text{10}}^{-\text{19}}\phantom{\rule{0.25em}{0ex}}\text{J}}\right)\\ & =& \text{1239.84 eV}\cdot \text{nm}\\ & \approx & \text{1240 eV}\cdot \text{nm}\end{array}$
Confirm the statement in the text that the range of photon energies for visible light is 1.63 to 3.26 eV, given that the range of visible wavelengths is 380 to 760 nm.
(a) Calculate the energy in eV of an IR photon of frequency $\text{2.00}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{Hz.}$ (b) How many of these photons would need to be absorbed simultaneously by a tightly bound molecule to break it apart? (c) What is the energy in eV of a $\gamma$ ray of frequency $3\text{.}\text{00}×{\text{10}}^{\text{20}}\phantom{\rule{0.25em}{0ex}}\text{Hz?}$ (d) How many tightly bound molecules could a single such $\gamma$ ray break apart?
(a) 0.0829 eV
(b) 121
(c) 1.24 MeV
(d) $1\text{.}\text{24}×{\text{10}}^{5}$
Prove that, to three-digit accuracy, $h=4\text{.}\text{14}×{\text{10}}^{-\text{15}}\phantom{\rule{0.25em}{0ex}}\text{eV}\cdot s,$ as stated in the text.
(a) What is the maximum energy in eV of photons produced in a CRT using a 25.0-kV accelerating potential, such as a color TV? (b) What is their frequency?
(a) $\text{25.0}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{eV}$
(b) $\text{6}\text{.}\text{04}×{\text{10}}^{\text{18}}\phantom{\rule{0.25em}{0ex}}\text{Hz}$
What is the accelerating voltage of an x-ray tube that produces x rays with a shortest wavelength of 0.0103 nm?
(a) What is the ratio of power outputs by two microwave ovens having frequencies of 950 and 2560 MHz, if they emit the same number of photons per second? (b) What is the ratio of photons per second if they have the same power output?
(a) 2.69
(b) 0.371
How many photons per second are emitted by the antenna of a microwave oven, if its power output is 1.00 kW at a frequency of 2560 MHz?
Some satellites use nuclear power. (a) If such a satellite emits a 1.00-W flux of $\gamma$ rays having an average energy of 0.500 MeV, how many are emitted per second? (b) These $\gamma$ rays affect other satellites. How far away must another satellite be to only receive one $\gamma$ ray per second per square meter?
(a) $\text{1}\text{.}\text{25}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{photons/s}$
(b) 997 km
(a) If the power output of a 650-kHz radio station is 50.0 kW, how many photons per second are produced? (b) If the radio waves are broadcast uniformly in all directions, find the number of photons per second per square meter at a distance of 100 km. Assume no reflection from the ground or absorption by the air.
How many x-ray photons per second are created by an x-ray tube that produces a flux of x rays having a power of 1.00 W? Assume the average energy per photon is 75.0 keV.
$\text{8.33}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{photons/s}$
(a) How far away must you be from a 650-kHz radio station with power 50.0 kW for there to be only one photon per second per square meter? Assume no reflections or absorption, as if you were in deep outer space. (b) Discuss the implications for detecting intelligent life in other solar systems by detecting their radio broadcasts.
Assuming that 10.0% of a 100-W light bulb’s energy output is in the visible range (typical for incandescent bulbs) with an average wavelength of 580 nm, and that the photons spread out uniformly and are not absorbed by the atmosphere, how far away would you be if 500 photons per second enter the 3.00-mm diameter pupil of your eye? (This number easily stimulates the retina.)
181 km | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 70, "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": 0.8910207748413086, "perplexity": 730.0203853191277}, "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/1534221209040.29/warc/CC-MAIN-20180814131141-20180814151141-00087.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=StudyGuides%2FCalculus%2FAppendix%2FExamples%2FA-7%2FExampleA-7-9 | Example A-7-9 - Maple Help
Appendix
A-7: Trigonometry
Example A-7.9
Obtain a Maple graph of the sine function where the argument is given in degrees, not radians.
For more information on Maplesoft products and services, visit www.maplesoft.com | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 28, "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": 0.994918167591095, "perplexity": 4114.394216199136}, "config": {"markdown_headings": true, "markdown_code": false, "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-2023-14/segments/1679296949093.14/warc/CC-MAIN-20230330004340-20230330034340-00398.warc.gz"} |
https://www.arxiv-vanity.com/papers/hep-ph/0612153/ | # Diffraction at HERA, the Tevatron and the LHC
C. Royon DAPNIA/Service de physique des particules, CEA/Saclay, 91191 Gif-sur-Yvette cedex, France
###### Abstract
In these lectures, we present and discuss the most recent results on inclusive diffraction from the HERA and Tevatron colliders and give the prospects for the future at the LHC. Of special interest is the exclusive production of Higgs boson and heavy objects (, top, stop pairs) which will require a better understanding of diffractive events and the link between and hadronic colliders, as well as precise measurements and analyses of inclusive diffraction at the LHC in particular to constrain further the gluon density in the pomeron.
