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http://mathhelpforum.com/algebra/14016-quadratics-inequality.html	# Math Help - Quadratics Inequality x^2 - x - 5 < 0 and... What is the equation of the porabola that goes through the points: (0,0) (1,-2) (4,4) 2. Originally Posted by georgiaaa. x^2 - x - 5 < 0 I like to complete the square for tis one. x^2 -x + 1/4 < 5 + 1/4 (x-1/2)^2 < 21/4 -sqrt(21)/2< (x-1/2) < sqrt(21)/2 (1-sqrt(21))/2 < x < (1+sqrt(21))/2 3. Hello, georgiaaa! . . x² - x - 5 .< .0 Think of it as: .y .= .x² - x - 5 We have an up-opening parabola. .When it is negative? . . When it is below the x-axis. When does that happen? . . Let's find the x-intercepts. . . . . . . . . . . . . . . . . . . . . . . __ . . . . . . . . . . . . . . . . . . 1 ± √21 Quadratic Formula: .x .= . --------- . . . . . . . . . . . . . . . . . . . . .2 The expression is negative between those two intercepts. . . . . . . __ . . . . . . . . . . . . . __ . . (1 - √21)/2 .< .x .< .(1 + √21)/2 What is the equation of the parabola that goes through the points: . . (0,0), (1,-2), (4,4) The general form of a parabola is: .y .= .ax² + bx + c Plug in the three points: . . .(0,0): .a·0² + b·0 + c .= .0 . . . c = 0 . . (1,-2): .a·1² + b·1 + 0 .= .-2 . . . .a + .b .= .-2 .[1] . . .(4,4): .a·4² + b·4 + 0 .= .4 . . . 16a + 4b .= .4 . [2] Divide [2] by 4: . 4a + b .= .1 . . Subtract [1]: . .a + b .= .-2 . . and we get: .3a = 3 . . a = 1 Substitute into [1]: .1 + b .= .-2 . . b = -3 Therefore, the parabola is: .y .= .x² - 3x	{"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": 0.938364565372467, "perplexity": 1474.5210313911712}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131304625.62/warc/CC-MAIN-20150323172144-00287-ip-10-168-14-71.ec2.internal.warc.gz"}
https://mersenneforum.org/showthread.php?s=3cd123a74fb7ef0617e51aedde9aad5f&t=23941	mersenneforum.org Density of norms (field theory) Register FAQ Search Today's Posts Mark Forums Read 2018-12-28, 00:09 #1 carpetpool "Sam" Nov 2016 5·67 Posts Density of norms (field theory) In this post here, I asked for the conditional probability for an integer N being prime given that all prime q dividing n are congruent to 1 modulo 2p (for some prime p). As a result, I also got the answer of how many integers N not exceeding x can be written as a product of primes only congruent to 1 modulo 2p. This is asymptotically D(x) = cx(log(x))^(1/(p-1) - 1) for some constant c, which seems to be decreasing significantly as p increases. How many integers N not exceeding x can (I) be written as a product of primes only congruent to 1 modulo 2p and (II) in addition to (I), N can be expressed as the norm for some integral element f in the ring of integers in K=Q(zeta(p)) where K is the field of p-th roots of unity (the p-th cyclotomic field) ? The condition for (II) can be restated as there is at least one ideal of norm N that is principal in K. I am hoping for a precise answer (as in my last thread) in an attempt to solve another problem related to this. Again, any information is helpful, and thanks for help. 2018-12-28, 00:19 #2 science_man_88 "Forget I exist" Jul 2009 Dartmouth NS 2×3×23×61 Posts Quote: Originally Posted by carpetpool In this post here, I asked for the conditional probability for an integer N being prime given that all prime q dividing n are congruent to 1 modulo 2p (for some prime p). As a result, I also got the answer of how many integers N not exceeding x can be written as a product of primes only congruent to 1 modulo 2p. This is asymptotically D(x) = cx(log(x))^(1/(p-1) - 1) for some constant c, which seems to be decreasing significantly as p increases. How many integers N not exceeding x can (I) be written as a product of primes only congruent to 1 modulo 2p and (II) in addition to (I), N can be expressed as the norm for some integral element f in the ring of integers in K=Q(zeta(p)) where K is the field of p-th roots of unity (the p-th cyclotomic field) ? The condition for (II) can be restated as there is at least one ideal of norm N that is principal in K. I am hoping for a precise answer (as in my last thread) in an attempt to solve another problem related to this. Again, any information is helpful, and thanks for help. (2kp+1)(2jp+1)=4jkp^2+2(k+j)p+1 = 2(2jkp+k+j)p+1 so as many as the natural numbers up to X/(2p) of form 2jkp+k+j for some natural numbers k and j. Last fiddled with by science_man_88 on 2018-12-28 at 00:20 2018-12-28, 19:05 #3 carpetpool "Sam" Nov 2016 5·67 Posts It seems to be that the answer is 0, although I don't know for sure because many of my previous posts seem to be getting lack of attention due to the little known information and research of these topics. However, I did find this article. Still, any other explanations of it are welcome. Similar Threads Thread Thread Starter Forum Replies Last Post Cruelty Proth Prime Search 158 2020-07-31 22:23 carpetpool carpetpool 24 2017-10-29 23:47 devarajkandadai Number Theory Discussion Group 11 2017-10-28 20:58 Nick Math 4 2017-04-01 16:26 xilman Science & Technology 85 2010-12-20 21:42 All times are UTC. The time now is 21:29. Tue Nov 29 21:29:16 UTC 2022 up 103 days, 18:57, 0 users, load averages: 1.55, 1.57, 1.37	{"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": 0.9101929068565369, "perplexity": 736.038291737667}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710711.7/warc/CC-MAIN-20221129200438-20221129230438-00673.warc.gz"}
https://www.jiskha.com/questions/294294/what-is-the-molecular-weight-of-a-gas-if-a-15-0g-sample-has-a-pressure-of-836-mm-hg-at	Chemistry What is the molecular weight of a gas if a 15.0g sample has a pressure of 836 mm Hg at 25.0 degrees C in a 2.00 L flask? a)167 b)1.35 c)176 d)11.1 e)none of the above Please explain... I need to understand how to do this. Thanks. 1. 👍 2. 👎 3. 👁 1. Use PV = nRT to solve for n. Then n = grams/molar mass. 1. 👍 2. 👎 2. I'm still confused... I know that: P=1.1 atm V=2.00 L n=? R= I think .0821 and T= ? I have no idea how to change 15.0g to moles, since I do not know what the element is to look at the atomic mass--I know that one mole is avagadro's number, but I'm still really not sure what to do here. 1. 👍 2. 👎 3. T- 25+273 = 298k n=PV/RT n= 1.1 atm2.00 l/ (0.0821 (Latm/ molk) 298k) n=0.0899mol MW= 5.0g/0.0899mol =167amu 1. 👍 2. 👎 Similar Questions 1. CHEM A) a picture shows a container that is sealed at the top by a movable piston. Inside the container is a ideal gas at 1.00atm , 20.0 *C, and 1.00L. (The pic is of a 2L container which has a piston pushed downward to the 1L level. 2. (12)Chemistry - Science (Dr. Bob222) Calculate delta E for a process in which 88.0 g of nitrous oxide gas (N2O) is cooled from 165 ˚C to 65 ˚C at a constant pressure of 5.00 atm. The molecular weight of nitrous oxide is 44.02 g/mol, and its heat capacity at 3. Chemistry A gas mixture consists of equal masses of methane (molecular weight 16.0) and argon (atomic weight 40.0). If the partial pressure of argon is 200. torr, what is the pressure of methane, in torr? 4. Chemistry A gaseous compound is 30.4% nitrogen and 69.6% oxygen by mass. A 5.25-g sample of the gas occupies a volume of 1.00 L and exerts a pressure of 1.26 atm at -4.00°C. Which of these choices is its molecular formula? 1. chemistry Question: Consider this molecular-level representation of a gas. There are: 5 orange gas particles 6 blue gas particles 3 green molecules of gas (two particles attached together) If the partial pressure of the diatomic gas is 2. Thermodynamics 3. A gas is confined in a 0.47-m-diameter cylinder by a piston, on which rests a weight. The mass of the piston and weight together is 150 kg. The atmospheric pressure is 101 kPa. (a)What is the pressure of the gas in kPa? (b) If 3. Chem A sample of gas of mass 2.929 g occupies a volume of 426 mL at 0degree Cel and 1.00 atm pressure. What is the molecular weight of the gas? Thanks! 4. Chemistry a sample of helium gas is in a closed system with a movable piston. the volume of the gas sample is changed when both the temperature and the pressure of the sample are increased. the table below shows the initial temperature, 1. Science 10. You analyze the gas in 100 kg of gas in a tank at atmospheric pressure, and find the Following : CO2;19.3%, N2;72.1%, O2;6.5%, H2O;2.1% What is the average molecular weight of the gas? 2. physical chemistry The system N2O4 2NO2 maintained in a closed vessel at 60º C & a pressure of 5 atm has an average (i.e. observed) molecular weight of 69, calculate Kp. At what pressure at the same temperature would the observed molecular weight 3. Chemistry A student performed an experiment to determine the molecular weight of a gaseous compound. Using the ideal gas law PV = nRT, and knowing the pressure, temperature, and volume of the vapor, the student calculated the number of 4. Chem 130 A sample of oxygen gas is collected by displacement of water at 25°C and 1.11 atm total pressure. If the vapor pressure of water is 23.756 mm Hg at 25°C, what is the partial pressure of the oxygen gas in the sample?	{"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": 0.8593828678131104, "perplexity": 1707.3238989451763}, "config": {"markdown_headings": false, "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-2021-43/segments/1634323588398.42/warc/CC-MAIN-20211028162638-20211028192638-00281.warc.gz"}
http://academicteaching.net/lib/partition_of_an_interval.htm	# partition of an interval Partition of an Interval A division of an interval into a finite number of sub-intervals. Specifically, the partition itself is the set of endpoints of each of the sub-intervals.	{"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": 0.8057070374488831, "perplexity": 644.4282013193397}, "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/1585370493120.15/warc/CC-MAIN-20200328194743-20200328224743-00156.warc.gz"}
https://www.physicsforums.com/threads/definition-of-tangent-space-on-smooth-manifolds.548146/	# Definition of tangent space on smooth manifolds 1. Nov 7, 2011 ### Alesak Hi, I'm having trouble understanding why is tangent space at point p on a smooth manifold, not embedded in any ambient euclidean sapce, has to be defined as, for example, set of all directional derivatives at that point. To my understanding, the goal of defining tangent space is to provide linear approximation of a manifold near certain point. Why not to just say that tangent space of n-manifold at point p is R^n? In the end, the set of all directional derivatives is isomorphic to it anyway. And after all, manifolds key characteristic is that it is localy similar to R^n, so why to not use it? But there has to be some reason it is defined the way it is, since everybody is using it, so I just wonder what I'm missing here... 2. Nov 7, 2011 ### lavinia At a point the tangent space is just R^n but globally the tangent spaces may fit together to form a new space that is not the same as R^n. If you just associate R^n with a point on the manifold you do not know how to identify the vectors with directions on the manifold. the key is interpreting the vectors as directions. There are other ways to associate vectors with points on manifolds that can not be interpreted as tangential directions. Last edited: Nov 7, 2011 3. Nov 7, 2011 ### homeomorphic It doesn't have to be defined that way. There are at least 3 definitions that I know of. The first is in terms of differentiation operators. The second is in terms of curves. And the third is actually in terms of R^n, but you need a little more than that. Essentially, it IS R^n, since in finite dimensions, anything is R^n, but only up to isomorphism. You shouldn't forget about the isomorphisms. Think about a vector field. You don't want to just think about tangent vectors at one point. Isomorphic. But how do you get the isomorphism? That depends on your charts. You don't want it to depend on charts. The similarity with R^n depends on your charts. 4. Nov 7, 2011 ### Deveno the way i've always thought about it is: the earth isn't flat. i mean locally, it is pretty flat, so much so that "geometry" (the euclidean plane kind) was originally thought to be a good abstraction of the earth. but when we started making maps of the whole world, we ran into problems; you can't embed a sphere into a euclidean plane faithfully (isometrically), you get some distortion no matter how you approach it. in manifold terms: you can paste together a set of charts in a consistent way (think of taping different maps together), but what you get when you do that globally (ooh, i made a pun) is something that can have properties the individual tangent spaces don't. a circle is only locally a line. if you cut it, and straighten it out, so that it fits nicely into R, you wind up breaking paths that were just one piece. as you get farther and farther away from the point the tangent space is defined at, you can wind up no longer having a proper normal or tangent vector. time to switch to a different local map, one without the distortion that creeps in (linear approximations aren't THAT good, especially on manifolds with a lot of twists and curves). 5. Nov 7, 2011 ### lavinia Using charts is not the only way to identify the tangent space. If the manifold is embedded in Euclidean space then the tangent space at a point is the hyperplane of all lines that touch the manifold at that point without cutting through it. This definition is geometrical. Each tangent line gives you two directions along the manifold. 6. Nov 7, 2011 ### Deveno and then you get a whaddayacallit...tangent bundle of all those hyperplanes, and that doesn't look at ALL like euclidean space (unless your manifold is euclidean space, but maybe that's not such a useful example), although it's kinda cool to try to picture it in your head. 7. Nov 7, 2011 ### homeomorphic I was going to mention that, but then I realized he specifically asked about manifolds that are not embedded in Euclidean space, so we want an intrinsic definition. The tangent bundle is a manifold, so it is, again, locally like Euclidean space. Well, at least with the appropriate topology on it, but without the topology, all bundles may as well be trivial, so there's nothing interesting from a bundle point of view. But you would still have the coordinate independence problem. You have a tangent vector sitting at some point in the manifold and you want to say it's in R^n, but which vector in R^n is it? Depends on your isomorphism. So, you still have to sort of mod out by coordinate changes, to make it a coordinate-independent concept. 8. Nov 7, 2011 ### Sina I misunderstood the question so my previous answer is down there. The real answer is mainly because of the usage. We define tangent vectors to curve as "infinitesimal operations" that describe translation along that curve. Well the amount of translation is infact given in terms of the derivative with respect to the curves parameter which will be d/dt or in coordinates some a^id/dx^i. So basically the reason is that probably the people building this sturucture had dynamical systems in mind and they wanted the vectors do describe translations on the curves. As I said the best operation for that is derivative along the curve. You can define the tangent plane in terms of germs, derivations or by embedding it in some ambient space even usual tangent vectors etc. In the end though the strucure is that the tangent space at p is indeed isomorphich to Rn. However this isomorphism helps you only when you are working right at that point. If you want to make more global statements about the structure of your manifold then you have to give the collection of all those tangent spaces some structure. For instance to define (0,n) tensor fields on your manifold (or the reverse I am not sure which is which) you really need to defined a collection of tangent spaces over open sets. If you do not give manifold structure to your tangent bundle, how will you know what is the results of change of coordinates to an arbitrary tensor field for instance? By giving the tangent bundle a manifold structure explicitly and by defining charts etc you can manipulate vector fields and tensor fields defined on the whole of manifold. Or else you will be restircted to making a change of basis at single points then patching them together etc etc. It is all basically done by giving it the manifold structure. Last edited: Nov 7, 2011 9. Nov 7, 2011 ### Alesak Thanks for all answers. I'm going through the Introduction to Smooth Manifolds by Lee, and this whole modern differential geometry complex involving vector\covector spaces\bundles and combining them into tensors\differential forsms is making my head spin. But hey, on wikipedia they say Einstein had learned about them "with great difficulty":) I think the tangent space is clear to me now: we want to have an arrow at a point, that exist in itself, without any reference to any coordinates. I like to imagine that whatever coordinate chart we choose, the same arrow just sits there glued, oblivious to whatever happens around it. So while tangent bundle is a vector bundle, it carries this additional interpretation, and it is entirely up to us how we establish correspondence between a point in tangent space and a point in a same-dimensional vector space(i.e. what coordinates it will have). Personaly I find the old definition from physicists most revealing, becouse it is straghtforward: we pick some coordinate chart around point p that maps p to 0. Now we automaticaly get basis $(x_{i})$ for tangent space, which consists of vectors tangent to lines, that map to canonical axes in R^n. So each tangent vector is uniquely represented as $v = a^{i}x_{i}$ (do I have the summation convention right?). If we pick another coordinate chart which induces tangent space basis $(\widehat{x_{i}})$ and construct change of coordinates matrix (while understanding $x_{i}$ as i-th basis vector, or i-th coordinate function\i-th basis vector of dual space as convenient) $$A = \begin{pmatrix} \frac{\partial x_{1}}{\partial \widehat{x_{1}}} & \frac{\partial x_{2}}{\partial \widehat{x_{1}}} & ... & \frac{\partial x_{n}}{\partial \widehat{x_{1}}}\\ \frac{\partial x_{1}}{\partial \widehat{x_{2}}} & . & . & .\\ . & . & . & . \\ \frac{\partial x_{1}}{\partial \widehat{x_{n}}} & . & . & \frac{\partial x_{n}}{\partial \widehat{x_{n}}} \end{pmatrix}$$ we can comfortably write $\widehat{v} = Av$, which is in fact the same tangent vector. I suppose it can be remembered as a matrix having rows as old basis vectors expressed in new coordinates. In other words, $\widehat{v^{j}} = v^{i}\frac{\partial x_{i}}{\partial \widehat{x_{j}}}$. I guess defining covectors and tensors won't be that different. 10. Nov 7, 2011 ### Sina in that case too it is very natural to denote the basis vectors as $\frac{\partial}{\partial x_i}$ because according to chain rule it just changes with the matrix you have given :) in the chapters on 1-parameter flows and lie derivatives, you will see that this definition is much much more physical then what you have given. For instance given a curve γ on your manifold and some function f, to see the rate of change of f along the curve you just apply the tangent vector of the curve to the function that is $\frac{d\gamma^i}{dt}\frac{\partial}{\partial x_i}f$. In your definition, there is no direct way of applying the vectors to functions etc. You have to take the vector $v_ie^i$ and map it to something like $v_i\frac{\partial}{\partial x_i}$ to make it a derivation along the curve but that is exactly the tangent vector we are talking about. And infact this mapping can be shown to be an isomorphism between derivations and usual R^n. Lee's book is a very good book, very pedagogical and easy to follow even though it handles some quite non-trivial topics in a completely rigorous way. No wonder lee has many teaching prizes :) Be sure to solve some questions as it has important results and covers the usage of techniques learned in the chapter. Last edited: Nov 7, 2011 11. Nov 23, 2011 ### mathwonk no matter what definition you take for the tangent space V of a manifold at the point p, you must answer this question: given a smooth curve passing through p at time t=0, which element of V is the velocity vector at p for t=0? I.e. you must give a surjective map from the set of all smooth curves passing through p to the vector space V, such that 2 curves have the same velocity vector in V at p, if and only if for every coordinate chart taking a neighborhood of p to a neighborhood of 0 in R^n, the corresponding curves through 0 have the same velocity vector in R^n, in the usual sense of calculus. Any definition of the tangent space V, and of velocity vectors in V of curves, that has this property will do fine. I.e. all the tangent space needs to do for you is tell when two curves should have the same velocity vector at p. E.g., if we let the tangent space be the space of differential operators on functions at p, then given a curve through p, we define for each function the tangent vector to be differentiation along the direction defined by the curve. Then by the chain rule, indeed two curves have the same velocity vector at p if and only if they give the same directional derivative for all functions at p. Last edited: Nov 23, 2011 12. Nov 24, 2011 ### mathwonk Each chart at p does indeed give a way to view the tangent space as R^n. But two different charts give two different ways to do this. Then if the vector v in R^n is associated to a given tangent vector at p by one chart, we must know which vector w in R^n is associated to the same tangent vector by another chart. This is done by using the derivative of the composition of the two charts, and letting w be the vector image of v under this derivative. Thus a tangent vector at p is not one vector v in R^n, but is a collection of pairs {(f,v), one for each chart f at p}, where v is a vector in R^n and f is a chart. And it must be true that for any two pairs, (f,v) and (g,w) in the collection, that the derivative of gof^-1 takes v to w. 13. Nov 26, 2011 ### mathwonk in my clueless opinion i have completely answered this question. but i have the sinking feeling that my answer is totally a mystery. ????? 14. Nov 27, 2011 ### lavinia Every vector bundle has coordinate charts. In each bundle there are functions called coordinate transformations which say how to identify vectors above one chart with vectors on an overlapping chart. In the case of the tangent bundle these transformations are the differentials of the change of coordinates. But with other vector bundles they are not. When the coordinate transformations are the differentials of the change of coordinate functions it makes sense to define the derivative of a function with respect to a vector in the bundle. This is because the Chain rule will automatically work so the derivative of the function in one chart will transform into the derivative in the other. With other bundles it is not possible to differentiate function precisely because the Chain Rule fails. Similar Discussions: Definition of tangent space on smooth manifolds	{"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": 0.9095257520675659, "perplexity": 355.12962558575595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814566.44/warc/CC-MAIN-20180223094934-20180223114934-00222.warc.gz"}
https://byjus.com/secant-formula/	# Secant Formula The length of the hypotenuse, when divided by the length of the adjacent side, becomes the secant of an angle in a right triangle. It is written as Sec, and the formula for secant is: ## The formula for secant theta Sec X = $$\frac{Hypotenuse}{Adjacent Side}$$ As we know there are six trigonometric functions and out of these, Secant, cotangent, and cosecant are hardly used. ## More about Secant angles formula Secant is Reciprocal of Cos, Sec x = $$\frac{1}{CosX}$$ ### Examples of Secant Math Formula Example 1: Find Sec X if Cos x = 38 Solution: As Sec X = 1/ Cos X =1/3/8 =8/3 So, Sec X = 8/3 ## Practice Question for Secant trigonometry formulas Q1) Find Sec x if tan x = 5/4. Q2) Find Sec a, if cos a = 1/5 To study other Trigonometric Formulas and its applications, Register on BYJU’S.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9655604958534241, "perplexity": 2500.749267959344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056120.36/warc/CC-MAIN-20210918002951-20210918032951-00421.warc.gz"}
https://quant.stackexchange.com/questions/51230/selecting-strike-prices-for-put-writing-strategy-based-on-z-scores	# Selecting strike prices for put-writing strategy based on Z-scores I'm trying to replicate the put-writing strategy of Jurek and Stafford from 2015 (The Cost of Capital for Alternative Investments, Jrl. Fin. SSRN). Their strategy writes index put options on the SP500, rebalancing each month and invest the proceeds at the risk-free rate. They select strike prices based on Z-scores, not moneyness as measured by Strike/Spot-ratio. Their formula is as follows: $$K(Z) = S_t * exp(\sigma_{t+1}Z)$$ , where $$\sigma_{t+1}$$ is the 30-day implied volatility, measured by the VIX. My problem is that I don't get similar results as an example I've seen, based on this paper. Example uses $$S_t = 636$$ per 31. January 1996, $$\sigma_{t+1}=12.5\%$$, and $$Z=-2$$. Then, $$K(Z)=\589.95$$. However, I'm not able to get the same result, as I get $$K(Z) = 636exp(0.125\sqrt{30/365}-2) = 592$$. I've tried using 252 days in a year as well, without results. Hopefully, someone here can point me in the right direction. • Are you sure the option has a maturity of 30 days? On 31 January 1996 the next maturity dates are 16 Feb 1996 or 15 Mar 1996 (assuming these are monthly SP500 options on the regular calendar, third friday of the month). These would be 16 or 44 days in the future (unfortunately neither of which gives your desired result, where did you get 589.95 ? ). – noob2 Feb 18 at 19:08 • Apologies, the holding period is supposed to be 30 days. So the option is written on 31 Jan 1996, repurchased on 29 Feb 1996. Effectively having a holding period of 29 days. The maturity, however, is 45 days, namely the 16 Mar, where the feasible strike is $590$. The selection rule is such that the strike should be below the calculated strike, and the maturity date should be after the roll date (30 days). – Mkl Feb 19 at 8:03	{"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": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8295148015022278, "perplexity": 1137.4627402207504}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400212959.12/warc/CC-MAIN-20200923211300-20200924001300-00410.warc.gz"}
https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_ChemPRIME_(Moore_et_al.)/13Chemical_Equilibrium	Skip to main content 13: Chemical Equilibrium • 13.0: Prelude to Equilibria The term chemical equilibrium is used to describe a chemical reaction in which the concentrations of the substances involved remain constant. Read on to learn how chemical equilibrium is defined. • 13.1: The Equilibrium State The term chemical equilibrium is used to describe a chemical reaction in which the concentrations of the substances involved remain constant. Read on to learn how chemical equilibrium is defined. • 13.2: The Equilibrium Constant The equilibrium constant represents the constant ratio between reactants and products when a reaction has reached equilibrium. Read on to find out more about how this ratio is calculated. • 13.3: The Law of Chemical Equilibrium Chemical equilibrium is attained when the concentration of reactants and products stops changing. Since reactants and products eventually reach a constant value (given constant temperature and pressure) a ratio called the rate constant can be used to describe the equilibrium. This section unpacks this ratio and how it is calculated. • 13.4: The Equilibrium Constant in Terms of Pressure In its most familiar form, the equilibrium constant is described in terms of the concentration of products and reactants. However, for gases it is often more convenient to relate the equilibrium constant in terms of pressure. Read on to find out more about expressing the equilibrium constant in terms of pressure. • 13.5: Calculating the Extent of a Reaction In chemistry, it's often convenient to predict what the outcome of a reaction will be in numerical terms. This section teaches you how to calculate the extent of a reaction - how much product will be formed. • 13.6: Successive Approximation An approximation is often useful even when it is not a very good one, because we can use the initial inaccurate approximation to calculate a better one. With practice, using this method of successive approximations is much faster than using the quadratic formula. It also has the advantage of being self-checking. • 13.7: Predicting the Direction of a Reaction Often you will know the concentrations of reactants and products for a particular reaction and want to know whether the system is at equilibrium. If it is not, it is useful to predict how those concentrations will change as the reaction approaches equilibrium. A useful tool for making such predictions is the reaction quotient, Q. Q has the same mathematical form as the equilibrium-constant expression, but Q is a ratio of the actual concentrations (not the equilibrium concentrations). • 13.8: Le Chatelier’s Principle Le Chatelier’s principle states that if a system is in equilibrium and some factor in the equilibrium conditions is altered, then the system will (if possible) adjust to a new equilibrium state so as to counteract this alteration to some degree. • 13.9: The Effect of a Change in Pressure In general, whenever a gaseous equilibrium involves a change in the number of molecules (Δn ≠ 0), increasing the pressure by reducing the volume will shift the equilibrium in the direction of fewer molecules. This applies even if pure liquids or solids are involved in the reaction. • 13.10: The Effect of a Change in Temperature Similar to a change in volume, a change in temperature forces a reaction to change in order to offset it's effect. • 13.11: Effect of Adding a Reactant or Product Just as varying temperature or volume can affect equilibrium, so can adding/subtracting a reaction/product. Read on to learn the specifics. • 13.12: The Molecular View of Equilibrium Chemical equilibrium can seem to be an unchanging phenomenon from a macroscopic perspective. Diving into the microscopic perspective, we find a different story. Read on to find out more.	{"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": 0.933221161365509, "perplexity": 567.4737222239963}, "config": {"markdown_headings": false, "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-2019-13/segments/1552912202804.80/warc/CC-MAIN-20190323121241-20190323143241-00171.warc.gz"}
http://www.minet.uni-jena.de/Math-Net/reports/shadows/98-25report.html	Marcinkiewicz-Zygmund-type inequalities for irregular knots and mixed metrics Preprint series: 98-25, Analysis The paper is published: Vestnik RUDN, Mathematik, 4,5 (1), 90-115, 1997/98 MSC: 42B05 Fourier series and coefficients 42B15 Multipliers Abstract: Marcinkiewicz-Zygmund type inequalities on the equivalence of a continuous norm of a real-valued trigonometric polynomial of $\; l \,$ variables and its discrete one are proved in the general case of mixed $\; L_{\overline{p}}$-metrics, where $\; {\overline{p}} = (p_1,...,p_l);\;\; 0 < p_i \le +\infty, \;\; i=1,....,l,\,$ and of non-uniform grids. A new representation formula for a trigonometric polynomial, which contains a parameter is used to prove the main results. Keywords: discrete and continuous quasi-norms	{"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": 0.8774508237838745, "perplexity": 4120.61768343886}, "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-2017-47/segments/1510934805023.14/warc/CC-MAIN-20171118190229-20171118210229-00695.warc.gz"}
https://www.scribd.com/document/87417798/Frequency-Response-Methods-in-Control-Systems	You are on page 1of 535 # OTHER IEEE PRESS BOOKS Programs for Digital Signal Processing, Edited by the Digital Signal Processing Committee Automatic Speech & Speaker Recognition, Edited by N. R. Dixon and T. B. Martin Speech Analysis, Edited by R. W. Schafer and J. D. Markel The Engineer in Transition t o Management, I. Gray Multidimensional Systems: Theory & Applications, Edited by N. K. Bose Analog ldegrated Circuits, Edited by A. B. Grebene Integrated-Circuit Operational Amplifiers, Edited by R. G. Meyer Modern Spectrum Analysis, Edited by D. G. Childers Digital lrnage Processing for Remote Sensing, Edited by R. Bernstein Reflector Antennas, Edited by A. W. Love Phase-Locked Loops & Their Application, Edited by W. C. Lindsey and M. K Simon Digital Signal Computers ark Processors, Edited by A. C. Salazar Systems Engineering: Methodology and Applications, Edited by A. P. 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Hodges Power Semiconductor Applications, Volume I I : Equipment and Systems, Edited by J. D. Harnden, Jr. and F. B. Golden Power Semiconductor Applications, Volume I: General Considerations, Edited by J. D. Harnden, Jr. and F. B. Golden A Practical Guide to Minicomputer Applications, Edited by F. F. Coury Active Inductorless Filters, Edited by S. K. Mitra Edited by , ## Professor of Control Engineering University of Cambridge A volume in the IEEE PRESS Selected Reprint Series, . prepared under the sponsorship of the f i ,4. > . IEEE Control Systems Society. ## The Institute of Electrical and Electronics Enaineers Inc NPW Vnrk IEEE PRESS 1979 Editorial Board A. C. Schell, Chairman George Abraham Clarence Baldwin Walter Beam P. H. Enslow, Jr. M. S. Ghausi R. C. Hansen Thomas Kailath J. F. Kaiser Dietrich Marcuse Irving Reingold P. M. Russo Desmond Sheahan J. B. Singleton S. B. Weinstein 73501 R. K. Hellmann E. W. Herold W. R. Crone, Managing Editor Isabel Narea, Production Manager Joseph Morsicato, Supervisor, Special Publications Copyright O 1979 by THE INSTITUTE OF ELECTRICAL AND ELECTRONICS ENGINEERS, INC. 345 East 47 Street, New York, NY 10017 All rights reserved PRINTED IN THE UNITED STATES OF AMERICA IEEE International Standard Book Numbers: Clothbound: 0-87942-125-8 Paperbound: 0-87942-1 26-6 ## Sole Worldwide Distributor (Exclusive of IEEE): JOHN WlLEY & SONS, INC. 605 Third Ave. New Yo'rk, NY 10016 ## Wiley Order ~ u m b e r s Clothbound: 0-471-06486-6 : Paperbound: 0-471-06426-2 2, i ' Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii viii Obituary Statement: Harry Nyquisl:, H, W, Bode (IEEE Transactions on Automatic Control, December 1977) Part I: The Development of Frequency-Response Methods in Automatic.Control ..... IX ............................ - The Development of Frequency-Response Methods in Automatic Control, A. G. J, AdacFarlane (IEEE Transactions on Automatic Control, April 19791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ## Part II: The Classical Frequency-Response Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :. . .:,. . ,!,. ' 7 .-' . ,, . . . . . . . . . ."" -- Regeneration Theory, H Nyquist (Bell System Technical Journal, January 1932) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability of Systems with Delayed Feedback, Ya. Z. Tsypkin (Avtomatika i Telemekhanika, #2-3, 1946) [Translated from the Russian original by L. Jocik and S. Kahnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control System Synthesis by Root Locus Method, W R. Evans (Transactions of theAIE,E, Part 1, 1950) . . . . . . . . . . . . . The Analysis of Sampled-DataSystems, J. R. Ragazziniand L. A. Zadeh (Transactionsorr the AIEE, November 1'952) . . . When Is a Linear Control System Optimal?, R. E. Kalman (Transactions of the ASME, Journal of Basic Engineering, March1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Formulation of the Nyquist Criterion, C, A. Desoer (IEEE Transactions on Circuit Theory, June 1965) . . . . . . Synthesis of Feedback Systems with Large Plant Ignorance for Prescribed Time-Domain Tolerances, I. M Horowitz and . M. Sidi (International Journal of Control, February 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Graphical Test for Checking the Stability of a Linear Time-Invariant Feedback System, F. M, Callier and C. A. Desoer (IEEE Transactionson Automatic Control, December 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Simplifying a Graphical Stability Criterion for Linear Distributed Feedback Systems, F. M Callier and C. A. Desoer . (IEEE Transactionson Automatic Control, February 1976) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modern Wienelr-Hopi Design of Optimal Controllers-Part I: The Single-Input-Output Case, D. C Youla, J. J. Bongiorno, . Jr., and H. A. Jabr (IEEE Transactionson Automatic Control, February 1976) . . . . . . . . . . . . . . . . . . . . . . . . . . . The Encirclement Condition: An Approach Using Algebraic Topology, R. DeCarlo and R. Saeks (International Journal of Control, August 1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I I:Extensionst o Nonlinear, Time-Varying and Stochastic Systems I .................................. A Frequency Response Method for Analyzing and Synthesizing Contactor Servomechanisms, R. J. Kochenburger (Transactions of the A IEE, Part 1,1950) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Stability of Nonlinear Systems of Automatic Control, V M Popov (Automation and Remote Control,,February . . 1962) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Stability of Nonlinear Automatic Control Systems with Lagging Argument, V. M Popov andA. Halanay, (Automation and Remote Control, January 1963) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Geometrical Criterion for the Stability of Certain Nonlinear NonautonomousSystems, K. S. Narendra and R. M, Goldwyn (IEEE Transactionson Circuit Theory, September 1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Frequency-Domain Condition ifor the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element, I. W. Sandberg (Bell System Technical Journal, July 1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Frequency Criterion for Absolute Process Stability in Nonlinear Automatic Control Systems, 5. N Naumov and Ya. Z. . Tsypkin (Automation and Remote Control, June 1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Domain Stability Criteria-Part I, R. W. Brockett and J. L. Willems (IEEE Transactions on Automatic Control, July1965) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - Frequency Domain Stability Criteria-Part II, R. W Brockett and J. L. Willems (IEEE Transactions on Automatic Control, . October1965) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems-Part I: Conditions Derived Using Concepts of Loop Gain, Conicity, and Positivity, G. Zames (IEEE Transactionson Automatic Control, April 1966) . . . . . . . . . On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems-Part II: Conditions Involving Circles in the Frequency Plane and Sector Nonlinearities, G. Zames (IEEE Transactions on Automatic Control, July 1966) . . . . . . . Frequency-Domain Instability Criteria for Time-Varying and Nonlinear Systems. R. W. Brockett and H. B. Lee (Proceedings of the IEEE. May 1967) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Parabola Test for Absolute Stability. A . R. Bergen and M. A . Sapiro (IEEE Transactions on Automatic Control. June19671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Off-Axis Circle Criterion for the Stability of Feedback Systems with a Monotonic Nonlinearity. Y-S Cho and K. S. Narendra (IEEE Transactions on Automatic Control. August 1968) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Domain Stability Criteria for Stochastic Systems. J. C. Willems and G. L . Blankenship (IEEE Transactions on Automatic Control. August 1971) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Describing Function Matrix. A . I. Mees (Journal of the Institute of Mathematics and Its Applications and Recent Mathematical Developments in Control. 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Describing F~nctions Revisited. A . I. Mees and A. R Bergen (IEEE Transactionson Automatic Control. August 1975) . . . '^ ## Part IV: Multivariable Systems ............................................................. Design. of Multivariable Control Systems Using the Inverse Nyquist Array. H H Rosenbrock (Proceedings of the IEE. . . November1969) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multivariable Circle Theorems. H H. Rosenbrock (Proceedingso f /MA Conference on Recent Mathematical Developments incontrol. 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modified Multivariable Circle Theorems. P. A . Cook (Proceedings of /MA Conference on Recent Mathematical Developments in Control. 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Design of Linear Multivariable Systems. D. Q. Mayne (Automatics. March 1973) . . . . . . . . . . . . . . . . . . . . . . . . . . Dyadic Expansion. Characteristic Loci and Multivariable-Control-Systems Design. D. H Owens (Proceedings of the IEE. March1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions for the Absence of Limit Cycles. P. A . Cook (IEEE Transactions on Automatic Control. June 1976) . . . . . . . Modern Wiener-Hopf Design of Optimal Controllers-Part 11: The Multivariable Case. D. C. Youla. H. A . Jabr. and J. J. Bongiorno. Jr.. (IEEE Transactions on Automatic Control. June 1976) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Generalized Nyquist Stability Criterion and Multivariable Root Loci. A . G. J, MacFarlane and I. Postlethwaite (lnternational Journal of Control. January 1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Frequency Functions and Characteristic Gain Functions. A . G. J. MacFarlane and I. Postlethwaite (International Journal of Control. August 1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain and Phase Margin for Multiloop LOG Regulators. M. G. Safonov and M. Athans. (IEEE Transactions on Automatic Control. April 1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Output Regulation and Internal Models-A Frequency Domain Approach. G. Bengtsson (Automatics. July 1977) . . . . . . Part V: Multidimensional Systems 318 326 347 353 360 366 373 393 440 454 461 475 ........................................................... A Nyquist-Like Test for the Stability of Two-Dimensional Digital Filters. R. DeCarlo. R. Saeks. andJ. Murray (Proceedings of the IEEE. June 1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 Multivariable NyquistTheory. R. A . DeCar1o.J. Murray.andR.Saeks(InternationalJournalofControl. May 1977) . . . . 480 Bibliography Authorlndex Subjectlndex i - ......................................................................... ......................................................................... ........................................................................ ## 499 517 519 Editor's Biography ..................................................................... 523 Preface vii 1889-1 976 ## Obituary Statement: Harry Nyquist HENDRIK W. BODE H ARRYHe was 87died Apriland had been ill for some Nyquist 4, 1976 in Harlingen, Texas. years old time. His passing is a landmark event for engineers in many fields to which he made major contributions. To control theorists, Nyquist is no doubt best known as the inventor of the Nyquist diagram, defining the conditions for stability of negative feedback systems. This has become a foundation stone for control theory the world over, applicable in a much wider range of situations than that for which it was originally enunciated. The so-called Nyquist interval, or Nyquist signaling rate, plays a similar fundamental role for modern communications engineering. Nyquist's calculation of the thermal noise level in a transmission band of given width is another fundamental for communication engineers, as is a variety of more specific inventions such as the vestigial sideband system now used universally in television transmission. Nyquist was born in Nilsby, Sweden, on February 7, 1889, and emigrated to this country in 1907 at the age of 18. He spent the next ten years completing his education, supporting himself in the mean time through teaching and summer jobs. (There is a strong IHoratio Alger flavor in all of this.) Nyquist finished his formal education with a Ph.D. degree in physics from Yale University in 1917. He was immediately employed by the A.T.&T. Company, and spent essentially the rest of his life with the Bell System, either with the A.T.&T. Company's so-called D&R Department, or with Bell Laboratories. He retired formally in 1954, but continued as a part-time consultant for the Bell System or for various federal agencies for several years thereafter. All told, his working lifetime covered more than half a century. Much of Nyquist's most important research stemmed originally from the practical needs of the telephone system. For several years after joining the Bell System, he was concerned with high-speed telegraphy, including systems based on metallic return paths and the then-new carrier technology. This led in due course to his fundamental work on the relation between signaling speed, pulse shape, and the probability of intersymbol interference, subjects of importance for data systems and the like even today. The high point of Nyquist's creative activities probably took place in the decade or so straddling the end of the 1920's and the beginning of the 1930's. This is the period The author is with Haward University, Cambridge, MA. when he made his principal studies of television transmission and other phase-related systems, including the basic invention of vestigial sideband transmission for television circuits. This is also the period when Nyquist made his fundamental studies of thermal noise in communication circuits, giving -communication systems engineers a fundamental point of departure in planning new systems. Nyquist's most conspicuous contribution in these years, however, was probably the enunciation of his stability criterion for feedback systems. This was indeed a theoretical contribution which came just in time. Both the communication and control fields badly needed the improvement in performance promised by whole-hearted exploitation of the negative feedback principle, but before Nyquist's formulation, the design task of actually realizing such improvements ill performance in a stable system had often been unmanageably difficult. Nyquist's formulation brought the whole subject within the scalpe of rational design practice. In his later years, Nyquist's attention turned increasingly to digitalized systems, including, in particular, digital coding and some secrecy and switching problems. This was an easy step for Nyquist to take, since so much of the digital universe is spiritual kin to the idealized telegraph Reprinted from lEEE Trans. Automat. Contr., vol. AC-22, pp. 897-808. Dec. 1977. ## IEEE TRANSACTIONS ON AUTOMATTC CONTROL, VOL. AC-22, NO. 6, DECEMBER 1977 systems on which he had worked earlier. It has had the happy result, however, that it made available the vast storehouse of design techniques, including many of Nyquist's own contributions, which has accumulated from earlier epochs. There has been no need for cornrnunication engineers to develop brand new answers for every new problem in the digital era. Any recital of Nyquist's theoretical contributions must include the fact that he was also an exceptionally prolific inventor. During his Bell System career, he was granted 138 American patents, not counting a substantial number of foreign filings. The subject matter of these many patents covers a wide range. Many of the patents, of course, are individually important and interesting as adroit solutions of isolated technical problems. A considerable number, however, are of special interest because they are all related to the general problem of measuring and correcting for distortion which may be exhibited by transmission media of various sorts, whether one is concerned with speech, TV, data, or whatnot. The distortion problem is, of course, a fundamental one which one must expect to encounter in one way or another in any transmission system. The rich collection of inventive devices represented by Nyquist's patents amounts, in the aggregate, to a developed design theory for approaching such problems, whatever the particular situation. Nyquist received many honors in recognition of his many contributions in the communication and control fields. These include, among others, the NAE Founders' Medal of Honor (1969), the Ballantine Medal of the Franklin Institute (1960), the IEEE's Medal of Honor (1960), and the ASME Oldenburger Medal (1975). Personally, he was quiet and reserved, but a very good associate for intimate friends and colleagues. Part I ## The Development of Frequency-Response Methods in Automatic Control Editor's Note: As readers of this journal will recall, in 1976 the Control Systenls Society named three distinguished control systems specialists as Consulting Editors. One of the charges to these men was to submit an invited paper on a topic of their choice for publication without the usual IDC review procedures. At the same time Professor A. G. J. MacFarlane, Professor of Control Engineering at Cambridge University, was invited by the IDC to prepare an IEEE Press reprint book of important papers on frequency-domain methods in control and systems engineering. The coincidence of these two decisions has led to the following paper. "The Development of Frequency-Response Methods in Automatic Control" is one part of the IEEE Press book, Frequency-Response Methodr in Control Systems, edited by A. G. J. MacFarlane and sponsored by the Control Systems Society. The book will appear in mid- 1979. The paper has been selected by Consulting Editor Nathaniel Nichols ,and should be of substantial interest to TRANSACTIONS readers. It also conveys some of the spirit and content of the 'book which may be purchased from IEEE Press when available. popular belief, most good engineering C ONTRARY tofrom work oln an important practical theory arises problem; this was certainly the case with Nyquist's famous stability criterion 11691. His attack on the problem of feedback amplifier stability produced a tool of such flexibility and power that its use rapidly spread to the wider field of automatic control. This fusion of the dynamical interests of the control and communication engineer has been immensely fruitful. In order to appreciate fully the far-reaching implications of Nyquist's 1932 paper, one must first consider the developments in automatic control and telecommunications which led up to it. Although automatic control devices of various sorts had been in use since the beginnings of technology [152], Watt's use of the flyball governor can be f taken as the starting point for the development of automatic control as a science [153], [154]. The early Watt governors worked satisfactorily, no d'oubt largely due to the considerable amounts of friction present in their mechanism, and the device was therefore widely adopted. In fact, it has been estimated that by 1868 there were some 75 000 Watt governors working in England alone [75]. However, during the middle of the 19th century, as engine designs Manuscript received August 30, 1978. The author is with the Control and Management Systems Division, Department of Engineering, Cambridge University, Cambridge, England. changed and manufacturing techniques improved, an increasing tendency for such systems to hunt became apparent; that is, for the engine speed to vary cyclically with time. This phenomenon had also appeared in governed clockwork drives used to regulate the speed of astronomical telescopes and ha.d been investigated by Airy. (when he had, not unnawas Astronomer Royal) [I]-[3], [75]. turally, attacked this problem with the tools of his own trade: the theory of celestial mechanics. He carried out his investigations with great skill and insight, and essentially got to the root of the mathematical problems involved. Unfortunately, his work was rather intricate and difficult to follow; it therefore did not become widely known, and the subject remained shrouded in mystery to engineers grappling with the problem of fluctuating engine speeds. This problem of the hunting of governed engines became a very serious one (75 000 engines, large numbers of them hunting!) and so attracted the attention of a number of outstandingly able engineers and physicists [153], [154], [76]. It was solved by classic investigations made by Maxwell [150], who founded the theory of automatic control systems with his paper "On Governors," and by the Russian engineer Vyschnegradsky [22!6], [227], who published his results in terms of a design n~le, relating the engineering paramelers of the system to its stability. Vyschnegradsky's analysis showed that the engine design changes which had b~eentalung place since Watt's timea decrease in friction due to improved manufacturing techniques, a decreased moment of inertia arising from the use of smaller flywheels, and an increased mass of Reprinted from IEEE Trans. Automat. Contr., vol. AC-24, pp. 250-2,65, Apr. 1979. ## MAC FARLANE: FREQUZNCY-RESPONSE METHODS engineer's viewpoint, however, was quite different. It was natural for him to regard his various bits of apparatus in terms of "boxes" into which certain signals were injected and out of which emerged appropriate responses. Thus it was a natural next step for a colmmunications engineer, in considering his system's behavior, to replace the actual boxes in which distinct pieces of physical apparatus were housed by abstract boxes which represented their effect on the signals passing through them. A combination of this "operator" viewpoint with the electrical engineer's flexible use of complex-variable representations of sinusoidal waveforms made the: use of Fourier analysisbased techniques for studying dynamical phenomena in communication systems virtuallly inevitable. Following the pioneering work of Heaviside [92] on operational methods for solving differential equations, integral transform methods and their application to practical problems were put on a secure foundation by the work of Bromwich [37], Wagner [229], Carson [ a ] , [47], Campbell and Foster [44], Doetsch [59], and others; thus by the late 1920's and early 1930's the integral-transform approach to the analysis of dynamical phenomena in conmunication systems was available for the study of feedback devices, given someone with the initiative and skill to use it. The role of positive feedback in the deliberate generation of oscillations for high-frequency modulated-carrier radio telegraphy emerged shortly after the development of the triode amplifying valve; a patent on the use of inductive feedback to produce a high-frequency alternating current using an amplifying valve was granted to Strauss in Austria in 1912; similar developments were credited to Meissner in Germany in 1913, to Franklin and Round in England, and to Armstrong and Langrnuir in the U.S.A. [191]. Armstrong, in 1914, devt:loped the use of positive feedback in his "regenerative receiver" [28]. The central role of the feedback concept in steam engine control systems had been considered by Barkhausen [9], and Barkhausen's ideas were discussed by Moller [I641 in his treatment of feedback effects in electrical circuits. Further development of the idea of a feedback loop of dependence in an oscillator circuit led Barkhausen [lo] to give a "formula for self-excitation" : KF(j w ) == 1 where K is an amplifier gain factor and F) is the frequency-dependent gain of an associated feedback loop in the oscillator circuit. The Barkhausen criterion which developed from this formula was orginally intended for the determination of the self-excitation frequency of ac generators for use in radio transmitters. Prior to the appearance of Nyquist's 1932 paper, however, the phenomenon of conditional stability was not understood and hence it was widely believed that, for a given frequency-dependent gain function F(jiw), there was only a single value of the scalar gain parameter K which separated stable and unstable regions of behavior. Thus, particularly in the German literature, Barkhausen's equation came to be used as the basis of a stability criterion for positive and negative feedback almplifiers [135], [97]. An important c early contribution to the development of frequency-response methods for the analysis of linear dynamical systems was made in 1928 by Kupfmiiller. In this paper [124] he gave a comprehensive discussion of the relationships between frequency transmission characteristics and transient response behav~~or. another paper published in the In same year, Kupfmdler [I251 dealt with the problem of closed-loop stability Here, however, he did not use a fully-developed frequency-domain approach. Kupfmiiller represented the syste:m7sdynamical behawor in terms of an integral equation, and hence developed an approximate criterion for closed-loop stability in terms of time-response quantities measured and calculatedl from the system's transient response. Kiipfmuller's technique of approximately determiining closed-loop stability from such time-response measurements seems to have remained relatively unknown outside Germany. In hls hstory of automatic control, Rorentrop [191] refers to further work done in Germany in the 11930's on frequency-response criteria for feedback system stability, and in particular he refers to the development by Strecker of a frequency-domain stability criterion of what we would now call Nyquist type. This work appears to have remained virtually unknown and was only described in the scientific literature availa.ble after the end of the Second World War [213]-[216]. In his book Strecker j[215] refers to having presented a frequency-response stability criterion at a colloquium at the Central Laboratory of Siemens and Halske in 1930, and to having presented his results to a wider audience at a seminar held by the Society of German Electrical Engineers in 1938. Rorccntrop [191] says that the manuscript of this lecture is still available and that in it Strecker considered the case of open-loop unstable systems. The truly epoch-nnaking event in the development of frequency-response methods was undoubtedly the appearance of Nyquist's classic paper [169] on feedback amplifier stability, which arose directly from work on the problems of long-distance telephony. In 1915 the Bell System completed an experimental telephone l n between ik New York and San ]Francisco which showed that reliable voice communicatior~ over transcontinental distances was a practicable propocrition. This link used heavy copper open-wire circuits (ureighmg half a ton/mi) and was inductively loaded to have a cut-off frequency of 1000 Hz. The attenuation over a 3000 mi distance was 60 dB and, with a net gain of 42 dB provided by six repeating amplifiers, this was reduced to a tolerable net attenuation figure of 18 dB overall. The use of carrier systems on open-wire circuits was soon well advanced and had resulted in a substantial economy in conductor costs with multiplex operation im a frequency range 'well above the audible. A change to cable operations, however, posed a number of severe technical problems. In particular, because the conductors were small, the attenuation was large and this required the use of many repeating amplifiers. Thus a crucial technical problem had to be overcome, that of ## MAC FARLANE: PRBQUENCY-RESPONSE-ODs repeatedly passing signals through amplifiers, each of which contained unavoidable and significant nonlinearities, while keeping the total distortion over transcontinental distances within acceptable limits. It required an effective amplifier linearity to within better than several parts in a thousand in order to maintain intelligibility of the transmitted audio signals. Such an acute difficulty could only be overcome by a major invention, and this was provided by H. Black of the Bell Telephone Laboratory when he put forward the idea of a feedback amplifier. Black's important discovery was that high gain in a nonlinear and variable amplifying device could be traded for a reduction in nonlinear distortion, and that an accurate, stable, and highly linear overall gain could be achieved by the suitable use of a precision linear passive component in conjunction with a high-gain nonlinear amplifer. By 1932 Black and his colleagues could build feedback amplifers which performed remarkably well. They had, however, a tendency to "sing," the telephone engineer's expressive term for instability in amplifers handling audio signals. Some "sang" when the loop gain of the feedback amplifier was increased (which was not unexpected), but others "sang" when the loop gain was reduced (which was quite unexpected). The situation was not unlike that associated with the hunting governors of around 1868-an important practical device was exhibiting mysterious behavior. Moreover, it was behavior whose explanation was not easily within the compass of existing theoretical tools, since a feedback amplifier might well have of the order of 50 independent energy-storing elements within it (such as inductors, capacitors, etc.). Its description in terms of a set of differential equations, as in the classical analyses of mechanical automatic control systems, was thus hardly feasible in view of the rudimentary facilities available at that time for the computer solution of such equations. Nyquist's famous paper solved this mystery; it opened up wholly new perspectives in the theory of feedback mechanisms and hence started a new era in automatic control. Prior to 1932 the differential-equation-based approach had been the major tool of the control theorist; within the decade following Nyquist's paper these techniques were almost completely superseded by methods based on complex-variable theory which were the direct offspring of his new approach. The background to his invention and its subsequent development have been described in a fascinating article by Black [28]. It is clear from this that Black used a stability argument of frequency-response type, saying there that " ...consequently, I knew that in order to avoid self-oscillation in a feedback amplifier it would be sufficient that at no frequency from zero to infinity should /.@ be real, positive, and greater than unity." The prototype Black feedback amplifier was tested in December 1927 and development of a carrier system for transcontinental cable telephony, its first application, started in 1928. Field trials of a system using a 25-misection of cable with 2 terminal and 68 repeater amplifiers were held at Morristown, NJ in 1930, and successfully completed in 1931. Black [27] de- scribed his amplifier in a paper which makes interesting references to the stability problem. In particular he mentions the phenomenon of conditional stability in the following words: "However, one noticeable feature about the field of p/3 is that it implies that even though the phase shift is zero and the absolute value of Icp exceeds unity, self-oscillations or singing will not result. This may or may not be true. When the author first thought about this matter he suspected that owing to practical nonlinearity, singing would result whenever the gain around the closed loop equalled or exceeded the loss and simultaneously the phase shift was zero, i.e., @ = I p/3 1 +jO > 1. Results of experiments, however, seemed to indicate something more was involved and these matters were described to Mr. H. Nyquist, who developed a more general criterion for freedom from instability applicable to an amplifier having linear positive constants." Nyquist himself has also briefly described the events which led to his writing the 1932 paper [170]. Nyquist's open-loop gain frequency-response form of solution of the feedback stability problem was of immense practical value because it was formulated in terms of a quantity (gain) which was directly measurable on a piece of equipment. This direct link with experimental measurements was a completely new and vitally important development in applied dynamical work. The application of Nyquist's stability criterion did not depend on the availability of a system model in the form of a differential equation or characteristic polynomial. Furthermore, the form of the Nyquist locus gave an immediate and vivid indication of how an unstable, or poorly damped, system's feedback performance could be improved by modifying its open-loop gain versus frequency behavior in an appropriate way. It seems clear that when Nyquist set out to write his 1932 paper he was aware that the fundamental phenomenon which he had to explain was that of conditional stability. The successful theoretical explanation of this counter-intuitive effect is given due prominence in the paper. It is also clear that Nyquist was fully aware of the great generality and practical usefulness of his stability criterion. He, therefore, attempted to prove its validity for a wide class of systems, including those involving a pure time-delay effect, and he took the primary system description to be a directly-measurable frequency response characteristic. The importance of the feedback amplifier to the Bell Laboratories' development of long-distance telephony led to a careful experimental study of feedback amplifier stability by Peterson et a/. [178]. These experiments fully supported Nyquist's theoretical predictions and thus completely vindicated his analysis. It is altogether too easy, with hindsight and our exposure to current knowledge, to underestimate the magnitude of Black's invention and Nyquist's theoretical achievement. Things looked very different in their time. The granting of a patent to Black for his amplifier took more than nine years (the final patent, No. 2,102,671, was issued on December 21, 1937). The U.S.Patent Officecited technical papers claiming that the output could not be connected back to the input of an amplifier, while remaining stable, unless the loop gain were less than one; and the British Patent Office, in Black's words, treated the application "in the same manner as one for a perpetual-motion machine." Nyquist's work had shown the great power of complexvariable theory for the analysis of feedback system behavior, and it was inevitable that a tool of such promise would be further developed for design purposes. It was a natural inference from the develop~ments presented in his paper that the closeness of approach of the Nyquist locus to the critical point in the complex plane gave a measure of closed-loop damping. This was investigated by Ludwig [I351 who gave a neat formula for estimating the real part of a pair of complex conjugate roots associated with the dominant mode in tlie case where the Nyquist locus passes near the critical point. Thus, it soon became clear that the key to making an unstable (or otherwise unsatisfactory) feedback system stable (or better damped) lay in an appropriate modification of the amplitude and phase characteristics of the open-loop gain function for the feedback loop involved. Extensive and fruitless experimental studies were made, particularly by F. B. Anderson in the Bell Telephone Laboratories, in attempts to build feedback amplifiers having loops which combined a fast cutoff in gain with sn small associated phase shift. It therefore became important to analyze the way in which the amplitude and phase frequency functions of a loop gain transfer function are related. In another of the classic papers which lie at the foundation of feedback theory Bode [30] carried out such an analysis, extending previous work by Lee and Wiener [128]. This paper, written in a beautifully clear and engaging manner, showed how there is associated with any given amplitude/gain frequency function an appropriate minimum-phase frequency function. Bode was thus able to give rules for the optimum shaping of the loop-gain frequency function for a feedback amplifier. He introduced logarithmic units of amplitude gain and logarithmic scales of frequency, and hence the logarithmic gain and linear phase versus logarithmic frequency diagrams which bear his name. The critical point in the gain plane was put at its now standard and familiar location of (-- 1 jO), and the concepts of gain and phase margin were introducecl. Bode's classic work appeared in an extended form in his book Network Analysis and Feedback Amplifier Design [3 11. : Nyquist's criterion is not easy to prove rigorously for the class of systems which he far-sightedly attempted to deal with in his classic paper and the need for a rigorous approach to a simpler class of systems soon became apparent. For the case when the open-loop gain is an analytic rational function, Nyquist himself had given a simple complex variable argument in an Appendix to his 1932 paper. This approach was soon realized to provide a simple route to a satisfactory proof for the restricted class of system functions which could be specified as rational functions of a complex frequency variable. MacColl [I401 gave such a proof using the Principle of the Argument, appearand this became the standard form of expositi~on ing in influential books by Bode [31], James let al. [loll, and many others. Such ;simplified presentations did scant justice to the far-reaching nature of Nyqunst's classic paper, but they soon made the stability criterion a cornerstone of frequency-response methods based on complex function theory. The treatment given in Nyquist's 1932 paper had specifically excluded sy:stems having poles in the closed right-half plane. A pure integration effect, hourever, often occurs in the open-loop transmission of servomechanisms incorporating an electric or hydraulic motor, and the appropriate extension to the Nyquist criterion to handle transfer function poles at the origin of the complex frequency plane was described in various wartime reports such as MacColl's [139] and in a paper by Hall [86]. Using the complex-variable approach based on the Principle of the Argument which hatd by then become the standard one, Frey [73] extended the Nyquist stability criterion to deal with the case where the feedback system may be open-loop unstable, ancl this simple first version of the Nyquist criterion finally assumed the form which became familiar in a multitude of textbooks. OF APPROACH THESPREAD THE FREQUENCY-RESPONSE By the beginning of the twentieth century the basic concepts of automatic c'ontrol and their analyitical discussion in terms of ordinary differential equations and their related characteristic algebraic equations were well established. These techniques were consolidated in review papers by Hort [98] and Von Mises [228], a:nd in early textbooks on automatic control by Tolle [219] and Trinks [220]. The further development of automatic control devices received great iimpetus from important studies camed out by Minorsk:~[161] on the automa.tic steering of ships, and by Hazen [90] on shaft-positioning servomechanisms. Minorsky proposed the use of a proportional-plus-derivative-plus-integral control action for the steering control. His work was of particular significance in being practically tested in a famous series of trials on the automatic steering of the USS New Mexico in 1922-23 [162]. Both Minorsky's and Hazen's work wars explained in terms of ordinary differential equations, and their success with practical devices led to the widespread use of this approach to the analysis of automatic control systems. In the chemical proce:ss industries the introduction of feedback control tended at first to develop in isolation from the developn~ents mechanical and electrical enin gineering. One very important difference in the process industries was (and still., to a large extent, is) that the time-scale of controlled-variable behavior was sufficiently slow on many process plants to make manua.1 feedback control action a feasible proposition. In the chemical industry the first step along the road to automatic feedback control was the iritroduction of indicating instru- ## MAC FARLANE: FREQUENCY-RESPONSEMETHODS ments to monitor plant operation, followed by the attachment of pen recorders to these indicators to secure a record of plant behavior. The natural development was then to go one step further and use the movement of the pen on the recorder to effect feedback action on control valves in the plant through the use of pneumatic transducers, amplifiers, and transmission lines. During the 1930's these pneumatic controllers were steadily developed, and the idea of using an integral action term, long standard in mechanical governing, transferred to this field of control. Here, however, it was called "reset action" since the behavior of the pneumatic controller with the integral control term added was analogous to that which would have been obtained if the reference input had been slowly adjusted (or reset) to the appropriate new value required to cancel out a steady-state disturbance. In the late 1930's and early 19403, derivative action (usually called pre-act in this context) was introduced for these pneumatic controllers to give the full 3-termy' controller or "PID" (Proportional, Integral, and Derivative) controller. A theoretical basis for applied process control was laid by papers by Ivanoff [I001 on temperature control, and by Callander et al. [42] on the effect of time-lags in control systems. It is interesting to note that this paper by CalIendar et al. probably contains the first published description of the application of an analog computer to an automatic control problem. Ziegler and Nichols [244] made an important study which led to formulas from which proportional, reset (integral) and pre-act (derivative) controller settings could be determined from the experimentally measured values of the lag and "reaction rate" of a process which was to be controlled. By the late 1930's there were thus two separate but well-developed methods of attacking the analysis of feedback system behavior. 1) The "time-response approach" which involved ordinary differential equations and their associated characteristic algebraic equations, and which was much used in mechanical, naval, aeronautical, and chemical engineering studies of automatic control systems; and 2) the "frequency-response approach" which involved Nyquist and Bode plots, transfer functions, etc., and which was used for studies of feedback amplifiers. The frequency-response approach had the appealing advantage of dealing with pieces of apparatus in terms of abstract "boxes" or "blocks" which represented their effect on the signals passing through them. This proved to be a very flexible and general way of representing systems, and it was found that when such "block" diagrams were drawn for different kinds of control systems the ubiquitous loop of feedback dependence, which is the hallmark of a feedback mechanism in a representation of this sort, sprang into sudden prominence. The power and flexibility of the tools developed by Nyquist and Bode were such that their spread to other fields in which feedback principles were used was inevitable. Some early work on using the techniques of the feedback amplifier designer for the analysis of more general systems was done by Taplin at MIT in 1937 [loll. A crucial step in the transference of the telephone engineer's viewpoint to the analysis of other kinds of system was taken by Hams, also of MIT, who made the fundamentally important contribution of introducing the use of transfer functions into the analysis of general feedback systems [87]. Hams's idea enabled a mechanical servomechanism or a chemical process control system to be represented in block diagram terms, and thus analyzed using the powerful tools available to the feedback amplifier designer. In 1938 Mikhailov gave a frequency response criterion for systems described by a known nth order constant coefficientlinear differential equation and thus having an explicitly known characteristic polynomial p(s) [159]. It was stated in terms of the locus of p(jw) in a complex p-plane and so bore a superficiaI resemblance to the Nyquist criterion. It is, however, an essentially different thing in that it requires that the governing differential equation of the system being investigated must be known, whereas the essential virtue of the Nyquist criterion is that the Nyquist locus is something which can be directly measured for a plant whose behavior in terms of a differential equation description may well not be available. A criterion of this form was also formulated by Cremer [5 1 and Leonhard [I311, independently of each other and 1 of Mikhailov. In the German literature the criterion is accordingly known as the Cremer-Leonhard criterion; in the French literature it is usually called the Leonhard criterion. In the Russian technical literature the Nyquist stability criterion is often called the Mikhailov-Nyquist criterion. Work on generalizing the Nyquist criterion to deal with neutrally stable and unstable open-loop systems was done by Mikhailov [160] and Tsypkin 12211. The 1939-45 world war created an urgent need for high-performance servomechanisms and led to great advances in ways of designing and building feedback control systems. From the point of view of the develop ment of automatic control design techniques, the chief result of the immense pooling of effort and experience involved was to spread rapidly the use of frequency-response ideas into the mechanical, aeronautical, naval and later the chemical fields, and to produce a unified and coherent theory for single-loop feedback systems. Important reports written by Brown and Hall [loll were circulated among defense scientists and engineers and soon, accelerated by the end of the war, a number of classic publications and textbooks became available which resulted in the widespread dissemination and adoption of frequency-response ideas. Herwald [95] discussed the use of block diagrams and operational calculus for the study of the transient behavior of automatic control systems, including the use of compensating networks. Ferrel [69] laid particular stress on the parallels between electromechanical control system design and electrical network design. He suggested the use of the n o w - f d a r a s p totic Bode diagrams. Graham [83] made notable use of these diagrams and discussed dynamic errors, the effects of noise, and the use of tachometric feedback compema- ## E E E TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-24, N . 