In these lectures, we describe the most recent results on inclusive diffraction at HERA, as well as diffractive results from the Tevatron. We finish the lecture by discussing the prospects of diffractive physics at the LHC.
## 1 Experimental definition of diffraction
In this section, we discuss the different experimental ways to define diffraction. As an example, we describe the methods used by the H1 and ZEUS experiments at HERA, DESY, Hamburg in Germany.
### 1.1 The rapidity gap method
HERA is a collider where electrons of 27.6 GeV collide with protons of 920 GeV. A typical event as shown in the upper plot of Fig. 1 is where electron and jets are produced in the final state. We notice that the electron is scattered in the H1 backward detector111At HERA, the backward (resp. forward) directions are defined as the direction of the outgoing electron (resp. proton). (in green) whereas some hadronic activity is present in the forward region of the detector (in the LAr calorimeter and in the forward muon detectors). The proton is thus completely destroyed and the interaction leads to jets and proton remnants directly observable in the detector. The fact that much energy is observed in the forward region is due to colour exchange between the scattered jet and the proton remnants. In about 10% of the events, the situation is completely different. Such events appear like the one shown in the bottom plot of Fig. 1. The electron is still present in the backward detector, there is still some hadronic activity (jets) in the LAr calorimeter, but no energy above noise level is deposited in the forward part of the LAr calorimeter or in the forward muon detectors. In other words, there is no color exchange between the proton and the produced jets. As an example, this can be explained if the proton stays intact after the interaction.
This experimental observation leads to the first definition of diffraction: request a rapidity gap (in other words a domain in the forward detectors where no energy is deposited above noise level) in the forward region. For example, the H1 collaboration requests no energy deposition in the rapidity region where is the pseudorapidity. Let us note that this approach does not insure that the proton stays intact after the interaction, but it represents a limit on the mass of the produced object GeV. Within this limit, the proton could be dissociated. The adavantage of the rapidity gap method is that it is quite easy to implement and it has a large acceptance in the diffractive kinematical plane.
### 1.2 Proton tagging
The second experimental method to detect diffractive events is also natural: the idea is to detect directly the intact proton in the final state. The proton loses a small fraction of its energy and is thus scattered at very small angle with respect to the beam direction. Some special detectors called roman pots can be used to detect the protons close to the beam. The basic idea is simple: the roman pot detectors are located far away from the interaction point and can move close to the beam, when the beam is stable, to detect protons scattered at vary small angles. The inconvenience is that the kinematical reach of those detectors is much smaller than with the rapidity gap method. On the other hand, the advantage is that it gives a clear signal of diffraction since it measures the diffracted proton directly.
A scheme of a roman pot detector as it is used by the H1 or ZEUS experiment is shown in Fig. 2. The beam is the horizontal line at the upper part of the figure. The detector is located in the pot itself and can move closer to the beam when the beam is stable enough (during the injection period, the detectors are protected in the home position). Step motors allow to move the detectors with high precision. A precise knowledge of the detector position is necessary to reconstruct the transverse momentum of the scattered proton and thus the diffractive kinematical variables. The detectors are placed in a secondary vaccuum with respect to the beam one.
### 1.3 The Mx method
The third method used at HERA mainly by the ZEUS experiment is slightly less natural. It is based on the fact that there is a different behaviour in where is the total invariant mass produced in the event either for diffractive or non diffractive events. For diffractive events if (which is the case for diffractive events). The ZEUS collaboration performs some fits of the distribution:
dσdM2X=D+cexp(blogM2X) (1)
as illustrated in Fig. 3. The usual non diffractive events are exponentially suppressed at high values of . The difference between the observed data and the exponential suppressed distribution is the diffractive event contribution. This method, although easy to implement, presents the inconvenience that it relies strongly on the assumption of the exponential suppression of non diffractive events.
### 1.4 Diffractive kinematical variables
After having described the different experimental definitions of diffraction at HERA, we will give the new kinematical variables used to characterise diffraction. A typical diffractive event is shown in Fig. 4 where is depicted. In addition to the usual deep inelastic variables, the transfered energy squared at the electron vertex, the fraction of the proton momentum carried by the struck quark, the total energy in the final state, new diffractive variables are defined: is the momentum fraction of the proton carried by the colourless object called the pomeron, and the momentum fraction of the pomeron carried by the interacting parton inside the pomeron if we assume the pomeron to be made of quarks and gluons:
xP = ξ=Q2+M2XQ2+W2 (2) β = Q2Q2+M2X=xxP. (3)
## 2 Diffractive structure function measurement at HERA
### 2.1 Diffractive factorisation
In the following diffractive structure function analysis, we distinguish two kinds of factorisation at HERA. The first factorisation is the QCD hard scattering collinear factorisation at fixed and (see left plot of Fig. 5) [1], namely
dσ(ep→eXY)=fD(x,Q2,xP,t)×d^σ(x,Q2) (4)
where we can factorise the flux from the cross section . This factorisation was proven recently, and separates the coupling to the interaction with the colourless object.
The Regge factorisation at the proton vertex allows to factorise the and dependence, or in other words the hard interaction from the pomeron coupling to the proton (see right plot of Fig. 5).
### 2.2 Measurement of the diffractive proton structure function
The different measurements are performed using the three different methods to define diffractive events described in the first section. As an example, the H1 collaboration measures the diffractive cross section using the rapidity gap method:
d3σDdxPdQ2dβ=2πα2emβQ4(1−y+y22)σDr(xP,Q2,β) (5)
where is the reduced diffractive cross section. The measurement [2] is presented in Fig. 6. We notice that the measurement has been performed with high precision over a wide kinematical domain: , GeV, . The data are compared to the result of a QCD fit which we will discuss in the following.
The rapidity gap data are also compared with the data obtained either using the method or the one using proton tagging in roman pot detectors. Since they do not correspond exactly to the same definition of diffraction, a correction factor of 0.85 must be applied to the ZEUS method to be compared to the rapidity gap one (this factor is due to the fact that the two methods correspond to two different regions in , namely GeV for H1 and GeV for ZEUS). It is also possible to measure directly in the H1 experiment the ratio of the diffractive structure function measurements between the rapidity gap and the proton tagging methods as illustrated in Fig. 7. Unfortunately, the measurement using the proton tagging method is performed only in a restricted kinematical domain. No kinematical dependence has been found within uncertainties for this ratio inside this kinematical domain (see Fig. 7 for the and dependence, and Ref. [4] for the dependence as well). Note that the ratio could still be depending on and outside the limited domain of measurement.
### 2.3 QCD analysis of the diffractive structure function measurement
As we mentionned already, according to Regge theory, we can factorise the dependence from the one. The first diffractive structure function measurement from the H1 collaboartion [5] showed that this assumption was not true. The natural solution as observed in soft physics was that two different trajectories, namely pomeron and secondary reggeon, were needed to describe the measurement, which lead to a good description of the data. The diffractive structure function then reads:
FD2∼fp(xP)(FD2)Pom(β,Q2)+fr(xP)(FD2)Reg(β,Q2) (6)
where and are the pomeron and reggeon fluxes, and and the pomeron and reggeon structure functions. The flux parametrisation is predicted by Regge theory:
f(xP,t)=eBPtx2αP(t)−1P (7)
with the following pomeron trajectory
αP(t)=αP(0)+α′Pt. (8)
The dependence has been obtained using the proton tagging method, and the following values have been found: GeV, GeV (H1). Similarly, the values of have been measured using either the rapidity gap for H1 or the method for ZEUS in the QCD fit described in the next paragraph [6, 2]. The Reggeon parameters have been found to be GeV, GeV (H1). The value of has been determined from rapidity gap data and found to be equal to 0.5. Since the reggeon is expected to have a similar structure as the pion and the data are poorly sensitive to the structure function of the secondary reggeon, it was assumed to be similar to the pion structure with a free normalisation.
The next step is to perform Dokshitzer Gribov Lipatov Altarelli Parisi (DGLAP) [7] fits to the pomeron structure function. If we assume that the pomeron is made of quarks and gluons, it is natural to check whether the DGLAP evolution equations are able to describe the evolution of these parton densities. As necessary for DGLAP fits, a form for the input distributions is assumed at a given and is evolved using the DGLAP evolution equations to a different , and fitted to the diffractive structure function data at this value. The form of the distribution at has been chosen to be:
βq = AqβBq(1−β)Cq (9) βG = Ag(1−β)Cg, (10)
leading to three (resp. two) parameters for the quark (resp. gluon) densities. At low , the evolution is driven by while becomes more important at high . All diffractive data with GeV and have been used in the fit [2, 6] (the high points being excluded to avoid the low mass region where the vector meson resonances appear). This leads to a good description of all diffractive data included in the fit.
The DGLAP QCD fit allows to get the parton distributions in the pomeron as a direct output of the fit, and is displayed in Fig. 8 as a blue shaded area as a function of . We first note that the gluon density is much higher than the quark one, showing that the pomeron is gluon dominated. We also note that the gluon density at high is poorly constrained which is shown by the larger shaded area.