2, APRIL O 1979 tion. Brown and Hall [39] gave a 'classic treatment of the -1ysis and design of servomechanisms and Hams [88] gave a wide-ranging and thorough treatment of analysis and design in the frequency domain. The work done at the MIT Radiation Lab was summarized in a notable book by James et al. [loll; the first use of inverse Nyquist diagrams is discussed in their book and credited to Marcy [148]. Gardner and Barnes [77] gave a widely used treatment of the mathematical background to these developments. British contributions were summarized in papers by Whiteley [2311, [232]. The historical background to British work by Daniel, Tustin, Porter, Williams, Whiteley, and others has been described by Porter [183] and Westcott [230]. Several of the wartime and post-war historical developments in Britain and America have been discussed by Bennett [24]. German work during and after the war has been summarized by Rorentrop [1911. Applications of the Nyquist criterion to feedback control loops were treated in the German literature in papers by Feiss [66][68] and an important textbook was produced by Oldenbourg and Sartorious [171]. Leonhard [130] discussed frequency-response design techniques and extended the Mikhailov criterion approach [131]. Tsypkin [221], [222] discussed the effect of a pure delay in the feedback loop. Among the many textbooks used by designers, the twovolume work of Chestnut and Meyer [49] had a notable impact. Since the rotating aerial of a radar system only illuminates its target intermittently, many of the fire-control systems developed during the Second World War had to be designed to deal with data available in a pulsed or sampled form. The biasis for an effective treatment of sampled-data automatic control systems was laid by Hurewicz whose work is described in [loll. In particular, in his contribution to this book, Hurewicz developed an appropriate extension of the Nyquist stability criterion to sampled-data systems. The development of digital computing techniques soon led to further work on such discrete-time systems. Digital control systems operating on continuous-time plants require analysis techniques which enable both discrete-time and continuous-time systems, and their interconnection through suitable interfaces, to be looked at from a unified standpoint. Linvill [134] discussed this problem from the transform point of view, including a consideration of the Nyquist approach to closed-loop stability. Frequency-response methods of analyzing sampled-data systems were studied by Tsypkin [223]. A "2-transform" theory for systems described by difference equations emerged to match the "s-transform" theory for systems described by differential equations [188] and was treated in textbooks by Ragazzini and Franklin [188], Jury [104], [105], Freeman [72], and others. The "equivalence" between continuous-time and discrete-time system analysis methods has been discussed by Steiglitz [2 111. The unique feature of the Nyquist-Bode diagram approach to closed-loop system stability and behavior is that it can make a direct use of experimental1:y-measurable gain characteristics. Using such data one can make inferential deductions about the behavior of the closed-loop system's characteristic frequencies. Nevertheless, there are many situations in whiich one does have a direct knowledge of the form of the plant transfer function and it then becomes a natura.1 ques,tion to ask: what direct deductions can be made from this of the way in which the closed-loop characteristic frequencies vary with a gain parameter? This question was answered in 1948 by Evans who brought the complex-variable-based approach to linear feedback systems to its fully developed state by the introduction of his root-180cus method [62]-[64].. The effect of random1 disturbances on automatic control systems was also studied during the Second World War [loll. In 1920 the autocorrelation function had been introduced by G. I. Taylor in his work on turbulent flow in fluids [217]; N. Wiener realized that this function was the link between the time and frequency-response descriptions of a stochastic process, and based his classic studies of random process analysis [233] and their relationships to communication and control theory 12351 on the generalized Fourier trransfona of this function. Wiener became deeply interested in the: relationships between control and communication probleias and in the similarities between such problems in engineering and physiology. In addition to his important wartime report on time-series analysis he wrote a seminal book on cybernetics [234]. His books had the important effect of :propagating feedback-control ideas in general, and frequency-response methods in particular, into the fields of stocha~stic system theory and physiology. The "harmonic balance" methods developed in studies of nonlinear mechanics by Krylov and Bogoliubov [122] led to attempts to exte:nd frequency-response methods to nonlinear feedback control problems. From these efforts emerged the describing function method which extended the use of Nyquist diagrams to the study of nonlinear feedback system stability. This was developed independently in a number of countries: by Goldfarb [80] in Russia, by Daniel and 'Tustin in England [225;],by Oppelt [I731 in Gennany, by Dutilh [60] in France, and by Kochenburger [120] in the United States. Although at first resting on rather shaky theoretical foundations, this technique proved of great use in many practical studies and its introduction marked an important consolidation in the use of frequency-response methods. Investigations by Bass [18] Sandberg [204], Bergen and Franks [25],, Kudrewicz [123], and Mees [156], [157] have subsequently placed the method on a sounder b,asis. Aizerman [4] greatly stimulated the study d nonlinear feedback problems by putting forward his fannous conjecture on the stability of systems incorporating a "sectorbounded" nonlinearity. This led to work on what is known in the Russian literature as the "problem of absolute stability." Despite the fact that Pliss [:I791 demonstrated by means of a counterexample that the Aizerman conjecture is not generally true, it became manifestly important to discover for what classes of system the 1979 -- ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-24,NO. 2, APRXL 1979 relevance of algebraic theories for fundamental studies of the linear multivariable control problem, and they were soon followed by a strong and sustained research effort on the algebraic approach. Surveys of work on algebraic systems theory have been given by Barnett [12] and Sain [203]. Kalman's work has shown the importance of using module theory in the algebranc approach to dynamical systems; from the mathematical point of view this leads to a particularly "clean" treatment [114], [116]. The central role of a system's state in discussing its feedback control had been established by its part in optimal control theory and by Wonham's pole-shifting theorem. Kalman and Bucy had shown how to estiinate unknown states from noise-co,rrupted system outputs. It was thus natural to seek ways of using system model information to recover inaccessible system states from the uncorrupted outputs of deterministic dynamical systems, and Luenberger [136]-[I381 introduced the use of observers for this purpose. The idea had been emerging of separating a feedback problem into the two steps of 1) working out what to do if a system's state was completely accessible, and 2) devising a means of estimating the system's inaccessible states from the information contained in its accessible outputs. In the stochastic linear optimization problem a certainty-equivalence principle had been established [102], [239], [236], [13] which had shown that the stochastic optimal control problem could indeed be solved in this way. A similar sort of "separation principle" was established for the problem of pole-shifting using an observer: the same closed-loop poles are obtained using an observer (constructed with perfect plant model information) as would have been obtained if all the system's states had been available for feedback purposes [52]. These results and ideas led naturally to a deeper study of the problems of dynamic compensation [36], [237] which further closed the gap between the classical frequency-response methods and those of what was (unfortunately) becoming known as "modern" control theory. A linear vector space approach to control problems obviously has geometrical as well as algebraic aspects. Wonham and Morse [240] ca~riedout a definitive and far-ranging study of the geometrical treatment of multivariable control problems, which culminated in Wonham's elegant and important book on this topic [241]. This definitive text opened up a whole new prospect for control studies. In this work the dynamical significance of certain classes of subspaces of the state space plays a key role, and investigations of such topics as decoupling is carried out in a crisp and intuitively appealing way. Independent studies of a geometrical approach were canied out by Basile and Marro [14]-[17]. It seems clear that the geometrical theory has a key role to play in bringing together state-space and frequency-response approaches to the multivariable case [147]. Yet another line of approach to the multivariable feedback problem arises from the observation that the classical Nyquist-Bode--Evans formulation of the single-loop case is based on complex-variable "theory. Surely, it was thought, complex-variable ideas must have a role to play in the multivariable context, particularly when the algebraic studies had shown how to extend to the multivariable case such basic complex-variable concepts as poles and zeros. An early attempt to extend Nyquist diagram ideas to the multivariable problem was made by Bohn [32], [33]. In a series of papers MacFarlane and his collaborators demonstrated that algebraic functions could be used to deploy complex variable theory in the multivariable feedback context [141]-[146]. It was shown that the poles and zeros associated with transfer-function matrices by algebraic means, via the Smith-McMillan form for a matrix of rational transfer functions, were related to the poles and zeros of anappropriate function of a complex variable. This line of investigation in turn led to a generalization of the classical Nyquist stability criterion to the multivariable case [141]. Following earlier heuristic treatments of this generalization, complex-variable proofs were provided by Barman and Katzenelson [l I] and MacFarlane and Postlethwaite [142]. The generalization of the Nyquist stability criterion to the multivariable situation was soon followed by complementary generalizations of the root locus technique [121], [142], [143], [185], [145]. Together with these counter-revoluti~onary developments of the classical frequency-respoinse approaches came an increasing interest in the existence of links between state-space nnodels and methods and the various algebraic, geometric, and complex-variable techniques and results. It was discovered that deep and important links existed betweten the poles, zeros, and root-locus asymptotes of the complex-variable characterizations and the basic operators of a state-space description [145]. These findings emplhasized the deep significance for control studies of the algebraic and geometric approaches which were being SCI rapidly developed. Since much of the motivation for work on frequency-response methods arose from the need to develop robust design methods for plants described in terms of models derived from sketchy experimental data, a number of different design approaches in the frequency domain began to emerge. Many of these techniques were conceived in terms of interactive graphical working, where the designer interacts with a computer-driven display [198]. As such, they placed great stress on the insight and intuition which could be deployed by an experienced designer; this was in great contrast to the specification-and-synthesis approach which had been the hallmark of the optimal control solution. Many of these approaches naturally sought to capitalize on the experience and insight existing for single-loop frequency-response designs. The most straightforward way to do this is to somehow reduce a multivariable design problem to a set of separate single-loop design problems. In Rosenbrock's inverse Nyquist array method [ 1941, [198], [I 651 this was done by using a compensator to first make the systenn diagonally-dominant. A careful use of this specific form of criterion of partial noninteraction enabled the stability and performance of the closed-loop ## MAC PARLANE: FREQUENCY-RESPONSEMETHODS system to be inferred from its diagonal transmittances alone, and hence enabled a multivariable design to be completed using single-loop techniques. Mayne's sequential return difference method [151] took a different line of approach to the deployment of single-loop techniques. It was built around a series of formulas for the transmittance seen between an input and output of a feedback system, having one particular feedback loop opened, when all the other feedback loop gains were made large. Providing that one could find a suitable place to start, this enabled the designer to proceed to design one loop at a time, and give it a suitably high value of loop gain before proceeding to the next one. Other investigators were less concerned with the direct deployment of single-loop techniques and looked for ways of using the generalized Nyquist and root-locus results as the basis of design methods. MacFarlane and Kouvaritakis [144] developed a design approach based on a manipulation of the frequency-dependent eigenvalues and eigenvectors of a transfer function matrix. This line of attack was later extended to handle the general case of a plant having a differing number of inputs and outputs by incorporating a state-space-based root-locus approach as an integral part of the overall procedure [145]. The interest and importance of the multivariable control problem generated a wide range of other investigations. Owens [174]-[176] studied ways of expanding transfer function matrices as sums of dyads and developed a design approach on this basis. Wolovich developed multivariable frequency-response approaches to compensation, decoupling, and pole placement [237]. Sain investigated design methods based on transfer-function matrix factorization and polynomial matrix manipulation [202], [78], [177]. Bengtsson used geometrical ideas in the spirit of Wonham's work to devise a multivariable design approach [23]. Davison made extensive investigations of the multivariable control problem and developed an approach which, although state-space based, was in the same engineering spirit as the more frequency-biased work. His studies emphasized the importance of robustness to parameter variation [55]-[58]. Youla and Bongiorno extended the analytical feedback design technique developed by Newton [I681 to the multivariable case [242]. As the broad outlines of the frequency-response theory of linear multivariable control systems began to emerge, interest rose in the appropriate extensions of nonlinear criteria such as the describing function and circle criterion, and work to this end was started by several workers [224], [157], [1971, [501, [81, [1891, [158l. the stability of multidimensional feedback filters, and this in turn led to an appropriate extension of frequencydomain techniques, including that of determining closedloop stability via Nyquist-type criteria. Jury [107] has given a very comprehensive survey of work in this area. The need to enhance the quality of pictures transmitted back to earth from exploring satellites and space probes resulted in work on the "multidimensional" filtering of video signals, that is on their simultaneous processing in more than one spatial dimension. This further generalization of the problem of dynamic filtering led to a study of From our present vantage point we can attempt to put frequency-response methods into perspective. Nyquist started a completely new line of development in automatic control when he analyzed the problem of closedloop stability from the signal-transmission viewpoint rather than the mechanistic viewpoint. In so doing he showed the engineers designing and developing feedback devices and automatic control systems how to use the powerful tools which can be forged from the theory of functions of a complex variable. His famous stability criterion had an immediate and lasting success because it related to quantities which could be directly measured and because it was expressed in terms of variables which could be immediately understood and interpreted in terms of appropriate actions to be taken to improve a feedback system's performance. The frequency-response concepts, and the i.mmensely popular and useful design techniques based upon them, satisfied a criterion of great importance in engineering work-they enabled engineers to quickly and fluently communicate to each other the essential features of a feedback control situation. Complex-variable methods are of such power and potential that their continued use and development is surely not in doubt. Even at the height of the "state-space revolution" the classical Nyquist-Bode-Evans techniques were the workhorses of many designers for their single-loop work. The real significance of the introduction of state-space methods is that it marked the beginning of a new, more general, more rigorous, deeper, and more far-reaching approach to automatic control. We are now beginning to see that automatic control is a vast subject, still in the early stages of development, and requiring a great breadth of approach in setting up adequate theoretical foundations. Its scope is such that no single approach, via the "time domain" or the "frequency domain" alone, is going to be sufficient for the development of adequate analysis and design techniques. What it is hoped will emerge clearly from the contents of this book is that Nyquist's ideas, and the frequency-response approach developed from them, are alive at the frontiers of current research, and that they will continue to play an indispensable role in whatever grand theoretical edifice emerges in time. Nyquist made truly outstanding contributions to engineering. He carried on a great tradition in the applied sciences going back to Fourier whose epochal work first appeared in 1811 [45], and in doing so transformed the arts of telegraph transmission and feedback systems development into exact sciences. May his spirit live on in the work -+ ? collected here, and in the future developments of feed- , back and control. 2 d~~~3+~f~&>fp4G - ip IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 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In writing his classic paper [44], Nyquist therefore sought to deal with a wide a class of dynamical systems a possible, and s s thus did not confine himself to those which could be specified in terms of a simple rational transfer function. The arrangement he considered is shown in Fig. 1. The network i s specified in terms of i t s impulse response G ( t )which is such that: 1 ) G ( t ) has bounded variation for 2 ) G ( t ) = 0, for -a,< t 0, Fig. 1. 3) 1 ; < -=< t < r , n I G ( t )I dt < a,. These conditions specifically exclude lopen-loop unstable networks. An associated frequency-response is defined by J ( j w )= along any stipulated path joining their end points. They are the familiar Bromwich-Wagner forms of integral used in the standard treatment of Leplace transform integrals, and the large semicircular arc S+ is simply chosen for convenience in handling the investigation of contour integral properties. Nyquist thus proceeds to consider the sum 1" ## ~ ( e-j'" dt, t ) and the limit w and it is explicitly stated that J ( j w ) may be obtained directly by means of computations or measurements, in which latter case J(-jw) is to be taken a the complex conjugate of J ( j w ) . s An auxiliary function s(t) = k=0 f k ( t ) = lim where where +jM [ . Idz = Mlimm l j M + I' dz He then poses the question: what properties of w(z) and hence of A J ( j w ) will determine !whether s(t) converges tlo zero or diverges a t +m? In answering it Nyquist first shows that s is introduced; this Cauchy representation extends the frequency-response characteristic to a function analytic in the open right-half complex plane. The quantity A i s a scalar gain factor. A disturbance fo( t ) is applied anywhere in the feedback path and thought of in terms of a response repeatedly traversing the feedback loop. After n trips round .the loop, this response signal is represented by Nyquist in the form where and S+ i s a large semicircular arc in the right-half plane joining the points -jM and +jM. Since F ( z ) is analytic in the open right-half complex plane, these contour integrals can be taken which simply says that the result of the applied disturbance i s to produce a response whose transform i s the disturbance transform divided by the loop return difference evaluated at the point where the disturbance enters. His next step is crucial: the question of closed-loop stability (in terms of s(t) converging to zero) is tackled by finding out whether or not the transform of s(t) has i3ny singularities in the closed righthalf plane. The resulting condition for closed-loop stability i s finally given in his famous Nyquist diagram rule:: "plot plus and minus the imaginary part of A J ( j o ) against the real part for all frequencies from 0 t o rn. I f the point 1 + j0 lies completely outside this curve the system is stable; if not it is unstable." Nyquist's line of argument, though deeply interesting to follow, is unfortunately not rigorous for the class of systems which he considered. This lack of rigor, however, should in no way be regarded as detracting from his great achievement; more than thirty years were to pass before a rigorous proof was supplied by Desoer [12]. The necessity that the returndifference function [ I - w(z)] should not vanish in the closedright-half plane obviously holds, but the proof that this i s also sufficient for closed-loop stability i s what i s hard to show for the general class of systems considered by Nyquist. The snags involved arise from the facts that: 1) when dealing with fairly general classes of system it i s imperative to give a careful statement of what is meant by stability; 2) there are known examples of functions which are analytic in the right-half plane yet do not satisfy the conditions which are regarded as defining a "stable" response. GAIN PLANT 1 + g(s) b Fig. 2 . for which the transform, expressed in terms of the error function erf, and given by is entire (that is, analytic in the whole finite complex plane). The informal definition of stability used by Nyquist is essen- Hence, in the case when a plant transfer function is not tially that which is now called "input-output stability" rational, the proof of the Nyquist stability criterion contains (Willems 1551, Desoer and Vidyasagar [I411. In a careful for- real technical difficulties and cannot be based on simply showmulation of this approach the system is regarded in terms of a ing that the closed-loop transfer function has no singularities mapping between normed spaces, and the basic requirement in the closed right-half complex plane. for stability i s taken to be the boundedness of this map. One In the simple case where the plant can be described by a ramethod of attack i s to take the input and output spaces to be tional transfer function, a sound proof can be based on the function spaces of a stipulated type (L1-spaces, L2-spaces, . . . , Principle of the Argument, since in this case it i s sufficient for LP-spaces, for example) and then to attempt to show that an closed-loop stability that the closed-loop transfer function be input of admissible type (let us say in the space L1) always analytic in the closed right-half plane. The essence of such an produces an output of admissible type (also in the space L1). approach was given by Nyquist himself inan appendix to his To see how difficulties then arise in tying such an approach in classic paper. Following the work of MacColl [39]and Bode with a characterization in terms of right-half-planeanalyticity [5], this soon became the standard form and was extended by can best be done by considering examples of the sort given by Frey [I51 to handle the open-loop unstable case. Thus, for Cook [lo]. Let h(t) be the impulse response of a single-input the system shown in Fig. 2, where single-output system whose response to an input r ( t ) is given by the convolution integral n(s) and d(s) are polynomials in the complex frequency variable s with no common factor and k is a scalar gain parameter, and letg(s) be the Laplace transform of h(t). Then a necessary one has the following classic result. Nyquist Stability Criterion ( ) I The feedback system shown : and sufficient condition for bounded-input bounded-output in Fig. 2 i s closed-loop stable if and only if the Nyquist diagram stability is that of g(s) encircles the critical point ( - I l k +jO) po times in an anticlockwise direction, where po is the number of right-half plane poles of the open-loop transfer function g(s). Note that the number of encirclements here (and subsequently) is taken which implies that g(s) i s holomorphic and bounded in the to be that made by the "complete" Nyquist diagram, that is, open right-half plane. However, and this i s where the technical by the appropriate map of the entire imaginary axis of the difficulties referred to really come from, the convergence of frequency plane (s-plane) plus a suitable closure when imagithe integral i s not implied by g(s) being analytic for Re(s) 0. nary axis singularities of g(s) are present. The first rigorous proof of the Nyquist Stability Criterion Counter examples are for a more general class of systems was given by Desoer 1121. He assumed that the plant was linear time-invariant and nonh(t) = sin t anticipative (causal) and that it satisfied the following conditions. for which 1) The output y, the zero-input response z, and the input u are related by > y(t) = r(t) + ## where g i s the response to a unit impulse. 2) For all initial states, the zero-input response is bounded as t + = where z, is a finite number on [0, m) and z(t) +z, which depends on the initial state. 3) The unit impulse response g is given by where the constant r i s nonnegative; I i s the unit step func(t) tion; g1 is bounded on [0,00), i s an element of L1(O,=), and g, +Oast+. Under these conditions he established the following version of the Nyquist Stability Criterion. 1) Nyquist Stability Criterion ( 1: I f the Nyquist diagram of @(s), where @(s) is the Laplace transform of g(t) does not encircle or go through the critical point then is a sufficient condition for closed-loop stability. This condition can be checked (for this class of plants) by drawing a in the usual manner, since it i s Nyquist diagram for @(s;) equivalent to the requirement that the map of the closed right-half s-plane into the \$-plane be bounded away from the critical point (-Ilk, 0). Desoer and Wu showed that the closed-loop stability resullting from a satisfaction of the Nyquist criterion has the desirable features that one would hope for: a bounded input produces a bounded output, an input ) with finite energy (ie one in the function space L ~ produces an output with finite energy, and a continuous input produces a continuous output. The class of admissible plants was further extended by Callier and Desoer [8] to include ones with a finite number of right-half plane poles. They showed that R e s>o a) The impulse response of the closed-loop system is bounded, tends t o zero a t + m, and i s an element of L' (0, a). s b) For any initial state, the zero-input response of the closed-loop system is bounded and goes to zero a t + =. s C) For any initial state and for any bounded input, the response of the closed-loop system is bounded. d) Let r be positive; then for any input which tends to a , constant u as t+m, and for any initial state, the output y tends t o u, a t + m. Let r be zero and u + 0 as t + W;then s for any initial state, the output y -+ 0. Furthermore, if the Nyquist diagram of @(s)encircles the critical point (- llk, 0)a finite number of times, then the impulse response of the closed-loop system grows exponentially as t + m. The closed-loop system1 transfer function i s given by H(s) = where kg^(s) 1 + k@(s)' Hes inf I 1 + kg^(s)I > 01 is necessary and sufficient for the closed-loop transfer function to be the Laplace transform of the derivative of i3 function of bounded variation on lo,,=), which in turn implies that the Callier closed-loop system is LP-stable for all p, 1 <p and Desoer also gave an appropriate graphical test for condition (2.1) to be satisfied. Unfortunately, the price paid for this degree of generality is that the graphical t~estdoes not simply consist of drawing a Nyquist diagram. A simplified form of this graphical test was later given by Callier and Desoer 191. A comprehensive discussion of the input-output treatment of feedback system stability is given in the book by Desoer and Vidyasagar [14]. <=. > kgM Mikhailov's criterion applies to systems described by a known nth-order constant coefficient linear differential equation and thus having an explicitly known characteristic polynomial p ( ~ (Mikhailov [41]). It is stated in terms of the locus ) of p ( j o ) in a complexp plane and so bears a superficial resemblance to the Nyquist criterion. It is, however, an essentially different result in that it requires that the governing differential equation of the system being investigated must be known, whereas the essential virtue of the Nyquist criterion is that the Nyquist locus is something which can be directly rneasured for a plant whose behavior in terms of a differential equation description may well not be available. A criterion of this form was also formulated in Germany by Leonhartj [351 and I. Cremer [ I I The Mikhailov criterion i s obtained by a simple application of the Principle of the Argument. Let p(s) be the characteristic equation of a linear dynarr~ical system, l e t C be a closed Jordan contour in the s-plane, and l e t be the image of C, under the mapping ~ ( s ) , the p-plane. Let p(s) have P zeros inside C. in Then the net phase change associated with one traverse of as s traverses C once in the anticlockwise direction i s 2nP. Hence on obtains the foll~wing. Mikhailov Stability Criterion: I f p ( j o ) is plotted for w increasing continuously from 0 to (the Mikhailov locus) then the dynamical system with nth-order characteristic polynomial p(s) will be stable if and only if the phase of p ( j w ) increases Since it follows from the Principle of the Argument that the + returndifference [I k#(s)l is #O for all Re s >, 0 if and only if the Nyquist diagram of @(s)does not encircle or go through the critical point (-Ilk. 0). it follows that Desoer's careful proof shows that the nonvanishing of the return-difference in the closed right-half plane is indeed both necessary and sufficient for closed-loop stability, interpreted in terms of conditions a)-dl inclusive above. The class of plants to which the Nyquist criterion could be legitimately applied was further extended by Desoer and Wu [I31 t o the case where the impulse response i s of the form , where r is a nonnegative constant, g is integrable on [O, w), and Zz lgil ,o For a plant of this sort, whose impulse response may have an infinite sequence of impulses (as may for example occur in systems incorporatiiig lossless transmission lines), they showed that <=. Res>o inf Il+k@(s)l>O, continuously from 0 to n d 2 . Further developments of this criterion have been made by Leonhard [361. ## II. BODE'SWORKAND THE FURTHER DEVELOPMENT OF NYQUIST-DIAGRAM-BASED TECHNIQUES The approach to feedback system stability based on Nyquist diagrams had a further striking advantage in addition to i t s direct relationship to experimentally measurable quantities: it gave an indication of what should be done t o an open-loop frequency response in order to improve closed-loop behavior. As pointed out by Ludwig 1381 and realized by Bode and others, the closeness of approach of the Nyquist locus to the critical point gave a measure of closed-loop damping. Hence the key to an improved closed-loop response was seen to lie in a suitable shaping of the open-loop frequency-response characteristic. Considerations of this sort led to Bode's classic studies on the relationship between gain and phase characteristics (Bode 141, [51) during which the critical point in the gain plane was put in i t s now-standard place at (- 1 + jO), and the gain and phase margin measures of relative closed-loop stability were introduced. By the end of the Second World War the Nyquist-Bode approach to the analysis and design of feedback control systems had been firmly established by the work of Harris [241, Brown and Hall [71, Leonhard [351, Cremer [ I 11, Strecker 1491, Oldenbourg and Sartorius [451, and others (see Part I). A good example of i t s state of development at that time i s given by Hall [23]; this first appeared in a wartime report. The Nyquist-Bode approach to feedback system design was further developed in an important book by Horowitz [26] on the synthesis of feedback systems. In this book Horowitz considered the effects of large plant parameter variations on the design problem, and further studied this problem in a series of papers (Horowitz and Sidi [27], Horowitz [281, Horowitz and Sidi [29]). The effect of time delays on the closed-loop stability of feedback systems was considered by Tsypkin [501. An extension of frequency-response methods to sampleddata systems was developed by Hurewicz in the early 1940's and is described in the book by James, Nichols and Phillips [30]. Hurewicz developed the appropriate extension of the Nyquist stability criterion to sampled-data systems. The increasing importance of digital devices in the 1950's led to a renewed interest in the use of frequency-response techniques in this area, and Linvill [37] and Tsypkin [51] gave suitable extensions of the Laplace transform approach for handling interconnections of digital and analog devices. An alternative approach of developing a "z-transform" for discrete-time signals to complement the "s-transform" for continuous-time signals was developed by Ragazzini and Zadeh [47], Jury [31], [32] and others (Barker [ I ] , [2]; Bergen and Ragazzini [3]; Helm [25]; Kuo [34] ; Tsypkin [52]; and Freeman [22] ). The striking success of the state-space approach to multivariable systems in the late 1950's and early 1960's sharply reduced the interest in frequency-response methods of the great majority of theoreticians working on automatic control. Nevertheless the basic frequency-response approach remained the workhorse of very many designers of industrial control systems. It therefore became a matter of great interest to explore links between the new state-space-based approaches and the classical frequency-response methods, and a notable step in this direction was taken by Kalman [331 which established certain frequency-response properties of an optimal single-loop feedback control system. ## III. THE EVANS'ROOT-LOCUS METHOD The closed-loop characteristic frequencies and the gain parameter of the feedback system of Fig. 2 are related by 1 + kg(s) = 0 ## then this characteristic equation can be rewritten in the form and the closed-loop frequencies for a particular value of gain parameter k' must be such that the expression has a phase of 180" and a modulus of llk'a. Evans [ I 91-[21] showed how the set of possible locations of closed-loop frequencies in the s-plane could be determined by simple graphical constructions given the locations in the s-plane of the poles and zeros of g(s). After the graphical construction of this set of "root loci," he further showed how they can be calibrated in the gain parameter k, again by the use of simple graphical constructions based on a knowledge of the pole and zero locations. Evans' brilliantly simple idea enabled design engineers to rapidly sketch the variation of closed-loop characteristic frequencies with gain for quite complicated transfer functions, and the method rapidly developed into one of the key tools of the control engineer. IV. ALGEBRAIC FUNCTIONS, ENCIRCLEMENTS, AND TOPOLOGICAL CONSIDE RATIONS It is instructive a t this point to look briefly a t the root locus method from the point of view of algebraic function theory which was later used (see Part IV) to extend Nyquist and root locus techniques to the multivariable case. I f ## on making the substitution Now consider (2.2). It gives both open-loop gain a a funcs , tion of frequency s and, for the feedback system of Fig. 2 closed-loop frequency a a function of the gain parameter y. s I f we plug a specific value of s = Z in (2.2) and solve for y we obtain the Wiener-Hopf approach since Wiener based his solution on the so-called Wiener-Hopf integral equation. Since Parseval's theorem can be used to turn quadratic performance-index integrals with respect to time into equivalent integrals with respect to a frequency variable, i t was realized that WienerHopf techniques of solving optimal filtering problems could be used to attack optimal control problems. In particular Newton used this approach to study an analytical formulation of the problem of designing feedback control systems under various forms of constraint (hlewton, Gould, and Kaiser [43] 1. This gave another line of attack for the frequency-response design of feedback systems which was further extended by Youla, Bongiorno, and Jabr [56]. and if we plug in a specific value of y = 7 and solve for s we obtain the n values of closed-loop characteristic frequency [ I ] R. H. Barker, 'The theory of pulse monitored servomechanisms and their use for prediction," Rep. 1046. Signals Research and {si: i = 1, 2, . . . ,n) which correspond to this value of the gain Development Establishment, Christchurch, England, 1950. parameter. Equation (2.2) thus defines two algebraic func[21 -, "The pulse transfer function and its application to sampling tions: a gain function y(s) giving open-loop gain a a function s servomechanisms," F'roc. IEE, 99(4), 302-31 7,1952. [31 A. R. Bergen and J. R. Ragazzini, "Sampleddata processing of frequency, and a frequency function s(y) giving closed-loop techniques for feedback control systems," Trans. AIEE, 73(2), frequency a a function of gain. In this case (a single-input s 236-247.1954. single-output plant), the gain function is a single-valued func141 H. W Bode, "Relations between attenuation and phase in feed. back amplifier design," Bell Syst. Tech. J,, 19,421454, 1940. tion of s, and the frequency function is an n-valued function , of y. The frequency function is thus defined on an n-sheeted 151 - Network Analysis and Feedback Amplifier Design. Princeton, NJ: Van Nostrand, 1945. Riemann surface; it is of considerable interest to note that 161 H. W. Bode and C. E. Shannon, "A simplified derivation of linear least square smoothing and prediction theory," Proc. IRE, 38, Nyquist's classic paper involved a discussion of this surface. 41 7-425.1950. There are two important aspects of the fact that an open-loop [71 G. S. Brown and A,. C. Hall, "Dynamic-behaviour and design of gain function and a closed-loop frequency function are related servomechanisms," Trans. ASME, 68, 503-524, 1946. [81 F. M. Callier and C. A. Desoer, "A graphical test for checking the via a single defining equation. Firstly, this provides a natural stability of a linear time-invariant feedback system, lEEE Trans. point of view for a generalization of these ideas to multivariAutomat. Control, A.C-17, 773-780, 1972. able Systems. Secondly. it shows that. from the complex[91 -, "On simplifying a graphical stability criterion for linear distributed feedback systems," lEEE Trans. Automat Control, variable analytic-function point of view, the two classical apAC-21,128-129, 19'76. proaches to studying the behavior of a feedback system, I101 P. Cook, "Stability of linear constant multivariable systems." namely in terms of open-loop gain a a function of frequency s Proc. IEE, 120.1557'. 1973. (Nyquist-Bode), or in terms of closed-loop frequency a a [ I 1] L. Cremer, "Ein netles Verfahren zur Beurteilung der Stabilitat s linearer Regelungs-systeme," Z fur angew. Mathematik und . function of gain (Evans' root locus), are simply different ways Mechanik, 25-27,5Aj, 1947. of looking at the same! physical situation. [I21 C. A. Desoer, "A general formulation of the Nyquist criterion," IEEE Trans. Circuit Theory, CT-12.230-234.1965. The implications of the fact that the Nyquist diagram "lives on" a Riemann surface have been investigated by [I31 C. A. Desoer and M. Y. Wu, "Stability of linear time-invariant systems," IEEE Trans. Circuit Theory, CT-15,245-250,1968. MacFarlane and Posdlethwaite [40], and by De Carlo and El41 C. A. Desoer and M. Vidyasagar, Feedback Systems: InputSaeks [I81 who have thus been led to consider the Nyquist Output Properties. lUew York: Academic Press, 1975. stability criterion from the viewpoint of algebraic topology. [I51 W. Frey, "A generalization of the Nyquist and Leonhard stability Review, 33, 59-65, 1946. criteria," Brown BovI?N' These studies shed an interesting light on the role played in [I61 W. B. Davenport ancl W. L. Root, An Introduction to the Theory Nyquist stability theory by encirclements of a critical point of Random Signals and Noise. New York: McGraw-Hill, 1958. (Saeks [48]). The elationsh ship between encirclements and [I71 J. H. Davis, "Fredholm operators, encirclements, and stability . criteria," SIAM J Control, 10,608-622. 1972. stability for a fairly general class of operators has also been [I81 R. De Carlo and R. Saeks, "The encirclement criterion: An apconsidered by Davis ['I 71. proach using algebraic topology," Int. J. Control, 26, 279-287, 1977 .. I191 W. R. Evans, "Graphical analysis of control systems," Trans. V. THE FREQUENCY-RESPONSE APPROACH TO opTIMlzATloN THE W~ENEfl-'HOpF AND DEslGN E c H N l ~ u E T Following Wiener's classical frequency-response studies of stochastic processes and of the problem of filtering stochastic signals from stochastic disturbances (Wiener 1531, 1541; Paley and Wiener [461) the frequency-response analysis of control systems behavior was extended to handle stochastic inputs (James, Nichols, and Phillips [30]; Davenport and Root [I61 ). Bode and Shannon [ l gave a sim~pletreatment of Wiener's 6 spectral factorization solution of an optimal filtering problem. This general approach to stochastic problems is often called AIEE, 67,547-551, 1948. "Control systern synthesis by root locus method," Trans. [20] -, AIEE, 69,l-4,1950. [21] - Control System Dynamics. New York: McGraw-Hill, 1953. ~ 2 H. 1 ~ Discrete-Timesystems. New York: Wiley, 1965. I231 A. C Hall, "Application of circuit theory t o the design of servo. mechanisms," J Franklin Inst., 242,279307,1946. . [24] H. Harris, Jr.. "The .frequency response of automatic control systerns," Trans. AIEE, 65, 539-546, 1946. I251 H. A. Helm, "The Z-transformation." Bell Svst. Tech. J.. 38(1). 177-196.1959. [26] 1. Horowitz, Synthesis of Feedback Systems. New York: Academic Press, 1963. [271 1. Horowitz, and M. Sidi, "Synthesis of feedback systems with ireeman ## large plant ignorance for prescribed time-domain tolerances," lnt. J Control, 16,287-309, 1972. . [281 1. Horowitz, "A synthesis theory for linear time-varying feedback systems with plant uncertainty," IEEE Trans. Automat. Control, AC-20,454-464.1975. [291 1. Horowitz and M. Sidi, "Optimum synthesis of nonminimum phase feedback systems with plant uncertainty," lnt. J Control, . 27,361 -386, 1978. [301 H. M. James. N. B. Nichols, and R. S. Phillips, Theory of Servomechanisms. New York: McGraw-Hill, 1947. [311 E. I. Jury, "Analysis and synthesis of sampleddata control systems," Trans. AIEE, 73(1), 332346. 1954. [321 Sampled-Data Control Systems. New York: Wiley, 1958. . [331 R. E. Kalman, "When is a linear control system optimal?," J Basic Eng., Ser. D., 86.5 1-60, 1964. [341 B. Kuo, Analysis and Synthesis o f Sampled-Data Control Systems. Englewood Cliffs, NJ: Prentice-Hall, 1963. [351 A. Leonhard, "Neues Verfahren zur Stabilitatsuntersuchung," Arch. der Elektrotechnik, 38.17-28.1944. [361 -, Die Selbsttatige Regelung. Berlin: Springer, 1957. I371 W K. Linvill, "Sampled-data control systems studied through . comparison of sampling with amplitude modulation," Trans. AIEE, 70, Part l I, 1779-1 788,1951. [381 E. H. Ludwig, "Die Stabilisierung von Regelanordunugen m i t Rohrenverstarkern durch Dampfung oder elastische Ruckfuhrung," Arch. Elektrotechnik, 34, 2 6 9 f 1940. [391 L. A. MacColl, Fundamental Theory o f Servomechanisms. New York: Van Nostrand, 1945. [401 A. G. J. MacFarlane and I. Postlethwaite, "The generalized Ny. quist stability criterion and multivariable root loci," Int. 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Control, AC-21.3-13. 1976. Regeneration Theory By H. NYQUIST Regeneration or feed-back is of considerable importance in mamy applications of vacuum tubes. The most obvious example is that of vacuum tube oscilla.tors, n.here the feed-back is carried beyond the singing point. Another application is the 21-circuit test of balance, in which the current clue to the unbalance between two impedances is fed back, the gain being increased until singing occurs. Still other applications are cases where portions of the output current of amplifiers are fed back to the input either unintentionally or by design. For the purpose of investigating the stability of such devices they may be looked on as amplifiers whose output is connected t o the input through a transducer. This paper deals with the theory of stability of such systems. ## HEN theoutput of an amplifier isconnected to the input through a transducer the resulting combination may be either stable or unstable.. The circuit will be said to be stable when an impressed small disturbance, which itself dies out, results in a response which dies out. I t will be said to be unstable when such a disturbance results in a response which goes on indefinitely, either staying a t a relatively small value or increasing until it is limited by the non-linea.rity of the amplifier. \17hen thus limited, the disturbance does not grow further. The net gain of the round trip circuit is then zero. Otherwise stated. the more the response increases the more does the non-linearity decrease the gain until a t the point of operation the gain of the amplifier is just equal to the loss in the feed-back admittance. An oscillator under these conditions would ordinarily be called stable but it \vill simplify . . the present paper to use the definitions above and call it unstable. Now, this fact as to equality of gain and loss appears to be an accident connected with the non-linearity of the circuit and far from throwing light on the conditions for stability actually diverts attention from the essential facts. In the present discussion this difficulty will be avoided by the use of a strictly linear amplifier, which implies an amplifier of unlimited power carrying capacity. The attention will then be centered on whether an initial impulse dies out or results in a runaway condition. If a runaway condition takes place in such an amplifier, it folloivs that a non-linear amplifier having the same gain for small current and decreasing gain with increasing current will be unstable as well. ## permission from BellS~lst.Tech. J., vol. I I. pp. 126-147,Jan. 1932. Copyright O 1932 by Am. Telephone and Telegraph Co. 23 ## REGENERA TION THEORY STEADY-STATE THEORIES EXPERIENCE AND First, a discussion will be made of certain steady-state theories; and reasons why they are unsatisfactory will be pointed out. The most obvious method may be referred to as the series treatment. Let the complex quantity A J(iw) represent the ratio by which the amplifier and feed-back circuit modify the current in one round trip, that is, let the magnitude of A J represent the ratio numerically and let the angle of AJ represent the phase shift. I t will be convenient to refer to AJ as an admittance, although it does not have the dimensions of the quantity usually so called. Let the current ## ID= cos w t = real part of efw' be impressed on the circuit. and the nth by (a) ## I I = real part of AJek"' (b) I, = real part of AnJnefwt. (c) The total current of the original impressed current and the first n round trips is In = real part of (1 + A J + A2J2+ - . AnJn)eiwt. (d) If the expression in parentheses converges as n increases indefinitely, the conclusion is that the total current equals the limit of ( d ) as n increases indefinitely. Now If AJ I < 1 this converges to 1/(1 - A J ) which leads to an answer which accords with experiment. When 1 AJl > 1 an examination of the numerator in (e) shows that the expression does not converge but can be made as great as desired by taking n sufficiently large. The most obvious conclusion is that when 1 AJl > 1 for some frequency there is a runaway condition. This disagrees with experiment, for instance, in the case where AJ is a negative quantity numerically greater than one. The next suggestion is to assume that somehow the expression 1/(1 - AJ) may be used instead of the limit of (e). This, however, in addition to being arbitrary, disagrees with experimental results in the case where AJ is positive and greater than 1, where the expression 1/(1 - AJ) leads t o a finite current but where experiment indicates ail unstable condition. ## BELL SYSTEM TECIINICAL JOURNAL The fundamental difficulty with this method can be made apparent by considering the nature of the current expressed by ( a ) above. Does the expression cos wt indicate a current which has h e n going on for all time or was the current zero up to a certain timle and cos wt thereafter? In the former case we introduce infinities into our expressions and make the equations invalid; in the latter case there will be transients or building-up processes whose importance may increase as n increases but which are tacitly neglected in equati0.n~ b ) - (e). ( Briefly then, the difficulty with this method is that it neglects the building-up processes. Another method is as follows: Let the voltage (or current) a t any point be made up of two components where b'is the total voltage, V is the part due directly to the impressed 1 voltage, that is to say, without the feed-back, and V2is the component due to feed-back alone. We have V2 = AJV. (9) ## Eliminating V2 between (f) and (g) V = Vl/(l - AJ). (h) This result agrees with experiment when ( A J I < 1 but does not generally agree when A J is positive and greater than unity. The difficulty with this method is that it does not investigate whether or not a steady state exists. I t simply assumes tacitly that a steady state exists and if so it gives the correct value. When a steady state does not exist this method yields no information, nor doe!; it give any information as to whether or not a steady state exists, which is the important point. The esperimental facts do not appear to have been formulated precisel!r but appear to be well known to those working with these circuits. They may be stated loosely as follows: There is an unstable condition whenever there is a t least one frequency for which A J is positive and greater than unity. On the other hand, ,when A J is negative it may be very much greater than unity and the condition is nevertheless stable. There are instances of 1 AJI being about 100 without the conditions being unstable. This, as will appear, accords closely with the rule deduced below. ## REGENERA TION THEORY NOTATION D RESTRICTIOSS AN The following notation will be used in connection with integrals: the path of integration being along the imaginary axis (see equation 9), i.e., the straight line joining - i and M iM; the path of integration being along a semicircle having the origin for center and passing through the points - iM, i A l ; JP, the path of integration being first along the semicircle referred to and then along a straight line from iAB to - i.lP. Referring to Fig. 1 it iM -i M Fig. 1-Paths Z -PLANE ## will be seen that The total feed-back circuit is made up of an amplifier in tandem with a network. The amplifier is characterized by the amplifying ratio A which is independent of frequencv. The network is characterized by the ratio J ( i w ) which is a function of frequency but does not depend on the gain. The total effect of the amplifier and the network is to multiply the wave by the ratio A J ( i w ) . An alternative way of characterizing the amplifier and network is to say that the amplifier is For physical interpretation of paths of integration for which x > 0 reference is made to a paper by J . K. ('arson, Xotes on the Heaviside Operational ('alculus," B. S. I . J., Jan. 1930. For purposes of the present discussion the semicircle is ' preferable to the path there discussed. BELL S Y S T E M T E C H N I C A L JOURNAL characterized by the amplifying factor A which is independent of time, and the network by the real function G(t) which is the response . caused by a unit impulse applied a t time 1 = 0 The combined effect of the amplifier and network is to convert a unit impulse to the function AG(t). Both these characterizations will be used. The restrictions which are imposed on the functions in order that the subsequent reasoning may be valid will now be stated. There is no restriction on A other than that it should be real and insdependent of time arid frequency. In stating the restrictions on the network it is convenient to begin with the expression G. They are ## G(t) has bounded variation, G(t) = 0, oo < t < a. <t<0. [/ ~ ( tIdt exists. ) - m ## (AI) (AII) (AIII) It may be shown that under these conditions G(t) may be expressed by the equation where These expressions may be taken to define J. The function may, however, be obtained directly from computations or measurements; in the latter case the function is not defined for negative vallues of w . I t must be defined as follows to be consistent with the definition in (6): J ( - iw) = complex conjugate of J(iw). (7) While the final results will be expressed in terms of AJ(iw) it will be convenient for the purpose of the intervening mathematics to define an auxiliary and closely related function ~ ( z )2irz,, fzw~-az d ( i ~ ) , = l where and where x and y are real. Further, we shall define 2--0 (8) ## RECENERA TZON THEORY The function will not be defined for x < 0 nor for It;(= a. As defined it is analytic8 for 0 < x < oo and a t least continuous for x = 0. The following restrictions on the network may be deduced: u+m ## J(ky) is continuous. w(iy) = AJ(iy). Equation (5) may now be written From a physical standpoint these restrictions are not of consequence. Any network made up of positive resistances, conductances, inductances, and capacitances meets them. Restriction (AII) says that the response must not precede the cause and is obviously fulfilled physically. Restriction (AIII) is fulfilled if the response dies out a t least exponentially, which is also assured. Restriction (AI) says that the transmission must fall off with frequency. Physically there are always enough distributed constants present to insure this. This effect will be illustrated in example 8 below. Every physical network falls off in transmission sooner or later and it is ample for our purposes if it begins to fall off, say, a t optical frequencies. We may say then that the reasoning applies to all linear networks which occur in nature. I t also applies to other linear networks which are not physically producible but which may be specified mathematically. See example 7 below. A temporary wave fo(t) is to be introduced into the system and an investigation will be made of whether the resultant disturbance in the system dies out. It has associated with it a function F(z) defined by F(z) and fo(t) are to be made subject to the same restrictions as w(z) and G(t) respectively. DERIVATIONA SERIES THE TOTAL OF FOR CURRENT Let the amplifier be linear and of infinite power-carrying capacity. Let the output be connected to the input in such a way that the a W. F. Osgood, "Lehrbuch der Funktionentheorie," 5th ed., Kap. 7, \$ 1 , Hauptsatz. For definition of "analytic" see Kap. 6, \$ 5. ## BELL SYSTEM TECHNICAL JOURNAL amplification ratio for one round trip is equal to the complex quantity A J , where A is a function of the gain only and J is a function of w only, being defined for all values of frequency from 0 to oc. Let the distur1)ing wave fo(t) be applied anywliere in the circuit. \Ye have (13) The wave traverses the circuit and on completing the first trip it becomes (/) = &1 : T.8(iw) ( i ~ ) ~ ~ ~ ( 1 ~ (14) ~ ## and after traversing the circuit ?t times Adding the voltage of the original impulse and the first :PI round trips we have a total of The total voltage at the point in question at the time t is given by the limiting value which (17j approaches as n is increased indefinitely l l r . Carson has called my attention to the fact that this series can also I)e derived fro111'l'heoreni IS. 1). 40, of his Flectric C'ircuit 'l'heory. \\'hereas the present derivation is analogt)us to the theory expressed in equations ((1)-(c) above, the ;~ltcrnattve tleri\.ntion woultl be analogous to that in equations (j')-(11). ' ## REGENERA TION THEORY We shaII nest prove that the limit s(t) exists for all finite values of t. I t may be stated as of incidental interest that the limit does not necessarily exist although the limit s(t) does. Choose ATo and : 7such that I Ifo(X)[~AIo. O 5 1 5 t . (21) I G ( ~ - x ) I . ' N o. \\'e may write (22) ~h(t)J'nroLvidt= n ~ r n ~ j z 15 ! Similarly (26) (27) !fn(t) I 5 MoNntnln! It is shown in almost any text dealing with the convergence of series that the series in parentheses converges to eN1as n increases indefinitely. Consequently, s,(t) converges absolutely as n increases indefinitely. ## RELATION BETWEEN ( t AND ~ ) Next consider what happens to s(t) as t increases. As t increases indefinitely s(t) may converge to zero, indicating a condition of stability, or it may go beyond any value however large, indicating a runaway condition. The question which presents itself is: Referrittg to (18) a~rd(19), what propert,ies of w(z) and further what proper&ies of 11J(iw) determine whether s(t) converges to zero or diverges as t increases 6 G. A. Camphell, "Fourier Integral," A. S. T J., Oct. 1928, Pair 202. 15.g,, \\.hittaker and \\.atson, "Slodern ;\nalysis," 2d ed., p. 531. ## indefinitely? From (18) and (19) We may write providedl these functions exist. Let them be called qo(t)and lim ,(t) 1 0 respectively. Then where By the methods used under the discussion of convergence above it can then be shown that this expression exists and approaches zero as n increases indefinitely provided qo(t) exists and is equal to zelro for t < 0. Equation (29) may therefore be written, subject to these conditions In the first place the integral is zero for negative values of t because the integrand approaches zero faster than the path of integration increases,. Moreover, [[F/(I I - w)Jezldz (34) exists for all values of t and approaches zero for large values of t if 1 - w does not equal zero on the imaginary axis. Moreover, the integral ~ [ F / ( I w)]ezlde C (35) exists because 1. Since Fand ware both analytic within the curve the integrand does not have any essential singularity there, 2. The poles, if any, lie within a finite distance of the origin because w 4 O as I s I increases, and 3. Ileee two statements insure that the total number of poles is finite. REGENERATION THEORY We shall next evaluate the integral for a very large value of t. It will suffice to take the C integral since the I integral approaches zero. Assume originally that 1 - w does not have a root on the imaginary axis and that F(z) has the special value w'(z). The integral may be written ## Changing variables it becomes where z is a function of w and D is the curve in the w plane which corresponds to the curve C in the z plane. Afore specifically the imaginary asis becomes the locus s = 0 and the semicircle becomes a small curve which spirals around the origin. See Fig. 2. The function ## Fig. 2-l<epresentative paths of integration in the w-plane corresponding to paths in Fig. 1. and, therefore, the integrand is, in general, multivalued and the curve of integration must be considered as carried out over the appropriate Riemann surface.' Now let the path of inte~r.:tion shrink, taking care that it does not shrink across the pole a t ; = 5 and initially that it does not shrink across such branch points as interfere with its passage, if any. This shrinking does not alter the integral because the integrand is analytic at all other points. At branch points which interfere with the passage of the path the branches stopped may be severed, transposed and connected in such a way that the shrinking may be continued past the branch point. This can be done without altering the value of the integral. Thus the curve can be shrunk until it becomes one or more very small circles surrounding the pole. The value of the total integral 2 ## ' Osgood. loc. cit., Kap. 8. Osgood, loc. cit., Kap. 7, 5 3, Satz 1 . ## (for very large values of t) is by the method of residues " where zj ( j = 1, 2 n ) is a root of 1 - zo = 0 and Y , is its order. The real part of zi is positive because the curve in Fiy. 1 encloses points with x > 0 only. The system is therefore sta1)le or unstal)le according to whether ... is equal to zero or not. Rut the latter expression is seen from the procedure just gone through to equal the number of times that the locus x == 0 encircles the point tv = 1. If F does not equal w' the calculation is somewhat longer but not essentially different. The integral then equals if all the roots of 1 - zcr = 0 are distinct. If the roots are not distinct the expression becomes A jkLk-lezil, (40) j=l k = l 52 where A,,i, at least, is finite and different from zero for general values of F. I t appears then that unless F is specially chosen ithe result is essentially the same as for F = w'. The circuit is stable if the point lies wholly outside the locus x = 0. I t is unstable if the point is within the curve. It can also be shown that if the point is on the curve conditions are unstable. We may now enunciate the following Rz~le:Plot plus and minzrs the imaginary part of AJ(io) against the real part for all frepzrencies from O to m . If the point 1 i O lies completely outside this czrrve the system i s stable; if not it i s unst~zble. In case of doubt as to whether a point is inside or outside the curve the following criterion may be used: Draw a line from the point (u = 1, v = 0) to the point z = - i m . Keep one end of the line fixed at (u = 1, v = 0) and let the other end describe the curve from z = - i m to z = i m , these two points being the same in the w plane. If the net angle through which the line turns is zero the point (u = 1, v = 0) is on the outside, otherwise it is on the inside. If A J be written ( AJl (cos 8 i sin 8) and if the angle always Wsgoud, loc.cit., IGp. 7. 8 11,Satz 1. REGENERATION THEORY changes in the same direction with increasing w, where w is reill, the rule can he stated as follows: The system is stable or unstable according to whether or not a real frequency exists for which the feed-back ratio is real and equal to or greater than unity. In case dO/dw changes sign we may have the case illustrated in Figs. 3 and 4. In these cases there are frequencies for which w is real and Fig. 3-illustrating case where amplifying ratio is real and greater than unity for two frequencies, but where nevertheless the path of integration does not include the point 1, 0. greater than 1. On the other hand, the point (1, 0) is outside of the locus x = 0 and, therefore, according to the rule there is a stable condition. Fig. 4-Illustrating case where amplifying ratio is real and greater than unity for two frequencies, but where nevertheless the path of integration does not include the point 1, 0. If networks of this type were used we should have the follo\ving interesting sequence of events: For low values of A the system is in a stable condition. Then as the gain is increased gradually, the system becomes unstable. Then as the g;iin is increased gradually still further, the system again 1)ecornes stable. As the gain is still further increased the system may again become unstal~le. ## BELL SYSTEM TECHNICAL JOURNAL The following examples are intended to give a more detailed picture o certain rather simple special cases. They serve t o illustrate the f previous discussion. In all the cases F is taken equal to A J so that fo is equal to AG. This simplifies the discussion but does not detract from the illustrative value. 1. Let the network be pure resistance except for the distortionless amplifier and a single bridged condenser, and let the amplifier be such that there is no reversal. We have ## where A and a are real positive constants. In (18) l o The successive termsfo, f l , etc., represent the impressed vvave and the successive round trips. The whole series is the total current. I t is suggested that the reader should sketch the first few terms graphically for B = a, and sketch the admittance diagrams for B < a, and 3 :> a. The expression in parentheses equals eDL and This expression will be seen to converge to 0 as t increases or fail to do so according to whether B < a or B 2 a. This will be found to check the rule: as applied to the admittance diagram. 2. Let the network be as in 1 except that the amplifier is so arranged that there is a reversal. Then The solution is the same as in 1 except that every other term in the series has its sign reversed: s(t) = '0 - Be'"-'j'f. - Be4'(l - Bt + B2t2/2!+ . ..) ## Campbell. loc. cit. Pair 105. REGENERATION THEORY This converges to 0 as t increases regardless of how great B may be taken. If the admittance diagram is drawn this is again found to check the rule. 3. Let the network be as in 1 except that there are two separated condensers bridged across resistance circuits. Then The solution for s(t) is obtained most simply by taking every other term in the series obtained in 1. + - .) (49) ## 4. Let the network be as in 3 except that there is a reversal. Then The solution is obtained most directly by reversing the sign of every other term in the series obtained in 3. s(t) = = - Be4'(Bt - B3t3/3! - - - ) + ## - Be-" sin Bt. (51) This is a most instructive example. An approximate diagram has been made in Fig. 5, which shows that as the gain is increased the ## BELL SYSTEM TECHNICAL JOURNAL feed-back ratio may be made arbitrarily great and the angle arbitrarily small without the condition being unstable. This agrees with the expression just obtained, which shows that the only effect of increasing the gain is to increase the frequency of the resulting transient. 5. Let the conditions be as in 1 and 3 except for the fact that four separated condensers are used. Then The solution is most readily obtained by selecting every fourth term in the series obtained in 1. ## s(t) = Be-'(B3t3/3! B7t7/7 ! = \$Be-"' (sinh Bt - sin Bt). + - a ) (53) This indicates a condition of instability when B a,agreeing: with the result deducible from the admittance diagram. 6. Let the conditions be as in 5 except that there is a reversal. Then The solution is most readily obtained by changing the sign of every other term in the series obtained in 5. s(t) = ## - + ~ e ( ~ / d ~ - * ' (~t/.\Is~ 1 4 ) . sin - (56) This example is interesting because it shows a case of instability although there is a reversal. Fig. 6 shows the admittance diagram for ## B-\I2 - a < 0 and for ~ f i > 0. -a 7. Let AG(t) = fo(t) = A ( l - t ) , 05t 1. (57) We have ## Fig. 7 is a plot of this case for A 1. Fig. 7-Illustrating Example 7 . 8. Let A J(iw) = ## A(l iw) (1 i2w) + + This is plotted on Fig. 8 for A = 3. I t will be seen that the point 1 lies outside of the locus and for that reason we should expect that the system would be stable. We should expect from inspecting the diagram that the system would be stable for A < l and A > 2 and that it tvould be unstable for 1 A 5 2 . We have overlooked one fact, however; the expression for AJ(iw) does not approach zero as w increases indefinitely. 'Therefore, it does not rome \\ithin rcstr-iction ( B I ) and consecluently the reasoning leading up to the rule (foes not apply. a T h e admittance in question can be made up by l~ridging capacity in series with a resistance across a resistance line. 'I'his admittance - PLANE Fig S-Illustrating ## Exatnple 8, without distributed constants. obviousl\- does not approach zero as the frecluency increases. I n any actual network there ~vould, however, I)e a small amount of distri1)uted capacity \\-hich, as the frequency is incre;ised indetinitely, ~votllclcause the transmission throtigh the net\\-ork to a~jproach zero. This is shown graphically in I:ig. 9. The etTect of the distri\)uted capacity is Fig. 9-Illustrating ## Esample 8, with distributed constants. essentially to cut a corridor from the circle in Fig. 8 to the origin, which insures that the point lies inside the locus. APPENDIX I Alternative Procedure In some cases AJ(iw) may be given as an analytic expression in (iw). In that case the analytic expression may be used to define w for all values of z for which it exists. If the value for A J ( i w ) satisfies all the restrictions the value thus defined equals the w defined albove for 05x < only. For < .v < 0 it equals the analytic continuation of the function w defined above. If there are no essential REGEATERATION TZZEORY singularities anyu~hereincluding at a ~ , integral in (33) may be the enluated by the theory of residues by completing the path of integration so that all the poles of the integrand are included. R e then have If the network is made up of a finite number of lumped constants there is no essential singularity and the preceding expression converges because it has only a finite number of terms. In other cases there is an infinite number of terms, but the expression may still be expected to converge, at least, in the usual case. Then the system is stable if all the roots of 1 - w = 0 have x < 0. If some of the roots have x 2 0 the system is unstable. The calculation then divides into three parts: 1. The recognition that the impedance function is 1 - w.ll 2. The determination of whether the impedance function has zeros for which x 2 0.12 ## Fig. IGNetwork of loci x = const., and y = const. 3. A deduction of a rule for determining whether there are roots for which x =' 0. The actual solution of the equation is usually too laborious. To proceed with the third step, plot the locus x = 0 in the w plane, i.e., plot the imaginary part of w against the real part for all the values of y, - oo < y < oo. See Fig. 10. Other loci representing x = const. and y = const. H. W. Nichols, Phys. Rat., vol. 10, pp. 171-193, 1917. "Wf.Thompson and Tait, "Natural Philosophy," vol. I , 5 344. " Cf. ## BELL SYSTEM TECNNICAL JOURNAL may be considered and are indicated by the network shown in the figure in fine lines. On one side of the curve s is positive and on the other it is negative. Consider the equation and what happens to it as A increases from a very small to a very large value. At first the locus x = 0 lies wholly to the left of the point. For this case the roots must have x < 0. As A increases there may come a time when the curve or successive convolutions of' it will sweep over the point w = 1. For every such crossing a t least one of the roots changes the sign of its s. \TTeconclude that if the point zu = 1 lies inside the curve the system is unstable. I t is now possible to enunciate the rule as given in the main part of the paper but there deduced with what appears to be a more general method.. Discz6ssion o Restrictions f The purpose of this appendix is to discuss more fully the restrictions which are placed on the functions defining the network. A full disccssion in the main text would have interrupted the main argument too much. Define an additional function ## n(iy) = lim n(z). -0 This definition is similar to that for w(z) given previously. It is shown in the theorem l3 referred to that these functions are analytic for x # 0 if AJ(iw) is continuous. We have not proved, as yet, that the restrictions placed on G(t) necessarily imply that J(iw) is continuous. For the time being we shall assume that J(iw) may have finite discontinuities. The theorem need not be restricted to the case where J(iw) is continuous. From an examination of the second proof it will be seen to be sufficient that can be slightly modified to include all cases where conditions (A1)(ATII) are satisfied. Now ## A J(iX)d ( i X ) (ih - zo - Az)(ih - 20)' (iX -z Az ~ ) ~ iX - zo +- - zo + etc. zX Az2 (65) if Az is taken small enough so the series converges. I t will be sufficient to confine attention to the first term. Divide the path of integration into three parts, In the middle part the integral exists because both the integrand and the range of integration are finite. In the other ranges the integral exists if the integrand falls off sufficiently rapidly with increasing X. It is sufficient for this purpose that condition (BI) be satisfied. The same proof applies to n(z). Next, consider lim w(z) = w(iy). If i y is a point where J(iy) is z+O ## + P(iy). n(iy) = - AJ(iy)/2 + P ( i y ) w(iy) = AJ(iy)/2 Ad(ih). 21r2 ,iX - i y (664 (66b) 1.s the integral Subtracting ## ( w ( and ( n 1 increase indefinitely as x Next, evaluate the integral + 0. (68) l4 ## E. W. Hobson, " Functions of a Real Variable," vol. I, 3d edition, \$352. where the path of integration is from x - ioo to x ioo along the line x = const. On account of the analytic nature of the integrand this integral is independent of x (for x > 0). It may be written then lim 20 2m,, = f 1 zif(z)eztdz = lim . Z0 - 2 + ~ iim 2%;. - lirn 2 ~ .\r+m i 2+1 [It::- + lI,]E +6 : 1 2 s~, ## 1 f 2- f f l Ji (hi h )ze z t d ( i h ) d z rz,, - ez~(ih)dz where 6 is real and positive. The function Q defined by this equation esists for all values oft and for all values of 6. Similarly, ## The first: integral is zero for t large. Likewise, the < 0 as can be seen by taking x sufficiently second is equal to zero for t > 0. Therefore, ## We may now conclude that a(iy)eivtd(iy) = 0, - oo < .t < oo (741 provided C(t) = 0, But (74:) is equivalent to n ( z ) = 0, < t < 0. (A111 (74') w(iy) = AJ(iy). ## (BIII) (74') taken (BII) (BIII) is, therefore, a necessary consequence of (AII). with (68) shows that J(iy) is continuous. It may be shown l5 that (BI) is a consequence of (AI). Consequently all the B conditions are deducible from the A conditions. Conversely, it may be inquired whether the A conditions are deducible from the B conditions. This is of interest if A J(io) is given and is known to satisfy the B conditions, whereas nothing is known about G. Condition A11 is a consequence of BIII as may be seen from (67) and (74). On the other hand A1 and A111 cannot be inferred from the B conditions. It can be shown by examining ( 5 ) . however, that if the slightly more severe condition lim y J ( i y ) exists, Y ' Lg (y > I), GO, (B Ia) (AIa) is satisfied then G(t) exists, - oc < t < which, together with AII, insures the validity of the reasoning. It remains to show that the measured value of J(iw) is equal to that defined by (6). The measurement consists essentially in applying a sinusoidal wave and determining the response after a long period. Let the impressed wave be ## E = real part of eiWL, t E = 0, t The response is real part of < 0. 0. (75) (75') 1' AG(A)eh('-"dA = real part of Ae'"' For large values of t this approaches real part of Aef"'J(iw). Consequently, the measurements yield the value AJ(iw). See Hobson, lec. cit., vol. I1,2d edition, \$335. It will be apparent that K depends on the total variation but is independent of the limits of integration. (77) ' ## Stability of Systems with Delayed Feedback YA. Z. TSYPKIN Translation by L. Jocii. and S. Kalme Abstract-Stability conditions are derived for systems with delayed feedback. Influence of the delay time and associated parameters is investigated for several simple control systems. A generalized criterion is given, which enlarges greatly the class of systems with delayed feedback under consideration. Finally, a comparison is made of the results, derived from the established criterion, with an approximation method often encountered in the literature. Fig. 1. Block diagram of a system with delayed feedback. I: Forward element. 11: Feedback path. DE: Delay element. FE: Feedback element. We refer to a system characterized by the property that the feedback variable at time t depends upon the system state at time t - T, T-constant, as a system with delayed feedback. Such a system can be represented as having a delay element in its feedback loop (Fig. 1). The essential characteristic of the delay element is that the input variable is repeated exactly at the output after a time period T, which will be called the delay time (Fig. 2). In fact, the delay element may be a part of the system itself (I), or located in the feedback loop (11). Electrical power systems which have their feedbacks closed through tie-lines, systems with acoustic feedback, etc., are examples of systems with delayed feedback. Many other control systems belong to this class after a suitable approximation is made in their description. The presence of a delay element introduces new qualitative characteristics into ;I system, whch may cause undesired instability. For this reason, the analysis of such systems has considerable significance. There exists a large number of papers dealing with this problem. Most of them, [I] -[1 I], have been applying an approximation method based upon the Hurwitz criterion for stability analysis. However, the method has not been derived precisely, and as we shall see later, it often leads to qualitatively wrong results. Other authors have used graphical approaches to treat delayed feedback systems [12], [1:3], or approximate graphical or numerical solutior~s the characteristic equation [14]-[16]. of Needless to say these numerically posed problems do not leave any space for general conclusions;, and they appear cumbersome. Finally, the work reported in [lo], [17] has raised a question regarding methods, which avoid the explicit solution of the characteristic equation. Nevertheless these methods are not generally applicable, and in some places these papers contain errors leading to questionable conclusions. Therefore no previous work thai we are aware of has established a sufficiently general and exact method for analysis of systems with delayed feedback. 'The paper presented here is dedicated to this goal. Stability criteria proposed below are The author is with the Research Institute for Aircraft Equipment, Ministry of Aviation Indnstry, USSR. Fig. 2. based upon considera~tions frequency characteristics of the of part of the system without time-delay. The criteria obtained simplify greatly the stability analysis of systems with delayed feedback. I. EQUATIONS AND FREQUENCY CHARACTERISTICS OF SYSTEMS WITH DELAYED FEEDBACK Let us form the equations describing systems with delayed feedback. In order to abbreviate further equations, we will use the time derivative operator ## It is possible then to write the differential equation corresponding to system I (Fig. 1) as where Q,(D) and PI (11) are polynomials in D. The feedback path I1 (Fig. 1) is composed of two elements: a feedback element and a delay element. The differential equation for the feedb'ackelement has a form similar to (1): Qz(O)U~ =P2(D)uZ3 and the delay element is described by (a After expanding the right-hand part of the previous equation in a Taylor series, one can rewrite (3) as follows u4 = e - 7 D ~ 3 . (4) Reprinted with permission from Avtomat. Telemekh., vol. 7, pp. 107-129, #23. 1946. By eliminating u, and u, from (1), (2), and (4), we obtain the open-loop system equation (Fig. 3) as Q(D)u4 = P ( D ) ~ - ~,~ u where Q(D> = Qi (D) Q2 (D), P(D> = PI(DIP2 (DlIt is easy to see that the equation for the closed-loop system (Fig. 1) can be obtained from (5) by setting u4 = ul ,to get {Q(D) - p(D)edTD) u1 = 0. (6) ## 4 Fig. 3. System with open feedback path. (5) is the transfer function of the open-loop system. It is obvious that the ratio w ~ = P(iw)lQ(jW) is the transfer function of ) the equivalent system. By separating the real and imaginary part, WQw) can be represented as follows When r = 0, this equation transforms into an ordinary differential equation of the closed-loop system without the delay element ## pdiw) = W@d) = - A(w) + jB(o) Q(jW) (9) t Q(D) - P(D)I u1 = 0. ## or W(jw) = wo ( a ) eje(W)7 where (10) (7) The system described by (7) will be referred to as the equivalent system. The equation of the system with delayed feedback (6) differs from the differential equation describing the equivalent system (7), in that it contains not only the function u1 (t), but also ul (t - 7). In the literature this type of equation is called a difference-differential equation [Is], [19], or the histerodifferential equation [20]. A general theory of such equations is undeveloped up to now. In order to obtain solutions of the histerodifferential equations one has to solve transcendental characteristic equations, which is by itself a cumbersome operation, possible for numerically specified systems only. Another way of solving these equations is, for example, by emplaying the Laplace transformation [19], although in that case the form of the obtained solution precludes any further investigations. Furthermore, for stability analysis it is not necessary to know exact solutions of system equations; it suffices to determine the character of roots of the characteristic equation. For this reason we will not be primarily concerned with the solutions of histero-differential equations, and we refer the interested reader to the literature 1181-[21]. stability analysis will be essential role in the played by the frequency characteristics of the open-loop s y s tem. We turn our attention now to the determination of these characteristics. suppose that the input of the open-loop system (Fig. 3) is sinusoidal: u1 = utmejwt, j = a . ## Nu) e(o) = arctan A(w) ' and therefore, according to (8), the open-loop transfer function of the system with time-delayed feedback can be represented as follows wT(jW) = wO(jW)e j [ e ( ~ )I - ~ ~ (1 1) It is easy to see from (10) that the transfer function represents the relation between the amplitude and the ~ h a s e the at output and the frequency of the input sinusoidal signal. ~ eus t observe that in order to determine the transfer function it is not necessary to start with the time-domain system equations, as we did earlier. The transfer functions of nonelectrical sysanalogies [231 with the tems can be obtained by means laws for quantities11. STABILITY CONDITIONS can be reduced to the Stability problems for linear analysis of roots of the characteristic equations. A given system is said to be stable if all the roots of the characteristic equation have negative real part, and unstable if there exists at with a positive part. least One The characteristic equation for a system with delayed feedthe 'peratorD with back can be derived (6) z and by equating to zero the expression in parenthesis: Q(z) - P(z)e-7Z = 0. (12) The necessary and sufficient conditions for stability are expressed, then, as follows: Re [Zk]< 0, where zk are the roots of the characteristic equation (12). In the case of r = 0, which corresponds to the equivalent system, the characteristic equation takes the form of this algebraic equation: Q(z) - P(z) = 0. The output variable u4 win be also sinusoidal in the steady state, with different amplitude and phase. It is well known 1221 that u4 can be found from (5) by replacing D with jo: PQw) -. u4 = - ITW . e ul The ratio W,(jw) = 3 - P ( j 4 ,-jrw Ikt (13) Qdio) (8) Since the characteristic equation (12) of the system with time-delayed feedback is a transcendental rather than algebraic equation, it is not possible to apply directly the well-known Hurwitz stability criterion. Nevertheless, it will be shown that by using the open-loop transfer function of the equivalent system one can establish simple and straightforward stability criteria for systems with delayed feedback. 111. STABILITY CRITERIA FOR SYSTEMS WITH DELAYED FEE:DBACK Let us draw a graph of the transfer function (10) in polar coordinates (W,, 6), for the different values of the frequency w in the interval between 0 and =. Then, according to Nyquist [24], the equivalent system is stable if the point (1, jO) is outside the graph (Fig. 4(a)), and unstable if the point (1, jO) is inside it (Fig. 4(b)).' It is easy to show that the relative position of the point (1, jO) with respect to the polar plot of the transfer function determines the signs of the real parts of the roots of the in characteristic e q ~ a t i o n : ~ the case of Fig. 4(a) the real parts of all characteristic roots are negative, while in the case of Fig. 4(b) a certain number of roots have positive real parts. If the polar plot passes through the point (1, jO); then the characteristic equation will have two pure imagicary -oats, and the remaining roots will have negative real parts. 'Therefore, crossing of the polar plot of W(j&)) through the point (1, jO) corresponds to the sign change or real parts of characteristic roots. Let us consider now a polar plot of the open-loop transfer function WT(jw) for the system with delayed feedback: Fig. 4. Frequency characteristics of open-loop transfer function W(jw) for equivalent system. a: Stable equivalent system. b: Unstable case. Fig. 5. Variations of the frequency characteristics (WT(jw)-plane)and of the characteristics roots (z-plane) as T increases. a: T < TO.b: T = T . c: 7 >TO. O This graph differs from the one obtained for equivalent system transfer function W(jw) = W ( o ) e j N w ) o by the fact that the radius-vectors W,(jw) are shifted in clockwise direction by additional angles proportional to a. The coefficient of proportionality is the delay time T. It is absolutely clear that the presence of the delay element, which transforms the plots of W(ja) as indicated above, may cause instability of the closed loop system. Let us assume that the equivalznt system is stable. This means that the curve Wow) does not encircle the point (1, jO), and all roots of the characteristic equation (13) are located in the left half of the complex z-plane (Fig. 5(a)). By increasing T monotonically from 0, the roots of the characteristic equation (12) will relocate in the z-plane; the polar plot of W o o) will change as indicated above. Folr a certain value T = T, , the curve W(jo) will pass through the point (1, jO). In this case the characteristic equation (12) will have two conjugate roots on the imaginary axis of the z-plane (Fig. 4(b)), whose magnitudes are equal to the frequency w = a,. (1, jO) is the correplane. sponding point on the WT (jo) Further increase in T may cause the encirclement of the point (1, jO) in which case the characteristic equation (12) The arrow indicates the direction of increased frequency. 2See Appendix, and also 1251. would have two roots in the right half of the z-plane (Fig. 5(c)) and, consequentl,~, system would be unstable. the Increase in T could also cause the plot of W(jo) to contain the point (1, jO), now and then, and to encircle it, i.e., the roots of the characteristic equation (12) could be passing back and forth from the left half plane. As a consequence, the system may become stable and again unstable, and vice versa. We will refer to T, and wo as the critical delay time and critical frequency, respectively. Critical delay time determines the transfer of a pair of characteristic roots through the imaginary axis j and therefore, system stability. It is obvious that the o critical delay time and critical frequency are determined by the equation ## This equation is satisfied if the following conditions are met: and where n is a positive integer We solve (16)with respect to w to get the critical frequencies in wOi. By substituting ooi (17) we find the critical delay If 702 <rol, further increase in T causes an alternation of stable and unstable behavior. System is stable for rO2(n- 1) < T < TO, (n), and unstable for rol(n) < T < T,, (n). On the other hand, if To2 > rol(1) the system will be unstable for all T > T ~ ~ - If a closed system with delayed feedback is unstable, then the growing oscillations have frequencies for which Wo(w) Fig. 6. Determination of critical frequencies woi, and critical delaytimes rOjfrom the graph of an open-loop equivalent system. a: There exists one critical frequency. b: Two critical frequencies are present. > 1, (19) times i.e., the frequencies determined by the parts of the plot outside the unit circle. The frequency of growing oscillations decreases as T(T > rol) increases. If the system is unstable for larger T, i.e., if the alternation of stability and instability domains is missing, then resulting oscillations occur at several frequencies lower than wo, . The number of these frequencies is always finite, and it increases with increasing T. If, for example, Conditions (16) and (17) have a straightforward interpretation in the W(jw) plane. Plot first a circle with center at the origin of the Wow) plane, and unit radius (Fig. 6). The intersection of the W(jw) curve with this circle determines critical frequencies woi, and by dividing the angIes 8(woi) by corresponding critical frequencies we get critical delay times rot. Stability boundaries are determined only by those critical delay times r0i, at which the W, (jw)-curve contains the point (1, jO), without encircling it. These points r,,i are called limiting values. Therefore, in order to establish regions of stability and instability it suffices to find the boundary delay times. As can be seen from (18), the boundary delay times are functions of system parameters. This offers an opportunity to analyze the influence of system parameters on system stability. Considering the arrangement of Fig. 6 one can draw several general conclusions regarding stability of systems with delayed feedback, and to formulate simple stability criteria. 1) Let the number of frequencies will equal n. In systems with delayed feedback (for example, in control systems; we try usually to keep the delay time T low. For this reason, the most important is the smallest limiting delay time r0 min. Consequently, for such systems one can formulate the following practical stability criterion: A system with delayed feedback is stable, if its delay time T is smaller than the minimum limiting delay time, i.e., From the above considerations, one can see that for stability analysis of systems with time-delayed feedback it suffices to know only the open-loop transfer function. The relations established here determine the limiting values for delay time, and permits one to construct domains of stability and instability, thus giving a full picture of the influence of system parameters on stability. IV. STABILITY ANALYSIS CERTAIN OF SYSTEMS for all w. Then, critical frequencies do not exist, and the system is stable for all T. This case is not particularly interesting, and it will not be pursued any further. 2) Let Wo(jO) < -1. Then, there exists a single critical frequency o0 (Fig. 6(a)) which corresponds to a limiting value of the delay time TO. The system is stable for T < T ~and unstable if T < TO. The re, maining rO(n) are greater than r0 for any positive n > 1, and their consideration is unnecessary. 3) Let Before we turn our attention to the stability analysis of certain typical systems with delayed feedback, let us make a few preliminary remarks. It is often convenient to express the open-loop transfer function in terms of the dimensionless quantity x proportional to the frequency x=-. wc. (20) In this case, (16) and (17) which determine the critical frequencies and delay times become I wo (10) l < 1 (or in particular Wo(0) = 0. Then, there exist two critical ) A. Let us consider now the simplest pressure control system frequencies ooland woz (Fig. 6(b)), with wo, > wo2. Hence, 701 < 702, and 701 is the limiting delay time. The system is [26]. By u, we denote the variation of pressure from its nominal value, and u, describes the behavior of the controlling stable for T < rol,and unstable if TO, > T > rol. mechanism. Then, we can write equations for the plant and the positional3 control mechanism as follows where Ta is the plant time constant, z is the load coefficient, and a is a positive coefficient. It is obvious from (24) that we assume that the action of the controlling mechanism is delayed by a constant time T, with the respect to the response of the sensor n~easuring controlled variable u, . The controller equation for the equivalent system is given by u4 = -a&. Noting that Fig. 7. a: Frequency characteristic of a simple pressure control system (system A). b: Normalized frequency diagram for system A. ## or according to (9), we get finally The polar plot of the equivalent system transfer function, given by (29), is a semicircle (Fig. 7(aj). By constructing the unit circle we get xo and. T o . This system is stable if T < To , Fig. 8. Stability boundary To = f (7) system A. Unshaded side corfor and unstable if T > T o . responds to' the stability domain. In order to investigate the influence of y on stability we have to redraw the W(jx)/y on Fig. 7(b) and instead of the unit circle we draw a circle of radius lly. Changes in y are reflected The stability limit, determined by (32), divides the plane then by changes in circle radius. (T, y) in two parts (Fig. 8). The lower portion corresponds to If y < 1, the xo does not exist and system is always stable. a stable system, and the upper part to an unstable system. For y > 1 the system is stable if T < To. Larger y corresponds We refer to a point with coordinates ! to a larger xo and smaller T o . When :' = To the system starts to oscillate at the frequency T = Z T y=-a , (33) Ta 21 ' can be derived ## (21) and (22) as (31) To = n - arctan 1 dFi (32) as the parameter point. It is obvious that to each parameter point corresponds a system described by (23) and (24), and vice versa. A change in system parameters forces a coordinate change of the parameter point. The given system is stable if the parameter point is located within the stability region, and unstable otherwise. For +y >> 1 the stability limit equation can be simplified as follows ## 3According to the terminology used by Oppelt [26]. xo x y arctan i = arctan y x 77 2' and hence, By increasing y (i-e., a) we decrease To, thus implying that as the regulator sensitivity increases the system becomes unstable for smaller T , i.e., the presence of the delay always deteriorates the exactness of control. It is interesting also to notice that system stability is almost independent upon load changes. B. Consider now the electrical system represented by Fig. 9. This system is the electrical analog of many control systems for regulation of temperature, voltage, etc. In particular it represents a system for pendulum control which was investigated in detail by Minorsky [17]. Equations describing this system may be written in the following form: ## Fig. 9. Electrical system modeled in example B. , By assuming that the anode current i is lagging behind the grid voltage, we get Fig. 10. Normalized polar plot of the transfer function W ( j x ) representing system B for different values of P. where S denotes the slope of the tube's characteristics. Hence, (36) can be rewritten M S D U ~ ~ =' - 4 - ~ (38) We open the loop in points 1 - 1, and obtain the following open-loop transfer function: where ## Furthermore, we introduce the following dimensionless quantities the direction in which x increases. Let us construct a circle of radius 1/y centered at the origin. For small values of fl there exist two critical values of x: xol and xO2. As increases the W(jw)-plots gather around the origin. Therefore, starting with a certain value of 0, the system becomes absolutely stable for all T. We reach a similar conclusion by considering the case of decreasing A with = const. The system is stable if T < Tol, and unstable for TO2> T > To,. If Tol (1) > To2, then a smooth increase in Twill cause the regions of stability and instability to alternate. In order to establish stability regions we need to find xoi and Toi(n). By equating the magnitude of the transfer function to one, we get after some elementary transformations, x4 - (2 - p2 + y2)x2 + 1 = 0, and hence, and write the transfer function in the form (lo), to get4 and according to (22), -R ## Toi(n) = mc.7oi = where 3 2 - arctan XOi 0x0i 1 - x& 2nn XOi t-. (42) It is easy to see that the number of critical values xoi is determined by the sign of the discriminant, i.e., for A family of amplitude-phase characteristics for different values of /3 and y = 1 is shown in Fig. 10. The arrows indicate 4When the argument of arctg passes through infinity (x = 1) the system phase decreases by r. <P there is none; y = 0 there is one: xo = 1; y > p there are two critical values. y Therefore, in the first case the system is stable for all T , in the second case there will be oscillations with frequency w = Fig. 11. Dependence of the normalized critical frequency upon 7 for different values of (3. Fig. 12. Stability boundaries To =f(r, (3) for system B. side corresponds to the stability domain. o0provided T = T o , and finally, in the third case the system is stable whenever T < ToI . Fig. 11 displays the dependence of xo in terms of y. Boundaries of the stability regions in the (T, y)-plane are specified by (42), and they vary for different fl (Fig. 12). The region on the shaded side of the boundary corresponds to unstable systems, the opposite to stable ones. As it can be seen from Fig. 12, the alternation of stability and instability regions appears for y greater than 0, but close to it, as T increases continually. The stability region is wider for larger P. By kcreassing y for constant fl the region of stability gets smaller. Fig. 12 represents by itself a cross section of the hyperplane To = f(y, 0) with P = const., which separates stability and instability regions. The parameter point is specified by means of the three coordinates The analysis of the influence of parameters is either reduced to the determination of the trajectory of the parameter point in the (T, y, 0)-space, or to the question of finding a trajectory in the (T, y)-plane, simultaneously taking into account the relocation of the stability boundary with different p. For example, by eliminating o, from the first two equations in (43) we get the equation of motion of the parameter point as follows: If t, a, 6 are given, then t h e parameter point moves from the origin along a straight line, as w, increases. From the third equation in (44) we see that the stability boundary moves towards the origin as w, increases. For a certain w, = G, the parameter point belongs to the boundary corresponding to /3 = 6/G,. Consequently, the system is stable if w, < G,, and unstable if w, > 6. It is not difficult to see that smaller G, corresponds to larger and a, and smaller y. In other words, the system becomes unstable for smaller 7's if a and w, are larger and 6 smaller. The parameter point belongs to the stability boundary; then oscillations with frequency wo = x o o , (xo is obtained from Fig. 11) will arise in the system. To the lower stability limit (Fig. 12) corresponds the upper branch of the curve x o (Fig. 1l), where x > 1 and, hence, o > a , ; to the second stability limit corresponds the lower part of the curve xo ,where x < 1, thus o <we; the third limit is again determined by the upper branch of x-curve, etc. If a parameter point belongs to the region of instability, then the frequency of the growing oscillations is determined by those values of x which are located between two branches of x o , specified by a given P. When y >>P the frequency of oscillations is higher than w,(x > 1). For y 2: the oscillations may have the frequency lower than o,(x < 1). In treating the problem, Minorsky [17] arrived at the conclusion that similar systems are able theoretically to sustain growing oscillations at infinitely many frequencies, though only the first frequency occurs in practice, and the remaining higher frequencies are not observed. According to the analysis presented above, this conclusion is false. The error appears as a consequence of the fact that the condition (19) Wo(o) > 1, which determines frequencies of the self-sustained oscillations, was not taken into account. C. As a final example, we consider the same system treated in B, under the condition that the feedback loop is closed directly through the network, rather than through the mutual inductance. In this case (38) should be replaced by the following equation T Fig. 13. Normalized polar plot W ( j x ) for system C for different values of 8. Fig. 14. Dependence of the normalized critical frequency upon y for different values of P . where a is a constant depending upon the tube transconductance and the amount of feedback. The frequency characteristic of the open loop system is given by where y = a , and the remaining notation is the same as for the system in B. A family of curves for y = 1 and different P is shown in Fig. 13. It is easy to see that for small p's it is possible to have two values for Xo, and thus, all conclusions made for system B hold again. By equating the amplitude of the open-loop transfer function (46) with unity, we get xol = d m , hence, there will be oscillations in the system only if T = To ; 3) for 1 > y > dl - (1 - f12/2)2, there exist two critical values: xol and xo2, and alternation of the stability and instability regions is possible; 4) if > fi and y > 1, then there will be a single critical value Xol , and the system will be stable for T < Tol, and unstable for T > Tol . The dependence of xo upon y, for different P's, determines the domain of frequencies which may occur, and it is represented in Fig. 14. By combining (47) and (22) we get a - arctan so that Toi(n) = wOroi(n)= Xoi +-. Xoi (48) ## By exploring the discriminant we obtain for 0 < fi.: 1) when y < ,/I - (1 - P2/2)2, then xo does not have any critical value, and the system is stable for all T; 2) when y = there exists a single critical value The regions of stability and instability constructed according to (48) are shown in Fig. 15. The type of stability boundaries coincides with the nature of stability boundaries for system A when P > 1/2;otherwise with the ones corresponding to system B. Influence of system parameters and delay time is determined as before. Trajectory of the parameter point in terms of o, is Fig. 15. Stability boundaries To = f(r,P) for system C. Unshaded side corresponds to the stability domain. ## the straight line parallel to the T - axis: y=a, T=w,.T. (49) We remark, however, that for y >> 1 and remaining conditions unchanged, the stability boundary To is located below the boundary for system B. As before, the system becomes unstable for smaller values of T, as a and o, increase. These examples illustrate how smple it is to apply criteria for the stability of systems with delayed feedback derived in Section 111. The formulas which determine stability boundaries and regions of stability and instability in the (T, y)-plane permit us to analyze the influence of system parameters and the delay time on system stability. and at the same time they suggest ways to eliminate instability. Based on these examples, one can conclude that the presence of the time-delay requires lower values for w, and a, and therefore it decreases system efficiency. which is obtained from (51) by substituting ,r from the relation (52). After the critical fre(quenciesare determined and substituted in (52), we find the following critical delay times: V. GENERALIZATION THE STABILITY OF CRITERIA In certain systems with delayed feedback the change of the delay time 7 has a consequence that the input signal is repeated at the output without distortion after the time 7, but in a 7-dependent scale; in this case the equation describing a delay element has the following form: =f(7)u3(t - r), (50) where f(7) is a continuous function of 7. Therefore, the modulus of the open-loop transfer function of the equivalent system is equal to Wo1(w)f (7). According to (1 6) and (1 8), the conditions which determine critical values wOi and rOi written as follows: are U4 The critical delay tirnes may be determined as follows: For all a ' s in the W(jw)-plane, plot vectors starting at the origin and having magnitude W o ( j o ) and phase O(w). The end points of these vectors form a curve which intersects the W(jo) plot at points corresponding to the critical frequencies o 0 i . The critical del.ay times are determined by the ratio O(woi)/woi at these intersection points. In order to analyze stability for a given delay-time r = T , one can use the previa~usly derived criteria. It. suffices to plot a circle or radii l/f(F) centered at the origin of the W(jo)plane, to find ## Critical frequencies are then found from where G is a point where the circle intersects the W(jw)curve. If ?<To, the system will be stable; the system is unstable if 7 > To. The previous reasoning was based upon the assumption that the equivalent system is stable. It is equally easy to establish stability critleria for systems with unstable equivalent systems. The polar plot of the unstable equivalent system transfer function will encircle the point (1, jO), (Fig. 16). By constructing a unit circle, it is easy to see that the existence of a single critical frequency wol (Fig. 16(a)) implies system instability for any T. W e n there exist two critical frequencies o o l and wo, (Fig. 16(b)), then the system will be stable only if W(jO)<-1 a n d ~ ~ ~ ( l ) > ~ > ~ ~ ~ . The generalized stability criteria presented above enable us to enlarge considerably the class of systems which can be analyzed, and to apply them, for example, to derive the conditions for self-excitement of decimeter wave generators [27]. Fig. 16. Determination of critical frequencies woi and critical delaytimes rOifrom the transfer function W(jw)of the unstable equivalent system. a: There exists one critical frequency. b: Two critical frequencies are present. VI. ON AN APPROXIMATION METHOD FOR STABILITY ANALYSIS The most popular technique for analysis of systems with delayed feedback is the approximation method. This can be explained by the fact that this method replaces a hysterodifferential equation by an ordinary differential equation, and, therefore, it enables one to use the well-known Hurwitz criterion. The essence of the approximation method can be summarized as follows. Instead of the characteristic equation Fig. 17. Comparison of the stability boundary for system A, obtained by our method (shaded line), and by means of the fist, second, and third approximations of the exponential term (1, 2, and 3, respectively). Therefore, one concludes that the system is stable if T < l / h , y < 0, and periodically unstable when T > 1/y and y > 0. The stability boundary is given then by ## one considers the equation The stability boundary remains the same, except that for T > lly, the system is oscillatory unstable. Finally, in the third approximation obtained from (56) by replacing the term e-" by an nth order polynomial, which is an (n t 1) order truncation of the Taylor series for e d r Z . Equation (57) is an algebraic equation, to which it is possible to apply the Hurwitz criterion. This approximation, however, cannot be justified, and the method may lead to wrong conclusions. Let us consider, for example, the characteristic equation corresponding to the pressure control system (A) ## by using the following notation we get In the first approximation the term e-TY is often replaced by the first two terms of the series expansion to get ## and, hence, 1+Y y=-l-yT which implies that the system is unstable for any arbitrarily small T. Comparison of this results with the ones derived in Section IV is given in Fig. 17. We see that, the higher the order of approximation, the worse are the results obtained. Furthermore, for constant T the location of boundaries and type of instability depends upon the order of approximation, which, of course, cannot happen in reality. It is obvious that (60) does not have positive real roots, and hence, as established earlier, this system cannot be aperiodically unstable. Replacing e-TY by a polynomial, one essentially changes the structure of the equation, which implies the qualitatively wrong conclusion regarding this type of instability. Similar results for system B (Section IV) are represented on Fig. 18. Besides the above-mentioned paradoxical conclusions, the approximation method precludes one from establishing facts regarding the alternation of stability and instability regions. In the above examples the stability boundary obtained by using the approximation method is located below the exact boundary. It is possible to have examples in which the approximation boundary intersects with the exact one in several points. We are not going to treat them here, however. j u -- -. \ \ D \ y\ I / \ \ z\ D I -1-f I q -5 G 5 Fig. 18. Comparison ol' the stability boundary for system B, obtained by our method (shaded line), and by means of the first, second, and third approximation of the exponential term (1, 2, and 3, respectively). 4' , / ---' -J" This comparison o~fresults derived by the approximation Fig. 19. method with the exact ones implies that the approximation method may lead to not only quantitatively, but also quali- which represents the ratio of the characteristic polynomials tatively wrong results, and hence its application for stability corresponding to the system (12) with delayed feedback and analysis is not appropriate. the open-loop system without the delay element, or f(D)= 1 - W(D)e-7D. We assume that the open-loop system is CONCLUSION stable, and hence all the poles o f f (D) are in the left half of In this paper we establish a criterion for stability of systems the ~ - ~ l ~ ~ ~ . with time-delayed feedback. The criterion is based upon constability conditions require that all zeros belong also to the sideration of frequency characteris1.i~~ the open-loop trans- left-half of the ~ - ~ lnlorder to~establish a stability of ~ ~ . us to terion we use the Principle of the Argument, fer function for the equivalent system- This choose for simplify greatly the stability analysis of systems with delayed integration the contour consisting of the imaginary axis feedback. and a semicircle of arbitrarily large radius, which belongs to The analysis of the influence of system parameters and the the right half of the D-planes (Fig. 19). delay-time is reduced to the problem of finding the trajectory According to the Principle of the Argument, the stability of the parameter point through stability andinstability regions, conditions can be rewritten in the following form constructed according to the derived equations. AL argf(D)= 0, Several concrete examples of the systems with delayed feedback are considered also. The indicated generalization i.e., the change in argument of the complex function f ( ~ ) of the stability criterion enabled us to greatly enlarge the along the contour L must be zero class of systems which can be treated. Towards the end, we ~h~ change in argunlent when D takes values the semion compare the results of the stability analysis based upon the circle of the arbitrarily large radius in the right half of the Dmethod formulated in this paper, and the approximation plane, is equal to zero since in this case method, which is widely accepted in the literature, to conD = lim Re"= lim R(coscp+ jsincp), clude that the latter can lead to wrong results. R-+R-+m where @ varies from -n/2 to +n/2, and hence The author considers as his most pleasant duty to express his profound indebtedness to Prof. K. F. Teodorchik for discussion regarding questions raised in this paper, and to Ing. Yu. I. Goldfain for his help in computing and drawing the graphs. APPENDIX In Section V we have assumed tacitly that the Nyquist criterion can be applied to systems with delayed feedback. Since this generalization of the Nyquist criterion is not encountered in the literature, we think that it may be useful to present it here. Consider the function f(D)= Q (D) - P(D) e-TD Q(D) 7 Therefore, the stability conditions can be rewritten as A arg j ( j o ) = A arg ((1- W ( j w ) e-jTw) = 0 where o varies between --x and +-x, or A -m < w < +- arg (W(jw)e-iTw-1)=0. By plotting the curve W7(jo) = W(jw) e-i7'" (Fig. 5) in the complex W-plane for all w 2 0," we see easily that W,(jo) - 1 'It is easy to see that function f (D) fulfills all the necessary conditions for application of the Principle of Argument (e.g., [25]). 6 ~ o r < 0, the curve W,(jw) is symmetric to the curve WT(jw), w w > 0, w t respect to the horizontal axis, and hence it is unnecessary ih to draw the part corresponding to w d 0 (e.g., [25] ). itself represents a vector starting at (I, jO) and ending on the W,(jw)-curve. Therefore, the argument variation of W,(jw) - 1 will be equal to zero only in the case when the point (1, jO) is not encircled by the curve W,(jo), which corresponds to the stability. If the point (1, jO) is encircled by the curve W,(jw), then the change in argument of W,(jw) - 1 is nonzero, and the system is unstable. [ I ] T. Stain, Regelung und Ausgleich in Dampfanlagen, Springer, 1925. Russian translation: Adjustment and Control of Steam Power Plants, GNTI, 1931, p. 188. [2] M. Lang, "Theorie des Regelvorganges elecktrischer Industrieofen," Elecktrowarme, 9,201 (1934). [3] W. Stablein, Die Technik der Fernwerkanlagen, Miinchen, 1934. Russian translation: Techniques .for Remote Control, GONTI, 1939, p. 89. 141 S. G. Gerasimov. "Phvsical foundations for dvnamic control of thermal processei," ONTI, 1934, p. 71, (in ~ u s i i a n ) . [5] D. A. Wicker, "Effect of delays in automatic control processes," Avtomatika i Telemekhanika, 6,59 (1937). [6] P. S. Koshchiev, "Towards theory of tracking systems," Avtomatika i Telemekhanika, 5 , 81 (1940). [7] N. I. Chistiakov, "On phase lag computation for an automatic frequency control amplifier," Electrosviaz, 7,27, (1940). [8] B. I. Rubin and Yu. E. Heiman, "Fundamentals of aircraft control," LII GBF, 27 (1940). [9] V. A. Bogomolov, "Power control of a hydroplant by the rate of flow," Avtomatika i Telemekhanika, 4-5, 103 (1941). [ l o ] N. Minorsky, "Control problems," J. Franklin Institute, 232, 6, 524 (1941). & . [ I l l M. Lang, "Physik and regeltechnik," Phys. ZS., 9-12, 209, (1944). [12] F. Reinhardt, "Selbsterregte Schwingungen beim Parallelbetrieb von Synchronmaschinen," Siemens ZS., 10,413 (1925). [13] D. A. Wicker, "Dynamics of resistive voltage controllers," Elecktrichestvo, 9, 26 (1934). [ 141 A. Callender, D. Hartree, and A. Porter, "Time-lag in a control system. I," Phil Trans. Roy. Soc. London, 235,756,415 (1936). [15] D. Hartree, A. Porter, A. Callender, and A. Stevenson, "Time-lag in a control system. I1,"Proc. Roy. Soc. London, 161,907,460 (1937). [16] F. Reinhardt, "Der Parallelbetrieb von Synchrongeneratoren mit Kraftmaschinenreglen konstanter Verzogerungzeit," Wiss. Veroffentlichungen aus den Siemens-Werken, 18,1,24 (1939). [17] N. Minorsky, "Self-excited oscillation in dynamical systems possessing retarded actions," J. Appl. Mechanics, 9 , 2 , (1942). [18] A. Heins, "On the solution of linear difference-differentid equations," J. Math. Physics, 19,2, 153 (1940). [19] R. Churchill, Modern Mathematics in Engineering. New York: McGraw-Hill, 1944, p. 23. [20] L. Silberstein, "On a histero-differential equation arising in a probability problem," Phil. Mag., 29,192,75 (1940). [21] E. Kamke, Differentialgleichungen. Losungsmethoden und Losungen, Leipzig, 1942, p. 493. [22] T. Fray, "Introduction to differential equations," GNTZ, 1933. [23] A. Harkevich, "Theory of electroacoustic apparatus," Sviazizdat, 1940. [24] H. Nyquist, "Regeneration theory," Bell Syst. Tech. J., 11, 1 , 126 (1932). [25] Ya. Z. Tsypkin, "Stability of feedback systems," Radiotechnika, 1 , 5 , 3 3 (1946). [26] W. Oppelt, "Vergleichende Betrachtung verschiedener Regelaufgaben hinsichtlich der geeigneten Regelgesetzmassigkeit," Luftfahrtforschung, 16,8, p. 448. [27] Ya. Z. Tsypkin, "Towards a theory of klystrons," Radiotechnika, 1,49,1947. ## Control System Synthesis Locus Method WALTER R. EVANS MEMBER AlEE by Root 00 - 0i KJw(s) ( s )= 1 +KFG,(S)KBGB(S) Synopsis: The root locus method determines all of the roots of the differential equation of a control system by a graphical plot which readily permits synthesis for desired transient response or frequency response. The base points for thG plot on the complex plane are the zeros and poles of the open loop transfer function, which are readily available. The locus of roots is a lot of the values of s which make this transfer function equal to - 1as loop gain is increased from zero to1 infinity. The plot can be established in approximate f0r.m by inspection and the siKllificant partsof the locus calculated accurately and quickly by use of a simple device. For multiple loop systems, one solves the innermost loop first, which then permits the next loop to be solved by ailother root locus plot. The resultant plot gives a complete picture of the system, which is particularly valuable for unusual systems or those which have wide variations in parameters. H E root locus method is the result of an effort to determine the roots of the differential equation of a control system by using the concepts now associated with frequency response methods.' The roots are desired, of course, because they describe the natural reseonse of the system. The simplifying feature of the control system problem is that the open loop transfer function is known as a product of terms. Each term, such as 1/(1+Ts), can be easily treated in the same manner as an admittance such as l / ( R + j x ) . I t is treated as a vector in the sense used by electrical engineers in solving a-c circuits. The phase shift and attenuation of a signal of the fonn eat being transmitted is represented by 1/ (1 Ts) in which s in general is a complex number. The key idea in the root locus method is that the values of s which make transfer function around the loop equal to - 1 are roots of the differential equation of the system. The opening section in this paper, Background Theory, outlines the over-all pattern of The tion on Root Locus Plot points out the great usefulness of knowing. factors of the in finding the open loop transfer The g.raphica1 nature of the method - requires that specific examples be Used to the methid itself under the Sing1e Multiple Loop System, and Corrective Networks. The topic Correlation with Other Methods suggests methods by which experience in frequency methods can be extended to this method. The topic Other Applications includes the classic problem of solving an nth degree polynomial. Finally, the section on Graphical Calculations describes the key features of a plastic device called a "Spirule", which permits calculations to be made from direct measurement on the plot. The problem of finding the roots of the differential equation here appears in the form of finding values of s which make the denominator zero. After these values are determined by the root locus method, the denominator can be expressed in factored form. The zeros of the function 80/8, can be seen from equation 1to be the zeros of G,(s) and the poles of Gg(s). The function can now be expressed as shown in equation 2 demonstrate The constant Kc and the exponent y depend upon ihe specific system but for control systems y is often zero and Kc is often 1. The full power of the Laplace Transform2 or an equivalent method now can be used. The transient response of the output for a unit step input, for example, is given by equation 3 ## The amplitude A i is given by equation 4 Background Theory The over-all pattern of analysis can be outlined before explaining the technique of sketching a root locus plot. Thus consider the general single loop system shown in Figure 1. Note that each transfer function is of the form K G ( s ) in which K is a static gain constant and G(s) is a function of -the complex number. In general, G ( s ) has both numerator and denominator known in factored form. The values of s which make the function zero or infinite can therefore be seen by inspection and are called zeros and poles respectively. The closed loop transfer function can be expressed directly from Figure 1 as given in equation 1 The closed loop frequency response, on tlhe other hand, can be obtained by substituting s = j w into equation 2. Fortunately, the calculation in finding A* or Oo/Oi(jw) involves the same problem of multiplying vectors that arises in making a root locus plot, and can be calculated quickly from the resultant root locus plot. Paper 50-11, recommended by the AIEE Feedback. Control Systems Committee and approved by the AIEE Technical Program Committee for presentation at the AIEE Winter General Meeting, New York, N . Y . , January 30-February 3,1950. Manuscript submitted November 15, 1948; made available for printing November 22, 1949. WALTER . EVANS with North American AviaR is tion. Inc., Downey. Calif. The author wishes to express his appreciation for the assistance given by his fellow workers, K. R. Jackson and R . M. Osborn, in the preparation of this paper. In particular, Mr. Osborn contributed the circuit analysis example. LOCUS OF S FOR ## Root Locus Plot The open loop transfer function is typically of the form given in equation 5. so that the angles in turn can be visualized. For any specific problem, however, many special parts of the locus are established by inspection as illustrated in examples in later sections. Surprisingly few trial positions of the s point need be assumed to permit the complete locus to be sketched. After the locus has been determined, one considers the second condition for a root, that is, that the magnitude of K,Gp(s)KBGB(s) unity. In general, be one selects a particular value of s along the locus, estimates the lengths of the vectors, and calculates the static gain K,KB = l/Gp(s)GB(s). After acquiring some experience, one usually can select the desired position of a dominant root to determine the allowable loop gain. The position of roots along other parts of the locus usually can be determined with less than two trials each. An interesting fact to note from equation 6 is that for very low gain, the roots are very close to the poles in order that corresponding vectors be very small. For very high gain, the roots approach infinity or terminate on a zero. The parameters such as TIare constant for a given problem, whereas s assumes many values; therefore, it is convenient to convert equation 5 to the form of equation 6. The poles and zeros of the function are plotted and a general value of s is assumed as shown in Figure 2. Note that poles are represented as dots, and zeros as crosses. All of the complex terms involved in equation 6 are represented by vectors with heads a t the general point s and tails at the zeros or poles. The angle of each vector is measured with respect to a line parallel to the positive real axis. The magnitude of each vector is simply its length on the plot. In seeking to find the values of s which make the open loop function equal to - 1, the value - 1 is considered as a vector whose angle is 180 degrees +. n 360 degrees, where n is an integer, and whose magnitude is unity. Then one can consider first the problem of finding the locus of values for which the angle condition alone is satisfied. In general, one pictures the exploratory s point at various positions on the plane, and imagines the lines from the poles and zeros to be constructed a value of s just above the real axis. The decrease in 40 from 180 degrees can be made equal to the sum of 4 and 4~if the 1 reciprocal of the length from the trial point to the origin is equal to the sum of the reciprocals of lengths from the trial point to 1/Tl and 1/TZ. If a damping ratio of 0.5 for the complex roots is desired, the roots rl and ra are fixed b y the intersection with the locus of radial lines a t 60 degrees with respect to negative real axis. In calculating K for it is convenient to consider a term (l+T1a) as a ratio of lengths from the pole -l / T l to the a point and from 8 to the origin respectively. After making gain K = l / [G(s)].=,,a good first trial for finding r, is to assume that it is near -1/Ts and solve for (l/Tz+s). After the roots are determined to the desired accuracy, the over-all transfer function can be expressed as given in equation 8. ## Single Loop Example Consider a single loop system such as shown in Figure 1 in which the transfer functions are given in equation 7. ## Multiple Loop System The poles of the open loop function are at 0, -l / T l and -1/Tz as represented by dots in Figure 3. The locus along the real axis is determined by inspection because all of the angles are either 0 degrees or 180 degrees. An odd number of angles must therefore be 180 degrees as shown by the intervals between 0 and - l / T l , and from - 1/T2 to - a. Along the jw axis, \$0 is 90 degrees so that \$2 must be the complement of &, as estimated at a= jw,. For very large values of 8, all angles are essentially equal so the locus for the complex roots finally approaches radial lines a t 60 degrees. The point where the locus breaks away from the real axis is found by considering Consider a multiple loop system in which the single loop system just solved is the forward path of another loop, as shown in Figure 4. t9,/Bi is given in factored fonn by equation 8 so the roots of the inner loop now serve as base points for the new locus plot. For convenience, however, neglect the effect of the term (1-s/ra) so that the locus for the outer loop is shown in Figure 5. The locus for the outer loop would be a circle about the -1/T point as a center if the effect of 8,/81 were completely neglected. Actually, the vectors from the points YI and r3 introduce net angles so that the locus is modified as shown. The ## Figure 2. Root locus plot Evans-Control System Synthesis 58 mum build-up rate, overshoot, natural frequency of oscillation, and the damping rate as effective clues in solving this ]problem. Other Applications Figure 4 (above). Multiple loo)p block diagram . . -111 10 ## Figure 5 (right:). Multiple loop root locus angle at which the locus emerges from rl can be found by considering a value of s close to the point rl, and solving for the angle of the vector (s - n ) . Assume that the static loop gain desired is higher than that allowed by the given system. The first modification suggested by the plot is to move the rl and rz points farther to the left by obtaining greater damping in the inner loop. If these points are moved far to the left, the loci from these points terminate i.n the negative real axis and the loci from the origin curve back and cross the ju axis. Moving the -1/T point closer t:o the origin would then be effective in permitting still higher loop gain. The next aspect o synthesis involves adding correcf tive networks. These examples serve to indicate the reasoning process in synthesizing a control system by root locus method. An engineer draws upon all of his experience, however, in seeking to improve a given system; therefore, it is well to indicate the correlation between this method and other methods. ## Correlation with Other Methods The valuable concepts of frequency response methods1 are in a sense merely extended by the root locus system. Thus a transfer function with s having a complex value rather than just a pure imaginary value corresponds to a damped sinusoid being transmitted rather than an undamped one. The frequency and gain for which the Nyquist plot passes through the - 1 point are exactly the same values for which the root locus crosses the j w axis. Many other correlations appear in solving a single problem by both methods. The results o root locus analysis can be f easily converted to frequency response data. Thus one merely assumes values of s along the jo axis, estimates the phase angles and vector lengths to the zeros and poles, and calculates the sum of the angles for total phase shift and the product of lengths for attenuation. The inverse problem o determining zeros and f poles from experimental data is the more difficult one. Many techniques are already available, however, such as drawing asymptotes to the logarithmic attenuation curve. For unusual cases, particularly those in which resonant peaks are involved, the conformal mapping technique originated by Dr. Profos of Switzerland is re~ommended.~ The transient response is described by the poles of the transfer function. The inverse problem in this case is to locate the poles from an experimental transient response. One might use dead time, maxi- Corrective Networks Consider a somewhat unusual system which .arises in instrument servos whose open loop transfer function is identified by the poles pl and f i 2 in Figure 6(A). As loop gain is increased from zero, the: roots which start from \$1 and move directly toward the unstable half plane. These roots could be made to move away from the jw axis if 180 degrees phase shift were added. A simple network to add is three lag networks in series, each having a time constant T such thart 60 degrees phase shift is introduced ait p,. The resultant locus plot is shown in Figure 6(B). The gain now is limited only by the requirement that the rtew pair of roots do not cross the j w axis. A value of gain is selected to obtain critical damping of these roots and the corresponding positions of all the roots are shown in Figures 6(A) and 6(B) as small circles. Actually, greater damping could be achieved for roots which originate at p1 and Pz if a phase shifting bridge were used rather than the 3-lag networks. Its transfer function is (3.- Ts)/(l+ Ts) and is o the "nonminimum phase" type o cirf f cuit. Since these types of correction are somewhat unusual, it is perhaps well to point out that the analysis has been verified by actual test and application. Many systems require a set of simultaneous equations to describe them and :are said to be multicoupled. The corresponding block diagrams have several in:puts to each loop so that the root locus :method cannot be applied immediately. One should first lay out the diagram so .that the main line of action of the signals :forms the main loop with incidental (coupling effects appearing as feedbacks and feed forwards. One .then proceeds to isolate loops by replacing a signal which comes from within a loop by an equivalent signal at the output, replacing a signal entering a loop by an equivalent signal at the input. One can and should keep the physical picture of the equivalent system in mind as these manipulations are carried out. The techniques of the r'oot locus method can be used effectively in analyzing electric circuits. As a simple example, consider the lead-lag network of Figure 7(A). It can be shown that the transfer function of this network is as given in equation 9 The denominator can be factored algebraically by multiplying out and finding the zeros of the resulting quadratic. As an alternative, it will be noted that the zeros of the denominator must satisfy equation 10 B system Figure 6. Evans-Control System Synthesis 59 ## Figure 8 (right). Spirule g3 The vectors in this expression are represented according to the root locus scheme in Figure 7(B). The two roots are thereby bounded as shown by the two dots and the cross. Their exact locations could be estimated or accurately determined by graphical methods. The locus of roots now is simply intervals along the negative real axis between the open loop zeros and poles as shown in Figure 7 (B). The exact location of the roots along these intervals is determined in the usual way. Note that the constant in equation 10 is of the form R1C1, which R is the effective value of in ' Rz and Rain parallel. In more complicated networks, the advantages of the root locus concept over algebraic methods becomes greater; its particular advantage is in retaining at all times n clear picture of the relationships between the over-all network parameters and the parameters of individual circuit elements. In the classical problem of finding roots, the differential equation is given in the form of a sum of terms of successively higher order. This can be converted to the form shown in equation 10 order term. Solve for the roots of the first loop which corresponds to the quantities in brackets above and proceed as before for the multiple loop system. If the roots close to the origin are of most interest, substitute s= l/x first and solve for root values of x Other combinations . are, of course, possible because a single root locus basically determines the factors of the sum of two terms. The root locus method is thus an analytical tool which can be applied to other problems than control system synthesis for which it was developed. But in attacking a new problem one would probably do well to try &st to develop a method of analysis which is natural for that problem rather than seek to apply any existing methods. Several procedures are possible, but the over-all purpose is to successively rotate the arm wlth respect to the disk through each of the angles of interest. Thus for adding phase angles, the disk is held fixed while the arm is rotated from a pole to the horizontal, whereas the two move together in getting aligned on the next pole. For multiplying lengths, the disk is held fixed while the arm is rotated from the position where a pole is on the straight line to the position where the pole is on the logarithmic curve. Rotations are made in the opposite directions for zeros than they are for poles. Conclusions The definite opinion of engineers using this method is that its prime advantage is the complete picture of a system which the root locus plot presents. Changing an open loop parameter merely shifts a point and modifies the locus. By means of the root locus method, all of the zeros and poles of the over-all function can be determined. Any linear system is completely defined by this determination, and its response to any particular input function can be determined readily by standard mathematical or graphical methods. Graphical Calculations The root locus plot is first established in approximate form by inspection. Any significant point on the locus then can be checked by using the techniques indicated in this section. Note that only two calculations are involved, adding angles and multiplying lengths. Fortunately, all of these angles and lengths can be measured at the s point. Thus angles previously pictured a t the zeros and poles also appear a t the s point but between a horizontal line to the left and lines to the zeros and poles. A piece of transparent paper or plastic pivoted at the s point can be rotated successively through each of these angles to obtain their sum. The reader can duplicate the "spirule" with two pieces of transparent paper, one for the disk and the other for the arm. References 1. PRINCIPLESOII SERVOMECHANISMS (book), G. S. Brown. D. P. Campbell. John Wiley and Sons, New York, N. Y., 1948. TRANSIENTS IN LINEAR SYSTEMS(book), M. F. Gardner, J. L. Barnes. John Wiley and Sons, New York, N. Y., 1942. 2. This corresponds to a block diagram with another loop closed for each higher 3. G R M ~ C AANALYSIS CONTROL L OF SYSTEMS, W. R. Evans. A I E E Transactions. volume 67, 1948. pages 547-51. Evans-Control System Synthesis 60 ## The Analysis of Sampled-Dat a SYstems I. R. RAGAZZlNl MEMBER AlEE ASSOCIATE M M E AlEE E BR HERE is an important class of feedback control systems known as sampled-data systems or sampling servomechanisms in which the data a t orle or more points consist of trains of pulses or sequences of numbers. Such systems may have a variety of forms, a common example of which is shown in Figure 1. I n the case illustrated, th~esampling is performed on the control error by a so-called sampler which is indicated as a mechanical switch which closes momentarily every T seconds. The data a t the output of such a switch consist of a train of equally spaced pulses of short duration whose envelope is the control error function. In some practical systems, the separation between successive pulses is controlled by some characteristic of the input signal and consequently is not constant. Such systems will not be considered in this paper. In a typical sampled-data system such as that shown in Figure 1, the sampler is followed by a smoothing circuit, commonly referred to as hold or clamp circuit, whose function is to reproduce approximately the form of the original error function by an interpolation or extrapolation of the pulse train. Following the hold circuit, there are the usual components of the feedback loop, shown in Figure 1 as H and G, comprising amplifiers, shaping networks, and the controlled member. I t is apparent that the insertion of a sampler into an otherwise continuous control system in general should result ill an inferior performance due to a loss of information in the control data. Yet, rampled-data systems have certain engir,eering advantages which make them preferable in some applications to continuous-data systems. The most important of these advantages is the fact that error sampling devices can be made extremely sensitive a t the expense of bandwi~dth. A example of such a device is the electron mechanical galvanometer and chopper bar. In this device a very sensitive though sluggish galvanometer is used to detect error and its position is sampled periodically by means of a chopper bar. The latter permits an auxiliary source of power to rotate a sizable potentiometer to a position determined by the clamped galvanometer needle. The process is carried out a t uniform intervals and a sampled and clamped output is obtained for use in the continuous part of the control system. Bandwidth is lost through the sluggishness of the unloaded galvanometer, but the power gain is enormous. Similar devices for measurement of pressure errors, flow, or other phenomena can he devised along the same general lines. In addition, there are some systems in which the data-collecting or transmission means are intermittent. Radars and multichannel time-division communication links are examples of this type of device. Such devices may be treated, in general, as sampled-data systems provided the duration of sampling is small by comparison with the settling time of the system. Despite the increasing use of sampleddata devices in the fields of communication and control, the volume of published material on such devices is still rather limited.'-lo The several different methods which have been developed for the analysis of sampled-data systems are closely related to the well-known mathematical techniques of solution of difference equations. I t is the purpose of this paper to unify and extend the methods described in the literature and to investigate certain basis aspects of sampleddata systems. Input-Output Relations h central problem in the analysis of sampled-data systems is that of establishing a mathematical relation between the input and output of a specified system. This problem has received considerable attention in the literature of sampled-data systems, with the result that several different types of input-outrelations have been developed, notand ably by Shannon,' Hure~vicz,~ Linvi11.3 Shannon's relation involves the Fourier transforms of the sampled input and output; Hurewicz's r~elationis based on the use of so-called generating functions, which in this paper are referred to as z-transforms and which are, in fact, a disguised form of the Laplace transforms; while Linvill's relation involves directly the Laplace transforms of the input and output. The principal difference between Shannon's and Hurewicz's relations on the one hand, and that of Linvill on the other, is the fact that the former yield only the values of the output at the sampling instants, while the latter provldes the expression for th~e output at all times, though at the cost of greater labor. The analysis presented in this section has a dual objective: to achieve a unification of the approaches used by Shannon, Hurewicz, and Linvill; and to formulate the input-output relations for the basic types of sampled-data systems. In the next section, the problem of establishing a relation between the input and output will be approached from a significantly different point of view. Specifically, a sampled-data system will be treated as a time-variant system and its behavior will be characterized by a system function which involves both frequency and time. The basic component O F sampled-data systems is the sampler, whose output has the form of a train of narrow pulses occurring a t the sampling instants 0, + T, * 2T, . . . , where T is the sampling interval; see Figure 2. 'The frequency o,,= f2?r/T is called the sampling frequency. For purposes of mathematical convenience, it is expedient to treat the output pulses as impulses whose areas are equal to the values of the sampled time function a t the respective sampling instants. (This is permissible provided the pulse duration is small compared with the settling time of the system and the gain of the amplifier following the sampler is multiplied by a factor equal to the time duration of the sampling pulse.) Thus, if the input and output of the sampler are denoted by r(t) and r(t) respectively, the Paper 52-161, recommended by the AIEE Feed. back Control Systems Committee and approved by the AlEE Technical Program Committee for pres. entation at the AIEE Summer General Meeting, Minneapolis, Minn.. June 23-27, 1952. Manuscript submitted March 21, 1952; made available for printing April 16, 1952. ## J. R. RAGAZZINI L. A. ZA.DEH both with the and are Department of Electrical Engineering, Colu~nbia University, New York, N. Y. ## Figure 1. Typical sampled-data control system 61 I Figure 2. \- PULSE TRAIN Pulse train at output of sampler An inspection of either of the alternative expressions for R(s) indicates that when s in R(s) is replaced by s-t.jmwo, where m is any integer, the resulting expression is identical with R(s).' This implies that R(s) is a periodic function of s with periodjwo; thus R(s+jmw,) = R(s) (9) ## where BT(t) represents a train of unit impulses (delta functions) where m =any integer. The infinite series expression for R(s) given by equation 5 readily can be put ) into closed form whenever ~ ( t is a linear combination of products of polynomials and exponential functions. For instance, the right-hand member of when r(t) = equation 5 is a geometric series which upon summation yields Equation 1 may be written equivalently as When expressed in the form given by equation 5, the transform of the pulsed output, R(s), is a function of 6". This suggests that an auxiliary variable z= esT be introduced and that R(s) be written in terms ,of this variable. When this is done, the function R(s), expressed as a function of z, is called the z-transform of r(t). For notational convenience it is denoted by R(z) although strictly speaking i t should be written as R(l/T log z). With this convention, the transform given in equation 10, for example, reads where the negative values of n are absent by virtue of the assumption that r(t) vanishes for negative values of t. Equation 3 furnishes an explicit expression for the output of the sampler. I t is more convenient, however, to deal with the Laplace transform of r(t), which is denoted by R(s) One expression for R(s) can be obtained a t once by transforming both sides of equation 3 ; this yields I n what follows, the symbols R(s) and R(z) will be used interchangeably since they represent the same quantity, namely the Laplace transform of r(t)8T(t). Needless to say, in cases where r(t) has the form of a sequence of impulses (or numbers equal to the areas of respective impulses), the z-transform R(s) is simply the Laplace transform of r(t), and not of r(t)ST(t>. It will be noted that the z-transform as defined is closely related to the generating function used by Hurewicz. However, the z-transform is a more natural concept since i t stems directly from the Laplace transform of the sampled time function. It is of historical interest to note that generating functions, which, as pointed out, are essentially equivalent to z-transforms, were introduced by Laplacell and were extensively used by him in connection with the solution of difference equations. In practice, R(z) is generally a rational function of z, and its inversion, that is the determination of a function r(t) of which R(z) is the z-transform, is most rapidly carried out by using a table of z- tra.nsforms such as the one compiled in Table I. (More extensive tables of closely related types of transforms may be found in references 12 and 13.) One use of this table is in finding the z-transform corresponding to the Laplace transfom of a given function. Despite its brevity, Table I is adequate for most practical purposes in view of the fact that both the Laplace transforms and the z-transforms can be expanded into partial fractions each term of which can be inverted in. dividually. I t is important to note that the inverse of a z-transform is not unique. Thus, if F(z) is an entry in the table and f(t) is its correspondent, then any function of time which coincides with f(f) at the sampling instants 0, T , 2T, 3T, . . . , has the same z-transform as f ( t ) . To put it another way, if G(z) is the z-transform of some function g(t), then the inverse of G(z), as found from the table, is not, in general, identical with g ( t ) , although it coincides with g(t) a t the sampling instants. Thus, from the z-transform of a function one can find only the values of the function a t the sampling instants. In this connection, it should be noted that the value of a time function at the nth sampling instant is equal to the coefficient of zPn in the power series expansion of its z-transform (regarded as a function of 2-1). In cases requiring numerical computations, this property of z-transforms affords an alternative, and frequently convenient, means of calculating the values of corresponding time functions at the sampling instants. I t will be helpful to summarize at this An alternative expression for R(s) can be obtained by expressing BT(t) in the form of a complex Fourier series Table I. ## z-Transform F(z) z -I) z -n 1 1-z-l Tz -1 ( 1 -rg -I)? and substituting this expression in equation l. Then, transforming the resulting series term by term there results .. .. (2).. cTs (3). 60) .. ..6(1-nT).. -1 .. .. ., (4). (51.. 6 . s+a where R(s) is the Laplace transform of r(t). It is of interest to note that the equivalence between the two expressions for R(5) .. #-at ## . a. ( l - e - Q ' ) . . ( t - - z - ' ) ( ~ - - c - a T z - ~ ) s(s+a) (1 - ~ - a T z - ~ ) z-l(l-c-T) . s2+al . 1 - ( 2 sin ~ o T ) z - ~ + z ~ ~ aTz-1 co (8). F(s+ a ) .. c-atf(t) .. . F(t-aTe) ( 9 ) .. r-sTF(s). . f ( t - T ) . . %-IF(%) ( 1 0 ) .. ea3F(s) . . f ( t + a ) . . en 'TF(t) (7). -%- . . sin al (ll).. 1 was discovered more than a century ago by Poisson, and that equation 8 is essentially equivalent to the Poisson summation rule. s - 5Tn Z +a2 s aT " a .. z-a 2 .. - cos at 62 ## stage the basic points of the foregoing discussion : 1 A sampler transforms a function r(t) . ## equation 17 assumes the following form C(s)= G(s)R(s) (19) Laplace transform of got), it follows at once from equations 7 and 18 that G(s) iis the z-transform of g(t), that is into a train of impulses, r(t)= r(i!)6~(t), where h ( t ) represents a train of unit impulses with period T. 2. The Laplace transform of r(t), R(s). is expressible in two different but equivalent formsgiven by equations 5 and 7. 3. R(s) is a periodic function of .r with period jwo, where wo is the sampling frequency. 4. The z-transform of r(t). R(z), is equal to R(s) with e" in R(s) replaced by z; that is with eST replaced by z. As a preliminary to the considerat.ion of sampled-data feedback systems, it will be helpful to establish one basic property of z-transforms. The property in question concerns the relation between the ztransforms of the output and input of the system illustrated in Figure 3. Denoting the input by r(t), the output by c(t), and the transfer function lby G ( s ) ,this relation reads where C(z) and R(z) are the z-transforms of c(t) and r(t) respectively, and G(c) is referred to as the starred transfer function. The importance of this relation derives from its similarity to the familiar relation C(s)= G ( s ) R ( s ) ,which would obtain in the absence of samplers. This similarity makes i t possible to treat ztransforms and starred transfer functions in much the same manner as the conventional Laplace transforms and transfer functions. The proof of equation 13is quite simple. From inspection of Figure 3, i t is evident that the Laplace transform of c(t) is given by C(s)= G(s)R(s) (14) which is equivalent to equation 13. I t is seen that the starred transfer function G(s), which is related to the transfer function G(s) by equation 18, may be regarded as the ratio of the z-transforms of the output and input of the sampled-data system under consideration. The mathematical essence of the foregoing discussion is the fact that a relation of the form C(s)=G(s)R(s) implies C(s)=G(s)R(s). This fact per se is very useful in the analysis of sampleddata systems involving one or more feedback loops. I n the sequel, the process of passing from equation 14 to equation 19 will be referred to as the z-transformation of both sides of equation 14. The tacit understanding exists, of course, that the quantities actually subjected to the ztransformation are the time functions corresponding to the two members of equation 14. Among the properties of the starred transfer function G(s) there are two that are of particular importance. First, suppose that an input of the form r ( t ) = eSt is applied to the system shown in Figure 3. This input is transformed by the sampler into an impulse train r(t) which in view of equations I and 6 may be written as Consequently, G(s) may be expressed in terms of the values of g(t) a t the sampling instants tn=nT or, alternatively, in terms of the system function G'(s) via equation 18. Needless to say, G(s) may be obtained directly from either g(t) or G(s) by the use of a table of z-transforms such as Table I. I t frequently happens that the system N consists of a tandem combination of two or more systems. In particular, if N consists of two networks N, and Na with respective transfer functions Gl(s) and Gz(s),then the transfer function of N is given by the usual relation ## Operating on this expression with the transfer function G(s) gives Sampling c(t) and taking note of equation 18 yields after minor simplifications the expression for the sampled response of the system to r(t) = erl; that is The important point noted here is that the starred transfer function G(s) of two cascaded linear systems Nl and Ne which are not separated by a sampler is not the product of the respective starred transfer functions Gl(s) and Gz(s)but rather a new transfer function given by equation 25. On the other hand, if N3 and N 2 are separated by a sampler, as shown in Figure 4, then from equation 19 it follows at once that the z-transform of the output (of N2 is Gl(s)G2(s)R(s). Hence, in this case the over-all starred transfer function is G(s)= G~(s)Gz(s) (27) and correspondingly, by applying equation 7 where 8T(t) denotes a train of unit impulses. I t is seen that the response has the form of a train of impulses whose envelope is the bracketed term in equation 22. More specifically, this means that G(s)dt represents the envelope of the response of the system to an input of the form c". Consequently, it may be concluded that the starred transfer function G(s) relates the input r(t) and the envelope of the sampled output c(t) in the same manner as the transfer function G(s) relates the input and output of N. Another important property of G(s) concerns the impulsive response of N which is denoted by g(t). Since G(s) is the ## Because of the periodicity of R(s), the followingidentity is noted R(s+jnwo)=R(s) (16) Consequently, it may be concluded that the over-all starred transfer function of two or more networks cascaded through samplers is equal to the product of the starred transfer functions of the individiual networks. The expressions for tht: z-transforms of the output of more complex structures such as those encountered in feedback equation l 5 reduces (17) ## Denoting the bracketed term by G(s) G(s)=- G(s+jnwo) Tn=-_ (Is) Figure 3. Pulsed linear system showing important variables and their trandorms NOVEMBER 1952 ## Figure 4. Cascaded linerr systems sewrated by srmpler C(z)= (1 - z - l ) ( l + K - e - a T z - l ) - . . . . (37) ## as) RXs) G;(s) = systems in general can be derived in a similar manner. Such expressions for several basic types of sampled-data systems are given in Table 11. In this table, the first column gives the basic structure, the second gives the expression for the Laplace transform of the output, and the third gives the z-transform of the output. I t will suffice to go through the derivation of C ( s ) and C(s) for a typical system, say number 6 in the table. On inspection of the block diagram, it is seen that the expression for the Laplace transform of the error is E ( s ) = R ( s ) -H(s)C(s) (28) . . . G;(s) ## C(z)= G(o)R(z) I +GH(z) K ( l + ~ - e - ~ ~ -z-1) )(l C(Z) = (34) ## Referring to Table I, the 8-transform of the unit step is 1 R(z)=-1-2-1 (35) finding the respective inverse transforms from Table I and combining the results yields c(t)= (l+~-e-"~) " -(CJG} 1+K (39) ## From the same table, the z-transform aswith G(s) is AG(4 = I - ~ - Q T ~ - I (36) This function coincides with the actual output at the sampling instants and hence a t the nth instant, tn=nT, the value of the response to a unit step is c ( n r )= A -UT n+l ## Substituting these in equation 34, there results after minor simplifications (l+K-c-aT){l-(lfR) } (40) where C(s) is the Laplace transform of the input to the feedback circuit or, equivalently, the z-transform of the output of the system. The Laplace transform of the output C ( s ) is related to E ( s ) by Table II. I-T~MS~OI~ System of Output C(s) ## Combining this relation with equation 28 gives C ( s )= G ( s ) R ( s ) G(s)H(s)C(s) (30) Applying the z-transformation (see equation 19 and following) to both sides of this equation results in C(s)= GR (s) -GH(s)C(s) (31) which upon solving for C(s) yields the ztransform of the output as given in Table 11. Finally, substituting this expression into equation 30 gives the Laplace transform of the output which is the expression for C ( s ) listed in the table. As an illustration of the use of the expressions given in Table 11, a typical problem involving the fourth structure shown in the table will be considered. Suppose that H ( s ) = 1 and G(s) = K / ( s + a ) , where K and a are constants, and that it is desired to determine the values of the response of the system to a unit step input a t the sampling instants. To this end, the z-transform of the output must be found. From Table 11, this is Ragazzini, Zadeh-The Analysis of Sampled-Data Systems 64 NOVEMBER 1952 A sequence of ordinates obtained by evaluating c(nT) for successive values of n yields a graph which, can be used to assess the transient performance of the system. It will be noted that the system is stable for all K and a such that a-aT<: 1+ K. A brief discussion of the question of stability will be given in a subseq,uent section. ## Variable Network Approach By employing the techniques discussed in the preceding section one can obtain, in most practical cases, an explicil. expression for the z-transform C(s) and, if need be, the Laplace transform C(r) of the output of a specified sampled-data system. The former can be used to find the values of the output a t the sampling instants. The latter may be used, in principle, to determine the output a t all times by calculating the inverse Laplace transform of C(s). I n practice, however, the inversion of C(s) is difficult because C(s) is a rational function in both s and eqT and no tables of inverse transforms for such functions are available. An alternative approach which works quite well in those cases where an approximate expression for the continuous output-and not just its values at the sampling instants-is desired, is based on treating a sampling system as a periodically varying linear network. This approach is developed in the sequel, following a brief introductory discussion of the frequency analysis technique of handling time-variant systems.14 In using the frequency analysis technique, a linear time-variant system N is characterized by its system function K(s;t), which is defined by the statement that ~(s;t)e" represents the response of N to an exponential input 6". If the system function of N is known, then the response of N to an arbitrary input r(t) can be obtained by superpositilon. More specifically, the output is given by where oo=2?r/T and the K,(s) represent the coefficients of the series. Thus, in the case of a periodically varying network, the problem of determination of K(s;t) may be reduced to that of finding the coefficients of the Fourier series expansion of K(s;t). I n practice, a few terms in equation 42 usually provide an adequate approximation to K(s;t), so that in many cases only two or at most three coefficients in the Fourier series expansion of K(s;t) need be determined. (Note that K-,(s), is the conjugate of K,(s) since K(s;t) is a real function of time.) Once K(s;t)--or, rather, an approximation to it-has been determined, the system function K(s;t) can be used in the conventional manner for the purpose of obtaining the response of N to a specified input, for the investigation of the stability of N , for the calculation of the mean-square value of the response of N to a random input, for the determination of the ripple in the output, and many other purposes that are not pertinent to the present analysis. The application of the general approach just outlined to the analysis of a sampleddata system is quite straightforward. For simplicity, the third system in Table I1 will be considered first. An input of the form 8' is transformed by the sampler into a series of exponential terms which may be written as Substituting these expressions in equation 41 and performing the inverse Laplace transformation with the help of a standard table of Laplace transforms, with- t treated as a constant, one readily obtains X a:t)=-(l-~-~~)+-T(a2+tro2) Ta [a cos q t + o o sin wet - a ~ - @ ~ (47) ] 1 which is the desired expression for the output. Turning to feedback systems, consiider the fourth system in Table 11. In this case, i t is expedient to obtain first the expression for e(t) corresponding to an e.xponentia1 input r(t) = ePL. In view of e'quation 22, this is where the second factor represents 6T(t). To deduce X(s;t) from this expression, it is sufficient to find the response of the forward circuit, characterized by G(s), to eac(t)and divide the result by es'. This yields which is in effect a complex Fourier series expansion of K(s;t), with the coefficient of dwd being ## The response of the network N (following the sampler) to this input is I t will be noted that, as should be expected, at the sampling instants t,= mT, K(s;t) reduces to where G(s) is the system function (transfer function) of N. Consequently, from the definition of the system function K(s;t) of the over-all system, it follows that where 2 - I represents the inverse Laplace transformation and R ( J )is the Laplace transform of r(t). The variable t in K (s;t) should be treated as if it were a parameter. This implies that in evaluating the inverse Laplace transform of K(s;t) R(s), one may use a standard table of Laplace transforms and treat t in K(s;t) as a constant. When N varies in time with period T, its system function K(s;t) is likewise a periodic function of time with periocl T. Consequently, K(s;t) may be expanded into a Fourier series of the form which is in effect a complex Fourier series expansion of K(s;t). I t is seen that the coefficient K, (s) of ein"o' is equal to l / T G(s-tjnwo). As a simple illustration of the use of this expression suppose that the input is a unit-step, that G(s)=l/(s+a), and that coo is such that K(s;t) is adequately approximated by the first two terms in equation 45. In this case, R(s)= l/s and which will be recognized as the starred transfer function of the over-all system. In the case under consideration, the determination of the responsle of the system to a given input is complicated somewhat by the fact that the denominator of K,(s) is a rational function in esT,rather than in s. For purely numerical computations this is generally not objectionable. HOWever, in analytical work it is usually necessary to approximate the term GH (s) in equation 50 by a few terms in its expansion In, this way, one obtains a rational function approximation to Kn(s). I n the final analysis, the results ob- ## :i,Zadeh-The Analysis of Sampled-Data Systems ,..- tainable by the method outlined in this section are also obtainable, although less conveniently, from the expression for the Laplace transform of the output C(s). The chief advantage of the system function approach is that the function K(s;t) is essentially a time-varying transfer function and as such provides a clear picture of the state of the system a t each instant. mean-square value of the output a t any sampling instant. Sometimes it is more convenient to express u2 in terms of the autocorrelation function2 of the input, rather than in terms of its power spectrum. It can be shown readily that, in terms of the autocorrelation function of the input, \$(T), u2 is given by the following expression ## Response to a Random Input The expression for the system function obtained in the preceding section has an immediate application in connection with the important problem of determining the statistical characteristics of the response of a sampled-data system to a random input. Since a general discussion of this problem is outside the scope of the present paper, the following analysis is limited to the case where the input is a stationary time series, and it is desired to obtain the expression for the mean-square value of the output a t a specified instant of time. This particular problem has considerable bearing on the design of sampled-data systems that are optimum in the sense of the minimum rrns error criterion. A general expression for the meansquare value of the output of a timevariant system is given in reference 14; it reads where k(t) is the inverse z-transform of K(z). This expression is useful chiefly in those cases where the autocorrelation function +(T) drops off rapidly with increase in T. sampling instants t, and t,+l, from its values at the preceding sampling instants t,, &-I, tn-2, ..., is essentially a problem in extrapolation or prediction. An effective, though not optimum, method of generating the desired approximation is based on the consideration of the power series expansion of r(t)in the typical interval from t,=nTto t,+l=(n+l)T ## The Hold System As stated previously, the function of the hold circuit is to reconstruct approximately the original time function from the impulse train generated by the sampler. I t is evident that if it were possible to realize a perfect hold circuit, a sampleddata system incorporating such a circuit would be identical with a continuous-data system. However, in general, a perfect hold circuit is not realizable because of the random nature of the time function which has to be reconstructed. Furthermore, a very important consideration in the design of hold circuits is the fact that a close approximation of the original time function requires, in general, a long time delay, which is undesirable in view of its adverse effect on the stability of the system. Consequently, the design of a hold circuit involves a compromise between the requirements of stability and over-all dynamic performance on the one hand, and on the other hand, the desirability of a close approximation to the original time function and the reduction of ripple content in the output of the system. It should be remarked that the hold circuits commonly employed in practice are generally of the so-called clamp type, which is one of the simplest forms of hold circuits. More sophisticated types of hold circuits based on the use of polynomial interpolating functions have been described by Porter and S t ~ n e m a n . ~ A complete treatment of hold circuits cannot ignore the random nature of the time function which the hold circuit is called upon to reconstruct. Such a treatment is outside the scope of the present paper. The brief discussion which follows is concerned chiefly with some of the more basic aspects of hold circuit design. The generation of an approximation to the original time function between two where the primes indicate the derivatives of r(t) a t t, = nT. To evaluate the coefficients of this series i t is necessary to obtain the derivatives of the function r(t)at the beginning of the interval in question. Since the information concerning r(t) is available only a t the sampling instants, these derivatives must be estimated from the sampled data. For instance, an estimate of the first derivative involving only two data pulses is given by rl(nr)=-(r[n~]-?[(la-l)~]) 1 (57) ## and the second derivative is given by where S(w)is the power spectrum function of the input; KCjw;t)is the system function with s replaced by jw; and u2(t) is the mean-square value of the output a t a specified instant t. To apply this equation to a sampled-data system it is only necessary to substitute the expressions for the system function of the system and the power spectrum of the input into equation 53 and carry out the necessary integration. I n the majority of practical cases, the integration in question is most readily carried out by graphical means. When the specified instant of time t coincides with a sampling instant, t=nT, the formula given above assumes a much simpler form. Thus, as was shown in the preceding section, for t =nT the system function K(s;t) becomes identical with the starred transfer function K(s) of the over-all system. Consequently, equation 53 reduces to where K(jw) is the starred transfer function with s replaced by jw; and a2is the Thus to obtain an estimate of a derivative of r(t) the minimum number of data pulses which must be considered is equal to the order of the desired derivative plus one. This implies that the higher the order, the greater the delay before a reliable estimate of that derivative can be obtained. For this reason, an attempt to utilize the higher order derivatives of r(t) for purposes of extrapolation meets with serious difficulties in maintaining system stability. Generally, only the first term in equation 56 is used, resulting in what is sometimes described as a boxcar or clamp circuit but which will be referred to here as a zero-order hold system. More generally, an nth order hold system is one in which the signal between suecessive sampling instants is approximated by an nth order polynomial. Considering the zero-order hold system, it is evident that its impulsive re- ## i, Zadeh-The Analysis of Sampled-Data Systems 66 NOVEMBER 1952 sponse h(t) must be as shown in Figure 5 . BY inspection, the Laplace transform of this time function is seen to be The margins by which such an enclosure is avoided constitute a measure of the dam^ing of the system. Other more sophisticated techniques may be used if a better dynamic performance is desired. In view of the fact that the starred The frequency response of this hold sys- transfer function is analogous to the conventional transfer function, the same tem is obtained by replacing the complex frequency s by j w , resulting in the follow- basic technique may be applied in the case. of sampled-data control systems. ing expression for the magnitude of the The function which is plotted is GH(s) or transfer function G(s)H(s) according as the denominator of the z-transform of the output is of the IHGUI! sin form l+GH(s) or l+G(s)H(s), see This relation is plotted in Figure 6. It is Table 11. As usual, the complex frequency s is varied along a contour in the sobserved that the hold system is essenplane consisting of the imaginary axis and tially a low-pass filter which passes the low frequency spectrum of the impulse a semicircle enclosing the right-half of the train and rejects the displaced high-fre- plane. Since the starred transfer funcquency spectra resulting from the sam- tions are periodic with period jwo, the loci of the functions GH(s) or G(s)H(s) repling process. An important property of trace themselves a t each cycle, so that s the zero-order hold system is that the ripneed be varied only from -jwa/2 to joo/2 ple a t the output is zero if the input is a in order to obtain the shape of the locus. constant. In a similar way, the first order Critical regions such as the vicinity of the hold system has zero ripple output for an origin in the s-plane are handled in the input function whose dope is a constant. same manner as in the case of continuousAs is well known, if the signal does not contain frequencies higher than one-half data systems. Since the auxiliary variable z is dethe sampling frequency, perfect reprofined as cTS, it is evident that as s is asduction of the signal is obtained with an signed imaginary values over one comideal low-pass filter (ithat is, one with plete cycle, z traces a unit circle in the zunity gain and linear phase shift up to its plane. Thus, the loop transfer function cutoff frequency) whose cutoff frequency GH(s) can be plotted directly by expressis equal to one-half the sampling frequency. For that matter, any low-.pass ing it in the form GH(z) and varying a along the unit circle. To demonstrate the network having roughly this characteristechnique, the GH(z) locus for the systic can be used to extract most of the usetem shown in Figure 7 will be plotted. ful spectrum from the impulse train. It is This system is seen to consist of a zeroeven possible, though not advisable, to order hold circuit and a simple linear dispense with the hold circuit altogether, component in the forward circuit. The and rely on the low-pass characteristic:^ of the forward circuit to perform the neces- feedback transmission is unity so that the loop transfer function is sary smoothing of the sampled data. ## which can be simplified to =: 1 \$1 This function is plotted in Figure 8 where it is seen that for the constants chosen, the system is stable; but if the loop gain is increased by a factor of 1.5, the system becomes unstable. If it were desired to improve the margin of stability or, for that matter, stabilize this system with a higher loop gain, the procedure would be to add lead networks just as in the case of a continuous-data system. The major difficulty encountered in this procedure is that the resdting starred loop transfer function is not related in a simple manner to the original function, for as was shown previously the starred transfer function of two networks in tandem is not equal to the product of respective starred transfer functions. Consequently, the insertion of a corrective network in the feedback loop requires the recalculation of the starred loop transfer function. The need for recalculation is inherent in the stabilization of sampled-data systems by the insertion of a corrective network irk the feedback loop. Thus, as the art exists a t the present time, the shaping procedure involves essentially a trial and error method with the plot of the Nyquist diagram for each trial set of system parameters used to assess the stability of the system. Conclusions The sampled-data feedback control system may be analyzed in a systematic m,anner by applying the z-transform method. Techniques of locus shaping similar to those commonly used in continuous systems may be applied by plotting the starred loop transfer function on the complex plane. This is done with no approximations other than those relating to the narrowness of the pulses constituting the pulse train. Once satisfactory loci are obtained by the addition of ap- System Design Using z-Transfornu The primary objectives in the design of closed-cycle control systems include the achievement of stability and an acceptable over-all dynamic performance. 