Another fit was also performed by the H1 collaboration imposing . While the fit quality is similar, the gluon at high is quite different, and is displayed as a black line in Fig. 8 ( is the equivalent of for quarks). This shows further that the gluon is very poorly constrained at high and some other data sets such as jet cross section measurements are needed to constrain it further.
### 2.4 QCD fits using diffractive structure function and jet cross section measurements
In this section, we describe combined fits using diffractive structure function and jet cross section data to further constrain the gluon at high . First, it is possible to compare the diffractive dijet cross section measurements with the predictions using the gluon and quark densities from the QCD fits described in the previous section. The comparison [2] shows a discrepancy between the measurement and the expectation from the QCD fit by about a factor 2 at high . This motivates the fact that it is important to add the jet cross section data to the inclusive structure function measurement in the QCD fit to further constrain the gluon density at high . The new parton distributions are shown in Fig. 9 as a blue shaded area. The comparison between the jet cross section measurements and the prediction from the QCD fits are in good agreement as shown in Fig. 10. The present uncertainty is of the order of 50% at high .
### 2.5 Other models describing inclusive diffraction at HERA
Many different kinds of models can be used to describe inclusive diffraction at HERA, and we will describe here only the results based on the two gluon model [8]. Other models of interest such as the BFKL dipole model [9] or the saturation model [10] are described in Ref. [6] as well as the results of the fits to the diffractive data. Due to the lack of time, we cannot describe them in these lectures.
The 2-gluon model [8] starts from the image of a perturbative pomeron made of two gluons and coupled non perturbatively to the proton. As shown in Fig. 11, there are three main contributions to the diffractive structure function, namely the transverse, (neglecting higher order Fock states) and the longitudinal terms. Contrary to the QCD fits described in the previous section, there is no concept of diffractive PDFs in this approach. The -dependence of the structure function is motivated by some general features of QCD-parton model calculations: at small the spin 1/2 (quark) exchange in the production leads to a behaviour , whereas the spin 1 (gluon) exchange in the term corresponds to . For large , perturbative QCD leads to and for the transverse and longitudinal terms respectively. Concerning the dependence, the longitudinal term is a higher twist one. Finally, the dependence on cannot be obtained from perturbative QCD and therefore is left free. An additional sub-leading trajectory (secondary reggeon) has to be parametrised from soft physics and is added to the model as for the DGLAP based fit to describe H1 data.
The 2-gluon model leads to a good description of both ZEUS and H1 data. As an example, the comparison of the ZEUS data [3] in different and bins as a function of with the 2-gluon model is given in Fig. 12 where we note the good agreement between the model and the data. Fig. 12 also describes independently the three components of the model, namely the tranverse one which dominates at medium , the one at low , and the longitudinal higher twist one at high .
## 3 Diffraction at the Tevatron
The Tevatron is a collider located close to Chicago at Fermilab, USA. It is presently the collider with the highest center-of-mass energy of about 2 TeV. Two main experiments are located around the ring, DØ and CDF. Both collaborations have accumulated a luminosity of the order of 1.5 fb with an efficiency of about 85%.
### 3.1 Diffractive kinematical variables
The difference between diffraction at HERA and at the Tevatron is that diffraction can occur not only on either or side as at HERA, but also on both sides. The former case is called single diffraction whereas the other one double pomeron exchange. In the same way as we defined the kinematical variables and at HERA, we define (= at HERA) as the proton fractional momentum loss (or as the or momentum fraction carried by the pomeron), and , the fraction of the pomeron momentum carried by the interacting parton. The produced diffractive mass is equal to for single diffractive events and to for double pomeron exchange. The size of the rapidity gap is of the order of .
### 3.2 How to find diffractive events at the Tevatron?
The selection of diffractive events at the Tevatron follows naturally from the diffractive event selection at HERA. The DØ and CDF collaborations obtained their first diffractive results using the rapidity gap method which showed that the percentage of single diffractive events was of the order of 1%, and about 0.1% for double pomeron exchanges. Unfortunately, the reconstruction of the kinematical variables is less precise than at HERA if one uses the rapidity gap selection since it suffers from the worse resolution of reconstructing hadronic final states.
The other more precise method is to tag directly the and in the final state. The CDF collaboration installed roman pot detectors in the outgoing direction only at the end of Run I [11], whereas the DØ collaboration installed them both in the outgoing and directions [12]. The DØ (dipole detectors) and CDF roman pots cover the acceptance of close to 0 and in the outgoing direction only. In addition, the DØ coverage extends for GeV, and in both and directions (quadrupole detectors). The CDF collaboration completed the detectors in the forward region by adding a miniplug calorimeter on both and sides allowing a coverage of and some beam showing counters close to beam pipe () allowing to reject non diffractive events.