'The standard technique of achieving these objectives consists in plotting the Nyquist diagram of the loop transfer function and adjusting the system parameters until the plot does not enclose the point -1.0. Expanding this transform into partial fractions, there results On considering each term separately and obtaining the corresponding z-transforms from Table I, the starred loop transfer function is found to be lHI(i~)l h I \ 7 (right). HOLD COt44:JEt\$s Figwe 8. ## Plot of HG(z) locus for system used in example the output can be estimated by use of the variable network analysis described in this paper. One complication i n t h e design of sampled-data systems is the relative difficulty of evaluating the effect of inserting corrective networks i n the conk01 loop. If this complication could be removed, locus shaping would be no more difficult than with continuous systems. Research now i n progress is directed toward the devising of practical methods of locus shaping i n the loop transfer function plane. Results t o date indicate t h a t the z-transform method, in conjunction with design techniques somewhat analogous t o those used with conventional servomechanisms, furnishes a powerful tool for the analysis of linear sampleddata systems. 4. ANALYSISOF DISCONTINUOUS SERVOYGCHANISMS, H. Raymond. Annaks de Telccommuni. F. cation (Paris, France), volume 4, 1949, pages 25056, 307-14,347-57. 6. THEORY INTERMITTENT OF REGULATION, 2. Y. Tsipkin. Avfomalika i Telemekhanika (Moscow, USSR), volume 10, number 3, 1949, pages 189-224. 6. THE DYNAMICS OF AUTOMATIC CONTROLS (book), R. C. Oldenbourg, H. Sartorius. American Society of Mechanical Engineers New York. N. y , , 1948, chapter 5. 7. A NEW APPROACH THE DESIGN PULSETO OF MONITORED SERVO SYSTEMS, Porter, P StoneA. . man. Journal, Institution of Electrical Engineers (London. England). volume 97, part 11, 1950. pages 597-610. 8. A GENERAL THEORYOF SAMPLING SERVO SYSTEMS, F. Lawden. Proceedings. Institution D. of Electrical Engineers (London, England), volume 98, part IV, October 1951. pages 31-36. APPL~CAT~ON OF FINITE DIFFERENCE OPERALINEAR SYSTEMS, M. Brown. ProceedB. ings of Ihe D.S.I.R. Conference on Aulomatic Control. Butterworths Scientific Publications (London, England), 1951. TORS TO 9. ## K. S. Miller, R. J. Schwarz. Journal of Applred Physics (New York, N . Y.), volume 21, April 1950, pages 290-294. 10. ANALYSIS OP SAMPLING SERVOMECHANIS.\I, pmpriate networks, the transient performance of t h e system can be assessed by obtaining the time functions from a table of z-transforms or b y expanding the z-transform of the output function into a power series which gives t h e ordinates a t the sampling instants. The smoothness of 11. THEORIE ANALYTIQVE DES PROBABILITES, PART 1: Du CALCUL DES FONCTIONS GENERnrRxcEs (book), P. S. Laplace. Paris, France, 1812. 1. FUNDLIENTAL THEORY SERVOMECHANISMS OF LAPLACE TRANS12. A LIST OF GENERALIZED (book), L. A. MacColl. D. Van Nostrand Comaonns, W. M. Stone. Journal of Science, Iowa pany, Inc., New York, N. Y., chapter 10, 1945. State College (Ames, Iowa), volume 22, April 1948, pages 215-225. 2. THEORY SERVOMECHANISMS H. M. OF (book), James, N. B. Nichols, R. S. Phillips. McGraw13. TRANSIENTSN LINEAR SYSTEMS(book), I Hill Book Company, Inc., New York, N . Y., M. F. Gardner, J. L. Barnes. John Wiley and chapter 5, 1947. Sons, Inc., New York, N. Y., 1942, pages 354-356. ANALYSIS OF VARIABLE NET3. SAMPLED-DATA CONTROL SYSTEMSSTUDIED 14. FREQUENCY WORKS, L. A. Zadeh. Proceedings, Institute of THROUGH COMPAR~SON OF SAMPLING WITH AMPLIRadio Engineers (New York, N . Y.), volume 38, TUDE MODULATION, K. Linvill. A I E E TransacW. March 1950. pages 291-299. fions, volume 70, part 11, 1951, pages 1779-88. References Discussion John M. Salzer (Hughes Aircraft Company, Culver City, Calif.): The objectives of this paper are: (a) to unify the z-tramform (Shannon, Hurewicz) and s-transform (Linvill) approaches to sampled-data systems; (b) to formulate input-output relations for various types of such systems; and (c) t o treat such systems by considering the sampler a time-variant element. The paper fills a clear need in this underpublished field, and it presents its topic concisely and illuminatingly. I t brings together two viewpoints, (a) and (c), to bear on the same problem and shows the way toward systematization (b) of the solution. Perhaps the most significant contribution of the paper is the variable network approach. Although the applicability of this approach was recognized before,' the authors' more recent investigations in the general field of time-varying systems are now made to bear directly on the problem of sampled-data systems. The importance of the generalization thus afforded should not be overlooked. It is interesting to note that the variable network approach leads directly to the same expressions as the s-transform method. Both methods are predicated on the concept of using an arbitrary, characteristic input function [r(t) = c S t ] , but by distinguishing a time-variant transfer function certain generalizations are made possible, as already noted. An interesting illustration of such a generalization is given in the section titled "Response to a Random Input." There are several points and results in this paper which bear interesting relations to some of the work done by the discussor in connection with digital-analogue system^.^ The use of starred s-transforms, such as R(s) , in place of infinite sums, is a welcome convenience.3 This notation facilitates the manipulations in dealing with sampled-data systems and makes the systematizations of Table I1 of the paper easier to comprehend. As noted in the paper, in the case of starred transforms (that is, in the case of sampled functions) the s- and z-transforms are exactly equivalent. Thus, the use of ztransforms involves no approximation, and it offers certain conveniences. One advantage is notational, for z is easier to write than sEt. The other is that the infinite number of poles and zeros of the starred stransforms are replaced by a finite set in the z-plane. Furthermore, the ambiguity a t the point of infinity in the s-plane (due to essential singularity) is circumvented by the use of z-transforms. In case the s-transform is not a starred transform, information is lost by the use of z-transform because this amounts to representing a continuous function, c ( t ) , by its samples, c ( t ) , just as it is done in numerical mathematics. But whereas in numerical work the sampling (or tabular) interval may be adjusted until such a representation is justified, in analyzing a given control system one is faced with a sampling rate already determined and the analysis must be made correct for the existing physical situation. For particular systems the sequence of output samples may not give a satisfactory picture so that it becomes necessary t o study the behavior of the output also between sampling instants. Furthermore, and this is important, it is not always a priori obvious whether this is or is not the case. Where the output behavior between samples is also of interest, the z-transform method is still applicable but must be augmented by separate investigation of the output during the sampling period. A separate solution based on initial conditions at the sampling instant may be used in analysis, but a synthesis procedure would hardly be fruitful along these lines. The alternative solution is the s-transform approach. As noted by the authors, this method leads to transforms which are products of rational functions of s and Z, and tables of corresponding transform pairs are not available. Nevertheless, the exact analysis is straightforward, even if sornewhat laborious, and moreover, the frequency characteristics obtained may give a hint as to the nature of compensation needed to improve the response. Whether the z-transform method is applicable or not in a particular case depends on the question of bandwidth. If the sampling rate is many times higher than the bandwidth of the input or of the system, then the z-transform solution is expected to be a suitable representation of the continuous output. I t is presumed that the applications with which the authors concerned themselves were of this type. However, In the design of certain systems it is often desirable t o use the lowest permissible Sampling rate consistent with the specifications. In finding this limiting rate, one does not ## Analysis of Sampled-Data Systems NOVEMBER 1952 get a complete answer by the use of only ztransforms. I t may be noted that as far as stability is concerned either the z- or the s-transf0r.m~ lead to exactly the same result, for only a divergent continuous function has a divergent sequence of samples. That this is so is demonstrated in Table I1 of the paper, where the corresponding denominators in both the s- and z-transform columns are identical. To the numerous examples of sampleddata systems mentioned by the authors, t:he discussor wishes t o add one: systems in which a digital computer is incorporated. As generally conceived today, digital computers operate on sampled data; therefore, their presence in the system requires sampling. Since the output of the digital computer. is sampled also, the comput'er fits in between the sampler and the holding unit. If the digital computer is instructed to perform a linear difference equation on its sampled input, it can be represented by a transfer function which is rational in z. This result ties in with the methods of the paper. For example, in the case of a digital computer equation 27 of the paper applies, because the data stay sampled through all numerical work. Thus, the transfer function of a composite digital program equals the product of the transfer functions of the component programs. Tbe implication of this fact in system design is tb be noted. Suppose that the stability of the system illustrated in Figure 8 of .the paper is to be improved by a digital compensator, W L ( s ) . inserted in either the forward or the feedback section. Since REFERENCES dNALyas AND OF SAMPLED DATA CONTROL SYSTEMS (thesis), W. K. Linvill. Project Whirlwind Report R-170, Massachusetts Institute of Technology (Cambridge, Mass.). 1949. AND They also have another important p r o p a t 9 of b'eing time-invariant, which means that if W ( t )= ~ ( t ) then @~(t+to) , =x(t+to). where tois any constant. Because of this, they are commutative. They can be, and are in NUMERICAL PROCESSES THE FREQUENCY IN DOMAIN (thesis), John M. Salzer. Digital Cornof and trains' A.nother operator S, the sampling operaputer Laboratory (microfilmed report), Massachusetts Institute of Technology (Cambridge, tor, now is introduced whose effect is t o ConMass.), 1951. veri: a continuous function into a train of 3. Compares with the notation R(s) used in reference 2; for example, pages 56 and 176. - ~.r B. M. Brown (Royal Naval College, London, England): An alternative technique is available for handling the general theory and problems of the type discussed in this paper. Instead of using Laplace transforms. which imply a limitation to input and output functions which are zero for negative time, this technique assumes functions of general type, and uses as transfer functions operators which are functions of the operator D =d/dt. Such a function F ( D ) is usually a fraction, in which numerator and denominator are polynomials in D. Thus if input and output, u and x respectively, are connected by the relation x = F ( D ) u , where F ( D ) =P ( D ) / Q ( D ) , then x is understood to denote the general solution of the differential equation pulses. I t is of course equivalent to multiplication by 8T(t) in the notation of the paper. I t easily is seen that S is linear, but not time-invariant and not in general cornmutative with functions of D , E, or A. It will be shown briefly how the various processes described in the paper can be represented operationally. Consider first the effect of sampling a function u ( t ) and then passing it through a network with transfer function F ( D ) . The resulting function is F(W)Su(t) which is in general a continuous function. If this is sampled, either physically or for the purpose of analysis, the result is the pulse train S F ( D ) S u ( t ) . To determine the relation between the two pulse trains, let f ( t ) be the response to a unit impulse input to the network. Then WHG(z)= W(z)HG(z) the new stability diagram is directly related to the old one so that the synthesis procedure is facilitated greatly. Of course, this is not meant to imply that digital compensation can do a better job than analogue; it only means that it is easier to see what a digital unit does. A somewhat academic example of digital compensation is work.ed out in chapter 4 of reference 2 of this discussion. There is one comment concerning notation which may be found of interest: namely, it may be preferable to define z as being equal to rather than when dealing with sampled-data systems. This is so because the latter corresponds to a time-advance operation, which has no physical meaning in a real-time application. In purely mathematical work: one definition is as good as the other, andl it is just unfortunate that in previous operational and transform work with difference equatiolns the advance rather than the delay (e-") operator was given a symbol. In consequence of this choice, z naturally will a.ppear raised to negative powers, thereby diminishing the manipulative advantage of its usage. Table I of the paper il1ustrai:es this point. Of course, multiplication of bath numerator and denominator by an appropriate power of z always can eliminate the negative powers of z (as was done in line 11 of Table I), but this is an additional step. There are further reasons for which should be regarded the fundamental variable. For instance, in the investigation of stability of digital programs by conformal mapping the use e W S T leads to much simpler rules that that of d S T . Such operators have provided a classical method for solving linear differential equations. If the coefficients are constant the operators in general can be manipulated algebraically, and this type of manipulation proves t o be a very powerful tool. If a linear system has a transfer function F ( D ) , then the stability is determined by the roots of Q(X)=O. A steady-state solution can be obtained by expanding F ( D ) in a series of ascending powers of D and operating on u . I n many cases an adequate approximation is given by the first term of this series, which always can be written down by inspection. Thus if F ( D ) is given, the main characteristics of the response can be inferred without the labour of evaluating particular solutions t o particular inputs, whether by Laplace transform or other methods. Operators of the type F ( D ) have parallels in the form of functions of the operators E and A, witha complete set of analogous properties. These operators are defined by = C Ckf n (k~)u[(n-k)~]~(t--n~), putting n-m=k The: operator in brackets is a function of E. If it. is denoted by F L ( E )we have the operational relation where and l/(D+a). ## Consider the special case when F ( D ) = Then f(t)=e-=I, so that where T is constant. The three basic operators are connected by the relations As in the paper, the general operator can be dealt with by using partial fractions and a table similar to Table I. I t is now apparent that the operator E plays a part analogous to that: of the transfornl variable z, the association being similar to that of D and s. I t will be found further that most of the equations involving starred and unstarred functions of s given in the f ~ s part of the paper can be expressed in t operational form. In particular, equation 1 of this discussion corresponds t o equation 13 the latter being obtained by Taylor's theorem. Functions of E and A can be used for solving difference equations and for discussing systems based on such equations. A short account of the appropriate methods is given in reference 9 of the paper. Now operators of the types F ( D ) , F ( E ) ; and F ( A ) are all linear, which is to say that, denoting such an operator by @ NOVEMBER1952 ## Analysis of Sampled-Data Systems 69 of the paper, while the equivalent of equation 27 in the paper would be written in the operational form I t is not easy to make a comparison of the relative merits of the two alternative approaches. So much depends on the notation with which a particular individual is familiar. I t was claimed earlier that the use of operators implies greater generality, but this is perhaps of small account from a practical point of view. However, it may be an advantage to represent the operation of sampling by a special symbol. As an example of the use of operators, consider system 4 in Table I1 of the paper. The circuit equation is easily seen to be ## G(D)S[r(t) H(D)c(t)J c(t) = Operating with H ( D ) and sampling (5 ## HG(E)[Sr(t) SH(D)c(t)] SH(D)c(l) = place transform methods are equivalent and that he can change from one t o the other as the situation demands is mathematically correct as well as convenient. Freedom in changing from the z to the s domain is often particularly convenient in obtaining time responses. For example, the result of equations 46 and 47 of the paper can be seen from elementary considerations in the frequency domain. The sampler is an impulse modulator. Since G(s)represents a low-pass filter, only the pure signal and the pair of lowest frequency side bands from the sampler result in significant output. The pure signal input to the filter G(s)is a step of amplitude 1 / T and the two low-frequency side bands combine to form a cosine wave: 1/ T(ej"o'+ l - jW Ot)=2/T cos oot for t>O. The output resulting from the step is ( l / a T ) ( I - E - ~ ' ) and the output from the cosine wave is calculated from elementary transient theory to be 2/T(a2+wo2) [a cos wot+wo sin wot] for the steady-state component and -2ae-at/T(a2+w~2)for the transient. Superposing all the output components gives so that ## 1 2 c(t) = -(1 -e-at)+ Ta T(a2+wo2) [a cos 0\$\$00 sin wd-ae-"'1 The statement that the Laplace transformation procedure requires more labor than the other methods should be modified. To calculate the exact continuous time response is more laborious than to calculate a sequence of samples regardless of the method. The Laplace transform approach embraces both the s-domain and the z-domain pictures and from it the engineer can calculate the sampled response as easily as he can calculate it by any other method, but in addition he can get continuous signals exactly a t the cost of considerable labor. Procedures have been worked out for obtaining simply an approximate continuous response and they will be described in a forthcoming paper. ## Substituting in equation 5 and simplifying gives and analysis based on the Laplace transformation which is best suited for applications in which it is sufficient to know the values of the output a t the sampling instants. BY contrast, Dr. Linvill's approach leads to the expression for the Laplace transform of the continuous output, and does not yield directly the values of the output a t the sampling instants. The method described in our paper achieves a unification of these approaches in the sense that it furnishes a systematic procedure for determining both the Laplace and z-transforms of the output, 1 as illustrated in Table 1 . The connecting link between the two approaches is contained in the statement that a relation of the form C(s)= G(s)R(s),where C(s) and G(s) are ordinary Laplace transforms and R(s) is a starred transform (that is, z-transform), implies the relation C(s)= G(s)R(s). As we have pointed out in the paper, the expression for the output obtained by the use of the variable network approach also may be obtained from the expression for the Laplace transform of the output, and, in simple cases, the same results may be derived from elementary considerations in the manner indicated in Dr. Linvill's discussion. The main advantage of the variable network approach is that it yields the expression for a time-varying transfer function R ( s ;t ) which constitutes a much more explicit and flexible means of characterizing a system than the expression for the Laplace transform, C(s) of the response to some particular input. Furthermore, K ( s ;t ) is, in general, moreconvenient to work with than C(s). For example, one can readily express the meansquare value of the response t o a random input in terms of K ( s ;t), but not in terms of I t is of interest to observe that the process of clamping can be represented by the operator (E-lA/D)S. To prove this it is only necessary to point out that a clamped function is the integral of the sequence formed by the first differences of the sampled function. William K. Linvill (Massachusetts Institute of Technology, Cambridge, Mass.): This paper makes a concise mathematical summary of the analysis of sampled-data systems. Table I1 is particularly helpful and illustrates the applicability of the analysis to a wide variety of system configurations. The easy interchangeability between the ztransform method and the Laplace transform method should receive more emphasis than the paper gives. When the sampler output is considered to be a train of modulated impulses rather than a sequence of ordinates of the sampler input, the sampler has all the properties of the familiar pulseamplitude modulator and can be treated by conventional Laplace transforms. When the whole system is treated in the frequency domain from this point of view, there is no difference between the z-transform approach and the "old-fashioned" Laplace transform approach other than a change in variable z=eST. When the signals a t any point are discrete samples, their transforms are periodic and the . z ( = \$ ~ ) variable is the convenient one to use. When the signals a t a point are continuous, their transforms are aperiodic and the s variable is the convenient one to use. The attitude on the part of the engineer that the z-transform and the LaI (7s). With reference to Dr. Brown's statement that the use of Laplace transforms implies limitation t o input and output functions which vanish for negative time, i t should be noted that, when such is not the case, it is merely necessary t o employ the Fourier or bilateral Laplace transforms in place of the unilateral Laplace transforms. Thus, the applicability of the methods described in our paper is not restricted t o input functions which vanish for negative t. The operational approach presented by Dr. Brown is related t o the z-transform approach in much the same manner as Heaviside's operational calculus is related to the conventional Laplace transformation. In particular, the operator E corresponds to z, the relation S F ( D ) S = F(E)S is equivalent to the relation C(s)= G(s)R(s),and the operation with a sampling operator S corresponds to the z-transformation. I t is of interest to note that the operators employed by Dr Brown may be regarded as special forms of so-called time-dependent Reaviside operators. A useful feature of the operational approach is that it places in direct evidence the operations performed on the operand. HOWever, working with operators is more difficult than with z-transforms, since the latter require only purely algebraic manipulations. This is indeed the chief advantage of the 2transform method. J. R. Ragazzini and L. A. Zadeh: The authors wish t o thank Dr. Salzer, Dr. Linvill, and Dr. Brown for their constructive discussions. With regard to Dr. Salzer's comment on the variable network approach, it should be noted that the main feature of this approach is the characterization of a sampled-data system in terms of a transfer function K ( s ;t ) which involves both frequency and time. Such transfer functions do not appear in the report referred to by the discusser.' In defining z as rather than ecST, we have been motivated first by a desire to avoid conflict with the notation used by W. Hurewicz and others, and second by the fact that the alternative choice would make it inconvenient to use the only extensive table of z-transforms now available, namely, the table of so-called generalized Laplace transforms compiled by W. M. Stone. Otherwise, we are in complete agreement with Dr. Salzer's suggestion that it would be preferable to define z as being equal to e-ST rather than E + " ~ . With regard to Dr. Linvill's statement to the effect that there is no difference between the z-transform and the conventional LaIjlace transform approaches, we believe that it would be more precise to say that the z-transform approach is a technique of 1. ## See reference 1 of John M. Salzer's discussion. 2. TIME-DEPENDENTHEAVISIDE OPERATORS, L. Journal of Malhemalics and Physics (Cambridge, Mass.), volume 30, 1951, pages 73-78 NOVEMBER 1952 ## IWhen Is a Linear Control System Optimal? R. E. KALMWN Research Institute for Advanced Studies (RIAS), Baltimore, Md. The purpose of this paper i s to formulate, study, and ( i n certain cases) resolve the Inverse Problem of Optimal Control Theory, which is the following: Given a control law,find all petformance indices for which this control law i s optimal. Under the assumptions of ( a ) linear c0nstan.t plant, (b) linear constant control law, (c) measurable state variables, ( d ) quadratic loss functions with constant coeficients, (e) single control variuble, we give a complete analysis of this problem and obtain varwus explicit conditions for the optimality of a given control law. A n interesting feature of the analysis is the central role of frequency-domain concepts, which have been ignored i n optimal control theory until very recently. The discussion is presented i n rigorous mathematical form. The central conclusion is the following (Theorem 6): A stable control b w is optimal if and only if the absolute value of the corresponding return difference i s at least equal to one at all frequencies. This provides a beautifully simple connecting link between modern control theory and the classical point of view which regards feedback as a means of reducing component variations. Reprinted with permission from J Basic Eng., vol. 86, pp. 51-60, IVlar. 1964. . lntroduclion Tl~cseprescriptions are usually obtained from experimental analog-computer studies. They are rigorously confirmed by the present investigation only in the case of high loop gains, see (3) above. When n high loop g:tin is not possible or not desired, the ol)ti111:~1 control law will nlways depend to some extent on the plant which is being controlled and no universal prescriptions can be made. In such cases, it is strongly recommended that the calculation of the control law proceed by the standard methods of opti~nal control theory, which take into account the properties of the plant in a quantitative way 15). matrices, lower-case letters vectors, and lower-case Greek letters scalars.) The control law is given by Acknowledgments The research, of which this paper forms a part began as a result of a conference with Dr. E. B. Stear of the Aeronautical Systems Division, Air Force Systems Command, Mr. P. A. Reynolds, of the Cornell Aeronautical Laboratories, and their associates. The idea of using frequency-domain methods arose during conversations with Professor S. Lefschetz. Many improvements in the paper were suggested by the author's colleagues a t RIAS, particularly Dr. A. A. Frederickson. The work was supported in part by the U. S. Air Force under Contracts AF 49(638)-1026 and AP 33(657)-855 and by the National Aeronautical and Space Administration under Contract NASr-103. where k is a real, constant n-vector of feedback coefficients, and (k, z) is the inner product. By abuse of language, it will be often convenient to talk of a "control law k" instead of the "confmkkw -(k, z)." Substituting the control law (2) into ( I ) we obtain the free differential system representing the plant under closed-loop control: Here the prime denotes the transpose. (The inner product (k, z) may be written also as k'z.) The abbreviation Fk = F gk', where k is any n-vector, will be used frequently. The resolvent of the matrix F will be frequently used. This of matrix-valued fi.~nction the complex variable s is defined by Q(s) = (sZ - F)-I ( I = unit matrix). (4) ## Subdivision of the Problem Inlplementing the plan outlined in the introduction will require a rather lengthy analysis. But the details of the arguments must not be allowed to obscure the ideas involved. For this reason, it is convenient to break up the discussion into six separate problems. The study of these problems will lead to the solution of the Inverse Problem of Optimal Control Theory under certain clearly defined conditions. Problem A. What optimization problems lead to a constant, linear control law? Problem B. How is this optimal control law explicitly computed? Problem C. When is the optimal control law stable? Problem D. What algebraic conditions are necessary and sufficient for the existence of a constant, linear, stable (optimal) control law? Problem E. What is the most convenient form of the preceding conditions? Problem F. What can be said about the pole-zero pattern of the closed-loop transfer function of an optimal system? These problems are intentionally vaguely phrased; merely a guide to, and not the main object of, the following analysis. Clearly every element of 9(s) is a rational function of s. 9(s) may be regarded also as the formal Laplace transform of the matrix function exp Ft (7). With the aid of (4), it is easy to express transfer functions in vector-matrix notation. For instance, let q be the scalar k'z. Then the transfer function from p to q is given by We shall repeatedly use the notation #(s) = det (sZ - F ) for the characteristic polynomial of the matrix F. I n analogy with F, = F - gk' we have then also the notations ak(s) = (81 - Fk)-I and gk(s) = det(sZ - Fk). It is frequently necessary to manipulate these frequencydomain expressions in the same way as one manipulates tranafer functions in elementary control theory. To illustrate the algebraic steps involved, let us note some common formulas: &(s) = del(sZ - F + gk') det[(sZ - F ) - ( I + @(s)gkl)l where the last step follows by a well-known matrix identity. Thus we have an explicit expression for the rational function ## Mathematical Description of the Plant I t is assumed that the plant (or control object) is a finite-dimensional, continuous-time, constant, linear dynarnical system [Ill. I t is also assumed that only a single input to the plant is available for control purposes, and that all control variables can be measured directly (61. (Assumptions (a-c) in the Introduction.) The purpose of control is to return the state of the plant to the origin after i t has been displaced from there by some extremal diiturbance: we are concerned with a regulator system. The control law is constant and linear. (Assumption (d) in the Introduction.) The precise mathematical form of the preceding assumptions is: The behavior of the plant is represented by the differential system where z is a real n-vector, the state of the plant; p(t) is a continuous, real-valued function of time, the control function; F is a real constant n X n matrix; and g is a real constant n X 1matrix, i.e., an n-vector. (Similar notations will be used throughout the paper without special comment. I n general, capitals will denote kf9(s)g ie the so-called return difThe quantity Tk(s) = 1 ference of classical feedback theory [20]. To express closed-loop transfer functions in terms of open-loop ones, we proceed as follows: It should be noted that transfer functions-which express input-output relations-are independent of the choice of the (internal) state variables. For instance, suppose the (numerical) state vector z is replaced by R = Tx, where T is a nonsingular constant matrix. Then the matrices F, g, k in (1) and (2) must be replaced by TFT-I, \$ = Tg, and = (T-l)'k. a The minus sign in (2) is a notational convenience and a reminder that we are usually dealing with negative feedback. MARCH 1964 ## Transactions of the ASME The transfer function (5) is invariant under this change of coordinates because tl\$(s)d = ktr-l(sI F)-'g ~ ~ ~ - i ) - i r ~ = k'r-l[y(aI = kf(sI - F)T-'1-1Tg = kl@(s)g. Remark ( 4 ) is of direct relevance here. For future purpcses, it is very important to have a strict correspondence between time-domain and frequency-domain methods of system description. The frequec-y-domain form of (1) and (2) is the transfer funrtion ic'+(s)g. However, knowledge of this transfer function will not uniquely identify the matrices F, g, and k appearing in (1) and (2) (eve; disregarding the lack of has a finite value. (Of course, 52 may be the empty set.) of uniqueness due to the arbitrary chc~ice the basis for x-see Rein We ask: For what control function p~c(t) D (if any) is V(XO, ; mark (41)unless certain renditions are satisfied. These are p ) a minimum? [ I l l : (a) the plant is completely controllahle and (b) the control a The minimum of V(xo, m ; p ) is written as V(XO, ) . law k is completely observable. I t seems to be very difficult to give explicit conditions on L The first condition means that all state variables can be affected which are equivalent to the linearity of X . For this reason, we by some suitable choice of the control function p(t). The second shall base all further considerations on a simple and well-known condition means that the control fu~lction p(t) given by (2) can be sufficient condition: L is a quadratic form in the n 1 variables identically zero onl:y if the state is identically zero; in other (x, p ) (111. Since we have required L ti L(t), the coefficients words, the control law k is picked i11 such a way that any change must be constants. In other words, the most general form of L from zero in the state variables is counteracted by some control that we shall study is given by action. There is no loss of generality (as far as the problem discussed in 2L(x, p) = x'Qx 2(rfx)p upz (Q, r, u constants). (10) this paper is concerned) in assuming that both of these conThis is Assumption (d) in the Introduction. ditions hold (81. With this preliminary analysis, we arrive a t the following preIn practically all the analysis which follows, we must assume cise definition of the central problem of the paper: that Given a con&Inverse Problem of Linear Optimal Control Theory. The plant is comp1ett:ly controllable. (-41) pletely controllable constant linear plant (1) and constant linear This condition can be expressed in concrete form by [I11 f control law (2). Determine all loss functions L o the f c m (10) such that the control law minimizes the performance index (9). rank [g, Fg, . . ., Fn-lg] = n. (A;') What conditions must be imposed on tl and L to assure that x is constant (not explicitly dependent on t) and linear in x? Note that constancy ;snd linearity of x are entirely different properties. I either tl < m or L depends explicitly on t, we have a nonf stationary situation and in general x will not be constant. We msume therefore that L # L(t) (101 and that tl == m . The latter assumption must be made precise. Let fl denote the set of all continuous functions p(t) defined on the iinterval [O, ) for which the functional On the other hand, the assumption that k is a completely obeervable cont.ro1law, i.e., that [II] rank [k, F'k, ## Review of the Optimization Problem Now we turn to Problem B. We seek an explicit expression for the minimum of (9) and the corresponding optimal control law. As the general theory of minimizing (7) is quite well known [3-41, we shall stress those aspects of the problem which result from letting t1+ o. This problem was solved in [3]. The loss function L cannot be completely arbitrary. We want to investigate what restrictions must be imposed on L for purely mathematical reaaons. We shall also examine the physical significance of L. The firat restriction am L is a consequence of Pontryagin's "Maximum Principle" [4,121. According to this principle, the variational problem (7) has a solution only if the so-called preHamiUonian function [4] (where p is an n-vector, the costate of (1)) has an absolute minimum with respect t o p for every fixed value of (1, z,p). Substituting (10) into If, it follows that H will have s minimum only if u 1 0. If cr > 0, then H has a unique minimum for all t, x, p. This is the so-called regular case. If a = 0, then H can have a minimum if and only if r'x = 0 and p'g = 0, in which case H ia independent of p. This is the singular case. In the latter event, the Maximum Principle furnishes very little information about the proper choice of p { 1 1 . 2 T o avoid complications which are of little interest in thie paper, we shall consider only the regular case. (The same assumption is made in most of the classical literature on the calculus of variations.) Hence u must be positive. We may set a = 1 without loss of generality. Then (10) takes the form: . . ., (P')n-lk] = L, ## Mathematical Formulation of the Optimization Problem Our firat objective ought to be to find all optimization problems law which result in a conetant, linear co~itrol (Problem A). However, the general solution oi this problem is not known a t present. Therefore we shall merely give some sufficient conditions. I n the next section we will see that theae corlditiona imply that the control law is constant and linear. Let us recall the mathematical definition of the dynamic optimization problem i n control theory ( 9 ) . We denote by x,(t; a ) the unique solution or motion of (1) corresponding to some fixed, continuous control function p(t) and the initial state a = x5(0; a . The loss function (or Lagrangian) L is an arbitrary smooth ) function oft, x, and p. Suppose a is a fixed initial state and tl > 0 a fixed value of the time. Then the integral V(xo, Pi P) = Jo L(t, x j t ; a ) , dt))dt (7) is a continuous functional of the control function p(t). AB usual, V is called the perforntance index of the control system (1-2). We mk: For what continuous control function p(t) (if any) k V(xa,tl; p ) a minimum? The minimum value of V(a, tl; p ) will be denoted by V(n, t,). The Principle of Optimality of tho calculus of variations shows [3-41 that the function p(t) (if it exists a t all) may always be generated by a control law of the type Thus the precise formulation of Problem A is the following: 2L(z, p) = z'(Q --w')x + ( p + rtx)"Q A second restriction is required to assure that the minimum value of (7) does not diverge to - as tl approaches aome finite value. (In other words, we want to rule out the possibility of MARCH 1964 = Q'). ## Journal of Basic Engineering 73 "conjugate points.") The standard way to avoid trouble is co assume that Q - rr' is nonnegative definite. I t is well known that a symmetric matrix Q - rrl is nonnegative if and only if there is a p X n matrix H (where p = rank Q - rrl) such that Q - rr' = H'H. Consequently we shall write L in the form Th~s limit is readily evaluated numerically by computing the - m. solution of (12) which starts a t P ( 0 ) = 0 and letting t (ii) The control law (13), with I I ( t ; ti, 0) replaced by P,, generates functions pm(t) (dependent on xo) which are always in 0 [3, Theorem 6.71. Thus if xm(t)is the solution of (1) corresponding to pm(t),then + Let us pause to examine the physical significance of (11). Clearly L is nonnegative. If L is positive, we incur a loss. The object of optimal control is to minimize this loss. The loss is zero if and only if both terms in L vanish separately. Therefore H is chosen in such a way that the state of the plant is "satisfactory" if and only if H z = 0 (131. The choice of r in L fixes the desired level p = -rlx of the control variable. In the rather intricate calculations which are to follow, i t would be unpleasant to drag along the parameter T. Actually, r may be eliminated without loss of generality. Let us change the control variable p to Since L is nonnegative, the matrix Pmis nonnegative definite. (iii) Of course, pm(t) is not necessarily optimal and therefore in general ## Then (11) becomes But in reality the strict inequality sign cannot hold for any XQ [3, Theorem 6.71. I n other words, p" = p t . Now we have obtained an explicit expression for the minimum of the performance index (9) ## while k in (2) is to be replaced by i=r. kIt is then clear that the problems (F, g, k, L ) and (PIg, k, Z) are equivalent. From now on, unless explicit mention is made to the contrary, all considerations will be restricted to the case r = 0, and the adjective "optimal" will always refer to minimizing (9) with respect to the lossfunction (lla). For the sake of simplicity, the bar on F, k, and L will be dropped. Returning to the problem of minimizing (7), let us recall the results of the analysis given in 13-41. Let II(t; tl, 0) = P(t) be the unique symmetric solution of the Riccati-type matrix differential equation -dP/dt = The preceding results may be summarized as Consider a completely controllable plant and the associated optimization problem (9), where L is given by (Ila). This problem always has a solution, which is obtained by solving the differential equation (12) und then eoaluating the limit (14). The minimum value o the performance index is given f by (15). The optimal control law is given by (16). THEOREM 1. (Solution of Problem B.) ## Stability of the Optimal Control Law Optimality does not imply stability! In fact, let x be any control law. Define L so that L = 0 when p = -x(t, x) and positive for any other value of p. Then the loss is identically zero if and only if the control law x is used. Since L is nonnegative, V(XO,m ) is also nonnegative for any XQ. It follows that the given control law X-which is entirely arbitrary- is optimal and even unique. To understand the difficulty raised by this example, let us examine more closely the physical significance of the condition L = 0. Evidently this can happen only when the control law is p = 0. Then all solutions of (1) and (2) are of the form x(t) = eFtxQ. Consider now the linear subspace X1 of the state space X = Rn defined by PF + F'P - Pgg'P + H'H (P = P') (12) ## and the corresponding unique optimal control law is pr(t) = -(no; t,, o ) ~~ ( t ) ) . , Up to t h i ~ point, the analysis holds for every finite t1 > 0. Additional arguments are needed to eolve also the problem of minimizing (9). (i) If the plant is completely controllable, it may be shown [3, Theorem 6.61 that lim II(0; tt, 0) = P, tl-cm exists and is unique. I n view of the constancy of the plant and of L, the choice of the origin of time in (11) is immaterial: II(0; t,,O) = II(-tl; 0, 0) for d t, l also I the state belongs to XI, then the vector x(t) = ePtxQ f belongs to XI for any t, since eA('+~)= eAteAr any matrix A for and any scalars t, 7. Thus XI ie invariant under the control law p = 0; in fact, X1 is the largest subspace of X such that L = 0 when XQ is in XI. Hence 0 is an optimal control law (conceivably not the only one) with respect to states in XT. Actually, the optimal control law is unique by Theorem 1. Thus the oplzmal conlrol h w through& the subspace XI consists in setting p = 0. Going back for a moment to the case r # 0, i t is clear that in general XI is given by > 0. (14) X, = 1; 2 = 01 (F. - gr'). ## Therefore P may be defined equivalently as , lim t1-P- -n(t; 0, 0) P. , 4 We use the notation 11 x 1 1 2 ~ .where A is any symmetric matrix, for the quadratic form 'Ax. As far as initial states in XI are concerned, the outcome of the optimization problem is determined a priori by the choice of r. It is clear from these observations that there is no loss of generality in requiring dimension XI = 0. MARCH 1964 ## Transactions of the ASME This abstract mathematical conditioil is equivalent to the control-theoretical statement: The pair [F, H ] is conzpletcly observable. (A2') In turn, this is equivalent to the concrete mathematical condition rank [HI, F', H', . . ., (F')"-'HI] n. (Az") These matters are discussed (with proofs) in Ill 1. In conventional language, (AZ') means the following: Suppose the p-vector y = H z denotes the outputs of the plant with respect to which the performance of th,e control system is to be optimized. Then y must not vanish identically along any free motion of the plant unless the initial state XG = 0. (The outputs y used for defining the performance index must be carefully differentiated from those outputs which are directly measured 1131. One might say that (At) assures the nondegeneracy of the statement of the optimal conk01 problem. If (Az) holds, then 1 cannot be iclentically zero along any , motion of (1). Then V* is positive definite (not merely nonnegative definite) and its derivative along optimal motions is -L. I n other words, (A2) implies thzt V* = (1/2)\\xllz~,is a Lyapunov function. By a theorem well known in the Lyapunov stability theory [14, Corollary 1.3; 15, Theorem 81 the desired result of this section follows immediately. THEOREM 2. If Ass~cn~plions (-41-2) are satisfied, then P, is positive definite nnd the optimal control Law is (asymptotically) stable (i.e., all eigcnvalurs o F - gk' have ncgative real parts). f This may be regarded as the central theorem of linear optimal control theory. I'nder slight additional hypothcses it is valid also for the nonconsistant case [3, Theorem 6.101. Our preceding analysis implies also :t much stronger result: (14) and (16), and therefore also (17) and (19). Replacing Pg by k in (17), i t follows that (17') and (19) together imp19 (20). By Theorem 2, P, is positive definite. Hence optimality implies the existence of matrix P which satisfies (18)-(20). (ii) Conversely, let k be a, fixed control law. Suppose that we are given a matrix H which satisfies (A2) and a matrix P which satisfies (18)-(20). Then P also satisfies (17). We want to show that P is equal to P, of Theorems 1-2, for then the optimality of k follows from (19), and stability follows from (18) and (20). Consequently i t is sufficient to prove LEMMA 1. Tf H satisfies (Az), then (17) has a unique positive definite solution. Proof of Lemma I . Let P be a positive definite solution of (17). Let k = Pg. Then the matrix F, is stable. This is proved by introducing the scalar functioli V('(5) = IIZllaPl which is positive definite by hypothesis. Its derivativ'e along the solutions of,the differential equation is given by v = x'[PFI + FhtP]x since P satisfies (17). C1ear:ly 3 is nonpositive and is nero only if glPx = 0, which implies that k'x = 0. It follows that V is identically zero along a solution eFktzoof (17) only if I I ~ e ~ ~ ~= o lIl~ e ~ ~ x o l0. x1 =l By (A2) this can hold only if xo = 0. Hence 3 can vanish identically only along the trivial solution x(t) = 0. The stability of Fk then follows immediately from the Lyapunov stability theory [14, Corollary 1.31. Now Auppose that (17) has two positive definite solutions Pi FkilPi = -HIH by (i = 1, 2). Let ki = Pig. Then P i F k i (17), and Under Assumption (Al), a conditionfor the stability o the optimal conf necessary and s~rficicnt trol law is that all ciqenvalues of F restricted to XI have negative real parts. THEOREM 3. (Solution of Problem C.) ## Algebraic Criterion for Optimality Now we seek an explicit necessary and sufficient condition for a given control law k to be optimal (Problem D). Unfortunately, conditions (14) and (16) provided by Theorem 1 are inadequate for this purpose because P, is defined only indirectly by the limit (14). Let us observe, however, that by (14) P, is an equilibrium state of the Riccati differential equation (12). Accordingly P, eatisfies the algebraic equation -PF PiFh ## + F'PI - (PzFk + F1P2)' = (P,- PZ)F, + F,' (PI - P2)' 0. It is well known [16, chapter VIII] that the matrix equation XA BX = 0 has a unique solution, namely X = 0, whenever + AJBI # 0 ## for any pair j, k. (22) - F'P = H'H - Pgg'P, (17) which is obtained by setting dP/dt = 0 in (12). This does not characterize P , completely because (17)-being a system of quadratic algebraic equations--does not have a unique solution in general. Even if P is required to be nonnegative definite, (17) may fail to have a unique ~~olution f 14). Fortunately, the difficulty may be removed with the help of Assumption (Az). This provides the solution of Problem D: THEOREM 4. (Algebraic Characterizatiot~of Optimality.) Consider a completely control2able plant and the associated variational problem with H satisfying (Az). Let k be a fixed control law. Then a (i) necessary and (ii) sufiient condition for k to be a stable optimal control law is thut there exist a matrix P which satisfies the algebraic relations Since both FIand FZ stable matrices, condition (22) is obviously are satisfied. Hence PI - Pz = 0. This completes the proof of Lemma 1. Finally, by the Lyapunnv stability theory, equation (20) together with (18) implies that k is a stable control law, i.e., that Fo is a stable matrix. Theorem 4 is proved. ## Frequency-Domain Characterization of Optimality Since F and g are usually known, Assumption (Al) and the relations (18) ma,y be regarded as constraints on the parameters H and k, the matrix P serving as a connecting link. We want to eliminate P. This is Problem E. It is quite remarkable that a simple relation connecting H and k can be found a t all. It is truly astounding that thia relation must be stated i n the frequency-domain if it is t o be reasonably simple. This is the main result of the paper from which everything eke will follow ( 15) . The solution of Problem E is given by THEOREM 5. (Frequency-Domain Characterization of Optimality.) ## P = P' is positive definite, -PF, Proof of Theorem 4. (18) .- FkrP= H'II + kk'. (20) (i) Suppose that k: = k* is a stable optimal control law corresponding to some L which satisfies (A2). By Theorem 1 there exists a symmetric matrix P, which satisfies Consider a completely controllable plant and the associated variational problem, with L satisfying (A2). Let k be a fixed control law. Then a (i) necessary and (ii) s u m e n t condition for k to be an optimal control law is that k be a stable control law and that the condition MARCH 1964 ## Journal of Basic Engineering 11 + ~ ; ' @ ( i w )= l1~ + l \ ~ @ ( i w ) ~ , i ~ ~ (I) and F, g have the form (26), then the vector q has the representation hold for all real w. Proof. (i) The assunlptions being the same as in Theorem 4 , there is a unique nlatrix P satisfying (18)-(20). We add and subtract s P from the left-hand side of ( 1 7 ) and obtain P(sZ 9 = ## - F ) + (-SZ - F1)P = H'H - Pgg'P. [y] Yn (28) (23) Then we multiply the right-hand side by @(s)g and the left-hand side by g'@'( - s ) g'@'( -s)Pg g t P @ ( ~ )= g'@'(-s)[H'H g = - Pgg'P]@(s)g. ( 2 4 ) Recalling that Pg [l k by (19), we get ## + kt@(-s)g] [ l + k'@(s)g] = 1 + gf@'(-s)HfH@(s)g. Setting s = iw gives (1)s Finally, by Tlleorem 4, k = Pg is necessarily a stable control law if P satisfies (17)-(20). (ii) suppose k is a given stable control law. Theorem 2 implies that given some H satisfying (A21 equation ( 1 7 ) has a unique positive definite solution P , and there is an optimal control law k* = P,g. Using Condition ( I ) , we will prove that the matrix P , so determined satisfies (19), in other words, that k = k. Then ( 2 0 ) will be also satisfied and it will follow by Theorem 4 that Condition ( I )is sufficient for optimality. We consider again (241, which is a consequence of ( 1 7 ) and let ~ s = iw. We replace II~@(iw)g(/2 its value given by (I). Inby troducing the abbreviations In other words, if F and g have the canonical form (26), we can identify the components of the vector p with the numerator coefficients of the rational function (27). Proof of Lemma 2. The canonical form (36) follows immediately from the definition of controllability [17]. Formula ( 2 7 ) can then be verified readily by elementary algebraic manipulations; for instance, by signal-flow-graph methods. For the significance and proof of this lemma see [ I l l . Another fact which we need here concerns the factorization of nonnegative polynomials. This is the main step in the method of spectra~factorization [ I S , 191. Let e(w2) be a polynonzial i n w2 with real coefiients LEMMA 3. that i s nonnegative for all real w. Then there exist q pol~nomials vn(iw)( 1 6 Q \$ n ) with real coe@ients szlch that C3(w2) = k=l f : vh(iw)vk(-W) k=l (vk(u))I1 (29) ## ?r(iw) = g'P,@(iw)g, ~ ( i w = kf@(iw)g, ) and simplifying the resulting expressions we obtain &foreover, we can alwal~s q = 1 and choose vl(s) so that all its let zeros have nonpositive real parts. W i t h these two restrictions we have the unique 'Ifactorization" e ( w 2 ) = jvl(iw)12. Everything follows easily by factoring Proof of Lemma 3. e(w2) according to classical algebra. Returning to the proof of the theorem, we recall equation ( 6 ) , 1 Hence ( 2 5 ) implies + kl@(s)g +k(s)/+(s). 11 + Since the is clear that ?rI2 11 + KIZ. Pm law ?r is k = Pmg it = K. ## In other words, the proof has been reduced to showing that In other words, the nonnegative polynomial e(w2) of Lemma 3 has two different factorizations. By assumption, the zeros of #k(s) are in the left-half plane. By optimality and Theorem 2, the same is true concerning the zeros of &(s). Hence k'@(iw)g = k* '@(iw)g. (30) By Lemma 2, we take F and gin the representation (26). Then that = k. (27) This the proof the I Theorem 5 is not quite convenient as stated, because Assumption (A21 is expressed in the-domain language. It is easy to see that Assumptions (A]-2) are equivalent to the frequency-domain condition : (1 + k ' @ ( i ~ ) ~=211 + k @ ( i ~ ) ~ ( ~ l (25) implies k = k. We recall the hypotheses that ( a ) the pair [F, g] is completely controllable and that ( b ) k is a stable control law. We need to know also Then one a n LEMMA2. Suppose [F,g] i s completely alwaysfind a basis in the state space X such that F and g have the representation F = -al . . . 0 -&-I * . - ffw 1, I] = ## Rational functions H@(s)ghave no common cancelable factors (B1-2) The equivalence of this condition with (Al-\$)-in other words, with the complete controIlabiIity of [F, g ] and the complete observability of [F,HI-is established in [ I l l . Theorem 3 holds if and only if all zeros of the common cancelable factors of H@(s)g have negative real parts. ## where det ( s I - F ) = # ( s ) = sn Moreover, if qf@(iw)g = + ansn-I + . . . + cul. (26) Implications of Optimality From Theorem 5 we can deduce a number of interesting relations between optimality and frequency-domain concepts. In this way we obtain a fully satisfactory solution of the Inverse Problem of Linear Optimal Control Theory. From Condition (I) it is clear that a stable control law may be optimal only if the return difference T k ( h )satisfies the condition 6 The quadratic form 11 x 1126 for z complex and A real is defined, as usual, by ## y,(iw)"-l (iw)" a,,(iw)"-l + . . . + YI + . . . + a,' (27) llz [\'A x i,i ITk(iw)l = 11 + k'@(s)gI2 > 1. (11) 32iaij~j. ## where fi is the complex conjugate of xi. This condition may well fail to hold { 17). If i t does hold, then H may be obtained according to Lemma 3 by factoring the nonnegative polynomial r ( w P ) : MARCH 1964 ## Transactions of the ASME 76 the control error (as represented by the term ll~x112) and deoiating ( 3 1 ) from the given control law ( a s represented by the t m , ( 1 T ' X ) ~ ) . = l\~@(iw)~lp. Combining these terns additively to form the loss function (11), it follws that the control law k i s '%ettei7'than the control law r if and Many different matrices H will satisfy ( 3 1 ) because there are only if the return differencessatisfy the condition many different factorizations of I'(w2). However, not a11 H'a obtained in this way define optimization problems which yield [ T , ( i w ) / ~ + ( i w ) l 1 for all real w. > (IIb) the control law k. Since k was assilmed to be stable, the situaProof. Condition (11) for the case r = 0 is given by (34). But = ! tion described in Theorem 3 must hold. Therefore (see remark following (B1-2)) the rational functions H@(s)g must not possess I T ~ ( ~ w )= I + k ( i ~ ) i J / ( i ~ ) I . I (35) any common cancelable factor which has a zero with nonnegative ' real part. H s which do not satisfy this requirement must be The same holds for T,(iw) and ( I I b ) follows immediately. discarded. Formula ( 3 3 ) implies also a useful arithmetic condition for A proper choice of H is always possible. Using the unique optimality, which we state only for the case r = 0: factorization described a t the end of Lemma 3, and Lemma 2, we THEOREM 9. A control law k i s optimal in the sense of Theorem 6 can write if and only i f (32) ( a ) \$,(8) satisfies the Routh-Hurwitz conditions; ( b ) q ( w 2 ) = l\$,(iw)12 - l \$ ( i ~ ) 1 s~a nonnegative polynomial i where all zeros of the transfer function hl@(s)ghave nonpositive real parts. in us. w e aaaume now (without loss of generality, see Section 3 ) that k In other words, it is possible to characterize optimality solely is a completely observable control law. This is equivalent to the in terms of the open and closed-loop characteristic equations! fact that the polynomials + k ( s ) and J / ( ~are relatively prime. ) This is related to the fact that the numerical values of k depend Since on the coordinate system chosen to describe the state variables, whereas and qhare independent of this arbitrary choice! 1 1 kl@(zw)g\~ 1 = \+,k(iw)/#(iw)\2 - 1 (33) Condition ( b )in Theorem 9 is equivalent to = lh1@(iw)g12, (b') (w2) has no real, positive root of odd multiplicity. the transfer function hl@(s)g has thten no cancelable factors, condition (B1-2) is satisfied, and therefore also ( A z ) . The matrix TO express (b') in the form of inequalities on the coefficients of H = h' so constructed satisfies the requirements of Theorem 5. (w? is a classical proble!m in the theory of equations. BY techniques based on Sturm's theorem, i t is poaaible in principle to Hence we have: derive such inequalities (and to prove the Routh-Hurwitz conTHEOREM 6. (Solution of the Inverse Problem of Linear Optimal Conditions, see [16, Chapter XVl ). This is a very dimcult exercise in Consider a completely controllable plant with a algebra; the general inequalities are not known today. fro! Theory.) stable, completely observable control law k. Then k i s a nondeWe shall give explicit inequalities corresponding to (a)-@) only generate optimal contra1 law if and only if Condition (11) satisfied. in the special cases = md is = 3. The deriva,tions are I n general, k will be optimal with respect to seoeral H's, which are mentav and therefore onlitted. determined from (\$1) and must satisfy in addition the noncancelUsing the notations lation conditions mentioned above ( 1 8 ) . The requirement I ~ , ( i w ) l> 1 is a celebrated result of classical # ( s ) = det (sZ - F ) = sn a,,sn-1 . . . all feedback theory [20]. It assures that the sensitivity of the system to component va,riations in the forward loop is diminished by the addition of feedback. The larger the value of IT,(iw)l, the we have: Necessary and sufficient conditions for optimality of k greater the effect of feedback in reducing sensitivity. By Theorem are: n = 2:' 6, the same condition also assures dynamic optimality. We have thus a beautifully simple relations'hip between the "classical" argument in favor of feedback and the "modern" concept of dynamic optimality. 8' - a 2 2 0 1 1 It is well worth recording also the consequences of Theorem 5 Pz' - az2 - 2(81 - O L I5: 0 ) (37) and 6 when r # 0. both equal signs cannot hold simultaneouely THEOREM 7. Given a n y stable con.tro1 law k , there exists a loss r ( ~ ~ ) / l \$ ( i w ) ~ l11 = + k'@(iw)g12 - 1 function ( 1 1 )with H nonsingular for iuhich k i s optimal. Proof. Of course, this result hinges on the possibility of choosing r in a convenient way. Recalling the discussion of equivalent problems in Section 5, i t follows that Condition (11) may be generalized to 11 \$ (k' - r')@,(iw)gIz > 1. (IIa) 8 1 " P " l We must show that r can be chosen in such a way that the strict inequality sign holds. Now 8' 8 - arf - 2(Pz - a1 2 2 0, Either ## PZ2 - az2 - 2(/31/31 - ~ I , C Y I ) B 0 or (39) 8 2 " - 2(/31& - L Y I ~ ~= ) - ~ ( P ~a , 2 ) ~ ~ a 1 L from which i t is obvious that the required r exists. Q.E.D. THEOREM 8. Consider a completely controllable plant with a given control law r. Suppose we wish to find a "bettar" control law k. I n determining k a n optimal compromise :nust be made between reducing - 2~3% az)~. - ## Asymptotic Properties of Optimal Control Laws Now we turn to Problem F : I f L i s given by 2L(x, 1) = p l l ~ x l l ~ ## cussion at the 1962 Joint Aotomatic Control Conference. 'These conditions were first presented by the writer in an oral disMARCH 1964 PI, (40) ## Journal of Basie En~ineering 77 where H i s jixed and p i s variable parameter, how do the optimal control laws behave as p -r w ? U7e assume of course that p > 0. The solution of this problem is greatly simplified by a simple observation based on Condition (I)of Theorem 5: The same control law i s obtained if H i s replaced by the matrix hh', where h i s determined by I l ~ @ ( i w ) ~ = f Ih1@(iw)gl~. ll (41 Since the left-hand side of ( 4 1 ) is a nonnegative polynomial in w: the existence of h follows from Lemmas 2-3. The vector h is not uniquely determined by ( 4 1 ) ; i t is again convenient to fix i t by using the unique factorization mentioned in Lemma 3 ( 1 9 ) . Replacing H by h' means passing from the loss function defined by ( 4 1 ) to that defined by ~ U X) ,= p712 + EL: EL where n (h, z) = i =1 hixi. If we choose the ~pecial coordinates described in Lemma 2, then and every component of hi is nonnegative since all zeros of hisi'l must have negative real parts. Hence we have = . has an increasingly sharp low-pass characteristic as ( n - m ) - W , the bandwidth being p1'2(n-m). ( Ahigh return difference requires large bandwidth.) Moderate overshoot and low-pass frequency-response have always been regarded as typical of good servomechanism design. This fact is now "proved" with a rigorous theoretical analysis and brought within the framework of optimal control theory (201It must be emphasized, however, that the influence of the plant parameters may be significant for moderate values of p. In such cases the machinery of optimal control theory (e.g., ( 1 4 ) ) may lead to a control law which is not easily obtainable by the usual intuitive engineering design methods. The essential idea of Theorem 11 is attributed to Chang [18191, who called i t the lLroot-square-locus" method ( 2 1 j . It is of some interest to give explicit formulas for the closedloop characteristic equation of a system which is optimal with respect to the loss function (42). We shall do this only when n = 2. In this case the characteristic polynomial is written in the usual form s2 P2s P I = s2 2{@# ~ k : where 5;, is the (closed-loop) damping ratio and wk is the (closedloop) undamped natural frequency. We w m t an explicit relationship between p, fk, and wk, given a fixed transfer function hr@(s)g. These relations follow from (36)-(37)as well as from (44),which we can write also as THEOREM 10. Consider a completely controllable plant with a single control variable. Without affecting the corresponding control law, every quadratic loss function ( I l a ) may be replaced by (@), in which 7 i s a linear combination with nonnegative coefiients of a certain state variable X I and its del-ilratives. Condition (I) is now of the form \$ .- + + (44) It was fimt noticed by Chang [18, 191 that this equation admits a revealing interpretation in terms of root loci. Writing [ ( a ) to denote the numerator of the transfer function hl@(s)g(a polynomial with real coefficients of degree m < n), it is clear that ( 4 4 ) is equivalent to (45) #r(s)#r( - s ) = #(s)#( - s ) p f ( s ) f (-8)w precisely 2 m zeros of ( 4 5 ) tend to the zeros of As p f ( s ) l ( - s ) . The remaining 2 ( n - m ) zeros tend to m and are asymptotic to the zeros of the equation S Z ( ~ - m )= p. (46) 11 + k'@(iw)g12 = 1 + Plh'@(iw)g12. I Case I . Subcase. -w2 (iw 82 + 2{@k& + wkZ' - sl)(iw - s2) +P I/& and p = wk4. (41) where s, and 0 , s2 = 1. Then = Case 1. 81 = > 0 and f k I 1 ) then d[ + 1 hki ## Subcase. If hl@(s)g bound, while p = w,'. l/s(s lkattains i b lower Since all zeros of the polynomial \$ k ( ~ ) must be in the leftrhalf plane, i t follows that m zeros of \$,(s) tend to the corresponding zeros of f ( s ) (which are in the leftrhalf plane, or on the imaginary axis by the definition of h ) while the remaining n - m zeros of #k(s) tend asymptotically to a Butterworth pattern [9,10, 191 of radius ~ ' / ~ ( n - ) . These observations may be summarized in the following form: THEOREM I I. (Solution of Problem F). Consider a completely controllable plant and the optimization prcblem ( 9 )corresponding lo the loss fundion (45). A s p + m , m closed-loop poles of the optimal control system tend to the m zero of hJ@(s)g, while the remainder tend to a Butterworth configuration of order n - m and radius p'/2("-m). The optimal closed-loop poles are asymptotically independent of the (open-loop) poles of the plant. This is a highly important result. Note that the number p'/2(m-m) is closely related to the loop gain. At large values of p the return difference \ ~ ~ ( i wbecomes )I very large so that the system becomes insensitive with respect to plant variations. At the same time, the dynamical behavior of the closed-loop system becomes independent of the uncontrolled dynamics of the plant. The step response of a transfer function G ( s ) with no zeros and a Butterworth pattern of poles has an overshoot from 0 to about 23 percent as ( n - m) increases from 1 to w [ l o ] . Moreover, the frequency response function but both equal signs cannot hold simultaneously. Subcase (a). If h'(@)(s)g = l / ( S P I), then p = COk4 - 1, wX> 1, and fk attains its lower bound. Subcase [b). If hl@(s)g = s / ( s then p = f k 2 - 1, f k > 1, and wk = 1. 2 20~2' taneously. d! Case . 4. sl = i and sz = - i. Then wh 1 and f k -1 but ## again both equal signs cannot hold simul- Subcase [ ) If hr@(s)g= l / ( s 2 a. I ) , then p = wk4 - 1, and f k attains its lower bound. Subcare (b). I hl@(s)g = s/(.s2 f I ) , then pZ = 2 ( 2 f k 2 - 1 ) t, > 0, and wk = 1. Only in Case 4 is it possible to have a damping ratio less than 1 / 4 2 This is to be expected. If p is small, the performance index defined by ( 4 6 ) will be optimized and stability is achieved by introducing a small amount of damping. A large amount of damping would require too much control energy. Thus in usual cases a second-order system can be optimal only if i t has a damping ratio a t least as high as 1 / 4 2 . This confirms a well-known "rule-of-thumb" used in designing instrument servomechanisms, which calls for f k r 0.7 - 0.8. It is interesting that the optimality conditions (39) do not put an upper limit on 3;t + + MARCH 1964 ## Transactions of the ASME 1 P. Funk, "Variationsrechn,ung und ihre Anwendung in Physik und Technik," Spring~r, Berlin, 1962 (Grundlehren Series 94). 2 R. E. Kalman, "On the Inverse Problem of Optimal Control Theory" (to be publ.ished). 3 R. E. Kalman, "Contributions to the Theory of Optimal Control," Bol. Soc. Mat. Mezicano, 1960, pp. 102-119. 4 R. E. Kalman, "The Theory of Optimal Control and the Calculus of Variations," Proc RAND-Univ. Calif. Symp. on Optimization Theory. 1960; Univ. of Calif. Press, 1963, chap. 16. 5 R. E. Kalman, T. S. Englar, and R. S. Bucy, "Fundamental Study of Adaptive Control Systems," ASD Technical Report 61-27. 6 A. E. Bryson and W. F. Denham, "Multivariable Terminal Control for Minimum Mean Squa,re Deviation From a Nominal Path," Proceedings of the Symposium on Vehicle Systems Optimization, Institute of the Aerospace Sciences, November, 1961. 7 E. B. Lee, "Design of Optimum Multivariable Control Systems." JOURNAL BASICENGINEIERING, ASME, Series D, OF TRANS. vol. 83, 1961, pp. 85-90. 8 C. W. Merriam, 1 1 "A Class of Optimum Control Systems." 1. Journal ofthe Franklin Institute, vol. 267, 1959, pp. 267-281. 9 D. Graham aind R. C. Lathrop, "The Synthesis of 'Optimum' Transient Response: Criteria and Standard Forms," Trans. AIEE. vol. 72, part 11, 1953, pp. 278-288. 10 D. T. McRuer, D. Graham, et al., "Performance Criteria for Linear Constant Coefiicient Systerns With Deterministic Inputs," Technical Report ASD-Technical Report 61-501, February, 1962. 11 R. E. Kalman. "R'lathematicril Description of Linear Dyn~mical Systems," SIAhf J. Control, 1963. 12 L. S. Pontrya.gin, Y. G.Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, "The Mathematical Theory of Optimal Processes," Interscience Pyblishers, New York, IV. Y., 1962. 13 K. J . Astrom, J. E. Bertram, et a].. "Current Status of Linear Optimal Conitrol Theory," S I A M J. Control. 1963. 14 R. E. Kalman and J. E. Beriram. "Control System Analysis and Design Via the 'Second Method' of Lyapunov." JOURNAL OF BASIC ENGINEERING, TRANS. ASME. Series D. vol. 82, 1960, p. 371. 15 J. P. LaSalle and S. Lcfschetz, "Stability by Lyapunov's Direct h4ethod." Academic Press, New York, N. Y.. 1961. 16 F. R. Gantmakher. "The Theory cf Matrices," Chelsea. 1959. 17 R. E. Kalman, "Lyapunov Functions for the Problem of Lur'e in Automatic Control." Proc. Nut. Acad. Sci. USA, vol. 49, 1963, pp. 201-205. 18 S. S. L. Chang. "Root Square Locus Plot-a Geometrical Method for the Synthesis of Optims~lServosystems," IRE Convention Record. 1960. 19 S. S. L. Chang, "Synthesis of Optimum Control Systems," McGraw-Hill Book Co.. Inc., New York, N. Y., 1961. 20 H. W. Bode, "Network AnaLysis and Feedback Amplifier Design." Van Nostrand and Co., Inc., New York, N. Y.. 1945. 21 D. A. S. Fraser, "Nonpnrametric Methods in Statistics," John Wiley & Sons, Inc.. New York, N. Y., 1957. 22 0. Bolza, "Lectures on the Calculus of Variations." Dover Press, New York, N. Y.. 1960, pp. 31-32. 23 J. Douglas. "Solution of the Inverse Problem of the Calculus o Variations." Trans. A m . &lath. Soc., vol. 50, 1941, pp 71-128. f 24 F. B. Hildebrand, "Methods of Applied Mathematics," Prentice-Hall, Inc., Englewood Cliffs, N. J., 1952. 25 A. A. Frederickson, to be published. 26 P. D. Joseph and J . Tou, "On Linear Control Theory," Trans. A I E E , vol. 80, part 11, 1961, pp. 193-195. 27 R. E. Kalman. "Canonical Structure of Linear Dynamical Systems," Proc. Nut. Acad Sci. USA, vol. 48, 1962, pp. 596-600. 28 E. G. Gilbert. "Controllability and Observability in Multivariabl . Control Systems." S I A M J. Control, 1963. 29 G. A. Bliss, "Lectures on the Ca culus of Variations," Chicago Univ. Press. Chicago, Ill., 1946. 30 L. Berkovitz. "Variational Methods in Problems of Ccntrol and Programming," J . Math. Anal. Appl., vol. 3, 1961, pp. 145-169. 31 V. A f . Popov, "Absolute Stability of Nonlinear Systems of Automatic Control," Act. i Telemekh., vol. 22, 1961, pp. 961-979. a defect in the conceptual foundations of the! theory; it is merely an argument for more basic research 6 extend the applicability of the existing methods. {2] The issue involved here is quite similar t o the objections raised against parametric statistical procedures. To assume that a given statistical model has certain well-defined but unknown parameters (for instance, a Gaussian distribution with unknown mean and variance) is highly restrictive, and the resulting conclusions must not be interpreted too widely. The remedy is to use nonparametric statistics [21]. Then one clan claim more general conclusions, but they will be necessarily less explicit than in the parametric case. {3) The precise definition of the inverse problem of the calculus of variations is the following: Given a family of curves which satisfy the vector di\$ereni!ial equation () = F(t, z, x) (5, F = n-vectors, . = (dl&). Determine all scalar functions L(t, z, z ) such that every solution of () is a regular eztrernal o the ordinary variational problem f i n other words, determine all L such that () i s identical with the Euler equations and L;; is positive dejnite along every solution of (). A solution of this problem always exists when n = 1 [22]. The case n = 2 has been investigated and settled by Douglas [23]. His results are much too complicated to be stated here. The problem is reduced to the solution of a system of firsborder partial differential equations for the elements of the matrix L;;, from quadratures. Not every family () is which L is determined 1 ~ y a regular extremal. For instance, there is no L (with L positive definite) for which the curves generated by \$1 -- 5 1 2 222, \$2 = z, are extremal. See also the interesting heuristic treatment of Hildebrand 124, Sect. 2.14-2.161. (4) It seems that fre~quency-domainconcepts are unavoidable in system theory when explicit conditions are desired. A classical example is of course Nyquist's stability criterion. While equivalent criteria can be stated in the time domain (all characteristic roots must have negative real parts) or even in terms of the system constants (the Routh-Hurwitz conditions), the frequencydomain Nyquist criterion is very elegant, simple to state, and often more useful than the others. The deeper significance of the frequency-domain formulas is that they provide "coordinate-free" system description: since transfer functions refer to inpuboutput relations, they are independent of the choice of the coordinates in state space. General results of optimal control theory must be of course independent of the arbitrary choice of coordinates. A full discussion of the relations between the transfer-function and state-variable methods of describing linear dynamical systems may be found in [ l l ] . {5) The practical significance of optimality conditions is explored in detail by Frederickson [25]. The reader should consult his paper also for a theoretical analysis of the oo-called ITE2 (integral of time X error squared) and IT2E (in~tegralof time squared X error squared) performance criteria [lo]. (6) These assumptions are of course highly restrictive. One obtains a hierarchy of problems depending on the number of control variables and the number of state variables which can be measured directly. If all statevariables can be measured, the optimal controller does not contain dynamical elements because the best control action a t any instant depends only on the values of the sta,te variables a t that instant. But if some control variables cannot be measured directly-which happens very often in practical problems-opti- APPENDIX (1) Optimal control theory must be regarded ae a mathematical tool. A rigid definition of the performance index cannot and should not be avoided, because the power of mathematical reasoning is available only when problems are precisely stated. (If we seek approximations, then i t must be precisely defined what constitutes a good or bad approximation.) I t is true that the optimal control theory of today can solve only a few problems (mostly in the realm of linear matl-lematics), and that it is not applicable t o many of the more complex and admittedly more important questions that interest the engineer. But this is not ## Journal of Basic Engineering MARCH 1964 ma1 control theory requires that the missing state variables be estimated from the known ones using Wiener filtering techniques [13,26]. The Wiener filter will contain dynamical elements which are to be regarded as a part of the controller. For example, if only one state variable can be measured then any optimal control loop can be specified by two transfer functions: the transfer function of the plant from control input to measured output and the transfer function of the controller from plant output to control input. Given the first transfer function, by the methods of this paper it is possible to obtain implicit conditions which must be satisfied by the second transfer function if i t is to be optimal. No explicit form of these conditions ie known (at present). (7) The reader is reminded that in most of control theory (and also in this paper) there is no need to introduce the mathematical concept of a Laplace transform. All one requires are certain functions of a complex variable s. Only in rare cases is i t useful to know that transfer functions can be interpreted as Laplace transforms, i.e., as an integral. I n that case special restrictions must be imposed on s. No such restriction is needed or implied in the definition (4). (8) I t follows from the author's canonical decomposition theorem [27] (see also [l1, 281 ) that only one "part" of the plant is included in the control loop, namely that which is completely controllable and completely observable. Only this part is of interest in studying the optimality of the control law. (9) I n the classical calculus of variations, this problem is called f the Problem o Lagrange or the Problem o Mayer or the Problem f o Bolza. All three are equivalent [29, 869.1 f (10) This assumption is not necessary. For instance, Frederickson [25] shows that every loss function of the form L(t, x, p ) = tkllxllza are both solutions of (17). PI is nonnegative definite and Pz is positive definite. PI is the one which corresponds to the limit (14). Condition ( A 2 )fails to hold, because eFf = C O S ~ sinh t t sinh t cosh t and if [ :1, then ~ ~ H ~ F ~ x O I !E 0. ( 1.51 The idea of using frequency-domain concepts in the present context is due to V. M. Popov [31]. By an extension of Popov's ideas, the writer has succeeded in obtaining a solution of the celebrated probleni of Lur'e [17]. In fact, Theorem 4 is a variant of the Main Lemma used in [17]; which has also other important applications in system theory. (16) The assumption that "k is a stable control law" is an important part of Theorem 5. To see why, let us write using Lemma 3. Suppose the rational function v(iw) # k19(iw)g; in other words, some zeros of v(s) have positive real parts. By Lemma 2 there exists a vector q such that + p2 (k positive integer, &I and we may replace k by q without any effect on (I). Thus ( I )is satisfied by many vectors. But only one of these corresponds tn the optical control law, namely that which is determined by the special factorization mentioned in Lemma 3. (17) Plotting the frequency-response function 1i1@(iw)g the in complex plane, it is easy to see that (11) can be satisfied only if lim liwkl@(iw)gl > 0. Using the canonical coordinate system i t follows that the latter condition is satisfied if and only if ki z 0 for i = 1, . . ., n, which means that every single state variable must be fed back! This is a reasonable consequence of the theory: since all state variables are assumed to be measurable, some information would be discarded if a coefficient ki were zero. Thus optimality requires using all available information. (18) If H # h' is a matrix consisting of more than one row, the condition [F, HI = completely observable (i.e., (Az')) is not suficient to guarantee that the optimal control law is completely observable. For instance, if H = diag [hll, . . ., h,,,,,1, hii > 0, and F, g have the canonical form (26), then lI~@(iw)gl/~[hn = (rrm nonnegative definite) is strictly equivalent (yields the same minimum for (9) and the same optimal control law) to a constant loss function of the type L(x, p ) = I ~ x ~ / ~ Q z + p2 ## (Qz nonnegative definite). (11) This is a very special assumption. For example, if L is a homogeneous polynomial of degree 2k in (z, p), then under certain additional conditions x will again be linear in x. The explicit form and significance of these additional conditions on L are not known a t present, however. { 12) The part of Pontryagin's maximum principle which is used here is equivalent to the classical necessary condition of Weierstrass. Historically, this condition was first stated for the ordinary problem (no constraints) of the calculus of variations [29, \$91. The extension of the Weierstrass condition to the problem of Lagrange [29, 78] has been shown [30] to be equivalent to Pontryagin's requirement that the pre-Hamiltonian H possess an absolute minimum with respect to p for every !t, z, p). The Weierstrass, Pontryagin theory is needed here only to establish that u > 0; the rest of the analysis proceeds in the spirit of the Hamilton-Jacobi-Caratheodory theory. { 13) Note that this requirement does not suffice to determine H uniquely. There is an essential arbitrariness involved here; in fact, the lack of a nice physical interpretation of the term ll~z112 been a serious difficulty in the applications of optimal has control theory. This difficulty is removed to a large extent in the present paper. See especially Section 10. {14) Consider the matrices + hz2wZ + . . . + h,,w2"-2]//\$(iw)12. By proper choice of the hii the numerator and denominator will have common factors. Then if ## It is easily verified that the two symmetric matrices dZ-1 -dZ+l P z = [3+dZ 1+d2 1+fi I+,,% where h' is the result of the special factorization mentioned in Lemma 3, hl@(i)g will also possess cancelable factors. Hence [F, h'] and consequently [F, k'] will not be completely observable. { 19) And then it follows further that P,(Q) - P,(hhl) is nonnegative definite for all Q. See [2]: (20) That frequency-domain methods are applicable just aa easily to low or high-order systems has always been claimed as a major advantage for these methods. The present analysis shows that, qualitatively speaking, similar claims can be made also for comtime-domain methods. When i t is a question of accu~alely puting an optimal control law, however, serious difficulties of a numerical nature may arise in dealing with high-order.sptem8. I we had general design methods of the frequency-domain typef none exists today which is really general-it, too, would be subject to the same numerical difficulties. (21) Chang's results were restricted to the optimization of the step response of the system. T i inessential restriction hs arose solely because Chang used the older (frequency-domain) version of optimal control theory. MARCH 1964 ## A General Formulation of the Nyquist Criterion Abstract-The Nyquist diagram tttchnique is examined under very general assumptions; in particular, the linear subsystem is represented by a convolution operator., thus, the case of any linear time-invariant distributed circuit is included. It is shown that if there are no encirclements of the critical point, then the impulse response of the closed-loop system is bounded and absolutely integrable on [O, m ) ;it also tends to zero as t + m. For any initial state, &e zero-input response of the closed-loop system is also bounded and goes to zero. If, on the other hand, there are one or more encirclements of the critical point, then the closed-loop impulse response tends asymptotically to a growing exponential. Manuscript received September 6, 1964; revised October 26, 1964. The research reported in this paper was supported by the National Science Foundation under Grant GP-2413. This paper was presented a t the Allerton Conference on September 28, 1964. The author is with the Dept. of Electrical Engineering, University of California, Berkeley, Calif. ## HE NYQUIST CRITERION is proved for the single-loop feedback case. The purpose of the paper is to demonstrate the extreme generality of the criterion by construct;ing a proof which requires the least number of assumptions.' The main result is stated in the form of a theorem. The hypotheses of this theorem include most cases of interest. The discussion of stability for the case where the transfer functions are not rational is far from trivial. Any reader who doubts this should consider the function defined for t 2 0 by e t in ( e t ) and note that its Laplace transform is analytic for all f i ~ t s. This example e shows that the discussion of stability cannot be settled by "looking a t the singularity that is the furthest to the right," which is a legitimate procedure with rational transfer functions. ## Desoer: General Formulation of Nyquist Criterion Following Nyquist [I], we consider the linear time-invariant single-loop feedback system shown in Fig. 1. It will be referred to as the closed-loop system. The block labeled k is a constant gain factor (i.e., independent of time and frequency), if its input is rl(t), and its output is kq(t) where k is a fixed positive number. The block labeled G is linear, time invariant, and nonanticipative (causal), and it satisfies the following conditions: (G.l) Its input-output relation relating the output y, the zero-input response z, and the input 5 is y(t) = c) For any initial state and for any bounded input, the response of the closed-loop system is bounded. d) Let r be positive; then for any input u which tends to a constant u, as t -+ co , and for any initial state, the output y tends to u, as t 4 a. Let r be zero and u -+0 as t 4 co , then for any initial state, the output y 4 0. ## If the Nyquist diagram of G(s) encircles the critical point (-l/k, 0) a finite number of times, then the impulse response of the closed-loop system grows exponentially as t-+ m. r(t) + St g(t - T)E(T) dr for all t 2 0. (1) (G.2) For all initial states, the zero-input response is bounded on [0, w ) and z(t) -+ z, as t + w , where z, is a finite number which depends on the initial state. Let Znf supt,,(x(t)l. (G.3) The unit impulse response g is given by where the constant r is non-negative; l(t) is the unit step function; g1is bounded on [O, a ) , is an element of L1(O, w ), and g1 4 0 as t -+ a . When r = 0, for all initial states, z, in (G.2) is zero. We write glg(t)l G(s) = f + G,(s). Let g, sup lg(t) 1. I20 Comment: It should be stressed that the only assumption that is made concerning the box G is that it fulfills the conditions (G.l), (G.2), and (G.3). Such conditions are often fulfilled by the impulse response of systems described by ordinary differential equations, difference-differential equations, and those whose input-output relation is obtained through the solution of partial differential equations. The latter is the case for distributed circuits and for many control systems. The analysis to follow applies to all cases where r 2 0. For many circuit applications it turns out that r = 0 and that for initial states z, = 0. The reader will have no difficulty in inserting the consequent simplifications in the proof. Analysis: Let u be the bounded input applied to the system and let U, sup,,,,\u(t)l. The response of the closed-loop system starting from an arbitrary initial state is given by y(t) = x(t) ## + k j t g(t - r)[u(r) - Y(T)I for all t 2 0. (3) Fig. 1 . Single-loop feedback system under consideration: the gain factor k is positive and the linear time-invariant subsystem G is characterized by a convolution operator [see ( I ) ] . For ease of reference, we state formally the main result of this paper: Theorem Suppose the linear time-invariant single-loop feedback system shown on Fig. 1 satisfies the conditions (G.l), (G.2), and (G.3). I the Nyquist diagram2 of G(s) does not f encircle or go through the critical point (-llk, 0), then a) The impulse response of the closed-loop system is bounded, tends to zero as t 4 a , and is an element of L1(O, a). b) For any initial state, the zero-input response of the closed-loop system is bounded and goes to zero as t 4 w. 2 The Nyquist diagram is the map under G of the imaginary axis from which the interval [ - j e , j e ] has been removed and replaced : by the semicircle (eie: - ~ / 2 0 I r / 2 ) ; here e is taken arbitrarily small. The theorem will be proved in several steps. First, in order to be able to apply Laplace transform techniques to the integral equation (3), we establish that the solution is of exponential order; second, well-known facts concerning Laplace transforms are used to establish the uniqueness of the solution of (3); third, various tools of complex function theory and Fourier analysis are used to establish the properties of the impulse response of the closed-loop system and those of the zero-input response. The proof of the remaining assertions of the theorem follow easily. f Assertion: I (G.l), (G.2), and (G.3) hold, and if u is bounded, then 1) The output y is of exponential order and its Laplace transform Y ( s ) is analytic for Re s > kg,. 2) The output y, the solution of (3), is ~~nique. ## Proof: From (3) and the definitions of x,, we get Iy(t)I g,, and u ~ , 5 (ZU k ~ M ~ l t )h M L t IY(T))Ir . d < ~y(t)l5 b(t) 0 o (4) ## multiply the nun~eratorand denominator by s/(s we get kt+c1(s)) H(s) = + kr) (8) where b(t) 4 Z , \$- kgMu,t. Equation (4) implies that y is of exponential order and that its Laplace transform Y(s) is an analytic function of s for Re s > kg,. To establish uniqueness, suppose there were two responses yl and yz. By subtraction we obtain from (3) yl(l) - Y Z ( ~ ) k S t g(t - ~)[Yz:T) YI(T)] = d~ for O' (5) Now gl is zero for t < 0 and is in L1(O, a ) ; therefore, the Laplace transform of g is analytic for Re s > 0 and goes to zero as Is/ -+ with 1 {sj 5; 7r/2 [4], [9]. From I), y, and yz are of exponential order; hence, taking Laplace transforms of (5) we get + k t + G) & 1 kr xrsTrGlb) . + ks = y+=s+ ks Icr GI (4 ## The denominator may be rewritten as kr + kGl(s) - k xr Gi(s). ' P,(s)- Y,(s)=kG(s)[Y,(s)-- Yl(s)] Res>kg,. Therefore, Yl(s) - Yz(s) = 0 for all 8 in their domain of definition. By the uniqueness theorem of Laplace transforms [5], yl and y, are equal for almost all t in [0, a ) . Since yl - y, is continuous, yl(t) = y,(t) for all t in [0, m ). This completes the proof. I t might be worth noting that since g, u, and z are bounded their restriction to any finite interval, (0, t), is a n element of L2, hence, the existence and uniqueness of the solution of (3) may also be established by iterative techniques [6]. Proof of the Theorem Observe that C1[kr/(s kr)] = l(t)kr e-"l, which is a function in L1(O, a ) . Since gleL1(O, a ) and since the product of the transforms of two L1 functions is the transform of an L1 function, the denominator is of the form "one plus the transform of a function in L ' ( ~ , a ) " . The denominator has no zeros in the closed right-half plane. The numerator of (8) is also the transform of a function in L1(O, m). Hence, by a theorem of Paley-Wiener [8], it follows that h is in L1(O, a ) . OW h is bounded because (6) implies h, sup [h(t)l 5 kgnI 120 + lcgMSmIh(r) d r < a. 0 -+ - ## observe that (6) ---kt - [g(t + 6 - r) t+6 - ( 1 - ~)]h(r) - k d g(t + 6 - r)h(r) dr. To Prove a ) we recall that, by definition, h is the zerostate response of the system to a unit impulse applied a t t = 0. By definitioin of g and from a n examination of the configuration of the closed-loop system, to apply a unit impulse a t the input of the closed-loop system is equivalent to having an identically zero input applied to the system but having G start from the state whose zero-input response is kg. Thus, h(t) = Therefore, if we remember the form of g specified by (2), for all t > 0 and all 6 > 0, I[h(t + 6, ## \$- 6)1 - [h(t) - kgl(t)ll < k h ~ j ~ I g ~ 6() i g l ( [ ) d t + k 6 g ~ h r r . -f (9) o (1 ## Let H be the Laplase transform of h; then H(s) = Note that the right-hand side of (9) is independent of t. Since g, L1(O, a ), it follows that the first term of the righthand side goes to zero as 6 -+ 0 [9]. The same is obviously true of the second term. Consequently (9) implies that h - kgl is uniformly continuous on [0, a ) . Since h and gl are in L1(O, m), so is h - kg,; therefore, the uniform continuity implies that lim,,, [h(t) - kgl(t)] = 0 [Ill. By (G.3) it follows that h tends to zero as t -+ a . Let us prove statement (b) of the theorem. The zeroinput response of the closed-loop system z, satisfies the equation z.(t) = "Go + 1cG(s) Kes > kg,. (7) Now, by the princilple of the a r g ~ m e n tthe denominator ,~ of H(s) is # 0 for all Re s 0 if, and only if, the Nyquist diagram of G does not encircle or go through the critical point (-llk, 0). By the assumption concerning the Nyquist diagram, the denominator of (7) has no zeros in the closed right-half plane. Let us rewrite (7) using (2). I we f > r(t) - St h(t - r)i(r) d r for all t 2 (10) Let C be a simple closed rectifiable positively oriented curve. Let C* be the union of C?, the interior of C, and of C itself. Let f(z) be meromorphic in C, (i.e., have no other singularities than poles in C). I f has no zeros nor poles on C, then, when z describes C, f the argument of f(z) increases by 2?r(Zf - Pf) where Zf, (Pf, resp.), is the number of zeros, (poles, resp.) o f f in Ci. since hL1(O, a ) and z is bounded, x, is bounded. I t remains to show that z, goes to zero as t -+ a.For this purpose we need only show that the convolution integral tends to Z, since, by (G.2)) z(t) - -x, as t -3 m . The properties of h imply thaL for any > there is a ~ ( < a, 1 such that t > T(E)implies lh(t)\ < e and ## Desoer: General Formulation of Nyquist Criterion ness of z, and the fact that h is in L1(O, a ) . Incidentally, by a previous reasoning, y - z, is uniformly continuous on [O, a ) . Thus, statement (c) is established. Since The properties of z inlply that Iz(t)l 5 zM < a for all t z,(t) -+ 0 as t -+ a , statement (d) is equivalent to the and that for any e > 0 there is a Tf(e) < m such that ~T assertion that u(t) -+ umimplies that f i h(t - T ) U ( T )+ t > T1(e) implies Iz(t) - zml < e. Rewrite (10) u,. This implication has been proved in detail in proving (b). Therefore statement (d) holds. Suppose now that the Nyquist diagram encircles ( - l/k, 0) a finite number of times. Since G(s) -+ 0 as Is1 -+ with 1 Csl 5 7r/2 and since G is analytic in the open righthalf plane, the principle of the argument [7] shows that kG(s) has a finite number of zeros in the open rightFrom these considerations, we get the following inequali- 1 half plane. For simplicity of notation, we shall write the ties: for any t > T(e) TI() following expressions assuming that each pole is simple. By a partial fraction expansion we get , . t +, 5 ( t-T'(c) -1 dr [h(~) 1 d~ + (1ziw[ Res, > 0, v = 1,2, - . . , n where H,(s) is analytic for Re s > 0. It can be easily verified that the behavior of H(a jw) as w -+ satisfies the conditions of Doetsch's theorem [lo]. Therefore, we conclude that Changing the upper limit of integration of both integrals T1(e)implies that to a , we conclude that t > T (c;) ## that is lim z.(t) - z(t) t-m + 2-1' = where s, is the zero of 1 kG(s) which has the largest real part. (If there are several such zeros, then the right-hand side must include the appropriate sum.) This con~pletes the proof of the theorem. For some applications it may be useful to be able to relate the norm of the output y to that of the input u and the zero-input response z. Corollary: Let (G.l), (G.2), (G.3) hold and the Nyquist diagram satisfy the condition of the theorem. If, for some 1, both z and u are elements of Lp(O, m), then number p > h(r) drl 0. m (12) Now, since h lim t-m l1 Y&I ## ~up,,~jy(t)j, then, using h(l) d l lim H(s) 8-0 5 (1 + Ilhlll>z,l+ l(hjlluiw. > t-m ## hence, (12) gives lim z,(t) t-m Proof: Observe that, for any p 1, if h is in L1(O, ) and z is in Lp(O, a ) , then h z is also in Lp(O, a ) and jlh * z/(, 5 \lhlIl . I \ Z \ \ ~ [12]. The inequalities above follow directly from the application of this fact to (10) and (12). CONCLUSIONS Under very general assumptions pertaining to the openloop systen~, have shown that if the Nyquist diagram we satisfies the nonencirclement conditions then the zeroinput response, the impulse response, and the complete response have all the usual properties associated with stable systems. The inequality (14) shows that if z is in L', then for all p 2 1 (including p = m ) the system is Lp stable in the sense of I. W. The results obtained here are essential extensions of Popov's criterion [Ill. 0. Consider now statement (c) of the theorem. The configuration of the closed-loop system a,nd (1) imply that the output y starting from an arbitrary initial state a t time t = 0 and responding to an input u is given by f where z, is the closed-loop zero-input response. I u is bounded then y is bounded; this follows from the bounded- I E E E TRANSACTIONS ON CIRCUIT THE0R.Y Doetsch, G., Ibid., p 72. Tricomi, I?. G., Integral Equations, New York: Interscience, 1957, p 11. Hille, E., Analytic Function Theory, Boston, Mass.: Ginn, vol 1, 1959, p 252. Paley, R. E. A. C., and N. Wiener, Fourier Transforms in the Complex Domain, New York: Am. Mathematical Soc., 1934, pp 60-61. Goldberg, R. R.,Fourier Transforms, New York: Cambridge, 1961, p 4; Doetsch, G., op. cit., vol 2, p.150. Desoer, C. A., A generalization of the Popov criterion, I E E E Trans. on Automatic Control, Short Papers, vol 10, Apr 1965, pp 182-185. Dunford, N., and J. T. Schwarz, Lineal Operators, New York: Interscience, vol 1, 1958, p 528, exercise fi Sandberg, I. W., AL frequency domain condition for the stabilitv of feedback svstems containine: a single time-varying nonGnear element, Be11 Sys. Tech. JT, vol 43, Jul 1964, pp 1601-1608. The author expresses his gratitude to Prof. R. W. Newcomb and to C. T. Chen for valuable comments on an earlier version of the manuscript. [l] Nyquist, H., Regeneration theory, Bell; Sys. Tech. J., vol 2, Jan 1932, pp 126-147. [2] Coddington, E. A., and N. Levinson, Theory o Ordinary Dif ential Equations, :New York: McGraw-Hill, 1955, problem 1, p 37. [3] Bourbaki, N., Fonctions d'une Variable Rdelle, Paris: Hermann and Cie, 1961, ch 4, \$1, no 4, Lemma 2. [4] Doetsch, G., Handbuch der Laplace Transformation, Basel, Switzerland: Verlag Birkhauser, vol I, 1950, p 162. Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances7 ISAAC M. HOROWITZ Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel, and Department of Electrical Engineering, University of Colorado, Boulder, Colorado and MARCEL S I D I Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel [Received 15 September 19711 There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at'the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission function L( j w ) , in the form of continuous curves on the Nichols chart. Properties of L ( j w ) which satisfy these bounds with minimum effect of sensor white noise are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary tradeoffs, a t every step in the design process. The same design philosophy may be used to attenuate the effect of disturbances on plants with parameter ignorance. 1. Statement of the problem This paper is devoted to the following problem. There is a single i n p u t output ' plant ' imbedded in a linear ' two-degree-of-freedom feedback structure '. The term ' plant ' denotes the constrained part of the system whose output is the system output. The designation ' two-degree-of-freedom feedback structure ' indicates a system wherein the command input, r(t) in fig. 1, Fig. 1 A canonic structure. and the system output, c(t), may be independently measured. In such a structure (Horowitz 1963) the system response to command inputs, and the t Communicated by the Authors. This research was supported by NASA under Grant NGR 06-003-083. Reprinted with permission from Int. J Contr., vol. 16, pp. 287309, Feb. 1972. . 86 ## I . M . Horou-itz and M . Sidi system sensitivity to the plant, may be, to some extent, independently controlled. The structure shown in fig. 1 is of course only one of many possible canonic two-degree-of-freedom feedback structure. The plant parameters are not known precisely. Only the ranges of their values are known. For example, the plant transfer function map be a known function of elements of the set X = {x,, x . . ., x,), but these elements are known only to lie in a given 2,' closed region m n-dimensional space. Strictly speaking, the design technique is applicable only to fixed parameter plants, but it is well known that for engineering purposes it is also applicable to ' slowly-varying ' plant parameters. It, is less well known that the feedback can be quite effective even for rather fa,st-varying plants (Horowitz 1963). However, this is a topic requiring considerable sepa,rate treatment, so it is assumed in this paper that the plant is fised, but there is on the designer's part bounded ignorance of the plant param~eters. Fig. 2 Bound on t 1ttt)l Tolerances on unit step response. The system sensitivity to the ' plant ignorance ' is to be characterized by the resulting ignorance in the system time response to a deterministic time input. A unit step is chosen here, but any other input may be used The problem is to guarantee that the output ignorance is contained within prescribed bounds ; for example, those shown in fig. 2, for the case of the unit step input. It can be shown (Horowitz 1963) that if the plant is minimum-phase, then any such specifications, no matter how narrow the tolerances, may be approached as closely as desired. From this, it follows that it is easy to overdesign in minimum-phase systems. The ' price ' paid is the large ' bandwidth ' of the loop-transmission function, L = GP(s),which in turn opens wider the ' window ' to the noise in the feedback return path, lumped as sensor noise N in fig. 1. In the high-frequency range where IL(jw)l< 1, but where I P(jw)l< IL(jw)l, the noise output a d X in fig. 1is amplified by IL(jw)/P(jw)l,which tends to be very la,rge over a large bandwidth in such systems. The highly amplified noise then saturates the output stages of G ( s )or input stages of P(s) for n large percentage of the time. This problem has been emphasized before. 1 . 1 . Optimizai!ion criteria This paper copes with the above problem by taking the response bounds as inviolate, but attempting to satisfy them with an L(jw) whose magnitude as a function of frequency is decreased as fast as possible. Another important reason for doing this in all feedback systems is the difficulty of having the paper design correspond to reality in a frequency range where plant parasitics and higher-order modes tend to dominate (Bode 1945). Another approach to Spthesis o feed6ack systems with large plant ignorance f optimization would be to minimize an index into which enter both the spread in the response and the effect of the noise. Statistical methods have been very usefiil for such indices, for many problems in which the plant parameters are precisely known. Attempts to do the same for the present problem have been i~nsuccessfi~l because of the need t o obtain expectations over P of expressions like I'/(I + PM). The practice (Fleischer 1962) has been to neglect the ignoranoe of Y in the denominator, and replace it there by some nominal Po, but this is obviously a very poor approximation for plants with significant ignorance I)ountls. I . 2. Principal steps in design procedure 1 hese are : ( I ) Translation of time-domain bounds on c(t) (such as those given in fig. 2) into bounds on I T(jw)l IC(jw)/R(jw)( fig. 1. of (2) 1I)erivationof bounds on L(jw)from the bounds on (T(jw)( on P(jw). and (:\$) b'orrnulation of the optimum L(s) from the results of Step 2. (1) Derivation of the prefilter F(s) of fig. 1. ( 5 ) Jlotlification, if necessary, of L(s) and F(s). 2. Design procedure 1.1. TI-anslation of time-domain bounds into bounds on In T(jw)l 1 1 a, minimum-phase system the magnitude of the frequency response 1 I T(jw)I coinpletely specifies the transfer function T(s), which in turn uniquely (letermines the system step response c,(t). Hence, bounds on In (T(jw)(are just as good as those on both the magnitude and phase of T(jw). But the rigorous translation of time-domain bounds into bounds on 1 T(jw)J is, as yet, 1 an unsolved problem. 1 1practice, however, it has not been difficult to achieve a translation suitable for any specific numerical problem encountered. One may begin, for example, by assuming a simple second- or third-order system model for T(s), and finding the bounds on the model parameters which correspond to the bounds on the time response. From the model parameter bounds one then determines the resulting bounds on In IT(jw)l. Suppose this step leads to the solid-line bounds B,,, B, of fig. 3. It is desirable, of course to increase the spread between B, and B,, but it will be seen that there is no advantage in doing so a t isolated points. There is benefit only if the spread increases, on the whole, with increasing w. One soon finds, with a little experimentation, that indicated modifications B,', B,' in fig. 3 are generally achievable. It is very helpful for such experimentation to have a computer programme for finding time response from the magnitude of the frequency response. Additional experimentation reveals that there is a definite limit to the permissible spread in the lower frequency range for a reasonably smooth curve of ]T(jw)J. Subsequent design details provide one with an appreciation of the frequency ranges in which broadening of the bounds niay or may not be important. Such ranges depend a great deal on the nature of the plant and of the ignorance of the plant. Hence it is best to obtain com1)aratively quickly estimates of B,,', B,' and proceed with the design. The designer will subsequently understand whether it is worthwhile to return for better determination of bounds on T(jclr)l. r 7 ## I . M . Horowitz arnd M . Sidi Fig. 3 I Derivation of bounds on IT(w ) [ . j 2.2. Derivation of bounds on L(jw) 2.2.1. Tem~la~tes P(jw) of It is assumed that the translation of time-domain into frequency-domain specifications has been accomplished and the latter are of the fdrm shown in Q. 3. In fig. 1, C = R T = R F L / ( l + L ) , L = G P andas there isno ignorance of F , G, A specific value of frequency is chosen, say w, r.p.s. The values of P ( j w , ) over the range of plant parameters are calculated and the bounds obtained. The procedunt is illustrated for the case This is conveniently done on the plane of In L(jw)=In ILI +j arg L, the abscissa in degrees and the ordinate in decibels (the Nichols chart). Thus, at w = 2 r.p.s., P(2j) lies within the boundaries given by A, B, C, D in fig. 4. Since ln L = l n G+ln P, the pattern outlined by A, B, C, D may be translated (but not rotated) on the Nichols chart, the amount of translation being given by the value of G(2j). For example, if a trial design of L(2j) corresponds t o the template of P(2j) a t A', B', C', D' in fig. 4, then IQ(2j)lx IL(2j)l- IP(2j)l= (- 2.0) - ( - 13.0) = 11.0 d~ ; arg G(2j) =arg L(2j)- arg P(2j)= ( - 60') - ( - 163.4') = 93.4'. ## Synthesis of feedback systems with large plant ignorance Fig. 4 Range of IP(j2)l and resulting bounds on L(j2) on Nichols chart. 2.2.2. Bounds on L(jw) in the Nichols chart The templates of P ( j w ) are manipulated to find the position ofL(jw) which results in the specifications of fig. 3 on In I T(jw)l being satisfied. Taking the w=2 template, one tries, for example, positioning it, as shown in fig. 4, a t A', B', C', D'. Contours of constant In ILI(1+L)I are available on the Nichols chart. Using these contours, it is seen that the maximum change in In J L / ( l + L)I which, from eqn. (1 a), is the maximum change in In IT1 is, in this case, very closely ( - 0.49) - ( - 5.7) = 5.2 d ~ the maximum being a t the point C', the , minimum a t the point A'. The specifications of fig. 3 tolerate e change of 6.5 d~ a t o = 2, so IL(j2)I is in this case more than satisfactory. One may shift the template lower on the Nichols chart until the bounds on A In ITI correspond This is achieved when the lower left corner of the template is a t A" to 6-5 d ~ . in fig. 4. The template corners are a t A", B", C", D", and the extreme values of In IL/(l+ L)I are a t C" ( - 0.7 d ~ )A"( - 7.2 d ~ ) . If L(2j) for condition A is , , chosen to be - 4.2 d ~arg - 60, then it is guaranteed that A In I T ( j ) l < 6-5 d ~ , over the entire range of plant parameter values. If arg LA(2j) - 60, then = 4.2 d~ is the smallest magnitude of LA(2j) which satisfies the 3.2 d~ specification for A In I TI. Any larger magnitude is satisfactory, but represents, of course, overdesign a t that frequency. The manipulation of the w = 2 template may be repeated along a new found. Sufficient vertical line, and a corresponding new minimum of ILA(2j)( ## I . M . Horouyitz and M . Ridi points are obtained in this manner to permit drawing a continuous curve of the bound on LA(2j) shown in fig. 4. The entire process may be repeated at other as frequencies. Figure 5 shows the outline of the templates and the resulting bounds on LA(jw) for a number of frequencies. The template outlines are drawn in any convenient regions on the Pu'ichols chart, since they are, in any ca,se, translated later. In each case the permissible region is to the right of the curve. It is important to note that although condition A (a = k = 1) was chosen to generate the curves, any other set of values of a, k could have beell chosen. However, once condition A was chosen at any one value-of w , this seme condition A must be used for all the contours. If the bounds on LA,are satisfied, then automatically those on all other sets of parameter values, as encompassed by the templates, are also satisfied. This means that A In I T(jw)/ will not exceed the specifications of fig. 3. Fig. 5 DEGREES ## Derived bounds on L ( j w ) on Nichols chart. Since P ( s ) is infinite at w = 0, the system is 'I'ype 1 , and the zero frequency specification on A In I TI (assuming it is zero) can be xut)isfied with any finite ## ~'Jynthesis feedback systems .with large plant ignorance of value for lim sL(s). In practice there will be a requirement on the velocity s+o 8 4 ## 2.2.3. System response to disturbances The system response to command inputs r(t) (of fig. 1 ) is not the only response function.of interest. There are, in most systems, also disturbances to be considered. The disturbance response (for D in fig. 1 ) is I t is necessary, of course, to choose L(s) so that the disturbances are properly attenuated, and the technique of this report lends itself very readily for this purpose as detailed in 5 5. For the present, only one aspect of this disturbance response problem will be considered, namely, that there will generally be a constraint on the damping factor of the pole pair nearest the jw axis. This damping factor can be related to the peaking in Thus, using the single complex pole pair as a model, a peak of 8 d~ corresponds closely to a damping factor 5 = 0-2 ; 2.7 d~ to 5 = 0.4, etc. The usual constraint on the damping factor can therefore be translated into a constraint on the peak value of IL/(1+L)I. Suppose this happens to be 2 d~ in the present example. If so, the contours in fig. 5 must be modified, as they now permit peaking greater than 2 d ~ .The required modifications are shown by the dashed lines. The parts of the contours rendered invalid because of the above requirement are to the left of the dashed lines. Note that a portion of the I T = 2 d~ locus I is common to all contours. 2.2.4. The single high-frequency boundary Sooner or later there is a frequency wh such that for a11 w 2 wh the boundaries become the contour V of fig. 5. The reason for this is that a t large frequencies + any rational function P ( s )= Ka(s +z , ) / ~ ( sp j ) degenerates into Ks4, where e is the excess of poles of P over zeros of P. In the present example P ( j w )+ k a / ( j ~a)t ~ large w. The plant template approaches a vertical line of length (Aku/,,,=40d~. Also, a t large frequencies J L J g 1and eqn. ( 1 a ) becomes Thus, the spread in In C ( j o )becomes equal to that in In P ( j w ) ,which is acceptable when the allowed variation in IT(jw)l exceeds the actual variation in IP(,jw)I,and there is then no need of feedback in this frequency range. However, there is still the requirement of acceptable disturbance response, noted in 2.2.3., which provides the boundary V for all w where this situation applies. In the present example, a t large w , A In (I>(jw)l =-to CIB, and fro111fig. such a permitted change in I T ( j w ) (is acceptable for w 2 60 r.1j.s. Hence, for all ## I. M. Horou-itz and M. Sidi w 2 60 = w,,, the boundary is V of fig. 5. This reveals that there is no point in increasing t,he spread in the bounds on ( T(jw)l for w > 60 in fig. 3, and provides some insight on the problem considered in 2.1. 2.3. Properties of optimum L ( j w ) The optimum design has been defined as that which satisfies the specifications and which, under a certain constraint, decreases as rapidly as possible with frequency. There is no limit on the latter if e k the excess of poles over zeros of I,@) is allowed to be infinite. In practice, e must be finite, so the constraint of :% fixed e value must be added. Section 2.2 in effect provides bounds on L(jw) at each to, although only a discrete number of w values are displayed. The problem is to determine that L(jw) which satisfies these boundaries and which at some very large w is as small as possible. This problem has only been partially solved. Some important properties of the optimum L(jw) have been derived. Thesle properties are stated here and their proof will be presented in a suh~sequent paper. 1. Case : Boundaries on L ( j w ) have the property that dlLldw < 0. In this case the optimum L lies at each w on its respective boundary. Also, such an L(jtw) exists and is unique. In fig. 5 the low-frequency boundaries have some regions of positive slope. This is always the case because at low frequencies the templates of P ( j w ) are almost vertical lines. However, if the system is Type 1 (L(s)has one pole at the origin), then the optimum L often tends to pass through the boundaries in their negative slope regions. 2. Case: Boundaries on L ( j w ) have portions of both positive slope and of negative dope. It is proven that a necessary condition for an optimum L which first crosses boundaries where the slopes are positive, followed by the crossing of boundaries where the slopes are negative, is that L(jw) must lie on its respective boundary at each frequency. Such an L(jw) function exists, but uniqueness has not been proven. This is the usual Ca5e. The above theorems, plus practice and inituition, lead to the conjecture? that the infinitely complex ideal L(jw) function with constrained e, and the additional cansrtraint that the maximum permissible phase lag of L is 90e degrees, has the properties shown in fig.6. Up to w, < whit lies on the boundary appropriate at each w . For w 2 w, up to some w , > whit lies on V as shown. At w, the phase abruptly jumps from - 0, to - 90e degrees. Hence /LA[goes to infinity on the vertical line 0 = - 0, and returns along the vertical line 0 = - 90e. The Nyquist plot of L(jw) is shown for e = 4. In practice, one can approximate the above by a rational function as closely as desired. Equation 4 and the discussion in 2,4give the designer a means of judging when the added complexity is justifiable. 2.4. Pradica% guides in shaping of L ( j w ) The boundaries shown in fig. 7 are used to illustrate the procedure with wh:= 8 r.p.s., i.e. for all o 2 8, L must lie to the right of the boundary marked V - ## Synthesis of feedback systems with large plant ignorance Fig. 6 (b) Conjectured ideal L ( j w ) on (a) Nichols chart, ( b ) complex plane. in fig. 7. It should be recognized that the phase a t zero frequency is fixed at 0, or - 90, etc., according as to whether the system is Type Zero, or One, etc. This may not correspond to the optimum L(jw) if there was no such constraint. However, the ' loss ' due to this constraint can be made very small, because it is possible, a t as low a frequency as desired, to have L(jw)at the more favourable point, wherever it may happen to be. The practical limitation is the inconvenience of having poles and zeros very close to the origin. In fig. 7, whatever may be the value of L at zero r.p.s., it is easy to arrange that L ( j 0.5) have any desired value. Consider the boundaries of fig. 7. Arbitrarily, suppose one tries to achieve, say, - 26 d~ at w =w,= 8. Why is this impractical ? Note that at w = 4, L approximately - 10 d~ is needed for ILI. In order that I I decrease to - 26 d~ at w = 8, L must have an average slope of - 26 + 10 = - 16 d~ per octave over this range of frequencies (between w = 4 and w = 8). But the average phase of L over this range is a t best approximately ( - 180- 130)/2= - 155". With such a phase there is associated an average slope of - 1551180 x I2 n - 10 d~ per octave which is significantly less than the assumed - 16 d~ per octave. A ## I . M.Horowitz and M. Sidi Fig. 7 DEGREES lExample of practical shaping of L(jw)-Nichols chart. value of - 18 c l for I I a t w = 8 seems more reasonable. This value is checked ~ L in the same manner. The average phase is, a t best, now - 130, so the average slope of I I is 130/180 x 12 = - 83 d~loctave. If IL(j4)I = - 10 d ~then using L , the above slope, \L(j8)1=- 10-8#, which is reasonably compatible with the assumption of - 18 d~ a t w = 8. It is seen that the value of - 10 d s a t w = 4 is a good starting point. It would be possible to use IL(j4)(= - 1 1 d s providing arg. L( j4) = - I 10". But then the average phase from w = 4 tow = 8 would only be ( - 1 10 - 130)/2= - 1 20, wit,h an associated average slope of - 1201 180 x 12 = -- 8 ds/octave, so that ) L ( j 8 ) x - 1 1 -- 8 = - 19, which is so close to - 185 as to make little difference. ] What of the other boundaries ? What points on w = 2, 11, etc., should one aim for ? On the w = 2 boundary the - 130" line is clearly the best, because the L boundary is almost flat there. I n any case the average slope for I I near w = 2 is certainly such that if IL(j4)I = - 10 d s , then IL(j2)I will be greater than - 8 d ~i.e. (Lj2) might as well be on V. Similarly, for L ( j ) . As for L ( j 0.5) , the designer need only make certain that arg L ( j 0 - 5 )is close to - 1 10" or so. Working backwards from IL(j4)(= - 10 ds, this means that (L(j2)Iwill be ap, , proximately - 1 d ~ IL(jl)l approximately 7 d ~and IL(j 0.5)1 approximately Synthesis of feedback systems with large plant ignorance 16 d ~providing it is arranged to have approximately - 130" phase for L from , w=8 to w = l , and about -110" phase at o=0-5. How can the above phase values be simply obtained ? An average phttae of - 130" may be obtained by alternating a lag corner frequency (lacf) with a lead corner frequency (lecf). Let 1 + a be the number of octaves under consideration, with a slope of - 6 d~loctave over one octave and a slope of - 12 d ~ / octave over a octaves. Then the average phase lag is which is to be - 130" in the above example. Solving for u gives a = 0.8. Thus, if one allows a slope of - 12 daloctave for one octave duration, he should allow a slope of - 6 d~loctave 110-8= 1-25octaves. for 2.4.1. Rational function approximation The above results are applied to the example of fig. 7 as follows. About - 110" is desired at w =0-5. This is a Type 1 system, so assign a lacf (lag corner frequency) at w,, a lecf (lead corner frequency) at 2w1, such that the is ( 2 ~such ~ ) asymptotic slope of - 12 d~loctave over one octave, a lacf at that the asymptotic slope of - 6 d~loctave over 1-25 octaves, etc. Try is various values of w,, until w, is found such that the net result of an infinite series of the above leads to - 110" phase at w = 0.5. To simplify the numbers, 2.6 was used instead of 21.25= 2-37 and w, = 1, which is a conservative choice. In accordance with the above procedure, the lacf at w = 1 is followed by a lecf at w = 2, a lacf at w = (2.5)2= 5, and a lecf at (2) (5)= 10 r.p.s. This procedure is halted at 10 r.p.s., because near w = 10 (see jig. 7) a gradual decrease in phase Fig. 8 40 I 1 1 I 1 Ill( I 1 1 1 1 Ill( 1 1 1 1 111-220 - 100.1 0(I I l 1 1 1 1 l1 I 1 0 W I 1 1 1 1 1 1 10 0 ## I . M.Horoulitz and M . Sidi is permitted. The level of IL( is moved vertically until IL(j4)I= - 10 d ~ . The phase lag of the design, so far, is Curve d in fig. 8. The next lacf will be set a t a somewhat higher frequency than would result from the above formula, because it will be followed by two more lacfs, in order to have a final asymptote of - 24 da/octaves, corresponding to an excess of 4 poles over zeros for L(s). The situation at w = 10 is examined. At present arg L(jl0):= - 114", and from fig. 7, it can be x - 14.0" if IL(j1O)l > - 26 dn. Hence 26" more lag is tolerable a t w = 10. A lacf a t w = 30 contributes - 18.5" abtw = 10, lea,ving 7.5". The latter permits a c o q l e x pole pair a t 4 r.p.s. if its damping factor is 0.2. The phase lag due to all poles (andzeros, excepting the last complex pole pair, is given by Curve B in fig. 8, while the total phase lag ie given by Curve C. The resulting L is also sketched in fig. 7. The boundaries &reslightly violated for 6 < w < 10. More significantly, for w > 12, L could have more phase lag, which would permit its faster reduction. The following relation (Bode 1945) gives one an idea of the magnitude reduction available by increasing the phase lag : Compare two designs, in which IL(O)I is the same in both, but in one region 0, < 8, by, say, an average of 20" over one octave. The difference between the two, for the loft side of (4)is ## I (z) 2 = 0.154 nepers = 1-34dn. ln 57.3 7r Equation (4) is useful for estimating whether, in any particular problem, it is justifiable to seek to improve a tentative design. Thus, in fig. 7, a t w = 0.5, 1, 2 the phase lag could be increased by 6", 13", Go, respectively, so that the left side of (4) is increased in magnitude by about 1.7 dn. However, most of the resultand In (L(co)( would be due to ing increase in the difference between In (L(O)( a,n increase in IL(O)I, rather than a decrease in IL(co)l. The reason is that L(j4)I cannot be decreased, and since the increase in phase lag is in the frequency range less than 4 r.p.s., the decrease in slope of In IL(jw)l will occur primarily there. For this same reason, the increase in phase lag possible in the range w > 12 approximately is more appealing, for it occurs in the region w > 4, so most of the effect will be in the higher frequency range. In any case, the detailed shaping of L(jw) tends to be somewhat onerous and it is very desirable to develop a computer programme for this purpose. The above ttechniqucs provide one with a first approximation. 3. Design example It is useful'& this point to apply the above steps to a significant design example in order to better appreciate the later steps in the design procedure. ## The plant transfer function is given by Synthesis o feedback systems with large plant ignorance f with 1 f k f 1000, and the complex pole pair of P ranging anywhere in the rectangle M, N, Q, U of fig. 9 (a). The performance specifications require that the step response, characterizable by a dominant complex pole pair, be such that this complex pole pair lies within the region A , B, D, E, F, G in the complex plane of fig. 9 ( b ) . This problem has been treated in the literature (Olson and Horowitz 1970) by the Dominant Poles Method. Fig. 9 -6 -4 -2 0 S plane (a) S plane (b) (a) Range of complex pole pair of P(s). ( b ) Acceptable range of complex pole pair of second-order model of T ( s ) . The first step was to translate the dominant-poles specifications of fig. 9 ( b ) into equivalent ones on I T(jo)l. A second-order system model was used, and the range of model parameters found which satisfies the domain specifications. In this example this is already available, in fig. 9 ( b ) . The next step is to find the resulting range of variation of I T(jw1. These are shown in fig. 10 with the labels corresponding to those in fig. 9 (a). I t is important to broaden the in permissible range of 1 T(jw)J fig. 10 as much as possible, for clearly this permits design by an L(jw) of smaller bandwidth. To broaden the permissible range of ## I . M . Horou-itz and M. Sidi 1 T(jw)l one may proceed by trial and error, trying IT(jw)l functions which lprogressiv~ely decrease faster as functions of frequency, until the time-response specifications are intolerably violated. Similarly, one tries I T(jw)l functions -whichprogressively decrease more slowly versus w, until again the time-domain :specifications are intolerably violated. In this way the significantly larger bounds shown by the dashed lines in fig. 10 were obtained. Fig. 10 Bounds on (T(jw)lcorresponding to fig. 9 (b) (solid lines) and larger bounds by cut and try (dashed lines). The pro~cedure the previous sections was then followed. Some of the of plant templates are shown in fig. 11. Note that for 2 < w < 10 they extend to infinity, because from fig. 9 (a) this range is included in the plant pole-range variation. The resulting boundaries on L and several alternative designs of L are shown in fig. 12. These were all done by hand calculation. It is interesting to compare the four L,'s of fig. 12. The larger phase lag of L,, results in a larger value of [In IL,(O)( - In I L,(oo) I] (recall eqn. (4) and the discussion there). However, since this extra phase lag is in the low-frequency range, and the value a t o = 10 is even higher than the rest, this difference means only a much larger (L,,(O)I than the rest. One should not make much of the small differences at w = 100 r.p.s., because the design was performed by hand calculations, which do not permit very precise optimum design. In any case the resulting design leads to the changes in shown in fig. 13 (a). Most of the responses lie outside the permitted boundaries, which is not surprising, because so far the design only guarantees that the change in T(jw)l is no larger than the maximum change permitted in fig. 10. Fig. 11 360 -320 -280 -240 -200 -160 -120 -80 -40 DEGREES ## Plant templates at various frequencies. Since T = FT' (fig. I ) , it is necessary to choose F so as to shift the spread in T' into the permissible spread of T. For example, at w = 10, T'I lies between 3.6 and - 3.0 d ~ .Since (TI must lie below - 7.5 d ~ I F ( j l O ) l = - 11.1 d ~ .If L , was properly chosen, it is guaranteed that T ( j l 0 )does not range below its lower boundary TI'. The prefilter F is thus determined. Note that J F Juniquely determines arg F and the result can be approximated as closely as desired by a rational function. The step responses resulting from fig. 13 ( b ) are shown in fig. 14. These responses fall within the time-response bounds corresponding to the complex pole range of fig. 9 ( b ) . However, there are some time responses whose behaviour, as a function of time, may not be acceptable. The reference is to Curve 8, in which the first overshoot is below the final value. The corresponding frequency response marked ' 8 ' in fig. 13 ( b ) has a minimum followed by a peak which does not reach the (T(O)(reference. This is denoted as the ' wobbling problem '. (It is interesting to note that reflection of the IT(jw)l frequency response about the y exis at o = 0 gives a roughly qualitative picture ## I . M.Horowitz and M.Sidi Fig. 12 Bounds 011 L(jw) and some L(jw) designs. of the system step response. This is especially true of the ' wobbling' plhenomenon.) The wobbling problem will be next considered. 4. The wobbling problem Case 8 in fig. 13 (b) is a good example of a ' wobble ' in ( T(jw)( Evidently, . at a low frequency (2 r.p.s.), Case 8 must be at the low end of the template of Y(j2) (see fig. 11). When this template is positioned to find the boundary of acceptable L(j2), Case 8 lies near the low extreme of IT(j2)I. On the other hand, at w = 3, if the template of P(j3) is calculated, it is found that Case 8 lies near the top of the template, towards the left side, which means that it will be near the high extreme of I T(j3)l. The occurrence of a wobble of this type may therefore be predicted when there are some plant conditions which exhibit this kind of behaviour . How can the ' wobble ' in the frequency response be eliminated ? The simplest but least economical way is to increase the level of (LI for all w. The corrective effect is as follows (see fig. 15). At w = 2 the position of L(j2)is, say, ## Synthesis of feedback systems with large plant ignorance Fig. 13 (b) Range of I T(w ) 1-first j design. ## I . M. Horowitz and M . Sidi Fig. 14 Step responses-first design. in the neightbourhood of Q, i.e. in the low end, while a t w = 3, L ( j 3 )is, say, a t L', in the high end. Increase of (LI a t the same phase improves matters a t both frequencies. It decreases (TIa t w = 3, and increases it a t OJ = 2, both effects helping to straighten out the wobble. This method was used in the above , design example, increasing ( L (by 2 d ~which almost completely eliminated the ' wobble '. Fig. 15 m 0 - 140 -100 60 DEGREES - 20 ## Effect of irlcreasc of II on wohllle. L It is certainly possible to eliminate the wobble in a more economical tnanner, i.e. with a smaller increase of I I a t high frequencies. The two principal freL quencies of the wobble, i.e. the minimum and maximum points, which were at w = 2, 3 far Case 8 in fig. 13 ( b ) ,are considered. Figure 16 (a) is used to present the argument, with extreme wobbling for Condition 1, between w, and w,. Suppose L is changed a t w , such that I T I (is increased and IT,( is lowered, from points A,, .A, to points B,, B,, respectively. This by itself alleviates the Fig. 16 DEGREES (b) a : ## ~5'ynthesis feedback systems with la.rge plant ignorance of ' loss ' as much as possible. It is certainly better to do this than t o allow the correction, i.e. this higher level of ILI, to stand as is over this higher frequency range where it is not needed. 5. Application to disturbance attenuation The same technique may be applied to the problem of disturbance attenuation accompanied by plant parameter ignorance. Consider the three disturbances of fig. 17, one by one, ## C 1 T~~ - = - with L = @ P I P , . = D, 1+L' Let I = 1/L and then The rotation of the Nichols chart by 180" is equivalent to the transformation L = 111. Hence, the Pl'ichols chart may be so used for TD3. Templates of P = P I P , are found in the same manner as before and so are boundaries of l ( j w ) to satisfy given frequency response specifications on TD3,etc. Fig. 17 D3 -I ## Three kinds of disturbances. Consider where Po is any nominal plant transfer function. I t may be more convenient to work in the complex plane rather than in the Sichols chart. At any given frequency the range of P , / P may be found. Suppose it is as shown in fig. 18 ( a )at w = w,. Suppose also that -L,(jw,) at point A I is under consideration. Then P where B may raiipc :~iiy~vIlcre inside or on (/',,'I')( j w ) . O I ~ !nay ~ I I ( - I Itind the l)ooutlary of the - ~ , , ( j w , ) nhich satisfic~s llc. fi'c.clrlc~~icy t rcs1)orist1s11cv.ifc.atior1n of'the tlesipli is strnigl~tfor~vard. on T,),.The ba1anc.e~ ## I . M . Horozcitz and M . Sidi Fig. 18 (b) (a)Design technique for D,. ( b ) Design technique for D,. Finally, consider ## The boundaries of' are obtained and sketched as in fig. 18 (b). The region A of yl maps into the S!/nti~esisof feedback systems with large plunt ignorance larger region A' of y,,, B into B' and parts of A', B' may overlap. This is all a t some specificfrequency w,. From eqn. (10) When the region A in y, is considered, then the appropriate range of iV, in fig. 18 (h), is A', etc. In this way boundaries of acceptable -Lo(jwl) may be obtained and the design continued by the methods of this paper. Finally, if there are specifications on both the system response to commands, and its response t o disturbances, it is possible by the means described to find the resulting boundaries of L ( j w ) for each separately, and then take segments of each, such that both requirements are satisfied. REFERENCES BODE, kV., 1945, hTettc.orkAt~alysis H. and Feedback Amplijier Design (Xew Yorlc : Van Kostrand), pp. 358, 472, 286, 338. FLEISCHER, 1962. I.R.E. Tranr. autom. Control, 7, 2. P., HOROWITZ, 1963, Sp)ttl~e.~isFeedback Systems (New York : Acatlrtllic I'rcss), I. M., o f pp. 546, 332, 441, 205. OLSON, E., and HORO~VITZ, 1970, Int. J . Control, 12, 545. D. T. M., A Graphical Test for Checking the Stability of a Linear Time-Invariant Feedback System FRANK RII. CALLIER AND ## CHARLES .A. DESOER Abstract-A graphicial test is developed for checking the condition 20 11 k\$(s)l : 0 where k is a nonzero real constant and \$ is > the sum of a finite number of right-half plane poles and the Laplace transform of an integrable function plus a series of delayed impulses. As a conseqence, 1 \$- k\$ is, in R.e s >_ 0, asymptotic to an almost , periodic function, say f for Is/-t m. Theorem 1 gives a necessary involving the curve {f(jw)Iw E R) to ensure and sufficient conditio:~ (s) that infa, , l o (f > 0 ; Corollarjr 1 gives a corresponding graphical test. Theorem 2 and Corollary 2 give a necessary and sufficient condition involving the curve (1 f k\$(jw)lw E R) and a graphical test to ensure iXlf~e 20 11 k\$(s)\ > 0, a condition that guarantees the L p stability of the feedback syr;te.m for any p E [I, m 1. considering only the almost periodic part s the open-loop f transfer function. Finally, Section I11 gives the solution of the problern. OF I . DESCRIPTION THE ## SYSTEM ASSUMPTIONS AND We consid'er a continuous-time scalar linear timeinvariant feediback system with input u, error e, and output y. The la~tterare functions mapping R+ into R and satisfy y=ye (1) INTRODUCTION T IS well known that the classical graphical test for stability [ I ] of linear time-invariant feedback systems is extremely important for two reasons: 1) it is based on experimental data that are easy to obtain with great accuracy, and 2) m case of instability it gives clear indications of the required design modifications. Recently, Willems [2], [3] cleveloped a graphical test for a scalar linear time-invariant feedblack system with constant feedback where Lhe open-loop impulse response g(t) belongs to the convolution algebra A [4] of integrable functions @ impulses, and contains equally spaced impulses. This paper generalizes Willems' result in that: 1) the open-loop transfer f~mction\$(s) is the sum of a term in A (i.e., the algebra of Laplace transforms of elements in A ) and a finite number of poles in the closed right-half plane, and 2) it dloes not require that the impulses of y(t) be equally spaced. As a consequence, the function s +-?(s) is asymptotically almost periodic in Re s 0 for I sl- a ,and the most elegant conformal mapping technique of Willisms does not work. We have to rely heavily on the theory of almost periodic functions [6]-[8]. Therefore, the paper is organized as follows. I n Section I, the problem is defined and the notation is laid out. I n Section 11, we concentrate on the solution of the problem, where denotes the convolution operation, g is a realvalued distribution with support on R+, and k is a real nonzero constant. Let ^g denote the Laplace transform of g. We assume that \$ has the following form: where the poles p, are either real with real residues or conjugate complex with complex conjugate (4) residues ; Rte p, > 0, for a 1,2,. . .,1; (5) g, belongs to the convolution algebra A [4], i.e., gT(t) == 0, g,(t) such that g,(. = fort <0 (6) > g,(t) - I i=O 2 gi6(t - ti), for t L 0 ## ) is a real-valued function belonging to LI (o, ) (7) (8) g, E R, for i 0,1,2,. . . , Manuscript received January 6, 1972; revised April 28, 1972, and August 4, 1972. :Paper recommended by J. C. Willems, Chairman of the I E E E S-CS Stability, Nonlinear, and Distributed Systems Committee. This work was supported in part by the National Aeronautics and Space Administration under Grant NGL-05-003-016 and in part by the Joint Services Electronics Program under Contract F44620-71-C-0087. F. M. Callier is with the Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, Calif. 94720, and is also an "Aspirant" with the Belgian National Fund for Scientific Research, Brussels, Belgium. C. A. Desoer is with the Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, Calif. 94720. Note t.hat g, belongs to A if and only if its Laplace transform ?, belongs to the algebra 2 with pointwise product. I t is immedia,te that I t can be checked that the ordering of the ti's in (10) can be replaced by t; > 0 for i = 1,2,. . ., both for the needed results from the algebra A and from the theory of almost periodic functions. ## 773-780, Dec. 1972. \$,(.) is analytic in Re s > 0, bounded in Re s 2 0 and each function w )+ \$,(a jw) with s j w is uniformly continuous for a 0. (11) > ## It follows, therefore, that \$(-) is meromorphic function in Re s set S c L is said to be relatively dense on L if, for some length I, S is I-relatively dense on L. Given a set D c R (or c C) and a complex-valued function h: D -t C, an element T of D is said to be an E-translationnumber o h on D iff f Ih(z > + T) - h(z) 1 5 t, for all z E D. A complex-valued function h of a real variable x is said to be almost periodic iff, given any c > 0, there exists a real (12) number 1 = I() > 0 such that the set of -translation A necessary and sufficient condition that the closed-loop numbers T = T(E)of h on R is I-relatively dense on R. impulse response of the system defined by (1)-(10) is in A The number 1 ( ~is called the density length. ) (this implies that the closed-loop system is IApstable for Let a! < @ i a . A complex-valued function h 1l p 5 a)is of a complex variable s, analytic in 'a (vertical) strip (a,@), said to be almost perioclic in a strip (a!,@ ( [ a ! , / 3 ] ) is inf lc\$(s)l > 0. (13) iff, given any E > 0, there exists a real number 1 = I() > 0 Re s 20 such that the set of imaginary -translation numbers The proof follows easily using methods of [ l l ]and [12]. j~ = ~ T ( E ) h on the strip (a!,@) ([a!,@])is I-relatively of It should be stressed that it has recently been established dense on the imaginary axis. that, for any convolution feedback system whose closedNote that this last definition requires that the functions loop impulse response is in A, \$(s) can at most have w (+h(a jw) be almost periodic on any V, for a E isolated poles in Re s > 0 [5], [13]. (a,@)[a!,@]) ( with an almost periodicity that is independent The problem is to develop a graphical test for (13) based of u, for a E (4) ([a,@]). on the curve f 1 k\$(jw)lw E R).Let Fact 1: The function .f defined by (14)-(15) is almost . . . m m periodic in the strip [0, ). f(s) k 1 + k C yie-szi Li C fiewszi. i=o i=o (I4) Proof: As already noted, f can be uniformly approximated in the strip [0, a ) by a finite number of terms of the Then f(s) is a Dirichlet series with Dirichlet exponents series (14), i.e., by an exponential polynomial fN(s) 6 - ti subject to (10) and Dirichlet coefficientsf i such that fiewSzi. is easy to show that thelatter is an almost It for i = 1,2,. . . fo = 1 kg,; f i = kg, (15) periodic function in the strip [0, a ) [S, p. 731. Therefore, f is almost periodic in the strip [0, a ) . where the coefficients y, satisfy (8)-(9). Let us now consider the distribution of zeros of f(s) in First we develop a condition expressed in terms of the the strip (0, a ) . 20 curve { f(jw)lw E R), ensuring that infaeS f(s)I > 0; Fact 2: If f(s), defined by (14)-(15), has a zero SO = and then we use this result to develop the condition a0 jwo in the strip (0, a ) , then f(s) has infinitely many involving f 1 k\$(jw)lw E R] that will ensure (13). zeros in any strip (a,@) (with 0 < a < /3 < a ) containing Given s = a jw, we denote by V , the vertical line in C so, and their imaginary parts are relatively dense on the (i.e., the complex plane) with Re s = a. RiIoreover, by imaginary axis. j(s) we mean the complex conjugate of f(s). Let Proof: Without loss of generality, we assume that f(s) is not identically zero in Re s 2 0. SinceS(s) is analytic np 4 the number of poles of j(s) counting multiplicities with Re p, > 0. (16) in Re s > 0, its zeros are isolated in Re s > 0; therefore, m > 0 on we can choose 0 < r < a0 such that If(s)l - sol = r. By Fact 1, for any 0 < c < m there exists 11. A NECESSARY SUFFICIENT AND CONDITION INVOLVING a set of -translation numbers jr = jr(e) of f on the strip {f(jw) w E R) TO ENSURE infR, 20 J(s) > 0 [0, a ) that is I()-relatively dense on the imaginary axis. Note that f,defined by (14)-(15), is in A as a conse- Hence, by f(s j r ) = f(s j ~ - f(s) ) f(s), it f o l l o ~ s quence of (8)-(10) and can be uniformly approximated in from (17) and Rouche's theorem (see [9, theorem 9.2.3, Re s '0 by a finite number of terms of the series (14). p. 2541) thatf(s) has a zero in any disk 1s - (so jT)/ < T, Hence, which proves Fact 2. Q.E.D. [9, p. 2171, well defined and continuous almost everywhere in Re s 1 0. < 1 + ELo + + I I IS > > (I7) We now state some standard definitions [6], 171 and facts that streamline the proof of Theorem 1. Given an infinite straight line L, a set S c L is said to be 1-relatively dense on L iff any open interval of length I on L contains a t least one point of the set; moreover, a > 0 and ## Definition oi the Argument O(s) of f(s) Bydefinition, \$(s) = arg f(s) Im log f(s), in Re s 2 0 (18) with two additional conventions. (A.l) Let L denote an oriented straight line in Re s 2 0. By cnov . take s F L as the right- &u. ## CALLIER AND DESOER: GRAPHICAL S'rABILITY TEST argument of f(s) on L, i.e., \$(s), s E L is an arbitrary branch of the argurnent, which is continuous except a t the zeros of f(s) on L, while it is discontinuous with a jump of +mn, when s goes through a zero of f(s) of order m in the positive direction of L. At any discontinuity point we assign t o 4 the mean value of its one-sided limits. The function 4(s), s E L is then well defined (mod 27r) because of (17). (A.2) Because j'(j0) 1s real and because it will later be assumed to be nonzero, we pick for w /+@(jw),w E R that branch of the argument such that i ( j 0 ) = 0 (or n), according to whether f(j0) is positive (or negative, respectively). Remarks (R.l) I t is important to oblserve that, by convention (A.l) and (17), the principle of the argument may be applied tof(s) on any rectangle in Re s 0 that is oriented in clockwise sense and that has no zeros of f(s) on its corners. in (R'2) Because of (I7) and (18), for any strip Re 2 O such that O 5 a < P 5 and inf@>RP 8 2 " > 0, m(s) is well defined (mod 2 ~ and uniformly continuous ) in the strip [a,P] and analytic in the strip (a,@). Since, by Fact 1, w I + f(jw) is almost periodic, we have Fact 3. t Fact 3 171: ~ ef be defined by (14)-(15). If ## any -translation number T(E) w I+-~(jw) of satisfies ( Wj ) - ( j-2 6, for all w E R (25) where c, is an integer depending on T; (d) the function w -++(jw) is almost periodic if and only if the mean angular velocity X of j'(jw) is zero, or equivalently, if and only if there exists an increasing sequence ( w,) ,fZ'-, satisfying ## . . . < w-, < . . . < w-I < wo W--, = -w,, lim w, n+ rn < wl < . = for n = 0,1,%,.. . (28) (29) > such that ( w ) = ( 0 ) for n . . .,-2,-1,0,1,2,. - .. (30) inf lf(jw)l IOER > 0, (19) Proof: Part (a) is a straightforward transcription of (7, p. 167, theorem I ] . Part (b) : (21) implies that the mean f i e - j ~ k are angular velocities of f(jw) and fN(jw) & the same; for the latter exponential polynomiai (22) and (23) are valid (see [7, pp. 170-1761). Part (c) follows from [7, pp. 168-1701. The first statement of part (d) is obvious from Part (a). The second statement is established as follows. =) Since. f(-jw) = then : 4(jw) = j(jw), ## for all w E R for all w (31) f h(jw) (20) E R. (32) h(jw) is almost periodic; where X is a constant and w the constant X will be called "the mean angular velocity of f(jw)" (in the literature, the term "mean motion of J(jw)" is used; however, this is borrowed from celestial mechanics. Note that X = lim,,, [(4(jw) - 4(-jw))/ 2~1); (b) if N is the least nonnegative integer such that m I+ So, unless 4(jw) =4(j0), then for some w' +(3wf) > 4(\$0 or > 0 either > 4(-jwf) N+ 1 lfil 5 K sin (6/2), for some 0 (21) ## of f(jw) may be written where the coefficients ho,hl,- - .,hN are integers with sum 1 and X = - roto - Then (27)-(30) follow by the continuity and almost periodicity of 4(jw) on R and by (32). implies that (=The existence of the sequence { w,] ,"= -, 4(jw) is boun~dedon R; hence, the mean angular velocity 4(jw) is almost of f(jw) is zero, and thus, by (20), w Q.E.D. periodic. Before we give Theorem 1, we give a last interesting result. f Fact 4: Let j' be defined by (14)-(15). I I-+ inf If(jw)j A K oER >0 r1tl - . . . - TN~N (23) where the coefficients ro,rl,- . . ,rN are nonnegative rationals with sum 1; (c) with C, K sin ( 6 / 2 ) , then, given any u > 0, there exists a positive-real number 6, depending on u such that \4(u 0, a* ## + jw) - \$(jw)( < Cc uniformly in w. < 7r (24) Proof: Because of (19) and (17) there exists a a* > < u such that inf,* I.f(s)\ > 0. SO,by remark (R.2) and convention (A.2), d(s) is well defined and uniformly continuous in the strip [O,a] such that there exists a positive constant C, depending on a* for which Ic\$(u* ## + jw) - cp~jw)]< C,* uniformly in w. Observing that [u,u] is a closed substrip of the strip [O,w), in which f(s) is almost periodic by Fact 1, it follows that there exists a positive constant C,-, depending on a - a* such that (4(u principle of the argument may be applied to each of these rectangles oriented in the clockwise sense. We show now that the sequence ( ~ 4 , ) (where A4, is the net increase in argument around the rectangle R,) is bounded. This follows easily if one observes 4(jw) is a) t,hat, by (19) and (35) and fact 3(d), w almost periodic, and hence bounded. b) that, by (19) and Fact 4, I-+ ## + jo) '- +(a* + jw)l < C,-, uniformly in w + 14(ui + jw) 4(jw)1 < Cult uniformly in w. [7, pp. 178-179, theorem 3(iv)]. Combining the two C,-, the fact is results, we obtain that with C, = C,* true. Q.E.D. Theorem I : Let f(s) be the Dirichlet series defined by (14)-(15). Under these conditions, Re s 2 0 c) that, by (40), any branch of w ~ ( U I jw) is bounded. I n view of this, it follows then that there exists a positive constant C such that N = lirn N, n * rn .-I nm lim / ~ 4 , l C . < (42) inf I f(s) 1 > 0 if and only if ## ii) inf (fjw) oER >0 (19) iii) the mean angular velocity X of f(jw) is zero. (35) Proof: (= Observe that, because of (14)-(15), (17), and (34), lim f(a a-+m + jw) fo # 0 uniformly in w. (36) ## Thus there exists a 8 > 0 such that inf If(s)/ Res>; > 0. (37) So, by remark (R.2), +(s) is well defined (mod 27r) and uniformly continuous in the strip [8, m). Hence, (36) implies, for any branch of 4(s), lim 4 ( ~ jw) = 4,, r+ rn uniformly in w (38) where 4, = arg f~ (mod 27r). R/loreover, we can pick a a1 > 0 so large that and I4(ai 4- jw) - 4-1 Finally, observe that f ( ~ + 0 in ) 01 < 1, uniformly in w. (40) > Re s > 0 =) ## inf \$(s) a~>Res>O I > 0. (41) This can be proved using contradiction and [S, p. 71, theorem 3.61. Therefore, in view of (19), (39), (37), and (41), we will have established (33) if we show thatf(s) # 0 in the strip (0,al). By contraposition of Fact 2, f(s) # 0 in the strip (0,al) if we show that N, the number of zeros of \$(s) in the strip (O,ul), is bounded. Consider, therefore, a sequence of rectangles {R,) ,"=1 defined by R, [Olal] X 1-n,n] for n = 0,1,2,-. and let the corresponding number of zeros of f(s) inside R, be N, for n = 1,2,-. .. Observe that, because of (19), (37), and remark (R.l), the =) First, observe that the first equality of (36) still holds, so (33) implies (34). Next, (33) implies (19); hence, +(jw) is well defined and satisfies by Fact 3(a), w (20). Furthermore, by (33) and (17) and convention (A.2), s 1 +(s) is well defined and uniformly continuous 4 in Re s 2 0. Hence, again (36) implies (38), and again we can pick a u1> 0 such that (39) and (40) are true. +(jw) is almost periodic. For We claim now that w this purpose, in view of (20), it is sufficient to show that +(jn) for n = 0,1,2,. remains bounded as n + c o . Consider, therefore, the sequence of rectangles { R,'] ,"= defined by R,' [O,al] X [O,n], n = 1,2,- - . . By (33) and (17), it follows that the principle of the argument can be applied to each of these rectangles; hence, the net change in 4 around each R,' is zero for n = 1,2,. . .. Now, 4(s) in Re s 2 0, there by the uniform continuity of s exists a constant C independent of n such that for any {s = a jn: 0 I I al) a horizontal segment H,' for n = 0,1,2, . : (4(al jn) - \$(jn)l < C. This fact, with (39), implies that the sequence {4(jn) - 4(j)0] i Z ; is bounded. Hence, because 4(j0) is 0 or r by (A.2), the sequence {4(jn)),"=o is also bounded. So our claim is true, and by Fact 3(d) the mean angular velocity of f(jw) is zero, which implies (35). Q.E.D. It is interesting t o observe that under the conditions of kgo). Hence, Theorem 1 sgn f ( 9 ) = sgn fo = sgn (1 if 1 kg0 > 0 (respectively, <O), then 4(jO) = 0 (respectively, a). Now we want to develop a graphical test involving {f(jw)lw E R) to ensure infa, ,>o If(s)I > 0. Here, again, it will be the almost periodicity of w f(jw) that will save us. We start giving some definitions and two facts. Let I ( ) be the density length for the -translation numbers of w f(jw). It is known that -translation numbers of f(jw) can be determined by diophantine analysis (see, e.g., [8, pp. 146-1491). From their pattern, a density length I() can be determined. Consider now (see Fig. 1) the path T(e) defined by I-+ I-+ I-+ 1- I-+ ## and its closed -neighborhood N(e) defined by N(t) {z CI lz - \$(jw)l c; w E [o,l(t)I ) . (44) ## Fig. 1. The path (E) ancl neighborhood N ( E ) . Now we prove Fact 5, which allows us by the simple knowledge of the path y(e) == (f(jw)lw E [~,l(e)]]to locate the closure of the set {f(jw)lw E R) , and Fact 6, which informs us about the minimal value of X if it is nonzero. t Fact 5: ~ ef(jo) be given by (14)-(15) (setting s = jw). Consider the path ~ ( e ) and neighborhood N(e) given by (43) and (44), respectively. Under these conditions (a) for any e > 0, N(e) contains the closure of the set tf(jw)lw E R} ; (b) as e 4 0, then N(e) tends toward the closure of the set {f(jw)(w E R } . This fact follows by the almost periodicity of f(jw). Fact 6: Let f(jw) be given by (14)-(15) (setting s = jw). Assume that inf jf(jw)l u'R is well defined. (c) If the mean angular velocity X of f(jw) satisfies < Amin, then X = 0. . Proof: (a) is an immediate consequence of (19). (b) admits a is a consequence of the fact that the set ( -ti} such jinite integral base, i.e., a set of real numbers that i) there exist no integers hj, j = I , . . .,M, not all hjPj = 0 and ii) each number - t, zero, such that can be expressed in a unique manner in the form -ti = hj(i)Pj for i = 0,1,. . .,N, where hj(i) are integers (see [7, p. 1461, [6, pp. 82-83]). Equivalently, the N 1 numbers -ti can be represented by lattice points (i.e., with integer coordinates) in RM-space; indeed each point -ti may be represented by the M-vector (hl(i),hz(i),. . ,hjM@)) . with integer coordinates. Now (45) merely expresses the fact that the numbers z cah be represented as a subset of lattice points in RM that are 1 lattice in the closure of the convex hull of the N points hci), i = O,l,. . .,N, in RM. Hence, the set X is finite. The set X - (0) is nonempty since 0 > -ti E X for i = 1,2,. . . ,N. Therefore, (b) follows. Concerning (c), observe that, because of Fact 3(b), the mean angular velocity X of f(Cjw) belongs to X and also that 0 belongs to Q.E.D. X ; hence, (c) is a direct consequence of (b). As a final remark, observe that, because the e-translation numbers are relatively dense on R, it follows that, as soon as (19) is satisfied, we can pick a translation number r(e) such that (&)El xz zE1 :& > 0. Under these conditions: (a) The set X , suggested by conditions (21)-(23) of Fact 3(b), is well defined. For convenience, we note that We are now ready for a graphical test ensuring infRe 20 lf(s)l > 0. Corollary 1: Let f(s) be the Dirichlet series defined by (14)-(15), and @)-(lo). Let y(e) and N(e) be given by (43) and (44). Under these canditions, Res 2 0 ## inf \f(s)/ > O C hit, i=o i=o x riti; hi are N if and only if i) fo = + kg0 # 0 (48) 2 inf if(jw)l, oR x hi rationals with x ri integers withi i=O 1; r i are nonnegative ii) the origin 0 of the complex plane is positioned with respect to {f(jw)lw E R} such that a) there exists an belong to N(E) E i=O n+l x 1fij 5 K sin (45) b) for a m ## > 0, with 0 < e < K where -ti and fi are the Dirichlet exponents and coefficients of f(s). (b) The set X has a finite number of nonzero elements and, if N 2 112 the positive number X,in then given by for which the corresponding e-translation number r(e) satisfies (47), where Xmin is defined by (45)-(46), I+(jr) - +(j0)1 < n must hold. (49) Proof: Because of Theorem 1, we need only to show that (48) and (49) are equivalent to (19) and (35). Clearly, by Fact 5 , (48) (=) (19). So we are left to prove the equivalence of (49) and (35) under the assumption of (19). =) We assume that (49) is true. Then e < K implies e 5 K sin (6/2) with 6 < n. So, immediately by Fact 3(c), l+(jr) -. +(jO) - c;2nl 5 6 < n. Thus, by (49), The case N = 0 is immediate since then X = 0 by Fact 3(b). For this reason, we assume N 1 in the sequel. > c, = 0, and hence by (26), 6 or (S/T) < if and only if (T/T). SO, by (47)) < X,i,, which by Fact 6(c) implies X = 0. Q.E.D. {= We assume that (35) is true. Let 0 < r < K, i.e., ii) inf 1 1 k\$(jw)l oER e K sin (6/2), 6 < r, and let ~ ( t satisfy (47); then, ) immediately from Fact 3(c), Ic, 2rI 5 6 < r, i.e., c, = 0. iii) the mean ang~llar velocity of Q.E.D. Hence, from (25), l \$ ( j ~ )- 9(j0)1 I S < r. IXT~ < /XI < < >0 iv) lim [O(jw) - \$(jw) ] = e(j0) (R.3) It is important to observe that the knowledge of the density length I() allows us t o locate the closure of the set {f(jw)lw E R) and that the knowledge of a ) translation number ~ ( e allows us t o replace the condition X = Q by a condition on the increase in argument. f(jw) is periodic with period UO, two im(R.4) If w portant simplifications occur, i.e., a) {f(j,) ;w E R) = {f(jw); w E [o,w01). b) if (19) is satisfied, then t\$(jo) = Xw h(jw) where w h(jw) is periodic with period wo. Hence, X = 0 (=) 9(jwo) = d(jO), and hence, for the f(jo) is periodic with period wo, part ii) case that w of Corollary 1 can be replaced by: the origin 0 of the complex plane is positioned with respect to {f(jw) w E R] such that a) 0 does not belong to {f(jw); w E [ o , ~ o ] ] b) +(jwo) = d j 0 ) . 04 m ## f(jw) is zero (35) 4(jO) I+ I+ I+ .+ where b is an integer. Proof: (a) Let us first study the asymptotic behavior of 1 kG(s). I n view of (7), the Riemann-Lebesgue lemma implies &(s) + 0 as Is/ + a, in Re s 2 0 ; hence, by (3), (611 (141, and (1511 I+ An important conclusion is that, because of (53) and Fact 1, w 1 kfj(jw) has an asymptotic almost-periodic 1 ki(s) has an asymptotic almostbehavior on R and s periodic behavior in Re s 2 0 (for Is1 + ). (b) (=. We first show that I+ I+ + Re s > O inf I(fi)I > 0. (33) ## Definition o the Argument B(s) o 1 f f subject to (\$)-(lo) By definition, + kfi(s) + Indeed (51) and (53) imply lim inf If(jw)l > 0 as l 1 + a . o Hence, since w (+f(jw) is almost periodic on V by Fact 1, o inf [f(jw)l wER > 0. So by (34), (19), and (35), it follows that (33) is true by Theorem 1. O(s) = arg [ l kO(s)] = I m log [ l ki(s)], Observe that, by Fact 3(d), (19) and (35) are equivalent in Re s 2 0 (50) to the existence of a sequence { w,] ,"= - satisfying (27)(30). Now choose w, with positive index from this sequence with two additional conventions : and a 5 > 0, both sufficiently large so that: (B.l) Let L denote a straight oriented line in Re s 0. (a) the open rectangle ABCD ( 0 , ~ )X (-w,,w,) By convention, we take B(s), s E L as the right argument (Fig. 2), i) has all poles of 1 k i with Re p, > 0 in the of 1 kG(s) on L, i.e., B(s), s E L is an arbitrary branch interior of ABCD, ii) has all poles of 1 k\$ with Re p, = of the argument, which is continuous except at the zeros 0 on AB, iii) neither A nor B is the location of a pole of and poles of 1 kg(s) on L, while it is discontinuous with a 1 k!?. jump of +mr, (-mar), when s passes, in the positive (b) in the complement of this rectangle with respect to kG(s) of order m, { s / ~s e> 01, 1 direction on L a zero (pole) of 1 kG(s) is sufficiently close to f(s) such (m,). At a discontinuity point we assign t o 0 the mean that 1 ki(s) is bounded away from zero by (53) and value of its one-sided limits. The function O(s), s E L (33) in this complement. The principle of the argument is then well defined (mod 2 r ) because of (12). can be applied t o ABCD. Denote by AOAB the net change (B.2) Because 1 k\$(s) is real for s = a 2 0 and in the argument on the oriented segment AB. By the meromorphic in R e s > 0, there exists an interval ( 0 , ~ ) principle of the argument along with (16), it follows that on which 1 kg(u) is real finite and different from zero. We pick for w 1- O(jw) the branch of the argument such that e(j0) 4 0 (or r ) accordingly as 1 @(a) is positive where nZis the number of zeros of 1 ki(s) inside ABCD. (or negative) on (0,a). Remember that f(s) is analytic inside ABCD, continuous Theorem 2: Given the system defined by (1)-(lo), let on the boundary ABCDA, and, by (33), bounded away f(s) be the Dirichlet series given by (14) and (15) and let from zero in Re s 0; hence, again by the principle of the np be given by (16). Under these conditions, argument, > > 1 inf 1 Re8 2 0 + kG(s)l > O (13) ## CALLIER AND DESOIER: (tRAPEICAL SrPADILITPTEST wn 8 I' 1' IRe ABCD, again &bAB = 0 implies # = J0. By the ~ ~ ~ ~ ~ construction of A ~ B C ~ = 0. Hence, by A (53), again beBCDAN 0 such that ACAB N np2?r. Thus, for w,, n > 0 sufficiently large because B(jw) - O(j0) = e(j0) - e ( - j ~ ) and (28), e(jw,) N e(j0) npr. This implies, by (29) and (30), lim,,, [O(jw) - 4(jw)] = e(j0) - 4(jO) -4- np7r, which, because of (53)) implies (52). ~ = np2r. Similarly, A -Wn = A IS, Q.E.D. I n order to establish a graphical test, it is interesting to observe that, because of (53), the validity of condition (52) can be determined in principle by considering 1 ki(jw) and )(jw) only over & finite interval. Moreover, given the neighborhood N(E),defined by (44), it follows by the asymptotic and symmetric properties that: o ## > 0 such that e Moreover, since w, and a have been chosen sufficiently large, it follows from (53) that Q(e) =) 1 1 = 1 ) ## + kg(jw) E N(2e), where E cl Ix - f(jw)l I 26; w E [o,l(e)l} (60) + k\$(jw) - f(jo>l I (59) N(2e) where % indicates that equality is reached as w, -t w and a -. w . From conditions (28), (301, (52), the fact that e(jw) - e(j0) == B(j0) - 6'(-jw) because 1 k@(jw)= 1 ki(-jo) and (51), (s (b) infUfE 1 1 + ki(jw)[ > 0 + Hence, (55)-(57) imply A4BoD,l N ABBCDA N 0, which, along with (58) and (54), implies nz = 0. Thus, for sufficientlylarge w, and a, 1 ki(s) has no zeros in ABCD. Furthermore, by csonstruction, 1 ki(s) is bounded away from zero in the complement of ABCD with respect e ki(s) is bounded away to ( s l ~ s > 03, and by (51), I. from zero in the complement of ABCD with respect to ( s l ~ s 2 0). Henee, (13) follows. e (c) =) . Immediately, (13) iimplies (51). Thus, because of (13) and (53), we can pick an w = ij and a a = 3 so large that the rectangle ABlCD = (0,a) X (- ij,ij) (see Fig. 2) is such that )(s) is bounded away from zero in the s complement of ABCD with respect to ( s l ~ e 2 0) except on AB. Since, by Fact X: w I-,)(a jw) is almost periodic on any line V,, a E [0,a ) , it follows then that )(s) is bounded away from zero on all these V,. Hence, (=)the origin 0 of the complex plane is positioned k~(jw)[o R) and E with respect to { 1 ()(jw) ( o E R]such that there exists an e > 0 such that i) 0 does not belong to N(2e) ii) 0 does not belong to { 1 k\$(jw) w E [O,Q(e, 11. From this discussion, Theorem 2, Theorem 1, and Corollary 1, we conclude with the following graphical test. Corollary 2 (Graphical Test): Given the system defined by (1)-(10). Let )(s) be the Dirichlet series defined by (14) and (15). Let ~ ( e ) N(2e), Q(e) be given by (43), , (60), and (59). Under these conditions, inf 1 1 Re s >O + ki(s)l > O if and only if ii) the origin 0 of the complex plane is positioned with respect to { f(jw)lw E R] and (1 k\$(jw)(wE R ] such that (a) there exists an e > 0 such that 0 does not belong to N(2e) and f 1 ki(jw)lw E [~,Q(e)l] which, by Theorem 1,implies (34), (19), and (35). Hence, (b) for an e > 0 with 0 < E < K infWER lf(jw)l, because of Fact 3 (d), there exists a sequence ( w,) ,"= - , for which the corresponding -translation number ~ ( e ) such that (27)-(30) hold. We show now that (52) holds. satisfies (47), where,iX is defined by (45) and (46), then , From now on, pick the parameters of ABCD so that l 4 ( j ~ - 4(j0)1 < ?r must hold. ) ij is an element of [;he above sequence with positive index (c) lim,,, [B(jw) - 4Cjw) 1 = e(j0) - d j o ) np?r and so that ij and a are so large that (53) holds; finally, = b27r where b is an integer. [Note that, by (16), n, is k;(s) with Re p, > 0 should be inside the number of poles with positive real parts.] all poles of 1 ABCDA and all poles of 1 ki(s) with Re p, = 0 should be on AB, but neither A nor B should be the location of a pole. Again, the principle of the argument can be applied a) It should be noted that the graphical test as given in with ABCDA oriented in clockwise sense. Hence, AOABcDA Corollary 2 requires the knowledge a priori of the asymp- ## On Simplifying a Graphical Stability Criterion for Linear Distributed Feedback Systems F. M. CALLIER ANII C. A. DESOER Abstmct-In an earlier publication [I], Vidyasagar proposed to derive a standard Nyquist-type stability criterion (I, theorem 11from a result about a graphical test developtxl by Callier and Desoer [2, theorem 21. The attempt suffers from two major weaknesses which are detailed in the Appendix. Basically, Vidyasagar's concept is correct: namely, it is possible to give a graphical test which r e l i exclusively on the graph of 4 (1+ ki(&)), the phase of the return difference. We give below a correct statement and proof of such a graphical test. We shall use the sanie notation as in our previous paper [2]. We consider a continuous-time scalar linear time-invariant feedback system with constant feedback k and return d.ifference 1+ ki(s). We assume conditions (1H10) of [2]. Thus Comments: I) All the comments of [I, p. 441, last paragraph of first column] apply. 11) As the example below suggests, the calculation of the averages in (6) is not difficult. Indeed, if i), ii), and iii) hold, then 1 ~,+a %(jw)dw=k + a , with k,=integer. A + = lim Q lo Of course, a similar result holds for A _ , the second expression in the left-hand side of (6). To see the truth of (7), note ;hat first, by the hence, Riemann-Lebesgue lemma, as Iwl+w, [I + ki(jo)] -f,(jw)+O; as Iwl+oo, %(jw)-cp(jw)+O (mod2a). So S(jo) 2 %(jw)-+(jo) is bounded and tends to an even multiple of a as Iwl+oo; hence, for any fixed oo> 0, where &(s) is the Laplace transform of an L1-function with support on R+,Repa>O for all a,'r:\$JgiJ<w, t,=O, ri>O for i>O, and k is a real number. We write the last two terms of (1) ,as (Note: we write here fup(s) instead off (s) as in 12, eq. (14)] in order to distinguish it from Vidyasagar's f (s) [I, eq. (I)].) Following the prescriptions in [2, pp. 774 and 7781, we define the phase curves and w ( j , for W E R. (3) where c is an even multiple of r. Second, by assumptions itiii), +(jw) is almost periodic in w ; hence, its is, Q ] mean value over [ I W ~ , W ~ +in the limit as Q+w, independent of oo and equal to its mean value over (- w , w ) [3]. Now if cP(jO)=O, w ++cp(jw) is an odd function of w, hence, its mean value is zero; if +GO)= a , w H +(jw)- a is an odd function of w ; hence, its mean value is a.These conclusions together with (8) establish the assertion (7) above. 