### 3.3 Measurement of elastic events at DØ
Due to the high value of the production cross section, one of the first physics topics studied by the DØ collaboration was the elastic scattering cross section. Elastic events can also be used to align precisely the detectors. During its commissioning runs, the DØ collaboration was able to measure the diffractive slope for elastic events using double tagged events. The DØ results together with the results from the previous lower energy experiments are diplayed in Fig. 13. The normalisation of the DØ data is arbitrary since the data were taken using the commissioning runs of the roman pot detectors in stand-alone mode without any access to luminosity measurements. These data show the potential of the DØ roman pot detectors and this measurement will be performed again soon now that the roman pot detectors are fully included in the DØ readout system. A great challenge is to measure the change of slope in of the elastic cross section towards 0.55-0.6 GeV predicted by the models. Many measurements such as the pomeron structure in single diffractive events or double pomeron exchange, inclusive diffraction, diffractive , and -jets are being pursued in the DØ collaboration.
### 3.4 Factorisation or factorisation breaking at the Tevatron?
The CDF collaboration measured diffractive events at the Tevatron and their characteristics. In general, diffractive events show as expected less QCD radiation: for instance, dijet events are more back-to-back or the difference in azimuthal angles between both jets is more peaked towards . To make predictions at the Tevatron and the LHC, it is useful to know if factorisation holds. In other words, is it possible to use the parton distributions in the pomeron obtained in the previous section using HERA data to make predictions at the Tevatron, and also further constrain the parton distribution functions in the pomeron since the reach in the diffractive kinematical plane at the Tevatron and HERA is different? Theoretically, factorisation is not expected to hold between the Tevatron and HERA due to additional or interactions. For instance, some soft gluon exchanges between protons can occur at a longer time scale than the hard interaction and destroy the rapidity gap or the proton does not remain intact after interaction. The factorisation break-up is confirmed by comparing the percentage of diffractive events at HERA and the Tevatron (10% at HERA and about 1% of single diffractive events at the Tevatron) showing already that factorisation does not hold. This introduces the concept of gap survival probability, the probability that there is no soft additional interaction or in other words that the event remains diffractive. We will mention in the following how this concept can be tested directly at the Tevatron.
The first factorisation test concerns CDF data only. It is interesting to check whether factorisation holds within CDF data alone, or in other words if the and dependence can be factorised out from the one. Fig. 14 shows the percentage of diffractive events as a function of for different bins and shows the same -dependence in all bins supporting the fact that CDF data are consistent with factorisation [13]. The CDF collaboration also studied the dependence for different bins which lead to the same conclusions. This also shows that the Tevatron data do not require additional secondary reggeon trajectories as in H1.
The second step is to check whether factorisation holds or not between Tevatron and HERA data. The measurement of the diffractive structure function is possible directly at the Tevatron. The CDF collaboration measured the ratio of dijet events in single diffractive and non diffractive events, which is directly proportional to the ratio of the diffractive to the “standard” proton structure functions :
R(x)=RateSDjj(x)RateNDjj(x)∼FSDjj(x)FNDjj(x) (11)
The “standard” proton structure function is known from the usual PDFs obtained by the CTEQ or MRST collaborations. The comparison between the CDF measurement (black points, with systematics errors as shaded area) and the expectation from the H1 QCD fits in full line is shown in Fig. 15. We notice a discrepancy of a factor 8 to 10 between the data and the predictions from the QCD fit, showing that factorisation does not hold. However, the difference is compatible with a constant on a large part of the kinematical plane in , showing that the survival probability does not seem to be -dependent within experimental uncertainties.
The other interesting measurement which can be also performed at the Tevatron is the test of factorisation between single diffraction and double pomeron exchange. The results from the CDF collaboration are shown in Fig. 16. The left plot shows the definition of the two ratios while the right figure shows the comparison between the ratio of double pomeron exchange to single diffraction and the QCD predictions using HERA data in full line. Whereas factorisation was not true for the ratio of single diffraction to non diffractive events, factorisation holds for the ratio of double pomeron exchange to single diffraction! In other words, the price to pay for one gap is the same as the price to pay for two gaps. The survival probability, i.e. the probability not to emit an additional soft gluon after the hard interaction needs to be applied only once to require the existence of a diffractive event, but should not be applied again for double pomeron exchange.
### 3.5 Survival probability studies in H1
We mentioned in the previous section that the concept of survival probablity is related to soft gluon emission. This process can also be studied at HERA using resolved photoproduction where events are sensitive to the hadronic structure of the photon (see Fig. 17, right plot). The resolved process is different from the direct one where the photon couples directly to the pomeron (see Fig. 17, left plot). In that case, we get an hadron hadron process like at the Tevatron since we are sensitive to the hadronic contents of the photon. In Fig. 18, we display the ratio between data and NLO predictions for DIS (red triangles) and photoproduction data (black points). We notice that we see a different of about a factor 2 between these two data sets which might be an indication of survival probability effects. However, no difference is observed between resolved or direct photoproduction where factorisation is expected to hold.