111) With these facts in mind, checking condition iv) of the graphical test is quite simple: since A + is independent of wo, consider only a, large enough so that the Nyquist plot is substantially almost periodic; then, in most cases, from the curve of %(jw) versus o [see, for example, Fig. 1, which treats 1 ki(s)= 1 +50s (s+0.5)-'(s2+ s + I)-' - 1.1exp(- 1.2s) +0.35exp(- 3.6s)], the values of A and A - are readily obtained. In cases where the oscillations of % are quite complicated, the left-hand side of (6) gives a completely unambiguous method for calculatingA + -A -. Proof: We show that our four conditions are equivalent to those of [2, theorem 21. First, our i) and ii) are identical to i) and ii) of (2, theorem 21. Second, our-present iii) is equivalent to requiring that the mean angular velocity off,(jw) be zero: indeed, we have asymptotically as Iwl+oo We now state the new graphical test. It can be thought to be a new version of [2, corollary 21; its most appealing feature is that it requires only the phase B (jw) of 1+ ki(jw). Graphical Test: Consider the system defined by 12, eq. (1)-(10)]. The m closed-loop system has a impulse response in the algebra A (hence is Lp-stable for all p E [l, oo]) or, equivalently, Res > 0 and inf Il+ki(s)l>O where A is a constant and h is almost periodic (see [2, p. 775, eq. (20)]). Thus, an equivalent requirement is A = O which is precisely iii) of Theorem 2. Third, we show that our iv) is equivalent to iv) of Theorem 2. 2 p Using in iv) of Th~eorem the symmetry properties of B and c , we obtain w-tm if and only if i) Rcs-rw lim { [ 9 l ( j w ) - + ( j w ) l - [ % ( - j w ) - \$ ( - j w ) ] ) =25a; (9) (4) lim O+a> ## L/-wo B ( j w ) d u = 2 ~ r (6) a -00-Q iv) of Theorem 2 also implies that as w++w (and as w+-oo), 0 (jw)- cp(jw) approaches a constant which is an integral multiple of 2a. Now is almost periodic, so its mean over [w, Q+ w,] and its mean over [ - O- wo, - wo] tend, as O+oo, to the same limit [3]. Hence, taking the mean of the expression of (9), we obtain our iv). Conversely, recalling p the properties of 8 and c , it is easy to see that our iv) implies the old iv). Thus, we have shown that our present four conditions, which only / / involve 1+ ki(s), ;we equivalent to the conditions of Theorem 2. , where wo is any nonnegative number and, as in [2], n denotes the number of poles of i(s) in the open right-half plane, counting multiplicities. Manuscr~pl recelved hugusil 11, 1975. Thrr work w<+ssupported by the Nanonai S c ~ n c c Foundat~onunder Grant GK43024X and by ltre &\@an Natroual Fund for Sclentrfr Research F M Callrer a wrtlr the Department 01 Mathcmat~cs.Facultes Unrven~tarrade Namur, Belwum and the Bdgran Natronal Fund for Sc~enllficResearch. Bmssclr. &Iyum. C A. Dcsp~ra w~th fipartmcnt of Elecmcal Enlynccnng and Computer S k . the eand the Elecrronrcr Research Laboratory. Ulnrverluiy of Cal~forn~a, Berkeley, CA U 7 a The first difficulty with [l] is that as Iwl+w, %(jw)tends modulo 2a to the almost periodic function @Go),defined in (3); hence, except when the latter function is a constant, the expression limw,,B(jw) is meaningless. It is easy to see that the same conclusion holds for %(jw)- %(-jw). Hence, [l, conditilon (9c)l has no meaning. As a consequence, the proof 1 of necessity of [l, theorem 1 is false. The second difficulty is in the development of the sufficiency proof of Reprinted from IEEE Trans. Automat. Contr., vol. AC-21, pp. 128-1 29, Feb. 1976. -, " a2 0.0 -180.0 3 .- -360.0 -540.0 -720.0 -900.01 0.0 5.0 10.0 15.0 20.0 25.0 Fig. 1. #Go) versus w ## for 1+'kg'(s)= 1+50s(s+0.5)-'(s2+s+ 0.35 exp (- 3.6s). I)-'- l.lexp(- 1.2s)+ the theorem of [I], namely, inequality (20), which can be written as (using notations of [I]) lB(jw)-B(-jw)-2nvpI <26.:E 2 (10) ACKNOWLEDGMENT Our thanks go to Y. T. Wang, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, for wmputing the example. for all Iwl sufficiently large. Clearly, this can be true only for some special cases; indeed, for large w, B is asymptotic to cp, i.e., B(jw)-Ww) = 4 j (jw) (mod2n). Since the only requirement on the sequence (A):, 49 is that it be in 1 1 , it is clear that the amplitude of the oscillations of B can be arbitrarily large. Hence, [I, eq. (20)] does not hold in general. Vidyasagar's proof can be used for those cases where our (10) holds instead of his (9c). Note, however, that in the example above, w-+@(jo) oscillates by more than r/2 above and below its asymptotic mean, so even (10) does not hold in general. [I] M.Vidyasagar, "Simplified graphical stability criteria for disVibuted feedback systems," IEEE Tram Automat. Contr. (Tech. Notes and Corrap.), vol. AC-20, pp. 440-442. June 1975. [Z] F. M. dallier and C. A. Desoer, "A graphical test for checking the stability of a linear time-invariant feedback system," IEEE Trans. Automat. Contr., vol. AC-17, pp. 773-780, Dec. 1972. [3] C. Corduneanu, Almost Periodic Functions. New York: Wiley-Interscience, 1%1. Modern Wiener-Hopf Design of Optimal Controllers Part I: The Single - Input - Output Case IIANTE C . YOULA, FELLOW, IEEE, JOSEPH J. BONGIORNO, JR., MEMBER, IEEE, E BR AND HAMID A. JABR, STUDENT M M E , IEEE Abstract-An analytical feedback design technique is presented here for single-input-output processes which are characterized by their rational transfer functions. The design procedure accounts for the topological sbuctnre of the feedback system ensuring asymptotic stability for the closed-loop configuration. The plant or process being controlled can be unstable and/or nonmiuimnm phase. ' k treatment of feedback sensor I noise, disturbance inputs, and process saturation is mother major contribution of this work. The cornerstone in the development is the selection of a performance index based on sound engineering considerations. It is these considerations, in fact, which ensure the existence of an optimal compensator for the system and make tbe performance index a natural one for the problem at hand A N ANALYTICAL feedback design technique is presented for single-input-output processes which are characterized by their rational transfer functions. The design procedure accounts for the topological structure of the feedback system and ensures the asymptotic stability of The plant or process being the closed-loop cor~figuration. controlled can be unstable alnd/or nonminimum phase. The treatment of feedback sensor noise, disturbance inputs, and process saturation is another major contribution of this work. The cornerstone of the development is the selection of a performance index based on sound engineering considerations. It is these considerations in fact which ensure the existence of an optimal compensator for the system and make the performance index a natural one for the problem at hand. The classical treatment of the analytical feedback design problem by Newton is described in [I]. With his approach, which is inherently alpen loop, it is first necessary to find the transfer function Wc(s) analytic in Res > 0 of the optimal equivalent cascade compensator. The transfer function C (s) of the corresponding controller for the feedback loop is then cal~culatedby means of the formula C(S)= w , ( s ) / [ ~ F(s)P(s) wC(s)]. Manuscript receiv January 9, 19'75; reyiyiried October 16, 1975. Paper ~f recommended by J. . Pearson, Ch~alrrna~ the IEEE S-CS Linear work was supported by the National Science Systems Committee. T h ~ s Foundatton under Grant ENG 74-13054 and is taken in part h a a Ph.D. dissertation subrn~ttedby H. A. Jabr to the Faculty of the Polytechntc Institute of New York. D. C. Youla and J. J. Bongiorno, Jr. are with the Dep-nl of Electrophys~cs,Pulytechn~cInstitute of New Electrical Engineen% York, Long Island Centg, Farmtngdale, NY 117% H. A. Jabr was yitb; +e Department of El,cg&al E n g i ~ ~ e r i a d Electmphyrics, P o l y t e Inshtute of New Y o 4 [\$,a& pwr, Farmingdale, NY It?_,, He is now with th.~. Weersity of P & Q and_ Minerals, Dhii\$iam~, u d i Arabia. 6 F(s) and P(s) denote the transfer functions of the feedback sensor and plant, respectively. Unfortunately, this procedure is flawed because it can, and often does, yield a computed C (s) which possesses a zero in Res > 0 coinciding with either a pole of the plant or feedback sensor. Clearly, if C(s) possesses such a zero, the closed-loop system is unstable and the design is worthless. To exclude such a possibility Newton restricts the plant and feedback sensor to be asymptotically stable from the outset. In fact, several extensions of this idea to the multivariable case have already been made by Bongiorno and Weston [2], f31. - The earliest researchers to recognize the difficulty with right-half plane pole-zero cancellations within a feedback loop worked with sampled data systems [4]. The analogous treatment for continuous-time systems was presented by Bigelow [5].' His argument for ruling out polezero cancellations in Res > 0 is based on the fallacious reasoning that exact cancellation cannot be achieved in practice. Although the observation concerning what can be achieved in practice is of course true, it is also true that even if perfect cancellation were possible the system would nevertheless still possess unstable "hidden" modes. Despite the error in physical reasoning these two papers succeeded in focusing attention on several meaningful engineering problems. The frequeincy-domain optimization procedure described herein is the first one to correctly account for the asymptotic starbility of the closed-loop system and to correctly treat plants which are not asymptotically stable. It also supplies significant insight into the essential role played by the classical sensitivity function in feedback system design. Although confined to single-input-output systems, these ideas can be extended to the multivariable situation. This extension is nontrivial and is the subject of Part 11. Just als in [6], the scalar solution provided the necessary insight and impetus required to effect the breakthrough in the multivariable case. It is, therefore appropriate that both cases be presented in the literature. Moreover, it is only in the single-input-output case that the unique role of the sensitivity function manifests itself so clearly. The limitations imposed by feedback sensor noise have been known for some time. Horowitz [7] has proposed a design philosophy for single-input-output minimum-phase ~ 'The paper by Bigelow was kindly brought to the attention of the authors by P. Sarachii. Reprir~ted from IEEE Trans. Automat. Contr., vol. AC-21, pp. 3-13, Feb. 1976. stable plants which is quite imaginative but appears limited since it is modeled around a Bode two-terminal r-A-interstage equalization scheme. On the other hand, our ~eedforwardl I compensator approach takes the lumped character of the controller as I an explicit constraint from the outset and nonminimumrI 1 phase and/or unstable plants offer no special obstacles. Plant A discussion of the relationship of our frequencyP(s) domain design procedure and some of the more popular r-----------1 F ( s ) = F,(s)F+ (s) I ! state-variable techniques [I41 is certainly in order and will Feedback ~ r a n s d u c e r p be given in Part 11. For now, we merely observe that the F+(s) I methods of this paper obviate the need to find statevariable representations and can handle stochastic inputs Fig. 1. Basic single-loop feedback configuration. which are non-Gaussian and colored, as well as step and ramp-type disturbances. In addition, it permits the modeling and incorporation of feedback transducers such as and tachometers, rate gyros, and accelerometers with nondynamical transfer functions. where P(s), Po(s), F(s), and Fo(s) are four real rational functions in the complex variable s = a j w . Moreover, by changing Po(s) into 1 In this paper attention is restricted exclusively to the it is also possible to envisage any desirable feedforward design of controllers for single-input-output finitecompensation Pf(s). dimensional linear time-invariant plants embedded in an Straightforward analysis yields equivalent single-loop configuration shown in Fig. l? Suppose yd(s), the desired closed-loop output is related to ui(s), the actual input set-point signal in the linear fashion via the ideal transfer function Td(s). The preflter H(s) and can be selected in advance once and for all, but irrespective of the particular choice of criterion that is employed.3 where is the best available linear version of y,(s). Any reasonable performance measure must be based on the difference F-I+S e=( I -S ) u - s P ~ ~ + ( ~ ) F ~ ~ (7? ## 1 s (s) = 1+ F (s)P (s) C (s) is the closed-loop sensitivity function. In process control, the actual choice of a reliable feedback transducer F,(s) is more or less dictated by the problem at hand. However, as explained in greater detail later, some low-power-level between the actual plant output y(s) and the actual preequalization F,(s) is almost always necessary to model smoothed input u(s) driving the loop. For a given plant delay in the feedback path, to improve stability margin and overall sensor F(s) the design of the controller C(s) and to assure zero steady-state error. In other words should evolve from an appropriate minimization procedure subject to a power-like constraint on r(s) to avoid We therefore assume that Po(s), Fo(s), F(s), and P(s) plant saturation. Plant disturbance d(s) and measurement noise m(s) are are prescribed in advance. Equations (5H7) reveal the possibilities for tradeoff in the various frequency bands. modeled in a perfectly general way by assuming that Observe, that with unity feedback ( F = 1)) e(s) is the sum of the two errors 'TO avoid proliferating symbols, all quantities are Laplace transforms, deterministic or otherwise, all stochastic processes are zero-mean secondorder stationary with rational spectral densities and (,) denotes ensemble average. 3Function arguments are omitted wherever convenient. and YOULA et al. : WEZNER-HOPP DESIGN whose different origins are betrayed by the prefactors S (s) and 1-S(s). The impossibility of making both S (s) and 1- S (s) arbitrarily "&d" over any frequency band is partly intrinsic and partly conditioned by the plant restrictions [8], [9]..This fundamental conflict is inevitable and largely respo~lsiblefor a great deal of the difficulty surrounding practiical feedba.ck design. 1 -4 is the usual quadratic measure of steady-state response. similarly: if Ps(s) represents the column-vector transfer matrix coupling the plant input r(s) to those "sensitive" plant modes which must be especially protected against excessive dynamic excursions, (I3) and ## is a proven .useful penalty functional for saturation [I]. More explicitly, where where each numerator polynomial is relatively prime to its respective denominator mate. It is well known [lo]-[12] that if the plant, controller and feedback sensor are free of unstable hidden modes, the closed loop of Fig. 1 is asymptotically stable iff the '"reduced" characteristic polynomial Thus, k, a positive constant, serves as a weighted cost combining both factors. Using (6) and (7) and assuming all processes to be independent, a simple calculation yields the compact four-term expression jm is strict Hurwitz; i.e., iff q(s) has no zeros in Res > 0. Hence, the pair 4(s), np(s) as well as the pair dp(s), nf (s) must be devoid of common zeros in Res > 0 in which case P(s) and F(s) are said to be admi~sible.~ Observe that once H(s), P(s), F(s), Po(:r), Fo(s) and the statistics of ui(s), n(s), d(s), and m(s) are specified, y(s), r(s), and e(s) are uniquely determined by the choice of sensitivity function S(s). Consequently, the following definition and its accompanying lernma have an obvious importance and are fundamental t'o our entire approach. Definition 1: S (s) is said to be realizable for an admissible pair P(s), F(s) if the closed-loop structure of Fig. 1 is asymptotically stable for some choice of controller C(s) and possesses the sensitivity function S (s). Lemma I (Apptwdix): The function S (s)8 0 is realizable for the admissible pair P(:s), F(s) iff 1) S(s) is analytic in Res > 0; 2) Every zero QF the polyr~omialdf(s)dp(s) in Res >, 0 is a zero of S(rr) of at least the same multiplicity; 3) Every zero of the polynolnial nf(s)np(s) in Re s > 0 is a zero of 1- S (s) of at least the same multiplicity. Let Gi(- s2), G,(- s2), Gu(- s2), Gd(- s2), and Gm(- s2) denote the rational spectral densities of ui(s), n(s), u(s), d(s), and m(s), respectively. Setting aside for the moment all questions of.convergen~e,~ 4When Fo(s), H (s), Ff(s), and Po@) represent distinct physical blocks, these blocks must be stable: their transfer functions must be analytic in Res > 0 On the other hand if Fo(s), H(s), P (s), and Po(s) are merely . part of the paper modleling it is often posslb3e to relax the analytlc~ty requirements. ' ~ fA-(s) is a real rational (or meromorphic) matrix in s, A,(s)=A1 (- s), the transpose of A ( - s). 2-E=a+2 Jjw where jm ## (F- l)(F- I)* FF* Guds, (22) Our entire p'hysical discussion revolves around the implications of (21), and our assumptions are as follows. Assumption I: Rate gyros and tachometers are examples of practical sensing devices which are not modeled as dynamical systems.' Yet almost invariably, sensors are stable and their associated transfer functions 4 ( s ) are analytic in Res > 0. For our purposes it suffices to restrict 6Column-vectors are written a, b, etc., and det A , A', X,A* ( G P ) denote the determinant, transpose, complex conjugate, and adjoint of the matrix A , respectively. Note that for A ( s ) real and meromorphic, A&w) =A(jw), w real. 'A system with transfer function A ( s ) is dynamical if A ( s ) is proper; i.e., if A(co) is finite. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL, FEBRUARY 1976 F(s) to be analytic on the finite jw-axis and to insist that the component of the cost a be finite. In particular, the integrand in (22) must be analytic on the jw-axis and O(l/w2) for w2+co. Suppose that parameter variations induce a change Acp(s) in the characteristic polynomial cp(s). Clearly, if the nominal design is stable and structural changes are precluded is proper and analytic in Res > 0. Invoking the standard Nyquist argument it is immediately concluded that cp(s) + Acp(s), the reduced characteristic polynomial of the perturbed closed loop, is strict Hurwitz iff the normal plot of ~ ( j wdoes not encircle the point - 1+j O in a clockwise ) direction. It is imperative therefore that at the nominal setting Icp(jw)l be comparably large over those frequency ranges where lAcp(jo)l is expected to be large. Unfortunately, it does not appear possible to translate any nontrivial stability-margin criteria directly into manageable integral restrictions reconcilable with E. However, once the formula for the optimal S (s) is available, the role played by F(s) in securing adequate stability margin will be clarified and further discussion along these lines is postponed until the next section. Assumption 2: A pole of P (s) in Res > 0 reveals true plant instability but a pole on the jw-axis is usually present because of intentional preconditioning and is not accidental. For example, with unity feedback (F=1) and d(s) =,m(s) = 0, a stable loop enclosing a plant whose transfer function possesses a pole of order v at the origin will track any causal linear combination of the inputs 1, t, - . ,t '- with zero steady-state error. Similarly, if s = jw,, w, real, is a pole of order Y of P(s), a unityfeedback-stable loop will track any linear combination of eJot, teJWo . . . ,t "- 'ejuo' with zero steady-state error. ', These generalized ramp-modulated sinusoids constitute an important class of shape-deterministic informationbearing signals and play a key role in industrial applications. In a nonunity-feedback loop this perfect accuracy capability is lost unless Ft(s) is also preconditioned compatibly. From (7) with d (s) = m(s) = 0, such as process control, the recovery of steady state under load changes is a requirement of paramount importance. As is seen from (26), if the shape deterministic component of P,(s)d(s) is envisaged to be the transform of a sum of ramp-modulated sinusoids, bounded zero steady-state error is possible iff e(s) vanishes at infinity and is analytic in Res > 0. Assuming SP, proper and SP,d analytic in Res 2 0 is evidently sufficient. In particular, reasoning as above, the jw-axis poles of Po(s)d(s), multiplicities included, must be contained in those of P(s). Summing up, (F- l)P, d,dp(uu) = dpG,dp* (27) and ## must be jw-analytic. Equivalently, in view of Assumption 1 and (23), is analytic on the finite jw-axis. Assumption 3: In general, the effects of parameter uncertainty on P(s) and F(s) are more pronounced as w increases and closed-loop sensitivity is an important consideration. This sensitivity is usually expressed in terms of the percentage change in the loop transfer function T (s) = p (3) C (s) 1 F(s)P(s)C(s) ' ## A straightforward calculation yields ' and once again S ( j o ) and 1 - S(jw) emerge as the pertinent gain functions for the forward and return links, respectively. Clearly then, to combat the adverse effects of high-frequency uncertainty in the modeling of F(jw) and P(jw) it is sound engineering practice to design S(jw) proper and equal to 1 at o = co. This requirement is easily introduced into the analytic framework by imposing the restrictions Gb(- s2)if 0 and Now according to Lemma 1, a finite pole s =jw, of P(s) of multiplicity v must be a zero of S (s) of order at least v. Thus, if s=jw, is also a zero of F(s)- 1 of order v or greater, (25) shows that the loop is again capable of acquiring any linear combination of the inputs ejwo', tejwo . . ,t '-leJ"~t with zero steady-state error. By setting ', U(S) m(s) = 0 in (7) we obtain = for large w2. Our final assumptions are fashioned for the express purpose of excluding from consideration certain mathematically possible 'but physically meaningless degeneracies. They are also motivated by Lemma 1, the structure of (21), and the requirement of finite cost. Assumption 4: For large w2, ## Assumption 5: Q is analytic on the finitejw-axis, has no purely imaginary zeros in common with np, and the constant k is positive. Assumption ti: L,et the pair P (s),F(s) is obtained from the formula 1- So(s) CO(s)= P (s) F (s) So(\$) and can be improper, unstable or both.' Nevertheless, the closed-loop structure is always asymptotically stable and So(s) is proper and analytic in Re s > 0. (Assumptions 1-6 actually force O(s) to be free of zeros in Res > 0.) With exact arithmetic the finite zeros and poles of P(s)F(s) in FLes 2 0 are cancelled exactly by the zeros of So(s) and 1- So(s), respectively. Thus, in any computer implementation of (41) and (42) it is necessary that all these exact arithmetic cancellations in Res > 0 be effected automatically by suitable preparation. Failure to do so will result in a nonstrict-Hurwitz stability polynomial cp(s) and a corresponding unstable closed-loop design. An examination of (41) reveals that the zeros of Q(s) and the poles of then (dpnf)G (d,n,)* is analytic and nonzero on tlhe finite s =jw-axis. For later reference we record the useful formula ## which drops out of (23), (24), and (34). Recall that any rational function A(s) possesses a Lawent expansion constructed from all its poles, finite or infinite and as is c:ustomary, ('4 (s)) denotes that part of the expansion associated witlh all the finite poles of A (s) in Res < 0. Thus, {A(s)) is analytic in Res > 0 and vanishes for S = 03. The remainder of the expansion is written { A (s)) - and of course, + constitute the poles of So(s). Since the poles of So(s) are all zeros of cp(s), the stability margin of the optimal design is ascertainable in advance. This important feature cannot be overemphasized. From the formula ## Theorem I (Appendix): Let and write ) The polynomial x,(s) absorbs all the zeros of ~ ( s in Re s < O and x,(s) all those in IRes > 0. Perform the spectral factorization xrxr(Ga+- C;,) = QQ (40) where Q(s) is free of zeros and poles in Res > 0. 1) Under Assumptions 1 4 , the optimal closed-loop sensitivity function So(s) associated with any admissible pair P (s), F (s) is given by it is seen that the zeros of x,xr,G and 1+ kQ/PP, in Res <O emerge as poles of So(s). The locations of these zeros depend on the choice of F(s), the spectral density G (- s2) and the value of k. Changing k means compromising saturation (and accuracy). A more detailed analysis shows generally that the negative images of the right-half plane poles of P(s) and F(s) are zeros of x,xr, and therefore poles of S,(s) unless G (- s2) is properly preconditioned. If solme of these poles lie close to the s =jw-axis, it may be impossible to attain adequate stability margin. This difficulty can be circumvented by simply incorporating the offending poles into G (- s2). Hence, the rule, any pole of P (s)F(s) in Res > 0 which lies "too close" to the imaginary axis must be made a pole of G(-s2) of exactly twice the multiplicity. Last, we mention that delay T in the feedback path can be simulated by introducing right-half plane zeros into F,(s) through one of the many available rational function approximations to e-"'. IV. EXAMPLE "+ . The theory developed in the preceding sections is now (41) Q used to design the controller C(s) for the system shown in f (s) a real polynomial. The; requirements E < m (finite Fig. 2. Since the theory is based on rational transfer cost) and So(s) realizable for P(s), F(s) determine f (s) uniquely. c(s) proper if the integer 1 in Assumption 3 equals the order of the is 2) The optimal controller Co(s) which realizes So(s) for zero of ~ ( s ) P ( sat infinity. This is often the case. ) so= ' Substitution of the given data into (23), (24), (34), and (35) yields Fig. 2. Example. functions, the first step is the selection of a suitable approximation for the ideal delay represented by F(s) = e-O.lS. Highly satisfactory results are obtained with the second-order PadC approximation 12 --(0.1)~ (0.1) s+s2 (52) and F (s) = +- - pf(-s) - Pf ( 4 ## (45) where P,(s)= 100~ O ~ S ~ + S ~ Note that F(s) and P(s) are an admissible pair and (45) and satisfies the condition F- 1= 0 at s = 0, the only pole of pZ(s)= 4)s2+ s4. P (s) on the imaginary axis. Because of the plant pole at the origin and the choice of rational approximation for F(s), the closed .loop is capa- It now follows from (40) that ble of following step inputs u with zero steady-state error when m = d = 0. The simplest example calls for fl P;'P:(~+~)P~(s) a= 1 GU=-:. (46) 1O+s (i + (56) I ## For the remaining spectral densities we choose (57) and We also assume that the plant input is the signal most are the factors containing all the zeros of p,(s) andp2(s) in likely to cause saturation and put Q = 1. The only remain- Res < 0, respectively. In addition, ing quantities needed for the calculation of the optimal controller are k, Fo, and Po.Comparing Figs. 1 and 2, it is seen that Fo= Po=1. With regard to k, we note that the performance index E is actually an auxilary cost function. The design objective is to minimize E, subject to the constraint E, < Ns, Ns a specified bound. Thus, k is a where Lagrangian multiplier chosen to meet the design obXrXr ko jectives. =-~ O + S (60) The first step in the design of the optimal controller + C(s) is the determination of So given by (41). For the fis2(2+ s)(4 - s2)pf(s)p: (s) determination of the optimal sensitivity function we need k,= (61) the quantities xr, 52, G,, and G,. From (45) and Fig. 2 it P~+(-s) s= - 10 follows that X=npd,nf df=s(l -s)(2-s)pf ( 3 )(- S) ~ ~ and (49) {T(Gb-zGu) 1 ## We now find that YO- et al. : WEINER-HOPF DESIGN in which the sixth-degree polynomial f(s) is uniquely determined by the interpolatory conditions 1- s =0 s2 1. large. (65) In (64), z , and 2, are the zeros of pf(- s). Due to roundoff error, the computed polynomial f (s) will, in general, not satisfy conditions (63)-(65) exactly. However, we know from the them{ that the poles of F(s) and P (s) in Re s > 0 nnust be zeros of S,(s) and the zeros of F(s) and P (s) in IZe s > 0 miuslt be zeros of 1- S,(s). The computations are therefore conditioned so that ## Fig. 3. Variation of performance integrals. and a 3 z n W 12- This is accomplished by dividing the computed h(s) by s(s-2), setting h,(s) equal to the quotient and ignoring the remainder? The same procedure is followed in obtaining (67), and (42) then yields t a w 3 v, 2 -4- 12 1 5 18 21 TIME IN SECONDS 24 27 30 for the optimal controlller. The design described above depends parametrically on k. The values of E, and E, have been computed for various values of k and the results are shown in Fig. 3. It is clear from the curves that the choice k = 4 leads to a suitable compromise between the desire to minimize E, while limiting E,. (Note that all transfer functions in the frequency domain and all signals in the time domain are taken to be dimensionless quantities but time is measured in seconds. It follows that E, ancl have the dimensions of seconds.) The transient response of the optimally designed system to a unl~tstep input has also been investigated. The error and plant input responses for several values of k are shown in Fig. 4 and Fig. 5. Since these responses are obtained with d = m = O they do not, and should not, reflect the optimality of the design. They do show, however, that reasonable transient performance is obtained with the choice k =4. It has been pointed out in Assumption I that I+(jo)l should be large or, equivalently, itls reciprocal should be small for good stability margin. We have in fact computed and plotted +-'(jw) for several values of w in the range zero to infinity. The results are shown in Fig. 6 and are highly satisfactory. With k = 4, I+-' '(jo)l< and the system remains stable no matter what the phase of A+(jo) 9The accuracy of the computations on a digital computer is such that this remainder is quite small (coefficients less than in this example). -16L i :I\// d -16L TIME IN SECONDS ## Fig. 5. Plant input responses. provided only that IA+(jo)l< 10 000. For completeness, plots are also shown in Fig. 7 of IS,(jo)l versus w for several choices of Fc. Evidently, in the light of these observations, the choice k = 4 makes engineering sense and the final step in the design is to compute the optimal controller transfer function with k = 4. Using (68) we get 1976 1 0 12 1 4 1 6 1 8 20 where ## TABLE I COMPARISON OF RESULTS Optimal C ( s ) Suboptimal C(s) F(s) = e-O.IS F ( ~ ) =- O . ' ~ e Approximation for F ( s ) Er 646.9 E, 986.7 646.1 957.6 676.8 952.2 The reader has probably noticed that a, and p, are very nearly equal. Although we have in fact verified theoretically that these two quantities cannot be equal, it is nevertheless natural to inquire whether any significant deterioration in performance results by putting a, =p2 and using the suboptimal controller instead of the Padt approximation and the delay is ignored. However, when the corresponding controller is employed it is found that the system remains stable only for 0 < r < 0.08 s. Thus, with an actual delay r=0.1 s the system designed optimally with F = 1 would be unstable. This facility to incorporate delay is of significant practical value. c (s) = ## 67.2(s - 0.015)(s2+ 60s + 1200) (s - 2.4)(s + 33.7)(s2+ 36s 549) (71) Another aspect of the design which should be clarified is the use of the Pade approximation (45) for the delay e-'.'" in the feedback loop. These points have been taken up and the results are presented in Table I. A comparison of the first two columns in the table reveals that the use of the Pad6 approximation is certainly satisfactory while a comparison of the last two shows that the use of the suboptimal controller is justified. This is gratifying since no analog controller can be designed to the accuracy demanded by the values in (70). We have also studied the stability margin with respect to variations in the delay r =O.I s. With the suboptimal controller the feedback loop remains stable for 0 < r < 0.155 s. This stability margin is clearly satisfactory and the example indicates that the design procedure is a practical one. One final point. It is quite obvious that the design equations are substantially simpler if F(s)= 1 . is used It appears from Fig. 4 that the transient performance of the optimally compensated loop in our example is poor. In fact, for k = 4 a peak error response of 10.7 is obtained. Is this poor transient performance a consequence of the design procedure, inherent limitations imposed by the plant, or both? To anqwer this question the example described in the previous section is considered once again but with G,,, = G, = k = 0. (Note that although the conditions k > 0 and G, (- s2)f 0 are violated, it is still possible to obtain an optimal solution. Optimal solutions can exist for cases which do not satisfy our assumptions on the data. Assumptions 1-6 are sufficient to guarantee the existence of an optimal controller and they hold in most cases of interest.) The optimal solution obtained for G,,, = G, = k =0 is the one which minimizes the integral square error with a unit step input. The optimal controller in this case is co = -z(S- 63)(~-~0)(~-3;0) (72) (S- P~)(S- 5 ) ~ YOULA et a[. : WEINER-HOPF DESIGN ## where so is given in (70) and k=: 1.136452161649 1 z - 12 The enor response to <aunit step input with this controller in the system is showi in Fig. 8,.'fie initial value of the error is e(O)= - 7.33 aind differs from unity because with G, = G = k = 0 the performance index is finite and S (oo) , # 1. In fact, the optimal sensitivi1.y function is TIME IN SECONDS ( r 0 ( r -16 I W LZ ## Fig. 8. Optimal error response. where It is clear from Fig. 8 that even in the best of circumstances, no disturbance inputs, no measurement noise, and no plant saturation constraints, the best possible transient performance is poor. The reason is that this particular nonminimunn-phase unstable plant is one of the most difficult to control irrespective of whether the policy is optimal or suboptimal. Indeed., since it is impossible to stabilize this plant P (s) = (s - l)/s(s - 2) by means of any dynamical stable compensation whatsoever [6], lead-lag methods are futile. In our opinion, a design methodology which can accomodate disturba.nce inputs, feedback sensor noise, rms res1.rictions on plant inputs, and also yield results as encouraging as those shown in Figs. 4 and 5, is a valuable engineering tool. then reveals that the multiplicity of s =so as a zero of 1- S (s) is less than its multiplicity as a zero of nf(s)np(s), a contradiction with Assumption 3. Q.E.D.'~ Proof of Theorem 1: In (21), the candidate functions S (s) must all be realizable for the prescribed admissible pair P (s), F(s), Hence, if So(s) is minimizing and E is any reai number, is a legitimate c'ompetitor provided 6(s) is analytic in Res > 0 and the resulting cost is finite. The proof of this assertion is simple. Both So(s) and S(s) must include all the zeros of df(s)dp(s) in Re s > 0 and so must the difference S- So.Again, all the zeros of nf(s)np(s) in Res > 0 must be zeros of 1- So(s) and 1- S(s) and therefore of (1 - So)- (1 - S ) = S - So. But df(s)dp(s) and nf (s)np(s) are relatively prime in Res > 0 and it follows that S (s) - So(s) is divisible by Proof of Lemma 1: The con.troller C (s) = nc(s)/ dc(s) is determined from the formula ## 1 (s)dc s (s) = 1+ P(s)F(s)C(s) - df (s)dp(8) ( ~ 1 (76) 'P This quotient 6 (3) is analytic in Res > 0 and the order of its zero at s = o must be sufficiently high to guarantee the o finiteness of E. To exploit the optimality of So(s), we set What must be shown is that q(s) is strict Hurwitz. According to Assumption 1, S(.s) is analytic in Re s > 0 and any zero so of q(s) in Re s > 0 must be a zero of df(s)d,(s)dc(s) and therefore of nf(s)np(s)nc(s). If it is a zero of df(s)dp(s),it cannot be a zero of nf(s)np(s) because P(s) and F(s) are assumed to be admissible. Thus, so must be a zero of nc(s) and cons~equently one of dc(s) not which is relatively prinie to nc(s). But this means that the multiplicity of s = so as a zero of S(s) is less than its multiplicity as a zero of df(s)dp(s) which contradicts Assumption 2. If instt:ad so is a.ssumed to be a zero of dc(s), it then follows th~at is a zer~o nf(s)np(s), but not it of a zero of nc(s). However, the expression ## and use the standard Wiener-Hopf variational argument [3]to obtain X(s) analytic in Res > 0. Performing the spectral factorization xrxr(Ga+ Gb)=QQ (81) where Q(s) is free of zeros and poles in Res>O and dividing both sides of (80) by Q/x, gives, after re'%e proof of necessity follows immediately from (76)+(78) and is trivial. 1976 ## arrangement, XrXr F- 1 QLyo-~r(Gb-~Gu)) s(j)G6(a)-a(b) (91) x =r rx* or, using (89), a(f)<a(xr)+1. (92) To guarantee that So(s) be realizable for P (s), F(s) every zero of (dfd,), must be a zero of So(s) and every zero of (nfnP), must be a zero of 1- So(s). Coupling this with (85) we get a total of 6 (x,) 1+ 1 interpolatory constraints on j(s) which is one more than its permitted maximum degree, 6(xr)+ I. Thus, j(s) is the.unique Lagrange interpolation polynomial [I31 satisfying these conditions and if a minimizing S,(s) exists, it must be the one given by (84) because the cost functional is quadratic in S(s). Now part 2 is obviously correct and to complete the proof of Theorem 1 it suffices to show that X,(s), as given by (84), is actually analytic in Res < 0. Rearranging (80) with the aid of (81) and (84) leads to IT(.XrXr* F- 1 - ## Using (35) and (38) it is seen that and invoking Assumptions 1, 5, and 6 we conclude that O(s) is actually free of zeros and poles in Res > 0. Thus, So(s)Q(s) must be analytic in Res > 0. But then the lefthand side of (82) is also analytic in Res > 0 and equals the right-hand side which is analytic in Res < 0. Being analy- 1.r where ## f (s) a real polynomial; or It is apparent that (93) is analytic in Res < O and it only remains to show that the same is true on the s =jw-axis. We observe first that the analyticity of (9) for s = j w is implied by Assumption 2, (82a), and Assumption 6. Thus, which is (41). Clearly, So(s) is analytic in Res>O. Since Gb(w2) (93) is analytic on the jw-axis if the purely imaginary zeros of xr are also zeros of f - (9) of at least the same = O(wZ'), l > 0, the convergence of (21) forces multiplicities. But this is automatic whenever f is chosen I - s0(jw) =0(l/w1+'). (85) to make So(s) realizable for the pair F(s), P(s). For suppose that s=jwo is a zero of xr of order v. Then, Write invoking Assumption 1, it is either a pole of P or a zero of FP of order v.I2 Suppose it is a pole of P. Since the only xrxr* F- 1 (86) jb-poles of Gb are either zeros of F or zeros of P, (94) shows that s =jwo is a zero of 9 of order at least v. It now + follows from the identity and that s =jw, is a zero of j- (9) of multiplicity v or higher [see (84)]. Suppose instead that s=jwo is a zero of xr which is a zero of FP of order v. A direct calculation yields h(s), g(s), a(s), and b(s) are four real polynomials. Then, as is easily checked, degree g Z degree h degree a = degree b + degree xr + I and (88) (89) ## To insure a proper So(s), we must impose the degree restriction" "6 (-)=degree (.). ## With the aid of (82a) and Assumption 6 it is seen that 'According to Assumption 1, F is jo-analytic. ## YOULA et a[. : WEINER-HOPF DESIGN s =jw, is a zero of the right-hand side of (97) of order v or more.-Sincef has been const~ructed make S, realizable to for F and P, s = jw, is a zero of 1- So of order at least v and therefore, in view of (96), a zero of f - {+)- of Q.E.D. multiplicity v or greater. [l] G. C. Newton, Jr , L.A. Gould, and J. F. Kaiser, Analytical Design of Linear Feedback Controls. New York: Wiley, 1957. [2] J. J. Bongiomo, Jr., "Minimuim sensitivity design of linear multivariable feedback control systems by matrix spectral factorization," ZEEE Trans. Automat. Contr., vol. AC-14, pp. 665-673, Dec. 1969. [3] J. E. Weston anti J. J. Bongiorno, Jr., "Extension of analytical design techniques to multivarria~blefeedback control systems," ZEEE Trans. Aufomat. Contr., vol. AC-17, pp. 613-620, Oct. 1972. [4] J. R. Ragazzini and G. F. Franklin, Sampled-Data Control System. New York: McGraw-Hill, 1958, pp. 155-158. [5] S. C. Bigelow, '"The design of analog computer compensated control system," AZEE Trans. (Appl. Ind.), vol. 77, pp. 409-415, Nov. 1958. D. C. Youla, J. J. Bongiorno, Jr., and C.N. Lu, "Single-loop feedback-stabilization of linear multivariable dynamical plants," Automatics, vol. 10, pp. 159-173, Mar. 1974. I. Horowitz, "Optimum loop transfer function in single-loop minimum-phase feedback systems," Int. J. Contr. vol. 18, no. 1, pp. 97-1 13, 1973. H. W. Bode, Network Analysis and Feedback Amplifer Design. New York: Van Nostrand, 1945. D. C. Youlla, "The modem design of optimal multivariable controllers via classical techniques," proposal to Nat. Science Foundation, Dec. 11973. - "Modern classical feedback control theory: Part I," Rome Air ~ e v e l o ~ m eCenter, Griffiss Air Force Base, NY, Tech. Rep. nt RADC-TR-70-98, June 1970. C. T. Chen, Introduction io Linear System Theory. New York: Holt, Rinehart and Winston, 1970. H. H. Rosenbrock, State-Space and Multivariable Theory. New York: Wiley Interscience, 1970. F. R. Gantmacher, The Theory of Matrices, vol. I. New York: Chelsea, 1960. C. D. Johnson, "Accommodation of external disturbances in linear regulator and servomechanism problems," IEEE Trans. Automat. Contr. (Special Issue on Linear-Quadratic-GaussianProblem), vol. AC-16, pp. 635-644, Dec. 1971. ## The encirclement condition An approach using algebraic topology'f R. DECARLOS and R. SAEKSS The usual proof of the Nyquist theorem depends heavily on the argument principle. The argument principle, however, supplies unneeded information in that it counts the number of encirclements of ' - 1 '. System stability requires only a binary decision on the number of encirclements. Using homotopy theory, a branch of alg.ebraic topology, we construct new prqofs of the classical Nyquist criteria along with the more recent linear multivariable results. I n both cases the proof is essentially the same. Moreover, we believe our point of view to be more intuitive and capable of generalization to systems characterized by functions of several complex variables. 1. Introduction The concepts delineated in this paper arose in part from an introductory study of Riemann surfaces. Associated with an analytic function is a Riemann surface. It has the property that the image of simply connected regions in the complex plane are simply connected on the Riemann surface. The point made here is that the Nyquist criterion is trivial for simplyconnected regions. Moreover, if one can work on the Riemann surface, this triviality carries over to the general case. To illustrate the point, let Fig. 1 (a) be the image of the right half-plane under an analytic map. The region is not simply-connected. Figure 1 (b) shows the ' same region ' as it might appear on an appropriate R,iemann surface. Here the region is simply-connected. Figure 1. Under the hypothesis that f is bounded at infinity, the boundary of the region in Fig. 1 (b) is the image of the imaginary axis as is the darkened line in Fig. 1 ( a ) . Now remove ' - 1 ' (this may be a set of points) from the Riemann surface. The essential argument we need is that the Nyquist contour in the complex plane is homotopic to zero if and only if ' - 1 ' is not in the interior of its image on the Riemann surface. Received 11 June 1976 ; revision received 3 August 1976. -/- Supported in part by APOSR Grant 74-2631d. \$ Department o Electrical Engineering, Texas Technical University, Lubbock, f Texas 79409, U.S.A. Reprinted with permission from Int. J Contr., vol. 26, pp. 279-287, AUg. 1977. . 130 ## R. DeCarlo and R. Saeks Although motivated by the intrinsic properties of the Riemann surface, this paper drops any further discussion of the concept so as to simplify the exposition. Instead, the paper exploits the fact that the Nyquist contour is a simply closed curve in the complex plane. Mathematically we draw only on the intuitive concept of homotopic triviality as found in algebraic topology. I n the sequel we prove the classical stability results via homotopy theory. In part,icular, we utilize covering space theory. We believe our analysis is clearer and more intuitive than has hitherto appeared. Moreover, we believe that this research indicates that the nub of the Nyquist criteria is in fact homotopy theory. In a future paper we will generalize these results to functions of several complex variables and their application to the stability of multi-dimensional digital filters. 2. Mathematical preliminaries Firstly, let QI,QI, and be the complex plane, the closed right half-plane, and the open right half-plane respectively. Let g- = Q: - C,. Basic to homotopy theory is the concept of a path. A path or a curve in the complex plane is a continuous function of bounded variation ,y : [0, 11-4. y is thus called a rectifiable curve. y is a closed path if y(0) = y(1). y is a simple closed path if y is a closed path and has no self -intersections. The image of I = [0, 11 under y is called the trace of y and is denoted by {y). Two closed curves yo and y, are homotopic (yo- y,) in QI if there exists a continuous funct,ion I' : I x I +QI such that 2 , Intuitively, yo is homotopic to y, if one can contiriuously deform yo into 7,. Moreover, i t is easily shown that the hornotopy relation is an equivalence rlelation (Hocking and Young 1961, Massey 1967). Another important property of a closed curve is its index or degree. The index of a closed curve, y, with respect to a point a not in (y) is n(y ; a)=-. Observe that 1 2~2. j (2-a)-ldz ## \$ (2-a)--ldz= j d [In (2-a)]= J d [In ( ( 2 - a ) ( ] + i j d [arg (z-a)] Y =i j Y d [arg (z -a)] This integral therefore measures i times the net increase in angle that the ray r of Fig. 2 acci~mulates its tip traverses the trace of 7. as Following the comments of Barman and Katzenelson (1974),for the integral to be well defined it is necessary to specify the appropriate branch of erg (z - a ) a t each palint of the integration. We will assume the choice of branch as outlined in the paper by Barman and Katzerlelson (19174). T h e encirclement condition Figure 2. Finally, we point out that this definition of index (encirclement)is a special case (i.e. in the plane) of the general topological concept of Brouwer degree (Hocking and Young 1961, Massey 1967, Milnor 1965). At any rate, intuition for the approach stems in part from the observation that n(y ; a,)= 0 if and only if y is homotopic to a point in C - {a) (cf. prop. 5.4 of Conway 1973). Simply then, a closed curve y does not encircle the point ' - 1 ' if and only if y is homotopic to a point in C - {- 1 ) . We will henceforth refer to such a y as being homotopically trivial. Conversely, y encircles ' - 1 ' if and only if y cannot be continuously deformed to a point in C-{- 1 ) . These ideas indicate that the Nyquist encirclement condition is fundamentally a homotopy concept. To further illucidate the point, let d(s) be a rational transfer function depicting the open-loop gain of a scalar single-loop feedback system. Suppose all poles of d(s) are in 6-and @(a) c m . Via the Nyquist criteria, the dM closed-loop system is stable if and only if h(s)=d(s)/[l +g(s)] is stable ; if and only if the image of the imaginary axis, under d(s) [the Nyquist plot of &s)] does not pass through nor encircle ' - 1 '. Specifically, the encirclement of ' - 1 ' by the Nyquist plot implies there exists at least one so in such that d(s,) = - 1 . Thus the Nyquist contour is homotopically trivial in C.+ - {#-I(- 1)) if and only if the Nyquist plot is homotopically trivial in \$(C+)- {- 1). Motivation behind this approach also arose from a close scrutiny of the classical proof of the Nyquist criteria which depends on the argument principle. The argument principle supplies unnecessary although specific information, in that it counts the number of times ' - 1 ' is encircled. This may account for the apparent difficulty in generalizing the Nyquist criteria. Nevertheless, the affinity between homotopy and encirclement ideas led the authors to a minor study of algebraic topology. I n our setting, algebraic topology establishes a topologically invariant relationship between a metric space, X, and an algebraic group called the fundamental group of X, denoted by n(X). The relationship is topologically invariant in that homoeomorphic spaces have isomorphic fundamental groups. Specifically, the fundamental group is a set of equivalence classes of closed curves. Each equivalence class consists of a set of curves homotopically equivalent. The group operation is ' concatenation ' of curves. For example, the fundamental group of Q: consists of one element, i, the , f identity, since all closed curves are homotopic to zero. I X = Q: - { - 11, then n(X) has a countable number of elements : ix (the identity) equal to the equivalence class of all closed curves not encircling ' - 1 ' and the remaining elements, p, (n= 1, 2, 3, ...) consisting of the equivalence class of all closed 6 , ## R. DeCarlo and R. Saeh curves encircling ' - 1 ', n times. Moreover, pi concatenated with pk is equal to the elemeilt pk++ Now let X and Y be metric spaces. Let f : X+Y be locally homoeomorphic. I11 particular, assume that for each point y in Y there exists an open neighbourhood G of y such that each connected component of f-l(G) is homoeornorphic to1 G under the map f. Under this condition X is said to be a covering space of Y (Conway 1973, Hocking and Young 1961). Also let r ( X ) and r ( Y ) be the fundamental groups associated with X and Y respectively. With these a,ssumptions, f effects a group isomorphism (i.e. a one-to-one onto mapping preserving group operations) between n(X) and a s~lbgroup n(Y) as in the of following diagram (Hocking and Young 1961, Massey 1967) : F is the functor which establishes the relationship between a topological space and its fundamental group. +f Figure 3. Before judiciously tailoring the complex plane so as to apply the above result, we distinguish between a critical point and a critical value. A point z, in C is a, critical point of a differentiable function f if f'(zo) = 0. A critical value of f is any point w = f(zo)whenever zo is a critical point. Now suppose f : C+C is a rational function whose set of poles is I'= {p,, ....,pn). Let Q = (q,, ..., q,) be the set of all points in QI such that f(q,) is a critical value of f. Note that there may be q,'s which are not critical points. To see this, consider g(z)= z2(z-a). gl(0)= O implies ' O ' is a critical value of g, but g(a)= 0 with gf(a) 0. Finally, define T = (ti\ti=f-l( - l), # i = 1 ... 8 ) Note also that since f is a rational function, P, Q and T are finite sets. Defiine X = C - { P u Q u T ) and define Y=f(X). Lemma 1 Under t,he above hypothesis, X is a covering space of Y. Proof For X to be a covering space of Y, each y in Y must have an open neighbiourhood G, such that each component of f-l(G,) is homoeomorphic to G,. Using the inverse function theorem (Rudin 1973) we construct such a neighbourhood. Let (x,, .. , x,) =f-l(y) where again the finiteness of this set is a consequence of the rationality of f . Let Wl, ..., W kbe disjoint open neighbourhoods of x, , ..., xk respectively. Since f is analytic on X and since f'(x) f 0 for all x in X, the inverse function theorem guarantees that there exist open neighbourhoods ## The encirclement condition U, Wi (i = 1, ...,k) such that Ui is homoeomorphic to Vi =f(Ui), where it c follows that Vi is an open neighbourhood of y. Thus f-l( V, u ... u Vk) = U1 u ... u Urn Define V = Vln ... n Vk. Clearly V is an open neighbourhood of y and f-l( V) c U, u ... u Uk. Since each Ui is homoeomorphic to V i 3 V, f-l( V) c U iis homoeomorphic to V. , Therefore each y in Y has an open neighbourhood G such that f-l(G,) has each of its components homoeomorphic to 0 . It follows that X is a covering , space of Y. Corollary The fundamental group r ( X ) of X is isomorphic to a subgroup N of rr(Y). This corollary says that a closed curve in X is homotopicaIly trivial if and only if its image under f is homotopically trivial. 3. The scalar Nyquist criterion I n this section we apply the above corollary to an ' ugly ' Nyquist contour. After proving the Nyquist theorem, using this ' ugly ' contour, we relate it to the usual Nyquist contour. This will establish the classical result. Let Q(s) be a rational function which represents the open-loop gain of a scalar, single-loop unity feedback system. We assume J(s) \$0. Thus the closed-loop system has a transfer function h(s)=Q(s)(l +i(s))-l. We will say that the closed-loop system h(s) is stable if and only if h(s)has all its poles in and ~ ( c o < CO. ) Let P = {p,, ..., p,) be the set of poles of d(s) and let Q = (q,, ..., q,) be the set of points q, such that Q(qi)is a critical value of Q. Define T = {tiIti = f-l( - I), i = 1, . . 8). Finally, let X = C- {PuQ uT) and let Y =f(X). Lemma 1 implies X is a covering space of Y under the mapping f . Assume for the present that Q(iw) # ' - 1 ' for - co < o < oo. The first task is to construct the ' ugly ' Nyquist contour as well as the classical contour. Define the ugly contour to be y,, where y, : I +X c C is a path whose trace is illustrated in Fig. 4 (a). Note that R is chosen strictly greater than max(Ipil, lqjI, Itkl) for l < i < n , l < j < m , and l < k < s . The indentations, along the imaginary axis into the right half-plane, are of radius 0 < E < E , . These semicircular indentations are made around all points of P lying on the imaginary axis and around all points qi of Q lying on the iw-axis with q,\$T. The other ' indentations ' (again of radius E, 0 < E < E,,) are slits into C+ which encircle all points of P and all points qi of Q (q,\$T) which are in C+ so as to eliminate these points from the interior of the contour. We have also labelled these slits pl, ..., pr where each pi maps an appropriate subinterval of I onto the specified subset of {yR). The parallel lines, connecting a pole in ?+with the semicircular portion of y,, are actually the same line segment (slit) traversed in opposite directions. Note that we have indicated the usual counter-clockwise orientation to the path. Thus the only points encircled by y, are points of T which are in C,. 0 0 t- Lemma 2 Under the above assumptions on @ and yR, h(s) is stable if and only if the path goy, does not encircle ' - 1 '. r: ## ' indicates a point of Q. I denotes the I + C U { ~ ) as indicated in ( b ) . I'roo f Since d(io) - 1 , - co 6 w 6 a, # there is a finite R such that yR encircles all This fact, together with g being analytic on X, implies points of T lying in C (., that the statements of the lemma are well defined. Suppose h ( s ) is stable. Then h(s)has all poles in 8-. Equivalently g ( s ) # - 1 Thus y does not encircle any points of T, implying that y, is , for all s in C ., homotopically trivial in X. By the corollary to Lemma 1 , g o y R is homotopically trivial in Y. Conversely, suppose that d o Y R does not encircle ' - 1 '. Then g o y R is homotopically trivial in Y. The same corollary implies that yx does not encircle any points of T. Thus all points of X which map to ' - 1 ' are in 8-. At this stage let us compare the information of the Nyquist plot, #or, with tlie ' ugly ' Nyquist plot, goy,. Lemma 3 Let n be the number of poles of fi in &+, then ## Proof Consideir that But J P o fir ( 2 - 1)-1 dz = j [\$(z) - 11-lg'f2) PX dz ## The encirclement condition If pk encircles a point of &, then [&x) - 11-l#'(z) is analytic in the region bounded by pk and thus the integral approaches zero uniformly for arbitrarily small E . Consequently the integral is zero at these points. If pk encircles a pole of &z), then since [&) - 11-ldl(z) is analytic in the region bounded by pk : for a suitable branch of the logarithm. The integral comes out as negative 2 ~ i since pk was traversed in the clockwise direction. The conclusion of the lemma now follows. At this point let us remove the restriction that d(iw)# - 1 for - co < w < co. We now give a proof of the classical Nyquist criterion using the above concepts. Theorem 1 Let d(s) be as above with the earlier restriction removed. Then h(s) is stable if and only if the Nyquist plot of \$(s) does not pass through ' - 1 ' and encircles ' - 1 ' exactly n times, where n is the number of poles of #(a) in 8,. Proof Suppose h ( s ) is stable, then all poles of h(s) are in 6- h(co) < co. Thv-s and &co) # - 1 and G(s)# - 1 for all s in C,. Therefore via Lemmas 2 and 3 the Nyquist plot encircles ' - 1 ' exactly n times. Conversely, suppose the Nyquist plot encircles ' - 1 ' exactly n times and does not pass through ' - 1 '. Thus &co) # - 1, which implies h(co) < oo. encircles ' - 1 ' n times and there are n poles of d(s) in Moreover, since 8+,we know that d o y , is homotopically trivial. Thus yR is homotopically trivial. Consequently there are no points ti in C+ such that g(ti) = - 1. Thus h(s) is stable. Matrix case Let the entries of an n x n matrix Q.3) be rational functions in the complex variable s. Suppose that Q(s) depicts the open-loop gain of the single-loop feedback system of Fig. 5. 4. Figure 5. \$(s) and \$8) are n vectors whose entries are also rational functions of s which represent the input and output of the system respectively. This article assumes that each entry of Q(s) is bounded at x= co. Thus o(s) is a mapping, L?(.) : C+CnXn, analytic on C except at a finite number of points, the poles of its entries. ## R. DeCarlo and R. Saeks For Fig. 5 to be well defined we require that det [ I + 6 ( s ) ] \$0. Thus there exists a closed-loop convolution operator, H, such that y = .Hx. Moreover the Lapla,ce transfalrnn of H, B ( s ) satisfies For the system of Fig. 5 to be stable, A ( s ) must havle all its poles in and h.ave all its entries bounded a t s = co . Under the aosumptions on G(s),the following factorization is valid : 8- where N ( s ) antd D(s) are right co-prime, polynomial matrices in s with +0. Moreover, so is a pole of 6 ( s ) if and o d y if it is a zero of det [l3(s)] det [13(s)](Wang 1971). Desoer and Slchulman (1972) have shown that the clojsed-loop operator H is stable if and only if det [ N ( s ) D ( s ) ] 0 for s in C+ and det [ I + G(co)] 0. + # # Using this fact, we state and prove the following. Theorem 2 H is stable .if and only if ( 1 ) the Nyquist plot of det [ N ( s ) D(s)]does not + encircle nor pass through ' O ', and ( 2 ) det [ I + 6(a ) ] 0. # Proof By hypothesis we require det [ I + G(co) ] # 0. Therefore we must only verify that det [ N ( s ) D(s)]# 0 for Re ( s ) 0 if and only if the Nyquist plot of + det [ N ( s ) - does not pass through nor encircle,' 0 '. iD(s)] Now the Nyquist plot of det [ N ( s ) D ( s ) ]passes through ' 0 ' if and only if + det [ N ( s ) D(s)]Jnas a zero on the imaginary axis-i.e. if and only if the closed+ loop system has a, pole on the imaginary axis. Finally, assunne that the Nyquist plot of det [ N ( s ) D ( s ) ] does not pass + D(s)] is a polynomial and thus a through ' 0 '. Observe that det [ N ( s ) + rational function. As shown in Theorem 1, appropriately define X and Y so that X is a covering space of Y under the map det [ N ( - + D ( - ) ] . The above ) + lemmas imply thah the Nyquist plot of det [ N ( s ) D ( s ) ]is h.omotlopicallytrivial if and only if there exists no point s in 8, such that det [ N ( s ) D(s)]= 0. The + assertion of the theorem now follows. Observe that if one assumes the open-loop gain to be stable [i.e. G(s)has all + poles in ?+I, then det [ I + 6 ( s ) ] can replace det [ N ( s ) D(s)] in the above theorem. This follows, since for all s in C ,, det [ N ( s ) D(s)] det [ I -t- Q ( s ) ] [ D ( s ) ] + = det with (let [ D ( s ) \$ 0 . Thus, in C,, det [ N ( s ) D(s)] has a zero if and only if ] + det [ I + 6 ( s ) ]has a zero. to Finally, it is ~rorthwhile cite the relationship between the above formulated multivariable Nyquist criterion and that formulated by Barman and Katzn.elson. For this purpose we let hj(iw); j= 1 , ..., n ; denote the n eigenvalues of C?(iw). I n general parameterization of these functions by iw is not uniquely d.etermined but one can always formulate such a function. The encirclement condition Moreover these functions are piecewise analytic and can be concatenated together in such a way as to form a closed curve which Barman and Katznelson term the Nyquist plot of i?(s). Now, since det [I+ e(iw)]= JJ [1 + Aj(iw)] j=1 and the degree of a product is the sum of the degrees of the individual factors and also equals the degree of the concatenation of the factors, the degree of the Barman and Katznelson plot with respect t o ' - 1 ' coincides with the degree of our plot with respect t o ' 0 '. As such, even though the two plots are different, their degrees coincide and hence either serves as a stability test. ACKNOWLED~MENTS The authors would like to acknowledge the contribution of Dr. John Murray (Department of Mathematics, Texas Tech University) whose continuous flow of counter examples helped to shape the ideas presented herein. REBERENCES BARMAN, JOHN and KATZENELSON, F., JACOB, 1974, Int. J. Control, 20,593. CONWAY, B., 1973, Functioions of One Complex Variables (New York : SpringerJOHN vkrlag). DESOER, CHARLES and SCHULMAN,D., 1972, Memorandum No. ERLM346, A., J. College of Engineering, University of California, Berkeley, California. JOHN and YOUNG, G., GAILS., 1961, Topology (Reading, Mass.: AddisonHOCKING, Wesley Inc.). MASSEY.WILLIAMS., 1967, Algebraic Topology: A n Introduction (New York : -. ~ a r c o u r tBrace & World 1nc.). , MILNOR, JOHN 1965, Topology from the Differential Viewpoint (University Press W., of Virginia). RUDM,WALTER, 1973, Functional Analysis (New York : McGraw-Hill Inc.). SPRINGER. GEORGE. 1957. Introduction to Riemann Surfaces (Reading,Mass.: AddisonwLsley 1nc.j. WANG, H., 1971, Memorandum No. ERLM309, Electronics Research Laboratory, S. University of California, Berkeley, California. ## Part Ill Extensions to NonlinearJime-Varying and Stochastic Systems The great success of frequency-response methods for the analysis and design of linear feedback systems inevitably led to frequency-response attacks on nonlinear feedback control problems. Work by Krylov and Bogoliubov on "harmonic balance" methods for prob~lems in nonlinear mechanics had shown the way (Bogoliubov and Mitropolsky [7] ; Minorsky [36] ). The describing fur~ction method-a quasi-linearization technique based on simple physical ideas-was developed more or less simultaneously ancl independently in Russia by Goldfarb [19], in England by Daniel and Tustin [481, by Oppelt [411 in Germany, by Kochenburger [301 in the United States, and by Dutilh [I61 in France. Cornprehensive descriptions of the use of the describing function method in control systems analysis and design are given in the books by Gelb and Vander Velde [I81 and Atherton [3]. (b) Fig. 1 solution of the nonlinear equations governing closed-loop behavior. The condition for first harmonic balance can be written a s The standard describing function method (Grensted [20], Gelb and Vander Velde ['18], Holtzrnann [23] i s applied to a nonlinear feedback loop such as that shown in Fig. l(a). Here N is a frequency-independent nonlinearity whose inputoutput characteristic has odd symmetry, and g(s) is the transfer function of a linear time-invariant: dynamical system with low-pass filter characteristics. Suppose that the feedback loop i s broken at the input to the nonlinearity, a shown in Fig. s I and an input (b), ei = a sin w t The describing function of the nonapplied at point # I . linearity gives i t s complex first-harmonic gain as N(a) = M(a) exp [j(a)l so that the output of the nonlinearity has a first harmonic given by en = M(a) a sin [ w t + (a)]. The first harmonic of the signal which is returned t o the other side of the break point at #2 is thus given by eo = Ig(jw)lM(a) a sin [cdt + n + (a) + phaseg(j w ) l . To see whether the closed-loop system of Fig. l(a) has a sustained oscillation one can now attempt to strike a balance between the input sinusoid at .#I and the first harmonic in the returned signal at #2. If 1:he linear dynarnical part of the system being considered severely attenuates higher harmonics, then it seems reasonable to hope that any balanc:e found in amplitude and frequency will closely approximate an actual periodic which is usually called the describing function method equation. I t s great attraction is that i t s solution may be investigated graphically using the Nyquist diagram for the linear dynamical subsystem involved in the loop. As shown in Fig. 2 the Nyquist locus g(jw), calibrated in frequency w, and the inverse describing-function locus -1/N(a), calibrated in amplitude a, are both drawn on the same complex gain plane and the result inspected to see whether any intersections exist. Such an intersection, like that shown in Fig. 2(a) would correspond to values of frequency w and amplitude a which would satisfy the harmonic balance equation (3.2), and thus indicate a possible closed-loop periodic behavior. I f no intersection exists, a in Fig. 2(b) and (c), an obvious extension of s the Nyquist stability criterion is invoked to predict stable or unstable closed-loop behavior according as any points on the inverse describing-function locus are or are not encircled an appropriate number of times by the complete Nyquist locus drawn for positive and negative frequencies. Furthermore, the way in which an intersecting Nyquist locus and inverse describing-function locus s i t with respect t o each other can be used to consider the stability or instability of a limit cycle associated with the intersection. For example, in Fig. 3 the limit cycles associated with the intersection points A and C would be judged to be stable and that associated with B deemed unstable. ## Intersection indicates o possible limit cycle Nyquist locus g (jo calibrated in w 1 4 I - (a No encirclement of points on inverse describing function locus indicates closed-loop stability. ---L ' , g ( j o 1 IS the Nyqulst locus for ## an open-loop stable plant Fig. 3. Encirclement of points on lnverso describing function locus indicates closed-loop instability. , - - t / / /-- ,\ where f (-x, - -f (x, x ) . i =) On normalizing in terms of a period T (if one exists) by putting where (c) Fig. ## he then considered the equation 2. First-order describing-function analyses naturally fail to give and broke this intolinear and nonlinear parts to obtain correct predictions under certain conditions. In general, one d2x dx may say that the predictions of the simple describing-function yz -+ y ~ - + czx l = (3.3) d02 dO approach tend to be optimistic in that they may indicate closed-loop stability when the system is actually unstable Taking x o ( ~ as the first-harmonic term in the ~~~~i~~ ) exrather than the other way about- Examples of the failure pansionof the vector x ( ~ )he then consideredthe equation , of the method are given by Willems [54]and Fitts [ I 71. That d2xo dxo great care i s needed in i t s application is clear from the fact y2 ?+ yc1 - + c z x o = @ xo, y (3.4) that an example of the failure of the first-order describing do dO function method has been given in a situation involving a and asked the question, under what circu_mstancesdoes the simple nonlinearity and a low-pass transfer function for which, fact that (3.4) has exactly one periodic solution imply that at first sight, it appears eminently applicable (Rapp and Mees (3.3) has at least one periodic solution? Bass showed that if [44] ). Investigations have therefore been made of the condi@(x,i ) is smooth, then under appropriate conditions (the tions under which a use of the describing-function method i s most important of which i s that the "low-pass filter" hypothejustified. Johnson [24] made a study of the accuracy of the sis of the standard describing function technique is satisfied) describing-function method based on earlier work by Bulgakov a periodic solution of (3.4) does indeed imply a periodic solu[13]; he considered the use of second-harmonic terms to cortion of (3.3). Although the restriction to smooth nonlinearirect first-order predictions and t o gauge accuracy. Bass [41 ties ruled out many of the forms of nonlinearity of greatest made the first attempt to give a rigorous treatment of the practical interest, Bass's work revived theoretical interest in mathematical validity of the method. He considered systems the technique and gave some moral support to the many enwhose behavior may be described by a vector differential equa- gineers using it. Sandberg [46] attacked the problem using tion of the form the tools of functional analysis applied to a space of periodic functions square integrable over a period, and found condi- ( ) and Nmam Cm = +; [ - I Fig. 4. where Nm consists of the first m rows and first m columns of the matrix N. The top left-hand element nll of N is the standard first-harmonic describing-function for the nonlinearity, and the matrix N has elements [ck(ai) - ck(aj-' )I nkj = - tions that guaranteed that a linearized operator of the type extracted by a describing-func:tion analysis is a contraction mapping in the whole space. The fixed point corresponding to such a contraction mapping 'would correspond t o a periodic oscillation of the closed-loop system. He also gave conditions under which subharmonics cannot occur. Holtzmann 1221 also made an investigation using a functional analysis approach, looking for contraction mappi~ngs the vicinity of the firstin harmonic approximate solution. Bergen and Franks [6] gave tests for the validitv of the metliod in which vague requirements such as "the linear plant, is a sufficiently good low-pass filter" or "the nonlinearity is not .too nonlinear" were replaced by precise statements which enabled appropriate numerical tests 1 t o be carried out 01 a system being investigated. Kudrewicz [32] made a similar type of study of the validity of the describing-function method. Some interesting work in this area was also done by Williarr~scln[56]. A good discussion of describing functiions for slop13-boundednonlinearities has been given by Mees and Berglern [35] who showed how t o give error bounds for oscillation predictions and how to find ranges of frequency and amplitude over which oscillation is impossible. An obviously important generalization of the simple firstorder describing-fur~ctionmethod i s t o replace the simple describing-function giving the effect of the nonlinearity in terms of the first harmonic in i t s output by a more comprehensive description giving the effect in terms of any desired finite number of harmonics. Ways of doing this for bias (zerofrequency) terms and a few harmonics, using what are usually known a "multiple-input desc;ribing-functions," are well des scribed in the books by Gelb and Vander Velde [I81 and by Atherton [3]. Mees [33] showed how this extension could be done in terms of a describing-function matrix which gives a straightforward algorithmic procedure for proceeding to incorporate terms of higher harmonic order in an analysis. In Fig. 4 let the input to the nonliinearity N be a i constructed in such a way that, at each stage in an investigation using the describing-function matrix, the incorporation of another harmonic only requires the calculation of another row and column of N. Hence, one may easily proceedfrom an investigation of harmonic balance of one order t o that at a higher order. For the system of Fig. let a diagonal matrix be defined as; Then the condition for there t o be an exactly periodic oscillation with periotd 2n/w is that the equation ## to the reduced equation and put where T denotes transposition. I f c i s the analogous vector of complex Fourier coefficients of the output y from the nonlinearity, then the describing-function matrix N is defined so that and assumes that if one exists it is close to the solution 9. Such an equation can, of course, no longer be solved by a simple graphical technique, and computer-generated solution methods using hill-climbing or other systematic search techniques are required. Mees [33] used the describing-function matrix in an investigation of mth-order harmonic balance methods and established the following pair of results. 1) I f a closed-loop oscillation exists, then some finite order of describing function solution will predict it. 2) Under certain conditions the existence of an mth-order describing-function equation solution will guarantee the existence of a higher order solution. He showed how to obtain bounds on the error of a solution of given order, and studied the problem of determining when a finite-order describing-function method can predict a specific periodic solution for a given system. Mees [34] has also used techniques based on the describing-function matrix to study the problem of limit cycle stability. Unlike earlier limit cycle stability tests using describing-functions, this approach has a built-in "reliability guide" and indicates when a higher order approximation needs to be used. The result of the investigations by Bass, Sandberg, Holtz- mann, Bergen and Franks, Kudrewicz and Mees has been to put the describing-function method on a sound basis and, in the describing-function matrix, to provide a flexible and powerful tool for the study of closed-loop behavior in terms of harmonic balances. Many important results in nonlinear feedback theory stem from work on what is called, in the Russian literature, the problem of "absolute stability." This concerns the stability of the class of systems, illustrated by Fig. 5 formed by associating a given linear dynamic system with a set of sectorbounded nonlinearities, that is of a type where the nonlinearity is confined to the sector bounded by lines of slope kl and k, with k2 kl. If the nonlinearity is replaced in turn by linear gains of k l then k 2 , the stability of the resulting pair of linear closed-loop systems can be easily checked by standard linear techniques. One is then led to ask, what can be inferred about the stability of the set of systems corresponding to the admissible set of sector-bounded nonlinearitiesfromthe knowledge that both of these "bounding" linear systems are stable? The system is said to be absolutely stable if it is in fact stable for all such sector-bounded nonlinearities. In 1949 Aizerman [ I ] put forward the conjecture that the system would be absolutely stable if, when the nonlinearity were replaced by a linear gain k, the resulting linear system was stable for > T h e nonlinearities in x m u s t have o u t p u t - input chorocteristics which lie between lines of slope k , and k 2 and which have continuity and differentiability properties sufficient t o guarantee a unique solution t o t h e differential equation describing closedloop behaviour. Fig. 5. Pliss [42] demonstrated that this conjecture does not generally hold by providing a counterexample. Further counterexamples have been provided by Hahn [ 2 1 ] , Fitts [I71 and Willems [ 5 5 ] . Aizerman's conjecture has also been discussed by Dewey and Jury [ 151. In 1957 Kalman [281 conjectured that if the slope of the nonlinearity were bounded by k: and k i , then the system would be absolutely stable if the corresponding linear system were to be stable for l-~)-l Fig. 6 . A counterexample to Kalman's conjecture has been provided by Willems [ 5 5 ] . The discovery of counterexamples to these conjectures shows that one cannot hope to merely reduce problems in nonlinear closed-loop stability to the investigation of a set of related linear problems. Nevertheless, they had an important influence, and work on the problem of absolute stability led directly to Popov's classic work on the frequency-response approach to the stability of nonlinear feedback systems. Po~ov STABILITY CRITERION In 1961 Popov [43] essentially discovered a class of systems for which the Aizerman conjecture is true; this is the class with the "multiplier property" described below. Popov's Stability Criterion marked an important advance in the application of frequency-response methods to nonlinear feedback system stability determination since it was a true sufficiency condition for stability. Unlike the first-order describing functi0.n method for certain situations, it never predicts that a system will be closed-loop stable if i t is really unstable. I n the gen- eralized form given by Brockett and Willems [ l o ] it may be stated as follows. The system analyzed i s shown in Fig. 6, and it is assumed that 1) p(s) and q(s) are polynomials without common factors; 2) that the degree of q(s) exceeds that of p(s); 3) that q(s) does not vanish in the closed right-half-plane; and 4 ) that the nonlinearity f is sufficiently smooth to ensure the existence of a unique solution to the governing differential equation. Popov Stability Criterion The null solution of the differential equation governing the behavior of the system shown in Fig. 6 is asymptotically stable in the large if 1 ) the nonlinear function satisfies f (0) 0 = and ## 2 ) there i s sonie constant a such that (1 + as) g(s) + km in the complex plane. I n terms of this plot the Popov stability criterion assumes the succinct form: It is sufficient for closed-loop stability that the Popov locus lie to the right of the Popov line. The relationship between the Popov stability criterion and the Aizerman conjecture is easily seen in terms of the Popov plot. Asymptotic stability of a corresponding linear system for all values of gain in the range is a positive real function. Positive real functions play an important role in frequencyresponse attacks on nonlinear feedback system stability. They were first used in the analysis of passive linear electrical networks and ere defined a follows. s ## Positive Real Function A rational function z(s) of the complex variable s which is real for all real s i s said t o be positive real if 1 ) Rez(j w ) 2 0 , for all real o, 2 ) Z ( S ) has no righit-half-planepoles, 3 ) all the imaginary-axis poles of z(s) are simple, and the residue of z(s) a t each one of them is real and positive. Positive real functions and their reciprocals are stable. particular In implies that the Popov plot does not intersect the negative real axis to the left of the point -Ilk,,-,, since the Nyquist plot and the Popov plot cross the real axis at the same point. Thus, for any system for which the Popov stability criterion is satisfied, the Aizerman conjecture will hold true. One can therefore regard the Popov criterion as singling out from the class of "sector-bounded nonlinear" systems a subclass of nonlinear systems for which the Aizerman conjecture is correct. Brockett [ I I ] has given an interesting nonlinear circuitbased physical interpretation of the Popov stability criterion. ## Parabola Test is positive real if The standard form of testing for closed-loop stability via the Popov Criterion is t o see whether the Popov locus lies t o one side of the Palpov line. This makes the investigation of the 1 ) Re g ( j a ) 0, for all real lw, 2 ) both the polllnomials p(s) and [p(s)+ q(s)] have all absolute stability of conditionally stable systems awkward. To deal with this situation, Bergen and Sapiro [5] showed their zeros in the left-half-plane Re s < 0. how t o replace the Popov line by a parabola in terms of which The basic reference for positive real functions is Wienberg the conditionally stable sector of absolute stability can be and Slepian [53]; they have also been discussed by Jury [271. found in a straightforward way. The term (1 + a s ) which occurs in the above statement of Popov's beautiful and powerful result aroused great interest the Popov stability criterion i s called a frequency-domain and enthusiasm and caused a surge of work on the frequencymultiplier. I t s presence leads t o the use of a modified polar response approach to nonlinear feedback problems. The most frequency-response plot or "Popov diagram" for the investigaimportant results to emerge from this activity were the various tion of closed-loop stability. For stability given an approforms of "circle criteria" which arose from the work of Rozenpriately sector-bounded nonlin~earity i s sufficient that it wasser [ 4 5 ] ,Bongiorno [81, [ 9 ] ,Narendra and Goldwyn [ 3 9 ] , Kudrewicz [ 3 1 ] , Sandberg [46] and Zames [ 5 9 ] . The basic circle criterion results will be given for the system shown in Fig. 7. I t s structure and the nature of the dynamical block for all real u. are the same a considered for the Popov criterion, but the s I f we put feedback block k ( x , t ) may now be a time-varying nonlinearity, where x is a suitable state vector for the dynamical R e g ( j w )+ j w I n i g ( j w ) = X + j Y , block. For such a system the following results can be estabthen we will have that lished. 1 ) The null solution of the governing nonlinear differRe [ ( I + orjw)g( jw)l = X - a Y , ential equation is uniformly stable if and so it will be sufficient for closed-loop stability that > for all x and t and if Now the straight line defined by 1 X-aY+-=Q, km which may be called the Popov line, is the equation of a straight line of slope l l a passing through the point -Ilk, on the real axis. The Popov locus is a plot of is positive real. 2 ) The null solution is uniformly asymptotically stable in the large i f there exists an E 0,however small, such that > ## and if F(s),defined in 1) above, is positive real. Fig. 7. 3 ) I f k ( x , t ) = k ( x ) is time-invariant and i f for all x and F(s),defined in 1 ) above, i s positive real, then the null solution is asymptotically stable in the large. Results of this sort are called circle criteria because of the following graphical interpretation. Put 2 ) I f sgn k l = -sgn k z , then the numerator will be positive inside the circle'C, and the Nyquist locus for g(s) must not have any point outside the circle C for (3.5) to be satisfied. In order t o interpret these stability criteria in terms of Nyquist diagrams for g(s), an additional condition must be invoked, since the satisfaction of (3.5) is not sufficient for the positive realness of F(s). The additional condition required is that the zeros of the polynomial p ( ~+)kq(s) must k k 2 . For the case where have negative real ,parts for k l kl 0 k 2 , this condition will be satisfied if the Nyquist plot of g(s) does not leave the circle C. I f k l and k2 have the same sign, then an application of the standard Nyquist criterion shows that the Nyquist plot of g(s) must encircle C a many times in an anticlockwise direction a there are s s right-half plane poles of g(s). Collecting all this together we 0, k z 0 , and have the following, stated for the case k l k2 > k l , and illustrated in Fig. 3.8. < < < < > > ## Circle Criteria for Closed-Loop Stability Let D ( k l ,k2 ) and D [ k l ,k2 I denote the open and closed disks in the g-plane having as a diameter the join of the points I f neither k l nor k2 is zero, then the numerator of this expression can be written a s numerator = k l k z { [ U + (-t+jO) and ( - ; + j 0 ) . (kyl 2 + ~2 - 5 (ky1 - k-I 1 1 2 + kzl )I k k l k 2 { [ u - v o I 2 + v2 - p : ) . Thus k ~ g ( j w )1 + Re [ k l g ( j w ) + >0, ## for all real w implies that I I and, provided that k l # 0 and k2 # 0, the numerator term obviously vanishes on the circle in the complex gain plane defined by 1 ) Then the system shown in Fig. 7 and detailed above will be stable in the sense that outputs are bounded for all initial ' conditions if the Nyquist locus I of g(s) does not intersect the disk D ( k l ,k 2 ) and encircles it po times in an anticlockwise direction, where po is the number of right-half plane poles of g(s). 2 ) This same system will be stable in the sense that all sets of initial conditions lead to outputs that approach zero as t + m, if does not intersect the disk D [ k l ,k z 1 and encircles it po times in an anticlockwise direction. The various possibilities for different signs of k l and k2 are described in detail by Narendra and Taylor [ 4 0 ] , who also discuss what happens when k l = 0 or k2 = 0 . In both of these cases the circle degenerates into a straight line. I f k l = 0, then this implies that ## , whose U-axis intercepts are (vo + P O ) where v + Po = -kyl o and vo - po = -k; 1 and the Nyquist plot of g ( j w ) must lie strictly to the right of a vertical line passing through - k i 1 0 . I f k2 = 0 , then we must have that < Thus the numerator term vanishes on a circle C with center at - ( k l + k2 )/2kl k2 and with radius Ikl - kz 1/2kl k z , that is, a circle having a a diameter that segment of the real axis s joining the points (:,O) and ( - i . 0 ) . The region in the gain plane (U, V - plane) which corresponds to a positive numerator will obviously depend on the signs of k l and k2. 1 ) I f sgn k l = sgn k2 (klk2 > 0 ) , then the numerator will be positive outside the circle. In this case for (3.5) to be satisfied, the Nyquist locus for g(s) must not have any point inside the circle C. and the Nyquist plot of g ( j w ) must lie strictly to the left of a vertical line passing through -k;' 0. The Popov criterion and circle criterion a described above s give sufficient criteria for a class of nonlinear feedback systems to be closed-loop stable. To complete the description of the behavior of these systems in frequency-response terms, it is important t o have the complementary conditions defining when the closed-loop system will be unstable. Such conditions have been given by Brockett and Lee [I21 who treat the circle criterion case in detail, and explain howtheir method can be extended to other stability criteria to derive appropriate instability counterparts. Their instability complements < is Nyquist locus o f an open-loop s t a b l e plant. N o encirclement or penetration of critical dtsc thus implies closed-loop s t a b i l i t y . Fig. 8 . I k k 1 l l I k.-k. Fig. 10. - 2 times is Nyquist locus of an open-loop stable plant. encirclement of critical disc thus implies closed-loop ~ n s t a b i l i t y . Fig. 9. to the circle criteria results quoted above are illustrated by Fig. 8 and Fig. 9 and as follows. Circle Criteria for Closed-Loop Instability 1) I f the Nyquist locus I? of g(s) does not intersect the disk D(kl, k2)and encircles it fewer tlianp, times in an anticlockwise direction, then the system of Fig. 7 will be unstable in the sense that one or more sets of initial conditions will lead to outputs which do not approach zero as t +rn. 2) I f does not intersect the disk D[kl, kzI and encircles it fewer than p times in the arlticlockwise direction, then the , closed-loop system i ill be unstable in the sense that one or more initial conditions will lead to outputs which grow without bound a t +=. s Furthermore, the voltage and current at the input terminals of a passive network consisting of linear constant inductors, capacitors, and resistors can be shown to be related by an impedance function which i s a positive real function. Thus, if in Fig. 10(b) z(s) is a positive real passive network impedance function, and f ( t )is always positive, we will have bounded solutions since we are dealing with a passive network. This gives an interpretation of the circie criterion for k 2 = 0 and kl = -. Now consider Fig. 10(c). I f f ( t ) is bounded between kz and kl, then the total resistance to the right of the dotted line is always positive. A suitable translation of this gives a network interpretation of the general circle criterion. The work of Zames 1581, [591, Brockett and Willems [ l o ] , and Narendra and Neuman [37] showed how sufficient conditions for nonlinear stability for feedback systems having monotonic and odd-monotonic nonlinearities could be expressed in terrns of the transfer function g(s) of the linear part of the system and a frequency-domain multiplier z(s) such that g(s) z(s) was a positive real function. Finding such multipliers by means of a geometric construction was investigated by Narendra and Cho [381 who established the following criterion. Off-Axis Circle Criterion I f any circle can be drawn which intersects the negative real axis of the complex gain plane at the points (-Ilkl + j O ) and ( - I l k 2 + jO) in such a way that the Nyquist plot for g(s) lies outside the circle and does not completely encircle it, then the nonlinear feedback system of Fig. 6, in which g(s) is open-loop stable, will be asymptotically stable for all real-valued functions f ( -) satisfying Circuit Analogy for rhe Circle Criterion Brockett [ I 1I has given the following interesting network interpretation of the circle criteri~on. The feedback system of Fig. 10(a) and the network of Fig. 10(b) can be shown t o have their behavior described by the same differential equation. where kl and k 2 are positive constants. (Note that these "off-axis" circles are not restricted t o having their centers on the real axis.) COMPARISON OF THE VARIOUS APPROACHES Jury and Lee 1 6 have discussed the extension of Popov's 21 method to a class of multinonlinear systems. Nonlinear sampled-data systems have been studied by 41 Tsypkin [50],[511 and Vidal [52]. Tsypkin 1 9 and Jury and Lee I 5 have discussed the extension of Popov's cri21 terion to nonlinear sampled-data systems. REFERENCES M . A. Aizerman, a problem concerning the stability i n the large of dynamic systems," Usu. Mat. Nauk.. 4.187-188.1949. [21 M. A. Aizerman, and F. R . ~antmacher, Absolute stability o f Regulator Systems, San Francisco: Holden-Day , 1964. [31 D. P. Atherton, Nonlinear Control Engineering, London : Van Nostrand Reinhold, 1975. [41 R. W. Bass. "Mathematical legitimacy of equivalent linearization by describing functions," Proc. First IFAC World Congress, Moscow, Butterworth Scientific Publishers. London, 20742083.1960. [51 A. 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[ l 2 1 R. w. Brockett and H. B. Lee, "Frequency-domain instability criteria for time-varying and nonlinear systems," Proc. IEEE, 55.604-619.1967. [13] B. v . Bulgakov, "Periodic processes in free pseudo-linear oscillatory systems," J. Franklin Inst., 235,591 -616,1943. [I41 P. A. Cook, "Describing function for a sector nonlinearity," Proc. IEE, 120,143-1 44,1973. [I51 A. G. Dewey and E. I . Jury, " A note on Aizerman's conjecture," IEEE Trans. Automat. Control, AC-10.482-483.1965. [I61 J. Dutilh, "Theorie des servomechanisms a relais," Onde Elec., 438-445.1950. -, I171 R . Fitts. "Two counter-exam~les t o Aizerman's coniecture." IEEE ~rans. Automat. Control, AC-1 1,553656,1966. [I81 A. Gelb and W. E. Vander Velde, Multiple-Input Describing Functions and Nonlinear System Design. New York: McGrawHill, 1968. [I91 L. C. Goldfarb, "On some nonlinear phenomena in regulatory systems," Avtomatika i Telemekhanika, 8 ( 5 ) ,349383.1947. to [20] p. E. W . ~rensted,"Frequency response methods nonlinear systems," in Progress in Control Engineering, Val. I, New York: Academic Press, 103-141,1962. [21 I W . Hahn, Theory and Application o f Liapunov's Direct Method. Englewood Cliffs, NJ: Prentice-Hall, 1963. [221 J. M. Holtzmann, "Contraction maps and equivalent linearization," Bell Syst Tech. J., 46,2405-2435.1967. [231 -,Nonlinear System Theory. New York: Prentice-Hall, 1970. [24] B. C. Johnson, "Sinusoidal analysis of feedback control systems containing nonlinear elements," Trans. AIEE, Part 1 1 , 71, 169181,1952. [251 E. I . Jury and B. W. Lee, "On the stability of a certain class of nonlinear samoled-data svstems." IEEE Trans. Automat. Control. AC-9,51-61.1964. [261 , " A stability theory for multinonlinear systems," Proc. Third World Congress IFAC, London, Session 28, Paper 28a, 1966. 1271 E. 1. Jury, innem and Stability o f Dynamic Systems. New York: Wiley, 1974. - ~ ~ [281 R. E. Kalman, "Physical and rrtathematical mechanisms of instability in nonlinear automatic c.ontrol systems," Trans. ASME, 79,553-566,1957. [291 -, "Liapunov functions for the, problem of Lur'e in automatic control," Proc. Nat. Acad. of Science USA, 49,201-205.1963. [30] R. J. Kochenburger, "A frequency response method for analysing and synthesizing contactor servomechanisms." Trans. AIEE, 69, 270-283.1950. [31] J. Kudrewicz, "Stability of nonlinear systems with feedback." Avtomatika i Telernekhanika,;!5,8,1964. [321 -, "Theorems on the existence of periodic vibrations based upon the describing function method," Proc. Fourth IFAC World Congress, Vllarsaw, Session 41, pp. 46-60,1969. [331 A. I. Mees, "The describing function matrix," J Inst. Math. . Appl., 10.49-67.1972. [341 -, "Limit cycle stability," J Inst, Maths. Appl., 11, 281. 295,1973. I351 A. I. Mees and A,. R. Bergen, "Describing functions revisited," IEEE Trans. Automat. Control, AC-20,473-478.1975. [361 N. Minorsky, Theory of Nonlinear Control Systems. New York: McGraw-Hill, 196!3. [371 K. S. Narendra and C. P. Neuman, "Stability of a class of differential equations with a single monotonic nonlinearity,"SIAM J Control, 4(2), 1966. . [381 K. S. Narendra aind Y. S. Cho, "An off-axis circle criterion for the stability of feedback systems with a monotonic nonlinearity," IEEE Trans. Automat. Control, AC-13,413-416,1968. [391 K. S. Narendra ar~d M. Goldwyn, "A geometrical criterion for R. the stability of certain nonlinear nonautonomous systems," IEEE Trans. Circuit Theory, CT-I 1(31,406-408,1964. [401 K. S. Narendra ar~d H. Taylor. Frequency Domain Criteria for J. Absolute Stability, New York: Academic Press, 1973. I411 W. Oppelt, "Locus curve method for regulators with friction," Z Deut Ingr., Berlin, 90, 179-183, 1948. . 1421 V. A. Pliss, "Necessary and sufficient conditions for the global stability of a certain system of three differential equations," Dokl. Akad. Nauk SSSR, 120,4,1958. 1431 V. M. Popov, "Absolute stab~~lity nonlinear systems of autoof matic control," Automation and Remote Control, 21, 961-979, 1961. [44] P. E. Rapp and A. I. Mees, "Spurious predictions of limit cycles in a non-linear feedback system by the describing function method," Int. J Control, 26,821429,1977. . [451 E. N. Rozenwasser, "The absolute stability of nonlinear systems," Avtomat. i. Telemekh ., 24(3), 283-294.1963. [461 1. W Sandberg, "A frequency domain condition for the stability . of systems containing a single time-varying nonlinear element," Bell Syst. Tech. J., 43, 1901-1908,1964. [471 -, "On the response of nonlinear control systems to periodic input signals." Bell Syst. Tech. J . , 43.91 1-926.1964. [481 A. Tustin. "The effects of backlash and of speed dependent fric. tion on the stability of closed-cycle control systems," J IEE, 94,Part ll, 143-151.1947, [491 Ya. 2. Tsypkin, "On the stability in the large of nonlinear sampled-data systems," Dokl. Akad. Nauk., 145.52-55, 1962. [50] Ya. 2. Tsypkin and Yu. S. Popkov, Theory of Nonlinear Sampled-Data Systems. Moscow: Publishing House "Science," 1973. 151I Ya. Z. Tsypkin, Relay Control Systems. Moscow: Publishing House "Science," 1974. I521 P. Vidal, Nonlinear Sampled-Data Systems. New York: Gordon and Breach, 1970. [531 L. Wienberg and P. Slepian, "Positive real matrices," J Math. . and Mechanics, 9.71-83.1960. [541 J . C. Willems, Proc. Fourth Allerton Conference on Circuit andsystem Theory, p. 836,1966. [551 -, The Analysis of Feedback Systems. Cambridge, MA: M.I.T. Press, 1971. [561 D. Williamson, "Periodic motion in nonlinear systems," IEEE Trans. Automat. Control, AC-20.479-485.1975. [57] V . A. Yacubovich, "Solution of certain matrix inequalities occuring in the theory of automatic controls," Dokl. Acad. Nauk SSSR, 143,1304-1307,1962. 1581 G. Zames, "On the stability of nonlinear, time-varying feedback systems," Proc. NEC, 20,726-730,1964. [591 -, "On the input-output stability of time-varying non-linear feedback systems," IEEE Trans. Automat. Control, AC-11, 228-238and465-476.1966. ## Frequency Response Method for and 'Y hesizing Servomechanisms RALPH J. K C E B R E OHNUGR ASSOCIATE AlEE HIS paper introduces a procedure for contactor Tsynthesizingthe frequencyservomechanisms in which response of the system is employed. In this respect it represents a radical departure from the techniques used previously to handle the analysis and synthesis of .such discontinuous servomechanisms. While these earlier methods all involve step-by-step procedures based upon the transient response, the method proposed here permits the analysis and synthesis procedure to be determined from a knowledge of the response of the components to sinusoidal signals o various amplitudes and fref quencies. The successful application of the proposed method is based upon an approximation which is valid for most of the physical systems encountered in practice. This approximation permits .the discontinuous contactor to be represented in terms of a linear describing function. The Paper 50-44, recommended by the .AIEE Committee on.Feedback-Control Systems and approved b y the AIEB Technical Program Committee for presentation at the AlEE Winter General Meeting, New York, N. Y., January 30-February 3, 1950. Manuscript submitted October 26, 1949; made available for printing December 1, 1949. limitations imposed upon the use of this approximation are discussed in the paper. Although the proposed method can be profitably applied to both the analysis and synthesis of contactor servomechanisms, it is the synthesis problem which is particularly facilitated by this approach. Instances are described in the paper where a synthesis procedure based on this method resulted in a selection of design constants which greatly enhanced the performance of a representative contactor servomechanism. Servomechanisms and feed-back control systems in general, may be classified as either continuous types or discontinuous types. Most of the existing literature concerning feed-back systems deals only with the-types in which control is continuous, particularly when problems of analysis and synthesis are discussed. The scarcity of published treatments dealing with systems in which control is discontinuous is due, in some part, to the difficulties encountered when such systems are subjected to a rigorous mathematical treatment. On the other hand, discontinuous controls represent a substantial number of the systems used in ERROR DETECTING MEANS REFERENCE INPUT practice, and methods for analyzing and, stiu more important, synthesizing them would be of decided practical interest. Figure 1is a block diagram representing a simple feed-back con, system. Such a system may be subdivided into the error-detecting means, the controller, and the servomotor and output-load combination as shown. I t is assumed that a direct feed-back path is employed, and that no additional component need be introduced to represent dynamic factors which modify the feed-back signal. The case when such an additional component need be considered in the feedback can, however, be handled as well by the general method described here. The symbolism used to describe the input and output signals of the various components is indicated in Figure 1. In the more familiar linear continuous systems, the relationship between the correction and error signals may be expressed as a continuous function. If the controller characteristic is plotted in terms of the quiescent response, or steadystate response to various nonvarying error signals, the relation between the correction signal D and the quiescent value of the error E, called EQ, may be plotted as in Figure 2(A). In discontinuous systems, this relationship cannot be expressed by such a continuous curve. In the discontinuous type of system discussed here, and specifically the contactor type of servomechanism, the D versus E, relationship is itself discontinuous, as shown in Figure 2(B). In other words, as the error of the system varies in some continuous manner, the correction signal applied to the servomotor varies in discrete jumps. ## RALPH KOCHBNBURGER J. is with the University of Connecticut, Storrs, Conn. The material described in this paper represents some of the results obtained as part of a doctoral thesis research program conducted by the author at the Massachusett?, Institute of Technology. The aid of various members of the Electrical Engineering Department and S?rvomechanisms Laboratory staffs and of H. K. Weiss of the Aberdeen Proving Grounds is gratefully acknowledged. The suggestions and criticisms furnished by Dr. Gordon S. Brown, Director of the Servomechanisms Laboratory, in his capacity as thesis supervisor were partitularly helpful. Acknowledgment is due to the United States Air Forces, Air Materiel Command, Armament Laboratory. Wright Field, who sponsored the project under which this work was done. ei +-yE = 9 j - 9 0 - 90 Figure 1. FEEDBACK PATH ## General form of single-loop servomechanism 148 CORRECTIClN SIGNAL, 13 ;/ 1 ' ( a ) CONTINUOUS SERVOMECHANISM +IOUIESCENT ERROR, ~q ItL (b) CONTACTOR SEHVOMECHANISIM CORRECTION SIGNAL, D OUIESCENT ERROR,^^ Figure 9. ## Typical controfler relations The type of controller charact.eristic shown in Figure 2(Bi) is found in those systems which emplsoy mechanica.11~ or hydraulically actuated electrical contactors, electromagnetic relays, elecFigure 3. Typical contactor se~omechanism for positional control tronic-control circuits of the trigger or flip-flop type, certain types of mechanical clutches, and some hydraulic and pnmmatic valves. These devices are clistinguished by an all-or-nothing characteristic. Their actual characteristics may be more complicated than shown, but will still possess the distinguishing feature of producing sudden jumps in the value of correction signal D. Because one of Ithe most frequent applications of contactor servomechanisms involves electromagnetic relays, this general class is sometimes described as relay servomechanisms. Contactor servomechanisms have found widespread employment in both military and industrial fields. They frequently possess advantages over continuous types in economy of weight, bulk, complexity, and cost of control equipment. One of the major practical objections to their use has been the nonlinearity of the response relationship. Only the simplest versions have been amenable to analysis. I t has been difficult to analyze their performance characteristics and to synthesize design constants and networks which will provide a specified performance. One of the earliest analyses of c o mn tactor-type servomechanisms was published by Hazenl in 1934. He emplo!yed the direct differential-equation method for analyzing performance and a number D.GLINE of characteristics peculiar to contactor types were demonstrated. This method becomes cumbersome when the system i s a t all complicated because a new equation must be established and solved for each switching cycle or correction internal initiated by the contactor. The phase-plane method of analysis provided a simple graphical approach. This method has been employed by various authors2J in connection with other forms of nonlinear systems. It was applied to contactor servomechanisms by Weiss4and MacCollband the analyses of certain types of SLI& servomechanisms were shown to be greatly facilitated. Unfortunately, the method is limited to systems possessing relatively few energystorage elements. A recent AIEE paper by Kahn,6 based on the dXerentialequation approach, proposes relatively simple graphical constructions to eliminate some of the computational labor associated with the analysis of such systems. There are two basic disadvantages to these methods of treatment, which might be characterized as solutions in the t&e domain. One disadvantage is the great difficulty of treating systems of apparently elementary physical configuration which have only a few energy-storage elements. The second and more significant disadvantage from. the standpoint -v +V VISCOUS FRICTION-f ## r = ARMATURE RESISTANCE OHMS . K J f ' MOTOR CONVIERSION CONSTJLN1.- GENERATED ' LOAD a MOTOR MOMENT OF INElRTlA RADIAN/SEC. = LOAD a MOTOR VISCOUS FRICTION COEFFICIENT +I UPPER (INCJ RELAY CLOSED 0 NEITHER RELAY CLOSED -I LOWER(DEC.) RELAY CLOSED eo(.) = ## R ~(1); S(Ta+l) h;V Kz+r(ra+r, eo(jw) = jw(l+jwTl D(Jw) WHERE: -RUNAWAY VELOCITY" ; 1950, VOLUME 69 Kochenburger-Synthesizing Contactor Servmechanisms ERROR DETECTING 7 I CONTACTOR MEANS I SERVOMOTOR AND I I 80 ## FEDBACK OUTPUT SIGNAL 80 FEEDBACK PATH of usefulness is the fact that the methods are not directly adaptable to system synthesis. It is the purpose of this paper to present an analysis and synthesis procedure in which the frequency response of the system is employed in a manner7 analogous to its use with continuous servomechanisms. Results obtained by the method have been compared with those obtained by more cumbersome exact methods and with test data* and the agreement has been found sufficient for most engineering applications. The method is not limited by any inherent dynamic complexity, provided the system can be described by a control loop as shown in Figure 1. Furthermore, the proposed method is particularly adapted to synthesis and permits the selection of compensating networks suitable for the attainment of given performance requirements. 'The contactor servomechanism simulator used to obtain the test data was purposely slow to provide ease mf measurement: the slow response indicated in the examples should not be considered as generally typical of contactor servomechanisms. ## General Configuration of a Contactor Servomechanism Figure 3 illustrates a typical contactor servomechanism for controlling angular position. In this example, the various signals are represented as voltage equivalents; that is, the voltage e& is an indication of the angular error &. A simple lead-type compensating network is shown in this example. The compensating network converts the error signal to a new form, here called the control signal, before application to the contactor means. When no compensating networks are employed, E and C are identical. In almost all cases, the introduction of a suitable compensating network results in improved performance. Networks of mechanical, hydraulic, or electrical forms may be used. In the contactor servomechanism of Figure 3, the controller includes the compensating networks and the succeeding contactor means. This is illustrated again in Figure 4 as a general block diagram of a singleloop contactor servomechanism. In some cases, compensating networks may be inserted between the contactor means and the servomotor. This is usually less practical than the arrangement shown in Figure 4 because of the el ## = ERROR SIGNAL VOLTAGE INDICATION Figure 5 (below). Typical con- p~ Q , ez = CONTROLSIGNAL VOLTAGE INDICATION tactor characteristics (right). trical phase-lead cornpensatin9 for caKade insertion in control system ## (a) SIMPLE PHASE-LEAD COMPENSATING NETWORK CORRECTION SIGNAL, 0 (0) ## NO INACTIVE ZONE CA-0 CORRECTION SIGNAL, D -SIGNAL. CONTROL c TI ## e R3Ca = RIG, REQUIRED CONDITION. ~ 2 4 Re CORRECTION SIGNAL, D ## (c) WITH INACTIVE ZONE. CA AND HYSTERESIS ZONE, Ch -+-1 Kochenburger-Synthesizing Contactor Servomechanisms A I E E TRANSACTIONS higher power level at. which the network is operated. When the network is directly cascaded with the servomotor and appears after the contactor means, its characteristics are generally lumped with those of the servomotor for purposes of analysis. The contactor may be considered as a discontinuous power amplifier. The relay shown in Figure \$: represents one particular type. The relationship between C and D might folloiw one of the characteristics shown in Figure 5. In these diagrams, symmetrical operation is assumed in that the correcti~reefforts associated with positive and negative correction are identical in magnitude. I t is comnvenient to express the correction signal D in dimensionless units, assigning it the value D = 1 when positive correctia~nis initiated by the contac~torand causes the servomotor to increase the outpilt 0, and D = - 1 when negative correction is initiated. In Figure 5(A), a relationship is shown where the contactor will initiate either positive or negative correction, depending on the algebraic sign of the control signal C. In this case, the inactive zone or range of error within which. no correctiive effort arises is zero. In Figure 5(B) an inactive zone, CA, is introduced and the control signal must have a magnitude: greater ~ than C A / before corrective effort is initiated; otherwise the correction signal will merely be expressed by D-0. An inactive zone may be unavoidable because of physical limitatio'ns, but often it is purposely introducetd because it contributes to dynamic atability. A system using a contactor possessing the characteristic of Figure 5(A) will always oscillate about some equilibri.um point while one having an inactive zone as in Figure 5(B) can be designed to remain in a state of rest when no significant disturbance is applied. Since the inactive zone represents the range o permissible error f within which no corrective action is initiated, it should not be excessively large. On the other hand, there is no pomint in making it smaller than the accuracy requirements of the application might stipulate. One of the basic prolilems of f system synthesis is the selection o compensating networks which will permit adequate dynamic stability for some specified range of inactive zone. This problem becomes more difficult as the specified range of inactive zone becomes smaller. Many contactors possess a hysteresis effect. This means that the control signal necessary to initiate correction is greater than that necessary to cease correction. In the case of electromagnetic relays, for example, the coil current of an alreadyclosed relay must usually be reduced to some value lower than that which originally caused the relay to close before the relay will reopen. Figure 5(C) illustra,tes the effect as designated by the symbol Lh. r Speed of response deals with the rapidity by which errors are reduced to a tolerablelevelfollowingsome disturbance. The fastest response may be obtained by: 1. Using a runaway velocity that is as large as the stability limitations and serw- It can be shown that a hysteresis effect generally adversely affects the dynamic stability. More complicated types of contactor characteristics are met in practice; for example, nonsymmetrical corrective action, or multistep contactors. These will not be discussed in detail but the general method of treatment to be introduced is applicable to them as well. The other components of the contactor servomechanism, as shown in Figures 3 and 4, are the servomotor and the errordetecting means. Both of these usually exist in essentially the same form as in continuous systems. In a contactor servomechanism, however, it is rarely necessary to design for continuously variable control of speed. This greatly simplifies the equipment. capabilitiespermit. 2. Having the frequencies associated with any damped transient oscillation as high as possible. Having adequate dggree o stability for f transient oscillations o amplitude greater f than the region o error tolerance. f 3. ## Performance Criteria of a Contactor Servomechanism The performance criteria of contactor servomechanisms are basically the same as for continuous types. Static accuracy is one of the major design considerations. It is the possible range of error which can exist when the servomechanism is in a state of rest and which will not be sufficient to initiate corrective action. The static accuracy is therefore met by keeping the range of the inactive zone within specified limits. Dynamic accuracy defines the extent to which errors are minimized when the system is responding to a disturbance. Such errors are best minimized by designing a system that will possess adequate speed of response and relative stability. CORRECTION SIGNAL, D Stability is the tendency of a system to approach an equilibrium condition without excessive oscillation. Absolute stability is descriptive of a system incapable of self-sustained oscillations. Degree of stability refers to the rate with which transient oscillations disappear after a disturbance. A distinction must be made between the stability requirements imposed on nonlinear systems and those imposed on linear types. For the former, continuous self-sustained oscillations of a small amplitude may be permissible. For the latter, the amplitude o any selff sustained oscillations would tend to increase to an undesirably high value. For contactor servomechanisms, performance requirements may be met by specifying a range of inactive zone consistent with the static accuracy requirements and a runaway velocity consistent with the desired speed of response. Once these are chosen, the major problem remaining is the selection of components which will provide adequate stability. The remainder of this paper is devoted primarily to the problem of stability. ## Mathematical Representation of Characteristics The assumption commonly employed in' the study of continuous servomechanisms, that all components possess linear characteristics, will be applied to all of the components of a contactor servomechanism except the contactor. Each of these linear components may then be described in terms of its transfer functions or ratios of output to input. SERVOMOTOR CHARACTERISTICS The transfer function of the servomotor, designated as G,(s), is defined by the ratio - Figure 7. Dimensionless representation of contactor characteristics (care involving both inactive zone and hysteresis) \-, where 0,(s) and D(s) are the Laplace transforms of the output go(>)and the 1950, VOLUME 69 Kochencburger-Synthesizing Contactor Servomechanisms 151 CONTROL SIGNAL, C Figure 8 (left). Relation of correction signal to control signal for a simple contrctor with hysteresis I,O-+- SIGNAL 00 :AVERAGE ## (0) A TYPICAL PERIODIC CORRECTION SIGNAL AS A FUNCTION OF TIME (01 ASSUMED SlNUSOlDAL SHAPE OF CONTROL SIGNAL ACTUAL CORRECTION S N LD I A, G 0.3 ## (b) HARMONIC DISTRIBUTION SPECTRUM OF CORRECTION SIGNAL. 0, WHEN u= l I (b) RESULTANT FORM. OF CORRECTION SIGNAL ## AND ITS BASIC COMPONENTS correction signal D(t), subject to zero initial conditions. When the polynomials of the transfer function are factored, it will appear as t, (c) OUTPUT SIGNAL RESULTING FROM CORRECTION SIGNAL OF (0) (CASE OF SIMPLE SERVOMOTOR ps= ) Idnl4 units of output quantity (2) The 7's and T's are time constants which may be either real or complex. Equation 2 applies to the most common form of servomotor which approaches the condition of a constant output rate of R output units per second. The quantity R is the runaway velocity for a given correction signal applied for a sufficiently long time. The linear factor s in the denominator of the transfer function represents a single-order pole at the origin of the complex s plane and identifies a system having a sustained rate for a constant input signal. The servomotor or equivalent may have other forms of transfer function; for example, in regulator applications, there may be no pole at the origin while, in other applications a higher-order pole may exist. Only transfer functions of the form given in equation 2 are treated herein although the method is generally applicable to any fo m . The transfer function also might be described in terms of its steady-state response to various real frequencies by substituting the imaginary quantity j w for the operator s (w = 2rf =angular frequency). Figure 9 (right). Comparison-exact response of servomotor with that obtained from fundamental harmonic approximation I 2 COMPONENT ## this case, G,(jw) becomes e, ~ i w , Gs =(jw) D(jw) where the relations for determining the runaway velocity R and the time constant T are given in Figure 3. For many engineering purposes, this simplified form of relationship is adequate for analysis and synthesis. Similar transfer functions arise in many other servomotors. One advantage of the frequency-function form is the fact that, when the component being considered is not practically subject to an analysis based upon its design constants, the complex transfer function G(jw) may often be expressed in graphical form on the basis of actual tests. The method of analysis and synthesis to be described here possesses the advantage that such graphical data may be directly employed. I t is not necessary to convert G(jw) to an analytic form. The type of servomotor and load combination shown in Figure 3 has an elementary transfer function if, for example, coulomb friction, annature reaction and armature inductance are neglected. In The characteristics of the compensating network may similarly be expressed in terms of a transfer function G,(jw) as (1+jo+,) (1+jwsb). . . CCjw) GJjo) = (l+joTal(l+j orb). , . = gG..) (5) I t will be noted that at zero frequency G,(jw) is unity. This is because both C and E are expressed in units of the output quanity ; C being described in terms of the equivalent value of E under quiescent ## Kochenburger-Synthesizing Contactor Seruomechanisms 152 AIEE TRANSACTIONS ## (a) AMPLITUDE RELiiTIONSHIP Figure 10 (left). Plot of the fun&mental harmonic transfer function go - s i m p l e contactor with hysteresis ratio h/A, for case of zero input velocity (ds/dt=O) and zero load torque ( 8 = 0) ~ ## (b) PHASE RELATIONSHIP tactor, considered as having no dynamic lag but having a discontinuous characteristic, is nonlinear and therefore dependent on signal amplitude. Any dynamic lag action occurring in a physical contactor is conveniently lumped with either the servomotor or compensating network. The contactor represents the amplitude-variant portion of the system, and its describing function GD,although independent of frequency, is dependent upon the control signal C. Other nonlinear effects such as backlash may be treated similarly in tenns of amplitudevariant describing functions. The over-all loop transfer function GCjw) or combined transfer function of all linear and frequency-variant components is particularly significant. With the arrangement shown in Figure 4, this would merely be C(iw) G ( j w ) =Gs(iw)G,(iw) = D ( j w ) el = constant conditions. Any a.ttenuation or amplification occurring within the controller is described in terms of its effect on the range of inactive zone. This range is the sole measure of the zero-frequency gain of the system. Figure 6 shows two common forms of compensating network for producing a phase-lead effect. For these networks, the transfer functions are given by G,(jw) =* a network of Figure 6(A) l+jwT (6) and simple phase-lead network, the transfer function given in equation 6 includes the attenuation factor a,computed as shown in the diagram. The higher the value of a the more effective phase-lead effect is obtained. However, a high value of a results in a relative increase in the transmission of high-frequency noise. Some engineering compromise concerning its value is therefore required in practice. Since the phase-lead effect of the circuit shown in Figure 6(A) cannot exceed 90 degrees, it may be inadequate for some applications. The resonant lead network shown in Figure 6(B) may then be employed. Equation 7 represents the transfer function of this latter network. Examples will be given where the use of these compensating networks materially improve system performance. THE OVER-ALL LOOP TRANSFER ## =loop transfer function (8) I t will be shown how the loop transfer function and the contactor describing function may be used together to determine the performance characteristics of the control system. Dimensionless Notation The mechanics of analyzing and synthesizing contactor servomechanisms are simplified and the results obtained acquire more general utility if a dimensionless form of notation is used. This procedure is employed in many engineering fields and the system of notation proposed here is an expansion and modification of that introduced by we is^.^ As a first step select a convenient time base tb seconds. Then divide all variables and constants having the dimensions of time by tb in order to obtain the corresponding dimensionless value; for example, 4=t/tp=elapsed time. A convenient time base is frequently obtained ## netwo!rk of Figure 6(B) (7) where FUNCTION ing ratios, respectively Since the components represented by G, and G, are essentially linear and have transfer functions which depend only upon the applied frequency, they are called the frequency-variant portions of the system. On the other hand, the conVECTOR LOCUS OF [ Q D ~ + O - I ( ~ ~ ) ] VECTORLOCUS ## VECTOR LOCUS OF [a. CRITICAL WINTD; u:O 1 I I OF INCREASING F E UN YU RO E C , I u-m ( 0 ) STABLE ua -D RESPONSE ## ARROWS INDICATE DIRECTION OF INCREASING FREOUENa U IU -m ( 0 ) STABLE u-0) RESPONSE ## (bl UNSTABLE RESPONSE 1950, VOLUME 69 Kochenburger-Synthesizing Contuctor Servomechanisms Table I. ## Dimensional and Dimensionless Symbols Used Dimensional Symbol fb Quantity Units Dimensionless Symbol Seconds Seconds . .d. = l / l b . . . . . . . . . . . . . . . p = d / d l . . . Seconds -' . .X = 1bP = d / d + Applied angular frequency. . . . . . . . . . . . . . w . . . Seconds-' . . u = w l b Signal base.. . . . . . . . . . . . . . . . . . . . . . . . . . . B = Rlb . . .Output-Units 80 . . .Output-Units. .r = @ d B Output.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B i . . .Output-Units. .6 = 8 i / B Error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E = 8 i - 8 0 . . .Output-Units. . 8 = G / B= d - a Control signal.. . . . . . . . . . . . . . . . . . . . . . . . ...Output-Units. .c = C / B C Correction signal.. . . . . . . . . . . . . . . . . . . . . . D .. .Dimensionless.. D (same) ## Time base.. ........................... ........................ ... ... / ~unawav velocitv.. . . . . . . . . . . . . . . . . . . . . R CA Ch . . .Outout-Units. .Unitv seconds / System ## . . . Output-Units. . A = CA/ B . . .Outout-Units. .h = Ca/ B Of servomotor.. . . . . . . . . . . . . . . . . . . . G I / 1 ( Of compensating network.. . . . . . . . . Transfer function properties: Time constant ~n numerator.. . " -5 C -'o . . 4=- =j ## . . .Output-Units . .8 = 8.8, . . . . . . . . . . . . . . . .. g g where D = g g ( c ) D =GD(C) .............. ... ... Seconds Seconds In denominator.. . ............ .. p = : t h T . .I =ti. Natural frequency. . . . . . . . . . . . . . . . . . . wl, a,. . . . Seconds-' . .UI = w ~ l b ,ug =w21b. .. Damping ratio associated with a natural frequency ........................... \$I. \$2,. . . . . .Dimensionless. . ( I , h .... (same) Attentuation factor of a compensating Dimensionless. .a (same) network ............................ a .. ... by setting tb= 1 ~ 1 1 where TI is the smal, lest time constant of the servomotor. In equation 4, for example, the only time constant T is naturally the lowest and and the resonant lead-controller network shown in Figures 6(A) and 6(B) now have the following dimensionless forms whether the system possesses absolute stability; that is, whether it is capable of maintaining self-sustained oscillations. By various rules-of-thumb, it also is possible to reach qualitative conclusions concerning the degree of stability which are adequate for most engineering purposes. This technique uses frequency loci ar\ I ranged in various forms to suit the convenience of the user; the well-known polar-locus plots are one example. In view of the successful application of the frequency-response method to linear systems, its application to contactor servomechanisms appears desirable. The nonlinear relationship describing the contactor operation has impeded such an application. In the preceding sections it was necessary to express this relationship in terms of a describing function; it could not be expressed in terms of a linear transfer function. The frequency-response method is therefore not applicable unless certain assumptions are made. In order to determine whether an adaptation of the frequency-response method might not be applied, let it be assumed that a sinusoidal control signal is applied to a typical contactor. Suppose that the contactor has the more general characteristic shown in Figure 7. If the control signal c is of the form c=co+lc,l cos u+ (9) t,=T. The next step is to select an output signal base B. All signals, such as the output O,, are divided by B in order to obtain the dimensionless equivalents, for example, the dimensionless output is expressed as a=@ J B . The inactive zone and hysteresis ranges, CA and Ch, are dimensionlessly expressed as: A = C a / B , h = Ch/B . The most convenient signal base B is the product of the runaway velocity and the time base, namely Rtb. Table I summarizes the dimensional and dimensionless notation used. In dimensionless form, the transfer functions of the servomotor and compensating network given in equations 3 and 5 become ## The Frequency-Response Method of Analysis and Synthesis The frequency-response method of analysis and synthesis has been found to be a practical means for treating essentially linear servomechanism^.^.^.^ By this method it is possible to determine the contactor will initiate a positive correction when c = A / 2 + h/2, cease correction when c = A / 2 - - h / 2 , initiate negative and correction when c= - A / 2 - h / 2 , cease correction when c = - A / 2 + h / 2 . These instants of time are designated by the angles, u+=al-/31,=a1+/31, = r + f f a -82, and =r+cun+&, respectively. This is shown in Figure 8(A). The angles 2/31 and 2Pz represent the duration of the positive and negative pulses, respec1 tively; the angles a and as represent the phase lags associated with the respective pulses. Figure 8(B) shows the rectangular form of the resultant periodic correction signal The transfer function of the simple second-order servomotor expressed in equation 4 now is Figure I 3 (right). Superposed frequency and amplitude loci & determine the stability of a contactor sewomechanism Kochenburger-Synthesizing Contactor Servomechanisms AIEE TRANSACTIONS 154 Figure 14 (left). Significance of various relative configurations of the loci Figure 15 (right). Vector relationships in loci diagram (used to determine relative stability) -25 -2.0 -1.5 -1.0 -0.5 !MA& AXlS (0) POINT 0 ATORIGIN APPROACHES tlm t IMAG. ## qo,(lC,lt FOR CONSTANT HYSTERESIS. (,dl -qo,(lcol) -gm,(lc4ll 1 / CUT-OFF IMAG. qlju) A juli+)u) AXlS ## (c\ A SYSTEM WITH A CONVERGENT EQUILIBRIUM POINT; ALWAYS OSCILLATE. WILL ## ( d l A SYSTEM WlTH A DIVERGENT EQUILIBRIUM POINT; IS C:ONDITIONALLY STABLE (FOR SMALL DISTURBANCES) Figure 16 (right). Values of Mp and up (used in determining relative stability and speed of response) 10 1.5 en eS 50 I . IMAG. AXIS. REAL AXIS ## (e) A SYSTEM WlTH A DIVERGENT 8 CONVERGENT EQUILIBRIUN1;WILL OSCILLATE CONTINUOUSLY FOLLOWING DISTURBANCES OF SUFFICIENT AMPLITUDE. ## If1 A BYSTEM WITH A DIVERGENT B CONVERGENT EQUILIBRIUM; WILL OSCILLATE CONTINUOUSLY AT FllllTE AMPLITUDE UNLESS DI!ITURBANCES ARE EXCESSIVE. D. The broken-line relation plotted in Figure 8(B) shows the average or zerofrequency component D and fundamental o l harmonic component D of this rectangularly shaped signal. Higher lhsunnonic components which contribute to the comers of the rectangular shape are not shown. I the correction signal consisted only f of Doand Dl,only sinusoidally varying signals would exist throughout the system when sinusoidal inputs were applied. The contactor would then appear as a quasilinear transfer device in that it would operate as a linear arnplifier for a.ny given constant amplitude of control signal. I t would not operate as a truly linear device because of the nonlinear relationship between input and output amplitudes. Considering the contactor as such a quasi-linear device will permit the frequency-response method to be used in determining the system stability for any given control signal amplitude. The ap- proximation which permits the higher harmonic components to be neglected8 so that the quasi-linear representation of the contactor may be employed must now be justified. Figure 9 shows the relative importance of the harmonic components of the signals in a typical contactor servomechanism. Figure 9(A) shows a typical rectangular correction signal resulting from a periodic control signal. Figure 9(B) represents the harmonic spectrum associated with this correction signal. I t is assumed, for example, that the servomotor has the typical transfer function given in equation 4, and that the fundamental :frequency of the control signal is designated by the relation u= 1. Under these considerations, an exact determination of the resultant servomotor output a obtained by a mathematical determination of the repeated transients, would appear as plotted by the solid-line curve of Figure 9(C). The harmonic distribution of this output signal is shown plotted in Figure 9(D). It may be seen that, while the original correction signal D possessed higher harmonic components of significant amplitudes (referring back to Figure 9(B), these higher hannonic components have relatively small amplitudes, compared with the fundamental, when measured as part of the output signal. The minor importance of the role played by the higher harmonic components also is demonstrated by the brokenline curve of the output response in Figure 9(C). This curve, designated as al,represents the output response which would have been obtained if only the average and fundamental components of the correction signal had been present. It may be seen that this latter response curve is a fairly good approximation of the exact respense represented by the solid-line curve. I the higher-harmonic components f may be considered negligible as far as the output signal is concerned, their contribution to the error and control signals will generally have minor significance. This in turn tends to justify the original assumption of a truly sinusoidal control signal. It is therefore proposed that the frequency-response method of analysis be used where the contactor characteristic is described in terms of a quasi-linear describing-function gD where the subscript 1 is used to indicate that the describing 1950, VOLUME 69 Kochenburger-Synthesizing Contuctor Servomechanisms function takes into account only the fundamental harmonic component. This function neglects the higher harmonics with the following justifications: average load torque and zero average rate-of-change of input. Under this latter condition, the input signal 6 may be of the form tion that the contactor exerts equal amounts of positive and negative corrective effort. ,Therefore, co is zero and equation 10 simply becomes c = lcll COS U + (15) 1. The normal frequency spectrum of a rectangular wave involves progressively smaller amplitudes for increasing orders of the harmonic components. 2. Most servomotorS serve as effective low-pass filters and minimize the importance of the higher-harmonic components. Justification (1) above is valid for most cases but becomes less applicable under conditions where the pulse widths associated with the correction signals are short compared with the duration of the intervals between pulses. A periodic impulse signal of zero pulse width has odd harmonic components of equal magnitudes. Justification (2) was demonstrated using the example of an elementary servomotor; more complicated servomotors should serve as still.more effective low-pass filters. On the other hand, it is conceivable that when servomotors exhibit marked resonance effects, the validity of the approximation may suffer under certain conditions where one of the neglected harmonic components might have a frequency near resonance and its effect on performance might therefore be particularly prominent. At any rate, the previously-mentioned approximation is proposed so that the frequency-response method can be employed. Some conclusions concerning the general accuracy of the method may be reached by the comparisons given herein between the results thus obtained with those obtained by more exact methods of computation or by actual test. I t should be kept in mind that the method proposed is an engineering approximation and that, even in the case of the better-understood continuous linear servomechanisms, conclusionsregarding stability that are accurate to within 10-20per cent are frequently satisfactory. In order to determine the relative stability of the system when subjected to such an input signal, only the steady-state response need be considered when the frequency-response method is employed. Under such steady-state conditions, since the average value of input is constant, the . average value of the output a also is constant. This can be the case only if the average component of the correction signal DOis zero, since from equation 2, the presence of an average component of D would result in an integrating action because of the s factor in the numerator of G,(s) and a D& term would appear in the corresponding time solution.for the response. The assumed sinusoidal nature of the control signal was expressed by equation 9. The average value of D or D is given by o Since any average component of the error E or & would result in a finite value of CO, the average error for the condition just considered will be zero. By means of a Fourier analysis of the rectangular corrective pulse shown in Figure 8, the fundamental harmonic component may be shown to be where 2 IDII (16A) ## =;dsin2~1+ ~ i n ~ ~ z + 2 s i n ~ ~ s i n ~ ~ X cos (el-(12) (16B) If the fact that a = a ~ = and 8=81=82 a~ is considered, these equations acquire the simpler form Since, for this case DO 00, =P2,and the = PI operation is symmetrical. But, from the trigonometric relations shown in Figure 8 where the values of a and /3 are given in equations 13 and 14. The contactor describing function, gD, may now be expressed in a manner similar to that used to describe transfer functions of linear components, that is, as the mathematical ratio of the output and input quantities, or Since PI =P2 the average value of control signal co must be zero, in which case 4 sin B (18B) ## The Quasi-Linear Representation of the Contactor Describing-Function The contactor may be approximately represented in the manner just described and cases involving the application of constant rates of input change or constant load disturbances may be taken into account. This entails additional mathematical complication because the duration of positive and negative corrective pulses will differ @1#Pz). For brevity, only the case where symmetrical corrective operation occurs (Dl =&) will be discussed. This implies the condition of zero Similarly, the phase-lag angles or* and or2 shown in Figure 8 will be equal and will be The foregoing results indicate that, for the case of an input disturbance represented by equation 9 and with zero average load torque, there will be zero average component of control signal. This also is based on the original assump- The above expression represents the describing function for the quasi-linear approximate equivalent of the contactor. The angles a and P given in equations 13 and 14 are functions of the inactive-zone range A, the hysteresis range h, and the control-signal amplitude \cl\. The amplitude and phase angle associated with the function gDl may be conveniently plotted as a function of the ratio lcll/A for various ratios of the quantity h/A representing the hysteresis effect. This 0 has been done in Figure 1 . Figure 10 shows the amplitude and phase angle associated with the contactor describing-function for various controlsignal amplitudes and various conditions of contactor hysteresis. I t may be noted that the describing function is independent of the frequency u. Howeve., a phase lag is involved because of the hysteresis effect. When the control amplitude lcl( is less than (A+h)/2, it i!j insufficient to actuate the contactor and, ns shown in the diagram, the magnitude of gDl remains zero. of equation 21 is independent of frequency and simply represents a scale factor. This has been done in constructing the typical locus of Figure 11. In this example Figure 11(A) represents a stable system which fullills the criterion. On the other hand the case represented by Figure 11(B) is unstable. In the case of the loci of g-'(juJ+g~, shown in Figure 11, the origin was the Stability Criteria for Contactor critical point whose location with respect Servomechanisms to the locus determined stability. HowIn view of the approximation proposed, ever, it is convenient to shift the coordinate axes along the vector g ~ , .These the contactor servomech~anismmay be translated axes are shown as broken lines considered as equivalent to a system involving all linear compon!ents if a single in Figure 11. In Figure 12, the coconstant signal amplitude lcll is con- ordinate axes have been translated in this sidered at a time. Under these concli- fashion and the point which was formerly tions the describing function ED, may lbe the origin is now the point Q located a t the tip of the vector -gD,. If such transtreated in the same manner as a convenlated axes are used, the stability criterion tional transfer function. The ratio of tlhe output respgnse of the servomechanism given may be revised as follows: I a polar locus of the inverse-loop f shown in Figure 4 to a sinusoidally varytransfer function g-'(ju) is drawn over the ing control signal of some dimensioinless range of frequencies, u, and if a critical frequency u and of fixed amplitude lcll point Q is located a t the tip of the dewould then be given by scribing-function vector -go, the system will be stable provided the point Q always appears to the left of the locus as the where latter is traversed from zero to infinite frequency. Figure 12 demonstrates the application of the above criterion. =loop transfer function The foregoing test for absolute stability may be made for any specified The error by definkion is value of control-signal amplitude 1 ~ 1 1 . It might in fact be found that the system will be stable for some amplitudes and If equations 19 and 20 are combined, unstable for others., The location of the the response ratio is conveniently given in point Q is dependent on the value of lfil. inverse form as In order to obtain a complete picture of the stability, it is necessary that a second locus be drawn connecting the points Q over the complete amplitude range behg where considered. This curve will be referred to as the amplitude locus to distinguish it from the frequency locus plotted for the function g-l(ju). The amplitude locus is drawn by plotA system having an inverse response ting the tips of the vector values of the ratio as in equation 21 vvill be stablle if, function - gD, and connecting them by a for a polar locus of 6/a(jz~)drawn for the curve. The two loci are superposed on complete range of frequencies, - rn <u< rn, the origin of the plot always ap- the same graph and their relative orientapears to the left of this locus as the locus tion determines the stability characteris traversed in the direction of increalsing istics of the system. Figure 13 is an example of the superfrequency. This criterion is merely a posed loci. The frequency locus shown is simplified version of the well-known Nyquist criteria for stability. It applies that representing the simple-loop transfer function given by the equation only to single-loopsystems. Since only the orientation of the loaus with respect to the origin is of interest, the criterion given could be applied as well to a locus representing merely the numerator The amplitude locus of Figure 13 repreof equation 21. Therefore only the ltocus sents the contactor describing function of g-l(ju)+g~, need be plotted, since the under conditions of a zero-average input quantity go, appearing in the denominator velocity (6 =0) and zero-average load torque. I t was obtained from Figure 10 andrepresents a hysteresis effect given by h/ A=O.2. In the case shown in Figure 13, the two loci intersect a t the amplitude lcll= 0.062. From the criterion just given, operating points on the amplitude locus corresponding to lower values of Icll, 0.060 <lcll< 0.062, are unstable. Oscillations existing a t these amplitudes will result in a tendency for the amplitude to increase. Operating points corresponding to higher values of lcl(, that is, lc11>0.062, are stable. Oscillations existing at these amplitudes will result in a tendency for the amplitude to decrease. At the intersection itself, where lcll =0.062, the oscillations will tend to be maintained a t this amplitude because this intersection corresponds to the borderline condition of stability. The intersection point of the two loci is described as an equilibrium point. Because of the fact that higher amplitudes of oscillation decrease and lower amplitudes increase, the system's operating point tends to converge at this equilibrium point, it is therefore described as a point of convergent equilibrium. The graphical construction shown in Figure 13 indicates that any initial disturbance sufficient to initiate the contactor, that is, Icl1>0.060, will result in oscillations which will approach a selfsustained condition with an amplitude and frequency corresponding to the intersection point of the two loci. Disturbances less than sufficient to trip the contactor are beyond the cut-off point of the amplitude locus, where go,=O, and naturally no oscillations result. From Figure 13, the self-sustained 0scsations will correspond to a dimensionless control-signal amplitude of lcll =0.062 (the error amplitude is the same since g, = 1) and a dimensionless frequency ratio given by u = 3.1. The graphical construction which provides this result depends upon the basic approximation mentioned previously. I t is therefore interesting to compare the predicted result with that actually observed or computed by more exact methods. Such calculations show that the actual conditions of steady-state self-sustained oscillations would be given by lcll = 0.066 and u =3.2. The agreements concerning amplitude and frequency are withir 61/2 per cent and 3 per cent respectively. ## Convergent and Divergent Equilibrium The preceding section illustrated an intersection of loci that indicated a con- ## NEGATIVE REAL AXIS Figure 17. Loci diagrams illustrating stability improvement resulting from use of a phase-lead compensating network Figure 18. ## Loci plots for conbctor sewomechanirm with critically damped sewomotor vergent condition of equilibrium. Figure 14 illustrates other possible configurations of the loci, some of which involve one or more intersections. In general, these intersections indicate convergent equilibrium if a slight increase in lcll from its value at the intersection causes the locus of - gDl to enter the stable region and if a slight decrease causes it to enter the unstable region. In this event, self-sustained oscillations may result. If the intersections possess the converse properties, the equilibrium condition is divergent; it then does not correspond to a condition of self-sustained oscillation, but merely represents an amplitude boundary. The operating point will always tend to shift away from such a divergent equilibrium point. I t may be noted from Figure 14 that the loci always intersect at the origin, corresponding to the condition that Is/-) and u-0. The origin equilibrium points are divergent in Figures 14(A), (C), and (E) and the amplitude of any oscillations which might arise will tend to decrease until either a convergent equilibrium point is reached or the contactor cut-off point, as marked by the abrupt termination of the amplitude locus is reached. The origin equilibrium points in Figures 14(B), (D), and (F) are convergent, indicating that if these systems are subjected to disturbances of sufficient magnitude, destructive oscillations will result. I t also might be noted that, in Figure 10, the amplitude of , is plotted in terms g of the quantity AgD,. This function is therefore inversely proportional to A for a given amplitude ratio Ic,l/A. The inactive-zone range A appears as a reciprocal scale factor affecting the amplitude locus. The scale of this locus may be varied by changing the inactive-zonerange, and the stability characteristics of the system may thus be altered. In the cases shown in Figures 13 and 14(A) and (E), the response can always be made stable for all conditions of operation by making the inactive-zone sufficiently large. ## Equation 21 may be inverted and rewritten as E(ju) = = 6 g - ' ( j u ) + g ~ , input output (23) Degree of Stability In many applications of contactor systems, a specification that requires merely a stable system is inadequate. The requirement that any transient oscillation following the correction of disturbance shall involve sufficient damping also is imposed. In other words the degree of stability is of importance. In the case of linear servomechanisms, a rule-of-thumb procedure has been developed7 which gives an indication of the degree of stability from the frequencyresponse loci. This procedure is to determine the peak value M p of the outputinput ratio M= 10,/Bil = lcl/16/. In general, it is found that satisfactory damping exists if M p <1.3, the exact limit imposed depending upon the application. This rule-of-thumb method must be used with caution. When it is backed up by the designer's past experience, successful system designs are obtained. It is proposed that a similar rule-ofthumb criterion be used to determine the degree of stability of contactor servomechanisms. Figures 15 and 16 show how this might be done. Figure 15 contains the two loci shown previously in Figure 13. However, the inactive-zone range has been increased to a value of A = 0.4. Under this condition, the system will possess absolute stability, as shown by the fact that the loci do not intersect. T h e peak ratio of output-to-input amplitudes M p may be determined for a given control-signal amplitude ]c1/ by graphical means now to be described. In Figure 15 the quantity g , is represented by the vector QO and the quantity gF1(ju) by the vector OP. The sum appearing in the denominator of 23 will therefore be represented by the vector sum (QO4-OP) or by the connecting vector QP. The output-input ratio is therefore expressed by the vector ratio ## The amplitude ratio or value of merely length of QO vector M is (25) For a given value of lcll the vector is constant. Therefore M p length or the peak value of Mover the frequency range is given by I Q O] ## length of QO vector minimum length of QP vector (26) The minimum length of the QP vector is found by drawing an arc of shortest possible radius, tangent to the frequencyresponse locus from the point Q as a center. The frequency u at the point of tangency of the QP,in vector with the frequency-variant locus is designated as the peak frequency u,. The values of M p and u p determined over the entire range of control-signal amplitude lcll provide a complete picture of degree of stability. This is shown in Figure 16. For the case considered in the examples, the values of M p and u, increase AIEE. TRANSACTIONS Kochenburger-Synthesizing Contactor Servmnechunisms 158 with a decrease in conr.ro1-signal amplitude. This indicates that during the beginning of a correction process involving high control-signal amplitudes, the system would act as though it possessed adequate response damping and had a relatively low natural frequency. On the other hand, as the correction process approached completion, the degree of stability would become poor and the frequency associated with the transient oscillations would increase. This is a common characteristic of uncompensated contactor servomechanism^,. When an a?tual servomechanism having the properties represented by Figures 15 and 16 was subjected to a large disturbance, the initial correction process appeared well damped but slow with a1 moderate initial overshoot, but the final portion of the correction process appeared poorly damped but more rapid. Eight posil.ive and eight negative corrective impulses were initiated by the contactor before the correction process ceased and the error finally remained within the inactive zone. The foregoing example represents a servomechanism possessing a low degree of stability, as evidenced by the value of M p approaching 4.5 before the at-off point is reached. It has been foundl from experimental tests on various types of contactor systems that an M , of 2.0 near the cut-off point and of 1.3 for higher control-signal amplitudes provides a satisfactory degree of relative stability for most applications. In the example shown, the degree of stability was poor in spite of the large inactive zone. I t is 01)vious that compensating networks are desirable to improve the response in even the most elementary types of contactor servomechanisms. dicate that the control-signal amplitude of oscillation (which also equals the error and output amplitudes since no compensation is assumed) is given by lc1l=0.040 at a measured frequency u=4.30. The discrepancy between calculated values and test values is only 12 and 10 per cent, respectively. A compensating network of the form shown in Figure 6 ( A ) was employed to improve stability. An attenuation factor a = 10 was specified since this represents a practical compromise between the problem o obtaining a large phase lead f and that of minimizing the amplification of noise signals. Plotting loci curves corresponding to various trial values of the time constant ratio pa, given in equation 6 A shows that the best stability characteristics are obtained when pa= 0.3. The dimensionless transfer function of the compensating network is then loci diagram, but also demonstrated that the addition of the compensating network resulted in a spectacular'improvement in the transient response. I t is interesting to note that the optimum value of the numerator time constant 7 , = 0.69 seconds selected on the basis of the loci diagram was also the optimum when experimentally selected by trial-and-error means. ## Application of a Resonant Phase-Lead Type of Compensating Network A contactor servomechanism possessing more complicated dynamic characteristics has also been investigated. The servomotor transfer function corresponding to equation 3 was W j w )= 30 jo(1+2.3j~)f degrees ## Application of a Simple Phase-Lead Type Compensating Network The example in the preceding section involved a low degree of stability in spite of the large inactive zone employed ( A= 0.4). In most piractical applications, a smaller inactive zone would be specified because of accuracy considerations and a typical value might be given by: A = 0.05. In that case, the amplitude locus would appear as in Figure 17. The uncompensated frequency locus is the same as in the preceding example. IBecause of the decrease in A, the loci intersect and self-sustained oscillations occur. The point of intersection indicates that these oscillations involve a control-signal amplitude l c ~ l =0.74A =0.35, at a fiequency u =4.70. Experimental tests in- The compensated frequency locus shown in Figure 17 is based on the network of equation 27. The frequency and amplitude loci no longer intersect and self-sustained oscillations will not occur. Furthermore, if the rule-of-thumb criterion for degree of stability is used as described previously, the value of M , is a maximum The uncompensated frequency locus when the control-signal amplitude Icll in Figure 18 corresponds to the loop is just at the cut-off point and is given by transfer function of the servomechanism Mp,,= 1.76. In view of the previous discussion on degree of stability, the corn.. without the compensating network. The amplitude loci curve of Figure 18 correpensated system should be adequately sponds to a contactor inactive-zone range stable. An experimental version of this servo- of A=0.07 and a hysteresis effect given by h=0.03 A=0.021. This ratio of h to mechanism confirmed the results preA represents a typical practical minidicted above. The test equipment involved a runaway velocity R of 30 de- mum for many applications. The two ' loci intersect a t a value of control-signal grees per second and a time constant I = 9.5A = 0.66 which is relamplitude of 2.3 seconds, giving a signal base of the atively large. This would correspond to dimensionless notation as B = RT = 69 deself-sustained oscillations with an error grees. The actual inactive zone was 3.5 degrees (or A = C ~ / B = 3 . 5 / 6 9 = 0 . 0 5 ) . amplitude of 46 degrees. I t would be necessary to increase the inactive-zone The compensating network had a numerrange to an impractically high value ator time constant T , of equation 6 given by r,=patb=paT=0.3 X 2.30=0.69 sec- A = 0.63 or a static error range of * BA/2 = 69XO.63/2 = 22 degrees in order onds. The electrical constants for the to suppress these oscillations. The need network of Figure 6 ( A ) are &=4,000 for a compensating network is obvious. ohms, RI 1.45 megohms, R2= 157,000 = Trial constructions of loci indicated ohms, and Cl = 1.0 microfarad. When this that suitable compensation was not obcompensating network was used, a sudden tainable with a simple lead controller of input change of 360 degrees resulted in an the type shown in Figure 6 ( A ) . A resoinitial error overshoot of 15 degrees and nant lead network similar to that shown then an undershoot of two degrees. The in Figure 6(B) was therefore used.* The error then remained within the inactivezone range. Such a response would be h he actual compensating network possessed the considered to have adequate degree of same characteristics as that of Figure 6(B); however, a circuit arrangement employing electronic stability for most applications. feedback was used. This was done in order to avoid the use of the impractically high values of This experimental test not only coninductance and capacitance which would otherwise firmed the conclusions reached from the have been required. The runaway velocity was R = 30 degrees per second as before. The basic time constant was again T z 2 . 3 0 seconds. However, the transfer-function term involving the time constant is now second order and the servomotor response is critically damped. The dimensionless transfer function of the servomotor then is reduced to 1950, VOLUME 69 Kochenburger-Synthesizing Contactor Servomechanisms characteristics were selected by trial-anderror, graphical constructions of the frequency locus being employed. I t was found that a resonant lead controller having a transfer function of the form stated in equations 7 and 7A provided the best response with the following constants wl = - =0.87tb ul ## radians u2 radians @=-=2.76 second la second w2 attenuation constant -= 10 w12 With a compensating network of this type the frequency-response curve is that described as the compensated-frequency locus in Figure 18. Since no intersections of the amplitude and frequency loci occur, the system is stable. The maximum value of M p is determined graphiThis value is somecally as Mp,,=2.5. what higher than that specified previously but indicates that a reasonable degree of relative stability might be expected. Experimental tests incorporating this compensating network with a critically damped servomotor having a transfer function of the form given in equation 29 were conducted. The transient response appeared satisfactory. A sudden input change of 360 degrees resulted in an initial error overshoot of 27 degrees, a subsequent undershoot of 13 degrees, an overshoot of seven degrees, and then an undershoot of two degrees before the correction process was completed. The total correction process described involved a time interval of 37 seconds; half of this time involved the initial correction process prior to the first overshoot when the speed of response is limited by the value of runaway velocity. In view of the inherently slow response of the servomotor (purposely made so to provide ease of measurement), the performance may be considered satisfactory. The results described above corresponded to an inactive-zone range of 4.6 degrees (A= 0.07). From Figure 18, a reduction in the inactive-zone range to a value given by & A =2.5 degrees ( A=0.038) would be just sufficient to cause the two loci curves to intersect and self-sustained oscillations to result. This change in inactive zone would be represented by a simple increase in the scale of the amplitude loci curve by the factor 0.07/0.038. The loci curves indicate that these oscillations would have a frequency given by u=5.9 or w=utb=2.56 radians per second or a period of 27r/w = 2.45 seconds, and a control-signal amplitude given by 11 = 0.027. 61 The corresponding dimenable design. The simulator was designed sional error amplitude would be so that the inactive-zone range and hys~lc1l/~,(j5.9) =0.76 degrees. teresis range could be adjusted, so that Actual tests indicated that i t was the servomotor transfer function could necessary to reduce the inactive-zone assume varied forms including those inrange to a value of & A = 2.7 degrees bevolving finite time delays, and so that fore self-sustained oscillations could ocvarious compensating networks could be cur. This is within 8 per cent of the preintroduced. The details of construction dicted value of 2.5 degrees. The obof this device and that of the electronicserved period of oscillation was 2.7 feedback type resonant-lead controller seconds, or 10 per cent longer than the mentioned in the preceding section are predicted value of 2.45 seconds, and the not discussed here since they are not error amplitude was 3/4 degree or equal directly pertinent to the subject matter. to the predicted value of 0.76 degrees as The instances given where the results closely as could be determined in view of predicted by t h e theory were compared the accuracy of measurement. with experimentally observed data repThe condition of self-sustained oscilla- resent only several examples of a large tions obtained with the smaller inactive number of such tests made for various zone might be acceptable for some aptypes of contactor servomechanism probplications of contactor servomechanism^. lems. In general, the percentage agreed n the other hand if the application's ment was the same as shown in these exspecifications prohibit such osciIIations, amples. I t was considered desirable that an 80 per cent increase in inactive-zone the results obtained by the simulator be range would provide the reasonably wellof a form which could be visually recorded damped transient response described preso that electrical recording instruments viously. I t is to be noted that the error would not generally be required. For ampIitude associated with self-sustained this reason, the relatively slow runaway oscillations can be considerably smaller velocity of 30 degrees per second and the than the inactive-zone range when a basic time constant of 2.30 seconds were phase-lead type compensating network is chosen. By employing proper scale employed. factors, faster servomechanisms may still The effectiveness of the compensating be represented by the simulator. The purposely slow response of the simulator network is best demonstrated by the fact should be kept in mind when evaluating that, in the case of the uncompensated the results described in the preceding secsystem, an inactive-zone range given by A =0.63 was required to prevent the octions; the dimensional performance figures given are not necessarily typical. currence of excessive self-sustained oscillations. With the addition of the compensating network an inactive-zone range Conclusions of A=0.038 was sufficient to prevent oscillation. This is a permissible improveThe investigation described in this ment in static accuracy by a factor of paper leads to the foIIowing basic conseventeen. clusions : I&]= ## Experimental Verification of the Theory Theexamplespreviously mentioned represented several cases where the predicted performance was compared with that obtained by actual experimental tests. These tests were performed on a model of a contactor servomechanism designed to simulate actual systems. This simulator employed a shunt-type d-c motor as the servomotor and operated essentially .as shown by Figure 3. The actual circuit details were, however, considerably more complicated than shown in that diagram. Various adjustable dynamic characteristics and contactor characteristics were provided and extraneous factors which might lead to incorrect results were minimized by suit- A frequency-response method of analysis and synthesis may be adapted to contactor servomechanisms by employing a simple approximation. This method is usually capable o providing sufficiently accurate f results for most engineering applications. The performance of contactor servomechanisms can be materially improved, in even the simplest cases, by the introduction of suitable compensating networks. The proposed frequency-response method of treatment is particularly adaptable to the selection of such networks. The approximate method of treatment used here in application to contactor servomechanisms also is generally applicable to the analysis and synthesis of other nonlinear systems. It might prove of particular value if the nonlinear effects, such as backlash, coulomb friction, saturation, and so forth, appearing in essentially linear servomechanisms were treated by this means. I t is hoped that the technique introduced here will contribute to the more effective employment of contactor servomechanisms in control applications. Kochenburger-Synthesizing ## snisms Contactor Servomech~ AIEE TRANSACTIONS 160 References R. L 1. THEORY OF SERVOME~:~IANISMS. . Hazen. ~ ~ pranklin ~~~~i~~~~(philadelphin, pa,), l ~ ~ ~ ~ volume 218, number : ,september 1934, pages 279! 330. 2. J. W. Edwards ## D A. Kahn. A I E E Transactions, volume 68, part . 11,1949, pages 1079-88. 4. ,ANALYSIS RELAY OF SERVOMECAANISMS, H. K. Weiss. Journal of the Aeronautical Sciences (New York, N. Y.), volume 13, July 1946. ## 7. PRINCIPLES OF SERVOMEC~IANISMS (book), G . S. Brown, Dam Campbell. John Wiley and P . Sons, New York, N . Y., 1948. THEORY OSCILLATION OF (book), A. A. Andronow, C. E. Chaikii. Moscow, 1937. English Edition edited by Solomon Lefschetz. Princeton University Press, Princeton, N . J., 1949. 3. INTRODUCTION TO NON-LINEAR MECHANICS 5. FUNDAMENTAL THEORY SERVOMECHANISMS 08 (book), L. A. MacColl. D. Van Nostrand Company, N~~ yo+, N, Y,, ,.hapeer I-VII, 1945. pany, New York, N . Y., appendix-Study of a AND SYNTHESIS LINEARSERVOOF 9. ANALYSIS Simple On-Off Servomechanism, 1945. MECHANISMS (book), A. C. Hall. Technology 6 AN ANALYSIS RELAYSERVOMECHANI~MS, OF Press, Cambridge, Mass., 1943. ~iO~,N~~~,"C~~~~,O Discussion George A. Philbrick (George A. Philbrick Researches, Boston, Mass ): Since many readers of this able and intelligent paper will not previously have come across its author, it is appropriate to mention an earl~er, and unpublished, contribution by him. While working with the Propeller Division of Curtiss-Wright Corpol-ation prior to 1946, Kochenburger prepared a skillful analysis of a compound automatic control problem in turbo-prop regulation. This analysis was happily received and applied in a related project by the present writer The paper under discussion treats a fundamentally nonlininar servomrchanism, which it appears to handle successfully by a harmonic method I t will thus give great comfort to frequency enthusiasts. If some of the comments which follow are a t best dispassionate, attribut-e them to the fact that this writer is an enthusiast from another camp altogether Items which it is felt belong in f he list of references include a paper by A. Ivanoff;' and the recently translated book, Dynamics of Automatic Control, by Oldenbourg and Sartoriu~.