### 3.6 Possibility of survival probablity measurements at DØ
A new measurement to be performed at the Tevatron, in the DØ experiment has been proposed [14], which can be decisive to test directly the concept of survival probability at the Tevatron, by looking at the azimuthal distributions of the outgoing proton and antiproton with respect to the beam direction.
In Fig. 19, we display the survival probability for three different values of as a function of the difference in azimuthal angle between the scattered and . The upper black curve represents the case where the of the and are similar and close to 0. In that case, only a weak dependence on is observed. The conclusion is different for asymmetric cases or cases when is different from 0: Fig. 19 also shows the result in full red line for the asymmetric case (, GeV), and in full and dashed blue lines for GeV for two different models of survival probabilities. We notice that we get a very strong dependence of more than one order of magnitude.
The dependence can be tested directly using the roman pot detectors at DØ (dipole and quadrupole detectors) and their possibility to measure the azimuthal angles of the and . For this purpose, we define the following configurations for dipole-quadrupole tags: same side (corresponding to degrees), opposite side (corresponding to degrees), and middle side (corresponding to degrees). In Table 1, we give the ratios and (note that we divide by 2 to get the same domain size in ) for the different models. In order to obtain these predictions, we used the full acceptance in and of the FPD detector. Moreover the ratios for two different tagging configurations, namely for tagged in dipole detectors, and in quadrupoles, or for both and tagged in quadrupole detectors [14] were computed.
The results are also compared to expectations using another kind of model to describe diffractive events, namely soft colour interaction [15]. This model assumes that diffraction is not due to a colourless exchange at the hard vertex (called pomeron) but rather to string rearrangement in the final state during hadronisation. In this kind of model, there is a probability (to be determined by the experiment) that there is no string connection, and so no colour exchange, between the partons in the proton and the scattered quark produced during the hard interaction. Since this model does not imply the existence of pomeron, there is no need of a concept like survival probability, and no dependence on of diffractive cross sections. The proposed measurement would allow to distinguish between these two dramatically different models of diffraction.
## 4 Diffractive exclusive event production
### 4.1 Interest of exclusive events
A schematic view of non diffractive, inclusive double pomeron exchange, exclusive diffractive events at the Tevatron or the LHC is displayed in Fig. 20. The upper left plot shows the “standard” non diffractive events where the Higgs boson, the dijet or diphotons are produced directly by a coupling to the proton and shows proton remnants. The bottom plot displays the standard diffractive double pomeron exchange where the protons remain intact after interaction and the total available energy is used to produce the heavy object (Higgs boson, dijets, diphotons…) and the pomeron remnants. We have so far only discussed this kind of events and their diffractive production using the parton densities measured at HERA. There may be a third class of processes displayed in the upper right figure, namely the exclusive diffractive production. In this kind of events, the full energy is used to produce the heavy object (Higgs boson, dijets, diphotons…) and no energy is lost in pomeron remnants. There is an important kinematical consequence: the mass of the produced object can be computed using roman pot detectors and tagged protons:
M=√ξ1ξ2S. (12)
We see immediately the advantage of those processes: we can benefit from the good roman pot resolution on to get a good resolution on mass. It is then possible to measure the mass and the kinematical properties of the produced object and use this information to increase the signal over background ratio by reducing the mass window of measurement. It is thus important to know if this kind of events exist or not.
### 4.2 Search for exclusive events at the Tevatron
The CDF collaboration measured the so-called dijet mass fraction in dijet events - the ratio of the mass carried by the two jets divided by the total diffractive mass - when the antiproton is tagged in the roman pot detectors and when there is a rapidity gap on the proton side to ensure that the event corresponds to a double pomeron exchange. The results are shown in Fig. 21 and are compared with the POMWIG [18] expectation using the gluon and quark densities measured by the H1 collaboration in dashed line [13]. We see a clear deficit of events towards high values of the dijet mass fraction, where exclusive events are supposed to occur (for exclusive events, the dijet mass fraction is 1 by definition at generator level and can be smeared out towards lower values taking into account the detector resolutions). Fig. 21 shows also the comparison between data and the predictions from the POMWIG and DPEMC generators, DPEMC being used to generate exclusive events [16]. There is a good agreement between data and MC. However, this does not prove the existence of exclusive events since the POMWIG prediction shows large uncertainties (the gluon in the pomeron used in POMWIG is not the latest one obtained by the H1 collaboration [2, 6] and the uncertainty at high is quite large as we discussed in a previous section). The results (and the conclusions) might change using the newest gluon density and will be of particular interest. In addition, it is not obvious one can use the gluon density measured at HERA at the Tevatron since factorisation does not hold, or in other words, this assumes that the survival probability is a constant, not depending on the kinematics of the interaction.
A direct precise measurement of the gluon density in the pomeron through the measurement of the diffractive dijet cross section at the Tevatron and the LHC will be necessary if one wants to prove the existence of exclusive events in the dijet channel. However, this measurement is not easy and requires a full QCD analysis. We expect that exclusive events would appear as a bump in the gluon distribution at high , which will be difficult to interprete. To show that this bump is not due to tail of the inclusive distribution but real exclusive events, it would be necessary to show that those tails are not compatible with a standard DGLAP evolution of the gluon density in the pomeron as a function of jet transverse momentum. However, it does not seem to be easy to distinguish those effects from higher twist ones. It is thus important to look for different methods to show the existence of exclusive events.
The CDF collaboration also looked for the exclusive production of dilepton and diphoton. Contrary to diphotons, dileptons cannot be produced exclusively via pomeron exchanges since is possible, but directly is impossible. However, dileptons can be produced via QED processes, and the cross section is perfectly known. The CDF measurement is pb which is found to be in good agreement with QED predictions and shows that the acceptance, efficiencies of the detector are well understood. Three exclusive diphoton events have been observed by the CDF collaboration leading to a cross section of pb compatible with the expectations for exclusive diphoton production at the Tevatron.
Other searches like production and the ratio of diffractive jets to the non diffractive ones as a function of the dijet mass fraction show further indications that exclusive events might exist but there is no definite proof until now.
### 4.3 Search for exclusive events at the LHC
The search for exclusive events at the LHC can be performed in the same channels as the ones used at the Tevatron. In addition, some other possibilities benefitting from the high luminosity of the LHC appear. One of the cleanest way to show the existence of exclusive events would be to measure the dilepton and diphoton cross section ratios as a function of the dilepton/diphoton mass. If exclusive events exist, this distribution should show a bump towards high values of the dilepton/diphoton mass since it is possible to produce exclusively diphotons but not dileptons at leading order as we mentionned in the previous paragraph.
The search for exclusive events at the LHC will also require a precise analysis and measurement of inclusive diffractive cross sections and in particular the tails at high since it is a direct background to exclusive event production.
## 5 Diffraction at the LHC
In this section, we will describe briefly some projects concerning diffraction at the LHC. We will put slightly more emphasis on the diffractive production of heavy objects such as Higgs bosons, top or stop pairs, events…
### 5.1 Diffractive event selection at the LHC
The LHC with a center-of-mass energy of 14 TeV will allow us to access a completely new kinematical domain in diffraction. So far, two experiments, namely ATLAS and CMS-TOTEM have shown interests in diffractive measurements. The diffractive event selection at the LHC will be the same as at the Tevatron. However, the rapidity gap selection will no longer be possible at high luminosity since up to 25 interactions per bunch crossing are expected to occur and soft pile-up events will kill the gaps produced by the hard interaction. Proton tagging will thus be the only possibility to detect diffractive events at high luminosity.
### 5.2 Measurements at the LHC using a high β∗ lattice
Measurements of total cross section and luminosity are foreseen in the ATLAS [19] and TOTEM [20] experiments, and roman pots are installed at 147 and 220 m in TOTEM and 240 m in ATLAS. These measurements will require a special injection lattice of the LHC at low luminosity since they require the roman pot detectors to be moved very close to the beam. As an example, the measurement of the total cross section to be performed by TOTEM [20] is shown in Fig. 22. We notice that there is a large uncertainty on prediction of the total cross section at the LHC energy in particular due to the discrepancy between the two Tevatron measurements, and this measurement of TOTEM will be of special interest.
### 5.3 Hard inclusive diffraction at the LHC
In this section, we would like to discuss how we can measure the gluon density in the pomeron, especially at high since the gluon in this kinematical domain shows large uncertainties and this is where the exclusive contributions should show up if they exist. To take into account the high- uncertainties of the gluon distribution, we chose to multiply the gluon density in the pomeron measured at HERA by a factor where varies between -1.0 and 1.0. If is negative, we enhance the gluon density at high by definition, especially at low .
A possible measurement at the LHC is described in Fig. 23. The dijet mass fraction is shown in dijet diffractive production for different jet transverse momenta ( 100 (upper left), 200 (upper right), 300 (lower left) and 400 GeV (lower right)), and for the different values if . We notice that the variation of this distribution as a function of jet can assess directly the high behaviour of the gluon density. In the same kind of ideas, it is also possible to use event production to test the high- gluon. Of course, this kind of measurement will not replace a direct QCD analysis of the diffractive dijet cross section measurement.
Other measurements already mentionned such as the diphoton, dilepton cross section ratio as a function of the dijet mass, the jet, , and cross section measurements will be also quite important at the LHC.
### 5.4 Exclusive Higgs production at the LHC
As we already mentionned in one of the previous sections, one special interest of diffractive events at the LHC is related to the existence of exclusive events. So far, two projects are being discussed at the LHC: the installation of roman pot detectors at 220 m in ATLAS [21], and at 420 m for the ATLAS and CMS collaborations [22].
The results discussed in this section rely on the DPEMC Monte Carlo to produce Higgs bosons exclusively [16, 17] and a fast simulation of a typical LHC detector (ATLAS or CMS). Results are given in Fig. 24 for a Higgs mass of 120 GeV, in terms of the signal to background ratio S/B, as a function of the Higgs boson mass resolution. Let us notice that the background is mainly due the exclusive production. However the tail of the inclusive production can also be a relevant contribution and this is related to the high gluon density which is badly known at present. In order to obtain a S/B of 3 (resp. 1, 0.5), a mass resolution of about 0.3 GeV (resp. 1.2, 2.3 GeV) is needed. A mass resolution of the order of 1 GeV seems to be technically feasible.
The diffractive SUSY Higgs boson production cross section is noticeably enhanced at high values of and since we look for Higgs decaying into , it is possible to benefit directly from the enhancement of the cross section contrary to the non diffractive case. A signal-over-background up to a factor 50 can be reached for 100 fb for [23] (see Fig. 25).
### 5.5 Exclusive top, stop and W pair production at the LHC
In the same way that Higgs bosons can be produced exclusively, it is possible to produce , top and stops quark pairs. bosons are produced via QED processes which means that their cross section is perfectly known. On the contrary, top and stop pair production are obtained via double pomeron exchanges and the production cross section is still uncertain.
The method to reconstruct the mass of heavy objects double diffractively produced at the LHC is based on a fit to the turn-on point of the missing mass distribution at threshold [24].
One proposed method (the “histogram” method) corresponds to the comparison of the mass distribution in data with some reference distributions following a Monte Carlo simulation of the detector with different input masses corresponding to the data luminosity. As an example, we can produce a data sample for 100 fb with a top mass of 174 GeV, and a few MC samples corresponding to different top masses between 150 and 200 GeV. For each Monte Carlo sample, a value corresponding to the population difference in each bin between data and MC is computed. The mass point where the is minimum corresponds to the mass of the produced object in data. This method has the advantage of being easy but requires a good simulation of the detector.
The other proposed method (the “turn-on fit” method) is less sensitive to the MC simulation of the detectors. As mentioned earlier, the threshold scan is directly sensitive to the mass of the diffractively produced object (in the case for instance, it is sensitive to twice the mass). The idea is thus to fit the turn-on point of the missing mass distribution which leads directly to the mass of the produced object, the boson. Due to its robustness, this method is considered as the “default” one.
The precision of the mass measurement (0.3 GeV for 300 fb) is not competitive with other methods, but provides a very precise check of the calibration of the roman pot detectors. events will also allow to assess directly the sensitivity to the photon anomalous coupling since it would reveal itself by a modification of the well-known QED production cross section. We can notice that the production cross section is proportional to the fourth power of the coupling which ensures a very good sensitivity of that process [25]. The precision of the top mass measurement is however competitive, with an expected precision better than 1 GeV at high luminosity provided that the cross section is high enough. The other application is to use the so-called “threshold-scan method” to measure the stop mass [23]. After taking into account the stop width, we obtain a resolution on the stop mass of 0.4, 0.7 and 4.3 GeV for a stop mass of 174.3, 210 and 393 GeV for a luminosity (divided by the signal efficiency) of 100 fb.
The caveat is of course that the production via diffractive exclusive processes is model dependent, and definitely needs the Tevatron and LHC data to test the models. It will allow us to determine more precisely the production cross section by testing and measuring at the Tevatron the jet and photon production for high masses and high dijet or diphoton mass fraction.
## 6 Conclusion
In these lectures, we presented and discussed the most recent results on inclusive diffraction from the HERA and Tevatron experiments and gave the prospects for the future at the LHC. Of special interest is the exclusive production of Higgs boson and heavy objects (, top, stop pairs) which will require a better understanding of diffractive events and the link between and hadronic colliders, and precise measurements and analyses of inclusive diffraction at the LHC in particular to constrain further the gluon density in the pomeron.
## Acknowledgments
I thank Robi Peschanski and Oldřich Kepka for a careful reading of the manuscript. | {"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": 0.9322327375411987, "perplexity": 884.0426696989506}, "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-2023-14/segments/1679296945292.83/warc/CC-MAIN-20230325002113-20230325032113-00557.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.