~ Ivanoff studied the combined influences of inertness and static friction, under sinusoidal performance, in a manner related to the Kochenburger approach; the compensatory effects of inertness, for example, were clearly shown by him. In the Oldenbourg book, which contains many pioneering elements, similar nonlinear phenomena also are introduced into otherwise linear control loops, and the stability and performance evaluated through consideration of the behavior of the fundam~rnt frea1 quency, as is done in the present paper. A recent noteworthy paper in this same field was prepared by. IF. A Rogers of the University of C a l i f ~ r n i a . ~ A more serious omission, from the writer's point of view, is the absence of any reference to the method of analogy as a tool of analysis and synthesis in the study of sys- Ralph J. Kochenburger: The author appreciates the constructive comments made by Mr. Philbrick in his discussion of this paper and wishes to thank him for calling attention to an earlier unpublished work in the field of feedback control systems. Several interesting points were raised in his discussion which warrant further clarification. Mr. Philbrick classifies this paper as belonging to the "camp of frequency-response enthusiasts." There has been an unfortunate tendency to group engineers studying feedback control systems into two groups those interested in the transient response of such systems and those interested in the frequency response. Such a distinction is fallacious since all of these engineers are interested in the transient response of the system they are studying. It however happens that, in complicated feedback control systems, the operational calculus necessary for a direct approach involves computations which become unduly cumbersome and the seemingly "round-about" method of first studying the response of the system to sinusoidal disturbances and then predicting the transient response from these results represents a more practical approach. This is particularly the case when the system contains components whose input-output relationships are not readily subject t o analytic representations but which can be described in terms of experimentally derived data. I t might be kept in mind that the Nyquist criteria, upon which the frequency response 1. THEORETICAL FOUNDATIONS THE AUTOOF method is based, are no more than means for MATIC REGULATION TEMPERATURE, Ivanoff. OF A. determining whether any objectionable Journal, Institute of Fuel (London. England), terms exist in the transient response relaFebruary 1934. tionship. I n its most frequently applied OF CONTROL (book) 2. DYNAMICS AUTOMATIC form, the locus of the transfer function is Oldenbourg, Sartorius. Translated by Dr. H..' L . plotted for various values of "real frequency," that is, the substitution is made that s = j w where s is the Laplace transfor-.LIMIT 1NDICATC)R RESPONSES mation operator. This form permits the question to be answered-is the system stable? I t is however possible to use this IN STEP-RESPONSE general method to determine how stable the TYPICAL FOR Gs system is as well. If it is not only specified COMPONENT GC ~l~l~;-l~l g,L that it be stable but that its response in(KEY) volve no oscillatory components with damping ratios less that some specified value, it Y L 7 A is merely necessary to draw a new transfer ADDER LAG ADOER COEFF. ADDER SERVO-RELAY LAG + locus using the substitution, s = IAollLollaolicollAqIs~ BOUNDS 0 1 j l r w over the complete range of fred-2, ~ ~ cAI c i(+ll/ h Figure 1. Anaquencies, w,, and then to apply the same ca 4 ((-1) 4 (-1) graphical test as before. If the criteria are o o log computing fulfilled for such a complex frequency locus, -.,' D 'INTERNAL assembly for the ::'SIGNAL CABLES -INTEGRATOR contactor S ~ N O then all damping ratios involved in the freI _ J mechanism sponse function will be greater than the tems of this general type. The current availability of analogue computing components, as a concrete fact, requires the qualification, even the revision, of many of the statements in the paper. Such components are regularly employed in our laboratory, and elsewhere, to represent regulatory systems of which those described in this paper form a special case. In a few minutes after receipt of this paper the first example described was set up, operated, and the conclusions corroborated quantitatively through oscilloscopedisplays. The computing assemblage is shown in Figure 1 of the discussion. Straightforward additions l~ead to the second and slightly more involved example. All the components exhibited are perfectly standard catalog items. An interesting outcome of our brief experience with the analog representation of this problem was the observation that substantially the same compensating characteristic is optimum when the nonlinear contactorunit is replaced by the simplest linear approximation. AS to the form of the time responses themselves, which are important for many applications, all will agree that these do not come readily from any frequency method. The analog of course, on the other hand, gives time responses directly for either discrete or random stimuli, and frequency responses as well for those vvho want them. In a more intricate nonlinear control or servo system, of which perhaps the mechanism discussed is a minor component, the applicability of the author's approach may be open to some additional doubt, whereas the abilities of the analog are inherently unaffected by the extent or complexity of the primary structure under study. Mason. American Society of Mechanical Engineers (New York, N. Y.), February 1948. 3. RELAY SERVOYECHANISMS. Rogers. Paper T. A. 50-S-13, Atnericaa Sociely of Mechanical Engineers ## (New York, N . Y.), 1950. OUT iwA0 - (tl)q] e., r, r~,, Kochenburger-Synthesizing Contactor Servomechanisms specified value 1 Sometimes this second . procedure is tedious and for this reason the engineer frequently determines the response function for the range of real frequencies only and then employs various well known rules-of-thumb to make approximate predictions covering the nature of the transient response. Mr. Philbrick mentions two earlier contributions where other problems in nonlinear dynamics employ a similar approximation by which all but the average and fundamental frequency components of a nonsinusoidal periodic function are ignored. Such an approximation method has been used frequently and is generally well-known. The author consequently did not cite all of the contributions where such a procedure is employed. The paper under discussion merely suggests a combination of this procedure and of the Nyquist method which is particularly applicable t o nonlinear servomechanisms. The reference given in the discussion to a recent paper by T. A. Rogers is of particular interest. The author has not yet had the opportunity to examine this contribution. The discussion points out the omission of the analogue method for studying feedback control systems. The author agrees that this method should have been accorded more emphasis since it constitutes a prac- tical means of attack in many instances. As a matter of fact, mention was made in the paper of a simulator or analogue device employed by the author to check the validity of the proposed method in the case of several specific problems. It is felt that the analytic and analogue approaches complement each other and that there is need for both in control system design. The analytic approach has an advantage in that it not only determines the nature of the response itself, but also shows which design factors are the cause of its various features. It consequently gives direct information leading to the synthesis of an improved system. Analogue methods show the nature of the system response but the synthesis problem still involves a trial-and-error variation of design constants. Furthermore, the experimental facilities involved in such an approach may be expensive. Their use may well be justified in a laboratory dealing with a number of related control problems. In the case of industrial concerns which encounter problems in feedback control systems a t only infrequent intervals, there is some question regarding their economic justification. It is appreciated that Mr. Philbrick used his analogue computor to check some of the results presented in the paper. More quantitative information concerning the correla- tion of the two methods would have been of interest. The discussion mentions that a prediction of the time response does not come readily from a frequency response method. Methods by which the general nature of the time response can be predicted from a knowledge of the frequency response have been mentioned. If an actual picture of the time response is desired, this too may be obtained by a method developed by Dr. George W. Floyd and described in Chapter 11 of reference 1 of the paper. In regard to elements of doubt regarding the applicability of the proposed method to more involved systems, it is true that, in view of space limitation, the paper discussed only the application to a basic single loop system. An extension of this method applicable to a multiloop system should follow directly and also should be valid provided the Nyquist criteria are correctly applied. The simple form of these criteria given in the paper holds only for single-loop systems with stable components. In any case, some check should be made in any given problem concerning the validity of the approximation upon which this method is based. In conclusion, the author wishes to thank Mr. Philbrick for his discussion and for the number of interesting suggestions he has made. ## Xochcnburger-Syntlzesizing Contactor Semmnechanisrns 1 62 AIEE TRANSACTIONS ## ABSOLUTE S T A B I L I T Y OF NONLINEAR S Y S T E M S AUTOMATIC: CONTROIL V OF . M . P o p o v (Buclharest) Translated from Avtomatika i Telemekhanika, Vol, 22, No. 8, pp. 961-979. August, 1961 Original article sublmitted January 17, 1961 The problem of absolute stability of an "indirect control" system with a single nonlinearity is investigated by. using a method which differs from the second method of Lyapunov. The main condition of the obtained criterion of absolute stability is expressed in terms of the transfer functioh of the system linear part. It is also shown that by forming the standard Lyapunov function -"a quadratic form plus the integral of the nonlinearity" it is not possible in the case considered here to obtain a wider stability domain than the one obtained from the presented criterion. Graphical criteria of absolute continuity arealso given by means of the phase-amplitude characteristic or by what is known as the modified phase-amplitude characteristic" of the system linear part. In the present paper Ithe absolute stability is investigated of nonlinear systems of "indirect control." The existing literature in this field (see [I] for example) deals exclusively with a direct application of Lyapunov'smethod. In this paper the solution is obtained by a different method, and this enables one to get new results. It is assumed that the reader is not familiar with the author's previous publications. Therefore, not only the most generalresults are given here but also a very simple example shall be considered. By using a new method the author has also investigated the absolute stability of other types of systems of differential equations (for exarnple,of the system of "direct control"), as well a.s of other classes of nonlinear functions (for example,of functions whose graph is contained within a sector). In all these cases the absolute stability of the system with several nonlinearities is also studied (the case of systems with many controlling devices). In his most recent papers now in press, the author has studied the stability in certain critical cases and also the stability of systems of differential equations with an "aftereffect". 1. S t a t e m e n t o f t h e P r o b l e m Systems of "indirect control" are considered which can be described by the following system of differential equations: ## and also the inequality introduced in [6]alwa svanishes. It should just be mentioned that if a \$0, the trivial solution of thesytem (1.1)-(l.S)cannot be asymptotical!y stable. Reprinted with permission from Automat. Remote Contr. (USSR), vol. 22, pp. 857-875, Feb. 1962. Copyright O 1962 Plenum Publishing Corporation. ## The system (1.1)-(1.3) admits the trivial solution whose stability is under investigation. It is assumed that the trivial solution of the linear system with constant coefficients where a l k are the same as in (l.l),is asymptotically stable, or (which is equivalent) that all real parts of the eigenvalues of the matrix (a1 k) are negative. The conditions are being sought which would be satisfied by the quantities a 1 k,b 1 ,c l , and y in order that the trivial solution of the system (1.1)-(1.3) be asymptotically stable, whatever the function q (a)of the class A (in other words, a condition of asymptotic absolute stability of the trivial solution). As we know to achieve this,it is necessarty that* y> 0. We shall therefore consider in the sequel the inequality (1.8) to be satisfied. 2.Introductory Definitions. F o r m u l a t i o n of C r i t e r i o n of A b s o l u t e S t a b i l i t y n (t) dt 2 a[&." k=:1 ## l (m= i1, ,22,. .... n = ..., n together with the initial conditions Slm(0)= 6lrn(l =1,2, where 6lm = 0 when 1 The functions W 1 # .. . . n ; m = l , 2 , . . .,n), (2.2) m, a n d 6 1 m = 1 when 2 = m. (t) form the fundamental system of solutions for the system (1.7). which satisfies the initial conditions xl (0) Let x1 (t), I, (t) be the solution of the equations (1.1)-(1.3). = xl, , I, (0) = I,, and let cp [o (t)] be a function of t obtained by substituting the function a (t) = clxl (t)- r (1) g ## It follows from (1.3) that (t) = 0 is not satisfied for I1 y < 0, the trivial solution is unstable when cp (a) = ha, h>O.If y = 0, the condition all solutions of the system. But their uniqueness is not The existence of solutions is a consequence of the assumptions made i n section 1. assumed in the sequel. A solutioncan always be extended inview of the conditions for stability as formulated below. -- where In agreement with our assumptions the trivial solution of the system (1.7) is asymptotically stable, and therefore two positive constants & and lKl can be found such that [see (2.1) and (2.2)] for all t 1 0 the inequality ## takes place. Therefore, for v (t) [see (2.6)] the inequality ( v (t)( is valid where <K,exCaf (t 01, We note that in view of the inequality (2.7), the Fourier transform of the function v (t) given by the formula ## must exist. We introduce the function which shall be called the transfer function of the system linear part (see Appendix 1). Now the following theorem can be stated. Theorem. If a nonnegative quantityqexists such that for all r e a l w the inequality -(or ImX) denotes the re.al (or,respectively, imaginary) part of a complex quantity X . * Rex takes place. then the trivial solution of the system (1.1)-(1.3) is asymptotically absolutely stable provided the assumptions made in Section 1 remain valid. 3. P r o o f o f t h e T h e o r e m i n S e c t i o n 2 Consider again the solution x l (t). (t), a (t) of the system 0.1)-(1.3) st well as the function responding to it. For any positive quantity T wedefine auxiliary functions r) (a (I )). cor- where -is the quantity occurring in the theorem. and v (t) is given by the formula (2.6); it follows that [see (2.211 q I t can easily be seen that the function XT (t) is bounded for all 0 Appendix 1) ## T. When t > T , the inequality (see I Ar ( t ) 1 < K,e-K.1 (KJ >0, t >T), takes place in which Kp is independent of t. This guarantees the existence of the Fourier transform ## There exists also the transform PT ( j w ) =~ 0 (t) dt.- (3.6) In view of (2.6). (2.l),and (2.7) the Fourier transform of the function dv (t)/ dt exists and can be written as [see (2.911 i 0 dv -dt = (I) dl ~ W Nj w ) ( - v (o), Therefore. by taking the Fourier transform of (3.2) [see (3.5). (3.61,and (3.7)] we obtain ## The following function of T is introduced: The Parseval formula can be applied in (3.9) because the function q p (t) is continuous for 0 5 t 5 T and it ~ t vanishes for t > T, and the function XT (t) is also continuous for 0 :5 T, and for t 7 T the inequality (3.4) takes place. We now obtain m where FT (W) is the conjugate complex of FT(jw). B replaci~ng (jw) by (3.8) in (3.10), and by taking into account that y LT we obtain p (TI 5 0 . ## B substitu1:ing (3.1) intol (3.9) we obtain y ~ B substituting the expression for r p (t) into (3.2) and by applying the formula (2.4) as well as the formula y ## B substituting this expression into (3.16) we obtain y Each term of the expression (3.19) shall be considered in turn. For the first term we have the inequality [see ( . ) 14 and ( . ) 15] The second term can be rewritten with the aid of the identity T 0 \ 9 ( u (t)) where dt = P (u (T)) - P (U (0)). 14 In the case when cp (a) belongs to the class A [see ( . ) and (IS)], the condition (O)ZO (3.22) is fulfilled, with the equality only taking place when a = 0 As q is nonnegative, one has. of course, the inequality . - qF (a (T)) 0. 5 The third term can be rewritten as follows [see ( . ) 123 (3.23) = 0 0 & (t) dt &' (2') - i. : 6 ## As y > 0 [see ( . ) we have 18J - p ( T ) < 0. (3.25) As far as the remaining terms are concerned we note [see Appendix 3 that there exists a positive quantity 3 such that the inequality: I<I ## ) q.~ (t))dt 1< (a takes place. K sop 1 (6) 1 01 = I l 2, . O<C<T n). Using the identities (3.21) and (3.24) as well as the inequalities ( . 6 . one is able to rewrite the inequality 32) (3.19) as We shall show that the trivial solution of the system (1.1)-0.3) is stable. To this end we combine the inequalities (3.27), (3.203.antd (3.23) i ## TEVT) K m=1.2. ....n( 1 %I 1o<W?'E (5) I <qp (a (0))+ -. d max SUP I i 75;- (3.28) Let t be an arbitrary positive quantity. The inequality (3.28) is valid for any T > 0. In particular, (3.25) remains valid for T (t) such that tlhe inequality OsT(t)st takes place, and also the equality (3.29) Such a quantity T (t), of course, exists. It is easy to see that in view of (3.29) the inequalities ## it (T (1)) I ,<LSUPI )I E (C) ( < sup 1 5 (5) 1. K <T( KC</ take place. By comparing (3.31) and (3.30) we obtain ## When considering (3.28) for T = T (t) and by using (3.32) w e obtain The polynomial of the variable ( E (T (t))l in the lefthand side of the inequality (3.33) has real roots of opposite sign, when made equal to zero. The inequality (3.33) is equivalent to the inequality which is valid for all t > 0. One can find two positive quantities Kg and Kg [see Appendix 43 such that for all t > 0 the inequality takes place. m=1.2.. + Ka sup 06C<l 1 & ( 5 ) I. ## By applying (3.30), (3.34).and (3.35) we finally obtain Let e be an arbitrary positive quantity. There exists than a 6 >0 such that when I x l o I< 6 and 1 E 0 1 < 6, the righthand sides of the inequalities (3.36) and (3.37) are less than ~f and therefore the inequalities 1 x (t) 1 < E and 16 (t) I < c. take place. consequently, the trivial solution of the system (1.1)-(1.3) is stable according to Lyapunov. Moreover, it follows from (3.36) and (3.37) that all the solutions of the system are bounded. This implies that all the solutions of the system (1.1)-(1.3) can be extended for all t > 0. I t shall be shown now that the trivial solution is also asymptotically stable for any cp (a) of the class A. W e combine for this purpose the inequalities (3.27), (3.23). and (3.25): ## By making use of (3.36) where fi (xmo. E ~is) a specified function of the variables Taking into account the reaction (see (1.1), (1.3)] Xmo and 50. cp (a ) [in the above formula xk and 6 satisfy the inequalities (3.37) and (3.36), a is determined by the equation (1.3) and is a continuous function of a], we obtain the inequality where fi(xmo, t o ) does not depend on t. By using (3.39) and (3.41) we obtain [see Appendix 51 lim (-roo (t) = 0 Here w e make use of the fact that F (a (0)) the equality a (0) = 2 c,zlo-~go I=1 ## is a continuous function of o (0), vanishing when a (0) = 0, and also of [see (1.311. and also lim cp (a (1)) = 0. f+cu because cp(o ) is a co~itinuousfunction. From (3.43) and from the eqluations (2.3) one obtains [see Appendix 61 lim st (1) = 0 . t+m ## The relation (1.3) implies [see (3.42) and '3.44)] The theorem has thus been proved. 4. C ri e R - so m fp aK nnog nt hK i n de s u l t s tion o w w i t h t h e Results O b t a i n a b l e w i t h t h e Aid of Lyapunov F u n c - It is of interest to compare the criteria (2.11) of absolute stability with the ones which can be obtained by constructing a Lyapunov function of the kind described as a "quadratic form plus the integral of the nonlinearity" 16-81. W e shall show that the latter are included in the criteria of the theorem in section 2, that is, if for the system under investigation there exists a Lyapunov function of the above kind, then a nonnegative quantity -also exists such q that the inequality (2.11) takes place. ## By differentiating the equation (1.3) we obtain a system equivalent to the (1.1)-(1.3): It is shown in Appendix 7 that the most general form of a negative definite Lyapunov function which is of the kind described as a "quadratic form plus the integral of the nonlinearity" and which can be formed for the system (4.1) takes the form rn,Cis ## 0 satisfying the conditions: The derivative of this function which, in accordance with the system (1.4). i a'v --= 2 d( n 0) = Xlrtm L l m=1 = n (2 li=1 amrxk + k(p (p (0)) + arv ( -2 I=1 c1 2 L)- ~ 1 ~ 1 ) \$(p (z (z 1=1 k=1 alkzk + '!)I ( ) -T(p ( ) a) u) (4.4) - that d We note dr must be positive definite for all functions cp (a) of the class A, and in particular when n -. 1=1 , - vcp (a). yl ( a ) = h u , h>O. (4.5) By putting (4.5) in (4.4) w e obtain a quadratic form of real variables xl and a which must be positive definite, This implies that for all complex values of x l and a not vanishing simultaneously the inequality is valid, the quantities 51 and d being conjugate to xl and o. That this is so can be seen by putEing in (4.6) xy = ut + jvt , a = p + jv, with u1 , vz p,and LJ real; the left-hand side of (4.6) assumes the form 4 (Wo , p ) + W, (ul (vf , v)), where Wo is the quadratic form (4.4)with cp (a) = ha. Thus, if u l , vl ,p,and u do not vanish simultaneously, we have the inequality (4.6). ## The inequality (4.6) is satisfied in the particular case of 6 = -1 h' (1 = I . 2, z = M I( j w ) where M l (jw) satisfy the system of simultaneous equations . . .. 4, The system (4.8) has a uniquesolution for any real was according to our assumptions (see section 1) the matrix (al k) has no purely imaginary eigenvalues. By taking Fourier transforms of the system (2.1) including (2.2) we obtain where m F (9, By comparing (4.9) and (4.8) we obtain n (t)) = 5 0 e-jWtb ( t ) dt. M I( j m ) = 2 F (\$,m (1)) bm = -1 25~ -10 n w j ~ ( t~ bmdt. h ) q Substituting (4.7) into (4.6) and using the relations (4.8) we obtain ## The inequality (4.12) can therefore be rewritten as or [see (2.1'0)] The a b v e consideration remain valid for any real w . Thus, it has been shown that for a Lyapunov function of the considered kind to exist it iis necessary that the inequality (4.16) be valid for all h>O and for any w. If a ; r ol y >O ## [see (1.811, from (4.16) must follow the inequality Indeed. should the inequality (4.17) be invalid for some w = wo, then a positive quantity h exists such that the inequality (4.16) will not take place for w = wo. The necessary condition (4.17) is identical with the inequality (2.11) if q = B/ay>O. ## I a = 0. we obtain firom (4.3) and (4.16) the inequalities f B > 0, Re joG (jo) > 0. W e shall now considrr certain properties of the function N(jw) [see (2.9)]. The function N(jw) is a continuous function of w in view of the inequality (2.8). By the Riemann-Lebesque Lemma lim N (jo) = 0. I * ITherefore, there exists a positive number PI such that for any u we lul- ## lim joN (j,)= v (0) = - 2 c[&(. 1 =1 note that Im (M, (lo) M, (io) 4- M [(jo) ~ i E 0. ) (o ) From therelations(2.6),(2.1) and(2.7) it is u s y m find that the integral 5I 1 OD dr m l a converge. ## It follows from (4.21). (4.22) and (2.10) that Re G ( j w ) = He N I ~l-roo I -1 ## > - PI, jwlV (jw) + r = - 2 clbr 4- ?' > 0. (jw) n 1 =1 The latter is a necessary condition and can be obtained from(4.6) and (4.18) by putting x l = 0, a = l / h , a = 0. It follows from (4.19). (4.24). and from the continuity of the function G(jw) that a positive quantity Pz can be found such that for any real w the inequality Re jwG (jw)>Pz. (4.25) takes place. By multiplying the inequality (4.25) by 24/ Pz and by adding the result to the inequality (4.23). we obtain The latter implies that the inequality (2.11) is valid (as a strong inequality) when q = ~ P I / Pso. ~ Thus it has been proved that the inequality (2.11) suffices for the trivial solution of the system described in section 1 to be absolutely asymptotically stable and also to be a necessary condition for the existenec of a Lyapunov function of the considered type. Remarks. (1) In order to construct a Lyapunov function of the "quadratic form" type (that is of the kind as in (4.2) with B = 0) it is necessary that the inequality (4.17) bevalid with B = 0, that is that Re G (jw) 2 0 for any real w. The condition (4.27) is sufficient for the absolute asymptotic stability of the trivial solution of the system under investigation, as in this case the inequality (2.11) occurs for q = 0. (2) M s of the Lyapunov functions so far constructed in [8] are of the kind as in (4.2), with a = 0. That is ot why it is necessary that the inequality (419) b e valid; the latter is also sufficient for the absolute stability of the system triviai solution. The condition (4.19) can also be written for any real positive w as Irn G (jw)< 0, (4.27) (4.28) ## and from the inequality [see (&lo), (3.7). and(l.8)I. (3) It would be interesting to find the general solution of the following inverse problem: If the condition (2.11) is satisfied is it always possible to construct a Lyapunov function of the kind as in (4.2) ? For some relatively Simple cases the answer is in the affirmative. 5. V a r i o u s A n a l y t i c a n d G r a p h i c F o r m s of t h e ( 2 . 1 1 ) C r i t e r i o n The function (1 + j wq)G(jw) can be written as (1 P +iw)~'(l.4 = (io) # (see Appendix 1) with P (jw) and Q (jw) being polynomials of jw. The condition (2.11)can now be written as where R (x) is a polynomial of the v a r i a b l e s The condition ( . 1 is reduced, therefore, to a polynomial o f 2 being 21) nonnegative for x 2 0. The solution can be obtained by using standard algebraic methods. An arbitrary nonnegative parameter qappears in the criterion, which can b e selected in a suitable manner in The algebraic methods for obtaining the optimal values of -are quite straightforward. q every specific case. The following graphicall criteria of absolute asymptotic stability are of special interest in practical applications. The locus of the points (u, v) in the plane of (u, v) such that ## ... which means that the M. P A C is in a half-plane. A graphical criterion (Fig. 1). If there is a straight line situated either in the first and the thM quadrants of the (u. V) plane or it is the ordi~iace axis* , and in addition it is such that the M.P.A.C. is "on the right of this straight line, then the trivial solution of the investigated system is absolutely asymptotically stable. One may add that the M.P.A.C. is "on the right' alf this straight line if any point of the M.P.A.C. is either on the straight line or it is in the half-plane bounded by tlhis straight line and containing the point (+.a, o),. or P.A.C. ## - that in the criteria1 We note Fig. 1. Fig. 2. (4.27) or (4.28) which described the results obtainable by the usual kind of Lyapunov function (see Remarks (1) and (2), there is no arbitrary parameter. * Such a straight line obvious;ly passes through the origin. Its equation is of the form u + qv = 0 with q 0. Or, otherwise, that the inequality (5.5) takes place. Fig. 3. Fig. 4 . The ordinary phase-amplitude characteristic can also be made use of in order to obtain simplified graphical criteria of absolute stability. The following graphical criterion is obtained from the condition (4.27). Simplified graphical criterion No. 1. If all the points of the ordinary (or modified) phase-amplitude characteristic are situated "on the right" of the ordinate axis, the trival solution of the system under investigation is asymptotically absolutely stable. (Fig. 2) It is necessary that this graphical criterion be satisfied in order that a Lyapunov function of the "quadratic form" kind may exist Lee Remark (I)]. From the sufficient condition (4.28) of stability another simplified graphical criterion is obtained. Simplified grapical criterion No. 2. If, when u > O , all the points of the ordinary (or modified) P.A.C. are situated in the t h i r d or the fourth quadrant or on the negative ordinate semi-axis, then the trivial solution of the system under i,lvestigation is asymptotically absolutely stable (Fig. 3). It is necessary that this criterion be fulfilled in order that a Lyapunov function6.3)with a = 0 may exist [see Remark (2)]. We should like to mention that no simple method exists to express the general graphical criterion (Fig. 1) by means of the ordinary P.A.C. Knowing the ordinary P.A.C.,one is able to obtain the modified P. A.C. by multiplying the ordinate of each. point by the corresponding value of the variable. 6. Concluding Remarks The majority of the arguments developed in the preceding sections can be applied with practically no alterations to more general cases mentioned in the introduction,and results of similar nature are obtained. The fact that in the criteria only the transfer function of the system linear part appears, apart from the simple assumptions of section 1, seems to constitute the main characteristic of the achieved results. The latter need not be evaluated with the aid of the formula (2.9) but can be obtained by more direct methods which have been developed for linear systems of automatic control. The graphical criteria of absolute stability developed above are also applicable when nothing but its linearity and independence are known about the linear block of the system, its phase-amplitude characteristic being determined experimentally. The author wishes to express his thanks to the collective of research-workers in the field of ordinary differential equations of the Mathematical Institute of the Academy of Sciences of the Romar~i;~riNational Republic, i n particular to Professor A. Khalan,for the interest they have shown and for their valuablc rt.~narks. APPENDIX 1 The investigated system can always be represented in the form of a block diagram as in Fig. 4 where the linear block is denoted by L, the latter described by the system of equations and N representing the nonlinear block. The input and the output quantities of the linear and nonlinear blocks are related by the equations q=-0, Let zo (t) be a known function such that the integral transform z=cp(a). m fl..2) 1, n 2 (1) ( ## exists, and let L {zO (t) ) be its Laplace Let further x &t). 6,' (t), 7 ~ ' (t) be the solution of 0.1) when z = zU (t) and when the initial conditions are all and L ( 7 ~ ' nul. I t follows from the asriumptions in section 1 that L{X! (t)) exists at least for Re s 2 0, and L {5"t)} (t) ) exist for R e s > 0. Taking lLaplace transforms of 0.1) w e obtain n sL ( x ; ( t ) ) . = X a,& k=l (1l0 (x: (1)) ## f b g L{zo (t)}, Re s >, 0 (1 = = i , 2 , . . . , n). ('1) 1=1 cIL{zj)(1)) + TI,{EO (t)), RO s > 0 . ( . 5) I The system of simu~ltaneous equations 0.4) has a unique solution when Re s > 0 (see the assumptions in section 1). Now taking Laplact: transforms of the system (2.1) with the initial conditions (2.2): ## By comparing 0.4) and 0.7)we obtain where T h e function Z (s) defined by the equations (I.lO)-fl.ll) is the transfer function of the system linear block. The function G (s) i!i obtained in the form of a rational function o f 5 The above considerations are only valid when Re s > 9 we shall say nevertheless that G (s) is a trafsfer function if it is a rational function defined in the whole s-plaw and obtained as the analytic continuation of the G (s) function. ~ If should be mentioned that the function L { w l , (t)} exists when R s = 0 and that when s = jw (w real),then e { (t)) = F {qLm (t))~ [see (4.10)]. By comparing (I.ll). with (2.6) and (2.9) we see that when s = jw the function q ~ 0.11) is equal to the function of (2.9). Therefore.the function (2.10) is equal to the above defined function G (s) with s = jw. APPENDIX 2 By using (3.1) and (3.2) we obtain for t > T ## Buteee (2.6). (2.1) and (2.711 n dv (1) I ~ ( ~ ) + ~ ~ CI ~ CI P~ I + Z 22 k=l m=l lakml)2 ill Therefore. l=1 ## This gives the inequality (3.4) ,where For a specified solution of the system and for a given T, KS is a constant. Therefore the Fourier transformation (3.5) exists. APPENDIX 3 From the eauation ( . ) we obtain 12 KC<T it follows that ## But [see (2.5). @.I),and (2.711 By substituting 011.4) and (l11.5) into (III.3) and by using the inequality ## we obtain the inequality (!1.26).where APPENDIX 4 By replacing cp (a (6; )) ## From (2.1) and (2.7) it follows that IZ, ( r ) 1 g K,, n (1 ZrnO1) n n 1 +~uPIE(UIKI I C I ( ~ +a ~ 1 S e - ~ d ~ - ~ ) 4 ) . 2 2 1 ~~ m=1 J=lk=l A 0 td < -4 e-K~(l-u ## I a - ,we obtain the inequality (3.35) xhere KO Kc nKt, APPENDIX 5 The result (3.42) could be obtained by making use of a lemma published by Barbalat 191. Below we shall give a more direct proof. We assume for this purpose that (3.42) is not true. Then a positive quantity 6 and a sequence of quantities ti must exist such that 10 (ti) I > 6, t i > O , lim ti = iua + a,. ## The sequence tk can be selected such that fa-ta-l>wl\$ 8 t ~ > ~ - . When I t - - t k ] < - ; we have 5v 0 ?' NtT) (6 ( f ) ) 'a+ t 8 - a ( 1 ) dl > k=1 fM' c (o (1)) p 8 (1) d t , k- -2 tMT, -1 6 Obviously 1.er inf I 4 0 ) ) = m. ## Then we obtain from (~.S)[see (1.511 I , <IJ<Ai* b n In view of (V.6) which contradicts (3.39). and so (3.e12)is proved. APPENDIX 6 In order to prove the relation (3.44) [see (2.3)J it is sufficient, in view of (2.7). to prove that ## follows from (3.44). According to (2.7) we have By applying de 1 'Hopital rule (see. for example, [lo]) we obtain [see (3.43)] I lim (-0-3 ## and hence follows (VI.1). J~"I.c(D)I~ e--Ktl-c) (G ({)) 1 ,i< 7 lim I+OD =- lim Icp (a ( 1 ) ) 1 = 0, &.r Kol +a, APPENDIX 7 I t can easily be seen that the most general negative definite Lyapunov function of the type "a quadratic form plus the integral of the nonlinearity" can be represented as with 11 , , a,6 , fi being constants. The derivative of the V function by virtue of (4.1) assumes the form .1 - 2 dt dl.' -- n n where W (xl , a ) is given by the expression (4.4). We shall show that the equalities fl = 0 (1 = 1 , 2 . . .. , n ) ## dV/dt is positive when (P (a) = eaa, where m i s one of the integers 1, 2, , n; 91 m (t) the functions introduced in section 2. and E an arbitrary (positive or negative) quantity. The function (VII.5) is of course of the class A. By integrating (2.1) we obtain, in view of Qlm(t) [see also (2.2)]. n 03 . .. - 91, Consequently [see (VII.4): (0) = - 4, = 2; al ykm( t ) d l . /;=I ## I t is easi1.y seen that for the values of (VII.4)-(VII.6) the function i dl; = n n dV/dtcan be written as !i=l (VII .9) in which 0 (6') denotes term!: with the property From (VI1.9) using (VII.8) we obtain 1 0 (r2)] < ## e with K ,being a constant. ~ If fm ;t O,t.hen for a sufficiently small E the expression (VII.lO) has the sign of - fmc; the latter, however, is arbitrary becausic the sign of E is also arbitrary. It is therefore necessary that fm = 0 be true. As - was taken as m arbitrary we see that the eclualities (VII.3). must be tNe. The function (4.2) is negative when x l = 0 , a = ' I , cp (a) = h o , h > 0. This gives the inequality ## -awhich proves finally the relations (4.3). p<O for any h>o, LITERATURE C I T E D A. I. Lur'e and E. N. Rozenvasser. Methods of constructing Lyapunov functions of the nonlinear control systems theory, Roc. of the First International Conference on Automatic Control, Acad. Sci. USSR hess (1961). Sci. Press (1961). V. M. Popov, Criterii de stabilitate pentru sistemele neliniare de reglare automata pe utikizarea transformatei Laplace. Studii si Ce:rcetarii de Energetica. Anul. IX. No. 4, 1959. V. M. Popov, Criterii suficiente de stabilitate asimptotica in mare pentru sisternele neliniare cu mai multe organe o e executie. Studii si Cevcetari de Energetica, Anul. IX, No. 4. 1959. V. M. P o p v , Nouveaux criteriums pe stabilite por les systemes automatiques non-lineaires. Revue d'Electrotechnique e t dlEnergetique, vol. V, No. 1 , 1960. V. M. Popov, Noi criterii grafice pentru stabilitatea starii stationare a sisternelor autornati nelin~are. Studii si Cerletari de Energetica. Anul. X , No. 3, 1960. A. M. Letolv, r h e stabillity of nonlinear systems under control, Gostekl~izdat(1951). A. I. Lur'e, Some nonlinear problems in the theory of automatic control, Gostekhizdat (1951). V. A Yakt~bovich,Nonlinear differential equations of automatic control systems with a single controlling de. vice. Bull. Leningrad Ilniv., No. 7 (1960). I. Barbalat,, Systemes d'equations diffirentielles d'oscillations non linkaires. Revue de mathematigues pures e t appliqukes. Acad. R. P. R.. voi. 4.,No. 2, 1959. Miron Nicolescu. ~ n a l i z % a t e m a t i c ' j l , ~ Vol. 11. Bucaresti Editura ~ e h r t i c g 1958. , ## ON THE STABILITY O F NONLINEAR AUTOMATIC CONTROL SYSTEMS WITH LAGGING ARGUMENT V. M. P o p o v . a n d A. H a l a n a y (Bucharest) Translated from Avtomatika i Telernekhanika, Vol. 23. No. 7, pp. 849-851. July, 1962 Original article submitted January 5, 1962 The present paper contains an application of the method of V. M. Popov [I] to the problem of stability of some systems with lagging argument. W e investigate systems of the form where A and B are square matrices. Z and c are vectors, \$ and a are scalars; w e assume that A, B. I and c are con9 stants. A special case of this system was studied in [2]. W e shall assume that the equation det(A + e'%- XI)= 0 (I is the unit matrix) has roots in the half-plane ReA s-a 5 0. It is known (for example, see [3]) that in this case the solution of the homogeneous System is estimated by I z(t. v)l S ~ e - ~ " vll a < a , where v is the initial function given in [-re 01. ll ' With respect to function 9 it is assumed that it is continuous and that there exist constants that hlQI ap (a)\< h,u8. c < ## Theorem. If there exists q> 0 such that for a11 real w, then the trivial solution of system (1) is asymptotically stable [under suirable assumptions with respect to the eiganvalues of the matrix (A + e-"B- XI) and with respect to flnctton V ] . Proof. - Let x(t) be an arbitrary solution of system (1).and let a(t)=(c, x(t)). Let us define the functions Reprinted with permission from Automat. Remote Contr. (USSR), vol. 23, pp. 783-786, Jan. 1963. Corporation. 182 ## and let us consider the s,ystem du -=Au(t) dt + B u ( t - r ) + Irl,(t - r ) . ## Let w(t) be the solution of the system satisfying zero initial conditions. We have w(t)= x(t)- u(t) when -r ditions as x(t). Let 5t =T. where u(t) is the solution of system (3) satisfying the same initial con- and 00 = 0 e-ht w ( t ) dt , 0 Then 011 the ## basis of the known formulas of Fourier transform theory we can write where ~ P T the complex: conjugate of VT. is Taking into account the system of equations for w, we obtain that --b -=w rjar (A + dwr j~ 1)" B- t(PT. ## and from the conditions in the theorem it ensues that p(T)S 0. But w(t):=x(t)-u(t), and (c. w)=(c, x)-(c, u)= 0-(c, u). We obtain the inequality it follows that 1 1<TC" ## Thus, the inequality T +US ( T ) + ha\ oa ( r ) dt 0 <q 'f~ 0 (u) do + p , ~ I v ~ l s u ~ ~ I u ( t ) st< (1vll.). I ## W e have the formula of "variation of parameters" (see 143. [5]): z (t) x ( t . 0 ) z (0) + + X -? (t, a + r ) B x (a)& \ 0 (t. a) lq Ia (a -r)l da, ## where ~ ( t a ) is the solutjon o system (2) such that X(t. , f 0 1 ~ when 0 0, 0- r s t < 0, X(0, o ) = l . I. n a t ) dt<C we obtain, as in X t+w Then, by using f+m ## ( t ) = 0, and the theorem h proved. 1. 2. 3. 4. 6. LITERATURE C I T E D V. M. Popov. "On the absolute stability of nonlinear automatic control systems." Avtomatika i telemekhanika, 22. No. 8 (1961). B. S. Razumikhin, .Stability of nonlinear automatic control systems with lag." lnzhenemyi sb., 2 (1960). N. N. Krarovskii, Certain Problems in the Theory of Stability of Motion [in Russian]. Fizmatgiz (1959). R. Bellman and K. Coolre, Stability Theory and Adjoint Operators for ILinear Differential-Difference Equations. Trans. Amer. Math. Soc:. Vol. 92, No. 3 (Sept., 1950). A. Halanay, "Periodic solutions of linear systems with lag." Revue de Math. pure et appliquees. Acad. R.S.R., vol. VI, No. 1(1961). All abbreirlrtions of perlodlcals in the above blbliogtaphy ere letter-by-letter trenslitec atiarns of the abbreviationm a s given in the original Rusmian journal. Some or all of Uais periodical literature may well be available in English translation. l Sina IX(r, a) 1<&s4'('+) when t >a. ## A Geometrical Criterion for the Stability of Certain Nonlinear Nonautonomous Systems In this communication the existence of "Common Liapunov Functions" (CLF's) for certain nonlinear systems is discussed and the results interpreted geometrically to provide a simple criterion assuring the stability of such systems. For a negative feedback system with a linear time-invariant part with transfer function G(s) in the forward path and a nonlinear gain k(u, t) in the feedback path, the criterion provides a simple method of obtaining a range k < k(u, t) < & within which the system is absolutely stable. Such a system is shown in Fig. 1, and for k constant the over all where transfer function is called G~(s) (1) G (s) -. (J k (a, t) ## Fig. I-Time-varying ay~tem. and ~g = q + +(i- i)h (9) It is assumed that for constant and a the system is stable. For convenience, with % constant (k 5 & 5 6) define l(a, and t) = k(a, t) -L (2) i=f-L>o !=L-420. with R 0, i.e., R positive sernidefmite. It is clear that the P of (8) must be positive definite if Q is positive dehite, i.e., Q > 0 , as it is now assumed that the system with 1 = 0 is asymptotically stable. (With (F, h') completely observable, one can show P > 0 even with R = 0 [2].) Eq. (8)insures that the V = X'PX a Liapunov is function and (9) insures that it is a CLF, i.e., independent of 1 . Eq. (8) may be written as Q = where > T'P + PT + gfT'-'Pg. (10) (11) With the above one is assured that H(s) = G&(s)is the transfer function of a stable system. It is assumed then that this system may be represented as ## (-&I ( real). - F'), w x = Fx u + gu t)a This is equivalent to = -&(a, (3) gfPT-lg (12) ## a = h'x where H(s) = hf(sI System (3) may be represented as + (1 - 0 Re H(iw) - Ii IH(iw)12 = IlT-1glIk+Iq'T-'g-112>0. F)-'g. (4) (13) x = \$(OX where (5) Hence (13) is necessary for the simultaneous solution of (8) and (9). One can also show [2], [5] that with (F, g) completely controllable, it is also a sufficient condition. Hence there exists a CLF in the range -1 to i(or k to &) showing stability. In terms of (1) and the configuration of Fig. 1, (13) may be written as @(1) = F - zgh'. (6) or 2 ( R - k ) -. 2 (14) The concept of a CLF [I], [2] may be stated generally in terms of (5). With 1 = 0 (k = k) the system is stable and it is required to determine a Liapunov function V(x) showing stability of a constant coefficient system with -1 5 1 i If one finds a pqsitive . definite V(x) independent of 1 such that for -I 1 5 (VX) is negative definite, then V(x) is called a CLF and shows stability for k(u, t ) 5 11. ).he approach used here is to -4 1 tt) , ( i (Or construct a CLF having a quadratic form v = x'Px. I,, the der of this paper CLF wiu refer only to quadratic-type Liapunov Functions. Razumikhin [3] and Roitenberg [4] have used a similar approach in attempting to pick the elements of a P matrix, but their work fails to give a recipe for constructing a P matrix or the entire range within which I (or k) may lie. Taking the Liapunov function V = xrPx, one can write < < (15) < Condition (15) may be interpreted geometrically and forms a simple, elegant, and effective method of determining a range of total stability of time-varying systems using the frequency response plot the open-1oop time-invariant system. GeometricalIntevretation The problem of stability may be stated in a variety of ways. 1) Given a plot of l/G(iw) and k and & determine whether a , CLF exists in the range k 5 & 5 &. 2) Given Gk(iw), determine the maximum range of k, i.e., determine A = 6 - &. 3) Given a mean value of k and G,,(iw), determine the range , & A/2. For I), if G(iw) and k are specified, the maximum range A can O), (-&, 0) as diameter lies outside the curve 1/G(iw) and is tangent it at one as shorn in fig.2. ( ~ h may lie inside ~ the Curve for cases in which points on the red axis lying within the v= -x'[R + (q + h ) ( q 4- h ) ' + (1 + - IW'Ix (7) ## if P and q simultaneously satisfy FfP + PF = -Q = - [R + qq' + jb'l (8) be obtained geometrically from the fact that the circle on (-_k, Manuscript received November 26, 1963; revised February 12, 1964. The work reported in thia communication wss supported in whole or in part by the Offiee of Naval Research. Contract NONR-1866(16). Reprinted from IEEE Trans. arcuit Theory, vol. CT-1 1 ,pp. 406-408, Sept. 1964. Correspondence t Fig. 2 4 e o m e t r i c iilterpretation of (15). Fig. I-Geometric interpretation of ( 9 . 1) limit of gain ## L is specified, the range A is given by Re - - GZ >_ 0, (wreal). [: In this case the ordinate through ( l / A , 0) is tangent to 6. For 3) one i given k and G,,,,(iw), where k is interpreted as , , , a "mean" value of gain. The range on k is now from k - A/2 to k, A/2, or the variation is fAD. For this case (11) becomes (18) ## F i g :I-Geometric interpretation of (17). curve correspond to !&able gains..) 'The Nyquist stability criterion if k were constant would demand that the points ( -b, 0) and ( -k, 0) not be encircled. (In I?ig. 2 it is clear that the system is unstable for all constant gains corresponding to the range a to b.) For 2) it is assumed the G is given. Using h G-"k it follows from (15) that = A 2 0 (16) This implies that the frequency response~/2G,,,, is tangent to the unit circle at the maximum value o A and lies completely within f it as depicted in Fig. 4. This special case of (19) has been given by Bongiorno [6] without using the simplicity of the Liapunov function derivation. The conditions obtained in the various cases insure the esistence of a quadratic-type Liapunov function in the various cases and are hence sufficient, but not necessary, for absolute stability. K. S. NARENDRA Harvard University Cambridge, hiass. R. hl. GOLDWN Rice University Houston, Tex. REFEREXCES Ill It. 11. Gol\$vyn and K. S. Narendra. "Stabil~tyof certain nonl~near differentla1 equations IEEE TRANS. AUTO MA TI^ C o m ~ ovol. AC-8, pp. 381-382; ON ~. ~rtnh-ri o ~ : < . ---- "-, ""-. [ l X. S., Narendra and R. M. ~ o l d ~ ~"E\$,stence of quadratic t>-pe Liapunov 2 yn, functions for a class of nonlmear sx-stems. Intern. J . Eturrrr. Sci... to be oub... lished. (W real). (17) H. S. "On t h e application of condition (17) may be interpreted in another way. F~~ example, I31 lems," Razumchin 'Mnt. hiekh., vol. 22, pp.J~iapunor'smethod to stability probI'rir~klad.. 33Wd49; May-June, 1958. J. N. noitenberg, "On Lipunov for if & = 0 so that Gk G, (17) is the oon&tion that I/,\$ + G(s) be a 141 systems with vmtriable a method of constructingid. Mekb..funct~ons linear coefficients," PrinHad. A vol. 22, pp. 167positive real function, e.g., a driving-point impedance function. 172;March-.4pril. 1958. 5 H. E. "Liapunov functions for in automatic conation (17) may be interprekd geometrically also. 1f G ~ ( is~ ~11)control \$alman. Nntl. Amd. Sci.. vol. 49, pp.the problem of Lur'e1963. I'roc. 201-205; Febnrury. 1 1 J. J, ono on pi or no. Jr.. "An extension of the N~qnlst-Barkhausen stability cri6 given, the ordinah through ( - 1 / ~ , which is just tangent to the 0) terion to linear lumped-parameter systems with time-vary~ng elements,'' curve Gk gives the maximum A. This is shown in Fig. 3. If the upper IEEE TRANS. A n m u ~ m c ON CONTROL. .iC-8. pp. 166-170; April, 1963 VOI. E ## A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single TimeVarying Nonlinear Element It i s protyed that a codition similar to the ,Vyquist rritericm gtrarantees the staldity (in an important sense) qf a large class o!f .feedback systems conlaining a &ngle time-torying nonlinear element. In the rase c\$ prillczllCZpal interest, the rorulilion i s satisfien iJ !he I~wttsof a certain co1np1e.r-tpaltled .ftinctirm ( a ) is 1n)trderl away jrcnn a partiettlar disk located i n the complex plane, and (b) dows not encircle the disk. Thc now well-kno\vu techniques introduced by Lyapunov have led to nlany very interesting results concerning the stability of tinle-varying nonlinear feedback systenls governed by systenls of differential equations. However, these nlethods have by no nleans Icd to a definitivch theory of stability for even the sinlplest no~ltrivialtinlc-varying now linear feedback systems. The general problenl is, of caonrse,one of cbollsiderable difficulty. The unparalleled utility of the R'yquist stability criterion for singleloop, linear, time-invariant feedback systenls is directly attrihutal)lr to the fact that it is an explicit frequency-domaincondition. Thc Sycluist locus not only indicates the stability or instability of a systcnl, it prcbsents the information in such a way as to aid the designer in arriving at a suitable design. The criterion is useful even in rases in which the systcm is so complicated that a sufficiently accurate analysis ic not feasihlc, since experimental nleasurenlents can he used to construct the loop-gain locus. The primary purpose of this article is to point out that sonicxrcv.c-ntly Reprinted with permission from BellSyst Tech. J., vol. 43, PP. 1601-1608, July 1964. Copyright @ 1964 Am. Telephone and Telegraph Co. 188 ## THE BELL SYSTEM TECHNICAL JOURNAL, JUL'Y 1Y64 obtained mathenlatical results,' not involving the theory of Lyapunov, inlply that a condition sindar to, and possessing the advantages of, the Nyquist criterion guarantees the stability (in an inlportant sense) of feedback systems containing a single time-varying nonlinear elenlent..t 11. THE PIHYSICAL SYSTEM AND DEFINITION OF ~&sTABWLITY Consider the feedback system of Fig. 1. We shall restrict our discus sion thro~iglhoutto cases in which gl , j, u, and v denote real-valued measurablle functions of t defined for t 2 0. The bloclc labeled \$ is assumed to represent a memoryless timevarying (not necessarily linear) element that introduces the constraint u()= \$V(t),t], in which \$(x,t) is a function of x and t with the ## Fig. 1- Nonlinear feedback system. properties that (O,t) = 0 for t 2 0 and there exist a positive constant /3 and a real constant a such that a=(-\$ ( i t ) tho for all real x # 0. I n particular, we permit the extreme cases in which #(r,t) is either independent of t or linear in z [i.e., \$ ( x , t ) = \$(l,t)x]. The block labeled K represents the linear time-invariant portion o f the forward path. It is assumed to introduce the constraint ## in which k and gz are real-valued functions such that * The reslulb of Ref. 1relate to feedback systems containing an arbitrar finite number .of time-varying nonlinear elements, but, with the exception of t i e case discussed here, they do not admit of a simple geometric interpretation. t For results concerned with frequency-domain conditions for the global as mp totic stability ( a sense of stability that 1s different from the one considered gerej of nonlinear eystems, see, for example, Refs. 2 4 . ## STABILITY OF FEEDBACK SYSTEMS The function g2 takes into account the initial conditions a t t = 0. Our assumptions regarding K are satisfied, for example, if, as is often the case, u and v are related by a differential equation of the form ## in which the a, and the b, are constants with anr# 0, and C anan # 0 n -0 for Re[s] 2 0. However, we do not require that u and v be related by a differential equation (or by a system of differential equations). Assumption: We shall aasume throughout that the response v is well defined and satisfies the inequality for alljinite t > 0, for each initial-condition function gz that meets the conditions stated above and each input g: such that Although this assumption plays an important role in the proof of the theorem to be presented, from an engineering viewpoint it is a trivial restriction (see Ref. 5). Definition: We shall say that the feedback system of Fig. 1is "&stable" if and only if there exists a positive constant p with the property that the response v satisfies for every initial-condition function gz that meeta the conditions ststed above, and every input gl such that In particular, if the system is &atable, then the response is squam integrable whenever the input is square-integrable. It can be shown* that the mqmnse v ( t ) appmache~ zero as 1 -+ oo for any square-integrable input gl,provided that the system is &stable, ## THE BELL SYSTEM TECHNICAL JOURNAL, JULY lMU In addition, it follows at once from the Schwar~inequality that the response v ( t ) is uniformly bounded on 10, oe ) for any square-integrable input 91,provided that the .system is gZ-stable,g2(t) is unifornll~ bounded on [0, ), and (3) is satisfied. 1x1. SUFFICIENT CONDITIOh'S FOR &-STABILITY Thefeedback system of Fig. 1 i s &,-stable i f one of the following three conditions i8 odi8fied: (i) c > 0;and the locus of K(h)for - oo < w < = ( a ) lies outa side tire circle C1of radius i(al 8') centered on the real axis of t1te complez plhm at [-4(u1 /3-'),0], and ( b ) does not encircle CI (.rtae + Fig. 8 ) (ii)la = 0, and Re[K( i w ) ] > -8'for all real (iii)a < 0 , and the locus of K ( i w ) for - oo < w < oo i s contain(,(/ within the circle Cz of radius ;(@-I - &') centered on the real axis of tire c o m p h plane at [ -)( &' @-I),o] (see Fig. 3 . ) I'roof: Note first that Fig. 2 - Location of the "critical circle" Ct in the complex plane (a > 0). The feedback system is 82-stable if the locus of K ( k ) for -00 < o < mJeaoHta;fs (!I and does not encircle C, --___--/ ## STABILITY OF FEEI>RACK SYS'I'EhlS Location of the "critical circle" (!2 in the complex plane (a < 0 ) ; T~if&~back system is rZ-stableif the luous of X ( L ) for -a < w < 00 is contained within C's . + g2) and f: ## and suppose that lm I + gl(t) g2(r) < a. .lccording to the results of Ref. 1, our assumptions* imply that there * In Ref. 1 it is assumed that I' I,. 12d. < ## THk: HELL SYSTEM TECHNICAL JOURNAL, JULY 1964 exists a positive cotistant pl (which does not depend upon g~ or g2) such that ## [1 f(t) ? dl < PI Im gdtl 1 g10) + provided, that, with and w = Ini[s], ( i) 1 :(a @)K(s)# 0 for Ite[s] h 0, and (ii) f (8 - a ) max I K(iw)[l \$ ( a ~)K(iw)]-'I -m<w<m < 1. Thus the feedback systenl of Fig. 1 is g2-stable if conditions ( i ) and (ii) are satisfied. ticcording to the \\.ell-kno\vli theorc~niof complex-function theory that l~~rt~dsthe Sycluist criterion, condition (i) is satisfied if (and to only i f ) the polar plot of K(iw) for < w < r; does not erlcirclc or patis through the poitit [-2(a 8)-',0]. I t can easily he verified that eoliditioti (ii) is niet if one of the following three conditions is satisfied. ( a ) a > 0, and the locus of K(iw) for - m < w < oc lies outside the circle C of radius \$(av'- @-I) centered in the coniplex plane at : l [-f(a1' 8-'),0]. ( b ) a = 0, and Re[K(iw)] > -8' for all real w. , (c) a < 0, a t d the locus of K(iw) for m < w < a is contained within the circle ( of radius \$(B' a-') centered in the coniplex ; plane a t 1-\$(a-' B'),o]. If a > 0, the point [-2(a 8)-',0] lies on the real-axis diameter of ('1 , while if eonditio~i (h) or (c). is niet, it is impossible for the polar 8)-',0]. Therefore, t h ~ plot of K(iw) to encircle the point [-2(a coilditions of the theoreill guarantee that the feedback systeni is Z2stable. Remarks With regard to the necessity of our sufficient conditions for g,stability, consider, for example, the case in which a > 0 and suppose, for siniplicity, that v and u are related by a differential equation of the t y p ~ mentioned in Section 11. Then, a moment's reflection shows that thew exists a \$(x,t), in fact a # ( x , t ) which is independent of t and linear in x , for all finite t > 0. Onr assumption that (2) ie satiefied for all finite t that this condition is met. > 0 implies ## STABILITY OF FEEDBACK SYSTE'MS that satisfies our assulnptions and for which the feedback systenl is not Zn-~tahl(?, providtxl that for some value of w, K ( i w ) is a point on . This clearly = .O. < 0 and a ## 1V. FITRTHER PROPERTIES OF TIIE FEEIJBACK SYSTEM OF FIG. 1 I t is possitde to say nluch more ahout the properties of the feedback system on the hasis of frequency-domain information if our assumpt,ions regarding \$(x,t) cre strengthened. For example, suppose that fold 2 0 and all rcal 11# 3 - 2 , arid that one of tht: three conditions of t our theore111is met. Let g~ and 8 dellotc t ~ v o 1 rtrl~itraryinput furlctions ?uch that ## for all finite t > 0, and l,ct v and 6, rcspcctively, denote thc (assulin!d ~velldc-fil~ctl) rcspoilscs due to g, and i1 . Tflitrl if ## > 0, and the assulliptions of Section I1 arc ~llct, follo\vs* it :~nd that there exists a positive constant h (which does not depend upon such or il) that ## T H E BELL SYSTEM TECHNICAL JOURNAL, JULY 1 W Suppow now that #(x,t) satisfies (4) and is either independent of 1 or jwriodic in t with period T for each x, and that one of the three conditioals of our theorelti is met. Assun~ethat the initial-condition futlction g2(t ) approaches zero as t -+ rn ,and that the input g l ( t ) applied at t == 0 is a houllded periodic function with period T. Then it can be shoivn* that there exists a bounded periodic function p, with period T, ivhich is illdependent of gz and such that the (assumed well defined) respollsthv ( t ) approaches p ( t ) as t --+ = , provided that the conditions of Sc~cthi are met, (2) is satisfied for all finite t > 0, and I1 Observe that the conditions of (5) are satisfied if u and v are related by a differential ecltiation of the form described in Section 11. HEFEBESCES 1. Sandberg, I. W., On the e2-Boundedness of Solutions of Nonlinear Functional Equations, B.S.T.J., this issue, p. 1581. 2. Popov, V. M., Absolute Stability of Nonlinear System of Automatic Control, Avtomatika i Telemekhanika, 49, Aug., 1961, pp. 961478. 3. Kalman, It. E., Lyapunov Functions For the Problem of Lur'e in Automatic Control, Proc. Natl. Acad. Sci., 49, Feb., 1963, pp. 201-205. 4. Itekasius, 2. V., A Stability Criterion for Feedback System with One h'onlinear Element, Trans. IEEE-PTGAC, AC9, Jan., 1964, pp. 46-50. 5. Tricomi, F. G., Integral h'qiiations, Intemience Publishing, Inc., New York, 1957, p. 46. fi. Santlherg, I. W., and Bene.?, V. E., On the Properties of Nonlinear Integral S Equations That Arise in the Theory of Dynam~cal p t e m , to be published. ## A FREQUENCY C R I T E R I O N FOR ABSOLUTE PROCESS S T A B I L I T Y IN NONLINEAR A U T O M A T I C C O N T R O L S Y S T E M S B . N. N a u m o v a n d Y a . 2. T s y p k i n Moscow Translated from Avtomatika i Telemekhanika, Vol. 25, No. 6, pp. 852-867, June, 1964 A frequency criterion is formulated for absolute process stability in nonlinear automatic control systems which is similar to the usual frequency criterion for linear automatic control systems. The proposed criterion is based on a generalization of V. M. Popov's absolute equilibrium state stability condition as a case of absolute process stability in nonlinear systems. Amplitude-phase and logarithmic frequency characteristics are used to investigate absolute process stability. Methods are also presented for the synthesis of stabilizing devices which provide absolute process stability and a degree of stability not less than a given one. A large class of nonlinear automatic control systems may be reduced to a structure which is a combination of a nonlinear element NE and a linear part LP (Fig. 1). Absolute equilibrium state stability in similar nonlinear systems is understood to be asymptotic equilibrium state stability in the Lyapunov sense for any continuous characteristics of a nonlinear element belonging to a specific class and for any instantaneous disturbances. The problem of absolute equilibrium state stability in nonlinear systems, first presented in 1944 by A. I. Lur'e and V. N. Postnikov [I], was studied on the basis of the Lyapunov direct method in the papers by A . I. Lur'e [21, A. M. Letov [3], I. G. Malkin [4], E. N. Rozenvasser [5], and others. Comparatively recently, the papers of V. M. Popov [6-81 appeared, in which to solve this problem the Parseval' formula and the theory of positive-definite functions were brought in, which made it possible to express the absolute equilibrium state stability conditions by functions related to the frequency characteristic of the linear part of the system. V. M. Popov's results were refined and extended for the critical cases in the paper by V. A. Yakubovich [9]. The relation between the V. M. Popov method and the Lyapunov direct method is established in papers by V. A. Yakubovich [9, 101, E. N. Rozenvasser [ll],and R. Kalman [12]. All of these results are summarized in the book by M. A. Aizerman and F. R. Gantmakher [13J. One must emphasize, however, that in many cases absolute equilibrium state stability is not sufficient to guarantee a normal operation of the'nonlinear automatic control system for various driving and disturbing influences. Therefore i t is very important to guarantee stability of the processes initiated by various outside influences together with equilibrium state stability of the nonlinear system. The aim of the present paper is to find the conditions for which in nonlinear control systems not only will the equilibrium state but also the processes initiated by outside influences be absolutely stable. and to state these conditions in the form of a simple frequency criterion suitable for practical applications. A frequency criterion for absolute process stability reduces the absolute stability problem to the investigatiori of KmIe linearized system,obtained from the nonlinear system by replacing the nonlinear element by a linear one, on the basis of investigating the amplitude-phase or logarithmic frequency characteristics. Using logarithmic frequency characteristics makes it possible to synthesize very simply a stabilizing device which guarantees realization of the required sufficient conditions for process stability and a degree of stability not less than the given one. The examples presented illustrate the practical use of the frequency criterion for absolute Process stability and the method for synthesizing stabilizing devices. Reprinted with permission from Automat Remote Contr. (USSR), VOl. 25, PP. 765-778, June 1964. Copyright @ Corporation. 196 1. ## Nonlinear A u t o m a t i c Control System Equation Fig. 1. (t) Let us assume that a b u n d e d outside influence f is applied to the nonlinear automatic control system under consideration for t 5 to. where it is possible that t ~ s (Fig. 1). Then its integral 0 equation with respect to the: error x(t) may be represented in the form z( 1 ) =j ' 1 t ## By app1yin.g a bilateral kaplace transform to both sides, we get In Eqs. (1.1) and (1.2). ~ ( t and W (p) denote the pulse response and corresponding transfer function of the linear ) part of the .system, O(x) is the nonlinear element characteristic. Depending, on the character of the poles of the transfer function W (p), the linear part may be stable, neutral. or unstable. In order to include all these cases, let us derive an equivalent transformation of Eqs. (1.1) and (1.2). Let us represent O (x) in the form: C (x) = (&(x) D + rx. (1.3) ## B substituting (1.3) i~n y (1.2). after the elementary transformations we get where Let us assume that there exists a minimum - for which all the poles of the transfer function WB(p) (1.6) have r negative real. parts. This means that the transformed linear part of the system is stable. If we go from the equation regarding representation (1.4) to the original and if we take into account that f t r ( t ) is applied at the momentt=toS 0, we get the integral equation1 of the transferred system in the form where, as it is easy to show, if we start from relationships (1.5) and (1.6) after going over to the original: w J t ) = L-'{~'''u(P) 1, t (1 .B) I t is apparent that when f"(t) is bounded and the transformed linear part is stable it follows from (1.9) that fm(t) is also bourtded. For r = 0 we have f n ( t ) = f(t), and Eq. (1.7) becomes Eq. (1.1). The block diagram of the transformed system corresponding to Eq. (1.7) is represented in Fig. 2. The integral Eq. (1.7) describes the variation process in the error x ( t ) due to the bounded outside influence a P plied to the system at the moment of time t = to. In particular, the case when t = m corresponds to an unbounded long-acting outside influence. In this case, if there exists the unique bounded process x(t) = xe(t): then it is possible, by analogy with linear systems, to call this a forced process. In Eq. (1.10), if we substitute t X for the variable of integration A , the equation for the forced process (1.10) may also be written in the following form: xO( t ) = j,(t) 00 1% (L)atr ( t - h ) ] d l . [zO Describing the processes by using integral equations is convenient because not only are systems with lumped parameters included, but also some classes of systems with distributed parameters for fixed boundary conditions. The Deviation Equation T o investigate the stability of processes x (t), let us assume that a t the moment t = 0 the initial conditions are disturbed in the linear part of the system. Let us designate the reaction of the transformed linear part to this disturbance by f i(t) (Fig. 2). Since the transformed linear part is stable: 2. On account of the influence of these disturbances, the process x (t) for t r 0 is changed by the value L (t) and in place of Eq. (1.11) we have where g ( t ) ~0 for t < O . Subtracting Eq. (1.7) from Eq. (2.2) we get an expression for the deviation E(t) from the process being investigated: t E (t) = fi where ( t )-\wa(t 1. - h ) Y . IE ( A ) : hl ah, Y [ E ( t ) ; tl = @&(t) + E(t)l - Q ) y [ x ( t I. ) ## Taking into account ( . ) l e t us write Eq. (2 -4) in the form 23, 5 ( t ) = f i ( t ) - \uL(t ; t. - 1) Y I5 (a): A1 dA. T h e block diagram represented in Fig. 3 corresponds to the deviation equation (2.6). This block diagram differs fromthe one in Fig. 2 in that the nonlinear element with stationary characteristic 9 t , ( ~ )is replaced by the nonlinear element with a n o n ~ t ~ t f o n a r y characteristic 9 [5 (t), t] which depends on the value x ( t ) and defines the process being investigated. It follows from (2.5) that the nonstationary characteristic I I l l . . '------- ## possesses such properties that YlO; t ] = 0. (2.7) Fig. 2 3. F r e q u e n c y C r i t e r i o n for A b s o l u t e Process Stability The conditions for absolute process stability, as shown in the Supplement, may be stated in the following way. In order that the process in the nonlinear system (Fig. I), initiated by a bounded outside influence f t r ( t ) s be absolutely stable, it is sufficient for a given r for the transformed linear part to b e stable, for the linear part frequency characteristic W(j w ) to satisfy the condition Fig. 3. to belong to the and for the derivative of the nonlinear characteristic @(XI region (r + E ,k E ), i.e., where is an arbitrarily small positive value and k is a positive value satisfying inequality (3.1). For the case when the: linear part is stable, having assumed in (3.1) that r = 0, we get ReW(jo)+1/k>O (O<w<oo) Geometrically, on the plane of the frequency characteristic kW(jw) of the open linearized system obtainedfrom the original nonlinear system (Fig. 1 ) by substituting for the nonlinear element a linear amplifier with an amplification factor k, this means that kW (jw) may be situated to the right of the straight line U(w) = -1. T o formulate the frequency criterion for absolute stability in the general case when r equality (3.1). If we multiply the left side of inequality (3.1) by A = k/r > 0, we get ## * 0, let us transform in- where Let us find on the U, V plane the locus of the points which correspond to a replacement of the inequality sign in (3.5) by an equality sign If we substitute (3.6) into rhe left side of (3.5). find the real part and set it to zero, we get the equation For the desired curves [ ~ ( o ) 1 +++ 1 1 ) ] 2 + ~ 2 ( o = ,(A ) (3.7) It is easy to show that (3.7) defines a family ofcircles passing through the point -1; jO, having a radiusR= (A-1)/2 and situated to the left of the straight line U(w) = -1 (Fig. 4). T o each circle corresponds its own v a l u e ~ = k / r x l . That inequality (3.5) will be satisfied outside of the ( A-circles is easily establishzd by assuming that U w) = V(w) = 0 for k > 0, r > 0. Condition (3.5) and the condition for stability of the transformed linear part will be satisfied if the frequency characteristic kW (j w) is found outside the corresponding A- circle. For A = 1 the A-circle degenerates into the point -1, j 0 and the sufficient condition for absolute stability of the nonlinear systern becomes a necessary and sufficient condition for stability of some linear system for k = r. Fig. 4. Fig. 5. Fig. 6. For A = = the A-circle becomes a circle of infinite radius and from (3.5) (after getting rid of the indeterminate form) we get condition (3.3). Now it is pssible to make the fdlowing simple statement of the frequency criterion for absolute pmcess stabiliw in nonlinear automatic control systems. SO that the processes in nonlinear systems are absolutely stable. it is sufficient for the derivative of the non- linear element characteristic d @/dx to belong to the region (r + & ; k 6 ), where & > 0 is an arbitrarily small number. and for the oDen linear system frequency characteristic kW (j W) which satisfies the Nyquist frqquency criterion, to be found outsid; the corresponding A-circle, where A = k/r. It is apparent that for realizing the tiequency criterion for absolute process stability, not only the proceues p o r sible in the system but also the equilibrium state will be stable. In this way the frequency criterion of process stability distinguishes a class of systems in which the equilibrium state stability also depends on process stability. In this sense the character of this class of systems is similar to the character of linear systems. The conditions for absolute process stability in nonlinear continuous systems are more rigid in comparison with the conditions for equilibrium state stability. On one hand. they impose additional restrictions on the derivative of the nonlinear element characteristic. On the other hand, they correspond to the special case of the V. M. Popov condition where q = 0 171. A condition similar to condition (3.3) which defines the condition for equilibrium state stability for nonlinear systems with a nonstationary characteristic was obtained by the more rigorous means of V. A. Y a k ~ ~ b o v i c h 101 and E. N. Rozenvasser [9, [Ill. For sampled-data systems a condition similar to (3.1)was obtained in 1141. Fig. 7. The paper by V. A. Yakubovich [15], with which the authors became familiar after writing the present article, contains a rigorous proof of the aiterion for absolute process stability which includes a wider class of systems with less strict restrictions imposed on the nonlinear characteristics. 4. lv Analysis of Absolute Process S t a b i l i t y The problem of absolute process stability consists of verifythe realization of the frequency aiterion. If the frequency characteristic of the linearized system kW (j w) is given, then it is easy to determine the set of regions (r + E ; k E ) to which the derivative of the nonlinear element characteristic must belong. For this purpose it is convenient to use the network of A-circles (Fig. 4). If we are given the& values 0 0.25 r1 6:s and if we represent on this network the frequency characteristic Fig. 8. kW (jw) of the open linear system, we shall determine the pararneter A = k/r of the A-circle t o which the frequency characteristic is tangent. If we know the values o f k and A it is easy to determine the corresponding value of -. In this way we r may obtain the relationship k = Q (r). Logarithmic frequency charaicteristics are usually used in automatic conuol system calculation, since in many cases it is considerably simpler to plot them than the usual frequency characteristics kW (jw). To find the relationship k = Q (r) in this case, instead of the circle diagrams in Fig. 5 it is necessary to have the same diagrams in Cartesian coordinates on the plane L(w) = 20 log ( k (jw) 1 and cp(w), which are shown in Fig.5. ~ Then, to find the relationship k = Q (r) it is necessary: 1. To plot the logarithmic amplitude- and phase-frequency characteristics for % = kkv = 1, where k,, is theamplification factor of the linear part of the system; 2. T o re-pl.ot them on the LC u),))cpw ) plane for various - and to determine the indices of the A-curves tangent ( k to the logarithmic characteristics; 3. Knowing the value of A, to find the value of 5 corresponding to various values of = kkv. L e t us note that the networks of A-circles (Fig. 4) and A-curves (Fig. 5) for various values of A coincide with the circle diagrams used for plotting the closed system real frequency characteristics with respect to the open system amplitude-phase characteristics. Let us consider an example. Let the transfer, function of the linear part of a servo system have the form '"" = p(T8p ## + I) (TmTyp2 + Tmp + I ) ' kkv (4'1) where kv = 10; T = 0.005 sec; T, = 0.1 sec; T = 0.001 sec. g Y Let us find the limiting value El,,, and for a given kv, l e t us determine the relationship k = Q (1). Let us plot the log amplitude-frequency characteristic L( w) and phase-frequency characteristic cp (w) for k = 1 (Fig. 6) and, with respect to these, the log frequency characteristic in Cartesian coordinates on the L(w), cp (w) plane, for Ti = 1 (Fig. 7). It is convenient to plot the characteristic L(w) = f [cp(w)] on transparent paper with the same scale on the axes as for the Acurves (Fig. 5). Let us shift the curve obtained up along the axis of ordinates until it is tangent to one of the A-curves. Let us determine the value of the upward shift in decibels which matches the increase in the amplification factor k by a corresponding number of decibels. Fig. 9. It is easy to determine the amplification factor limiting "alueXlv of the region (0, klv) from the condition that L = f ( v ) is tangent to the logarithmic A-curve corresponding to A = 5 which will be equal to 1 9 dB or 8.90. Knowing the numerical value of A and kv = 10, we compute the corresponding values of k and r, with respect to which the curve k = Q (I) is plotted (Fig. 8). Synthesis of S t a b i l i z i n g Devices which P r o v i d e Absolute Process Stability In a number of cases i t may be shown that for the nonlinear system being investigated it is necessary to have the values of k and r be somewhat different from the corresponding values of 'Ti which define the sectors and regions (k E , r + E ) of the nonlinear characteristic @(x) and its derivative 4'(x), whi,ch are computed with the aid of the methods stated in the previous section. 5. In this case the logarithmic frequency characteristic represented in Cartesian coordinates will intersect the Acurve (for A = QJ)a t the points wl and w2 (Fig. 9). This means that in order to fulfill the sufficient conditions for absolute stability, i t is necessary to introduce a stabilizing device which changes the log frequency characteristic plotted for a given value o f T to the frequency band wl, w,. In the frequency band w < w, and w> wl, the sufficient conditions for process stability are satisfied. Then the log frequency characteristic shaded in Fig. 9 may be considered to be the desired one Ld = f (vd). In practice, it is more convenient to find the log frequency characteristics of the stabilizing device by using ] Ld( W! [ ' P ~ ( w ) ~ ( w ) for the log amplitude-frequency characteristic, having determined it beforehand fromLd= (cpd) (Fig. 10). Now the problem reduces to finding that stabilizer (series or shunt), for which the log amplitude-frequency characteristic of the synthesized system would coincide with Ld(w) or lie inside the shaded region (Fig. lo). After finding the logarithmic characteristic of the stabilizer it is necessary to. plot the phase-frequency characteristic or the characteristic L = f (cp) of the synthesized system and to verify that the sufficient condition for absolute Process stability is satisfied. If it is necessary to provide certain required values of and II_,then instead of considering the A-curve for A = for synthesizing the stabilizer, it is necessary to consider the A-curve for A = Al (Figs. 9, 10). Let us look a t an example. Let the transfer function of a nonlinear system for the automatic control of motor Speed have the form QJ ## = 0.1 sec; T = 0.001 sec. Fig. 10. By using the log frequency characteristic, l e t us verify that the sufficient conditions for absolute process stability are fulfilled for 5 = 100 and r = 0, rl = 9.1, and in the case when they are not fulfilled, l e t us specify the necessary stabilizing device. The log amplitude- and phase-frequency characteristics corresponding to kW (p) f o r x = 100 are shown inFig. 11, and in Fig. 5 the same log frequency characteristic L = f (cp ) is represented in Cartesian coordinates (curve 2). From studying the relative position of the curve L = f ( 9 ) and the A-curves (Fig. 5) for A = 100/0 = -, Al = 100/9.1.= 11, it follows that the necessary and sufficient conditions for stability of the linearized system having a transfer function kW (p) are satiLsfi.ed, but the sufficient conditions for absolute process stability in a nonlinear system are not satisfied (for A = -, Al = 11). In Fig. 1 are also represented the desired log amplitude-frequency Ld(w), Ldl(w) and phase-frequency charac 1 teristics cp d(w), cpdl(w) corresponding to A = -, Al = 11, in addition to the characteristics of the original system. Now it is easy to determine the log amplitude-frequency characteristic Lk( w) of the stabilizer. which has a transfer function where For A = m, TI= 0.435 sec; T2= 0 1 sec; T3= 0.005 sec; TI = 0.0016 sec. . Let us verify that the sufficient conditions for process stability are satisfied im a nonlinear system containing the stabilizer found with transfer function (5.2)and with parameters corresponding to A, = -. Let us plot the log frequency Fig. 11. characteristic of the synthesized system in Cartesian coordinates (Fig. 5, curve 3). I t is easy to see that after introducing the stabilizer, the sufficient conditions for stability for Tt = 100 and A = * are satisfied, and also, there even exists a safety factor equal to +AX = 6 5 dB. For this, k = 211.3. . One should note that for rhe stabilizer obtained the sufficient conditions for absolute process stability will also be fulfilled for k = 100 and Al = 11. Similarly, the stabilizer parameters may be found which guarantee that the sufficient conditions for process ) stability are satisfied for Al = 11 (see curve 4, Fig. 5. These parameters will be: Tl = 0.284 sec; T2 = 0.1 sec; T3= 0.005 sec; T = 0.0011 sec. , 6 . Synthesis o f S t a b i l i z i n g D e v i c e s which P r o v i d e a D e g r e e o f S t a b i l i t y n o t Less t h a n a Given One Above, a frequency criterion was formulated for absolute process stability in nonlinear systems. Let us find the sufficient condition for which, in order to be satisfied, the system will have a degree of stability not less than the given one, i.e., the process deviation will satisfy the condition I E ( t ) ( <:1ioe-."~', where Mo is a constant, 60 is the given degree of stability. If we multiply the left- and r i ht-hand sides of (2.6) by edot and the integrand by e'b 0e o b X , we get ' g, E ( t ) = fi ( t ) -- \&(t - I ) q ) I [x( I ) ;h ] d h . (6.2) where f (t) -E ( t ) e4', (6.3) (6.4) r ( t ) = h (t) eS"t i ## wa(t - I ) = y ( t - A ) esn(l-AJ, , -Y [ E ( h ) ; I ] = \r 15 ( I ) ; h] eS' (6.5) (6.6) and bo is less than t h e degree of stability of the transformed linear part. In h i s case ## lirn fr (t) = lim fi (t) egt t-Kx, = ; 0. (6.7) t If the system described by Eq. (6.2) has absolute equilibrium state stability, then t-Mo l-rcx, (6.8) Fig. 112. ## and, consequently, inequality (6.1) will b e satisfied. It is easy to, show that colndition (S.21) is satisfied, and for the function \$16 (t);t] Actually, if we substitute in (6.9) the appropriate expressions from (6.3) and (6.6) we get (S.21) ## P < ' [E (( tt));tl < I ; - r - e . ------E (6.10) If we compare the integral equation (6.2) to Eq. (2.6). we see that the only difference in them is contained in the main part, whose spectral function has the following form m ( t )e' c-J-' ~r;, ((, ;I -6,) + rrV ( j o - 6 , ~ ' - iL7 ( - 6,)) ~ Therefore, the sufficiel~t condition, for which the degree of stability of the nonlinear system .will not be less than the given one, is represented in the form Also. the degree of stability of the uansformed linear part must not be less than do, and the derivative of the must nonlinear characteristic @'(XI belong to the region (r + E ; k E ). If we use thc log frequency characteristic method and plot, as follows from (6.12). the displaced amplitude- and phase-frequency characteristics L 6 ( w ) and cp6(w), and next Lb = f (cp6 ), we may use the methods presented in the previous section for the synthesis of a stabilizer for a system having a degree of stability not less than the given one. For discrete systems, a condition similar to (6.12) is established in 1171, and for continuous systems, in 1151, both for bounded and for vanishing influences. For the latter case in 1151, more general conditions are obtained. In Fig. 11 are shown the displaced amplitude- and phase-frequency characteristics L6 ( u ) and ~ ~ ( 0 ) when 6, = 6 for the system investigated as the example in the previous section, and also the log amplitude-frequency char, acteristic of the synthesized stabilizer which provides a degree of stability not less than & = 6. The log frequency characteristics L6 = f(v 6 ) for the original (curve 1 ) and synthesized (curve 2) systems are shown in Fig. 12. The authors express their deep appreciation to V. A. Yakubovich for his discussion of the present paper and for a number of valuable observations, and also to V. I. Dimkov and N. N. Popova for having taken part in computing the examples. APPENDIX Proof o f t h e S u f f i c i e n c y C r i t e r i o n f o r A b s o l u t e P r o c e s s S t a b i l i t y The process x(t) being studied and, in particular, the forced process will be absolutely stable if the deviation t ( t ) is asymptotically stable (in the Lyapunov sense) for any disturbances satisfying the condition (2.1). and also if to any outside influencefu(t) bounded fort@ t < there corresponds a bounded process x (t). The problem of absolute process stability in automatic conuol systems (Fig. 2) amounts to the problem of absolute equilibrium state stability in the system (Fig. 3) which contains a nonstationary nonlinear element, and to h e problem of establishing the fact that x(t) is bounded for to s t <=. In order t o determine the sufficient conditions for absolute process stability, let us investigate the integral equation related to the deviation Since the transformed linear part, whose transfer function equals Wu(p), is stable, then linl w 1- (t) = 0 . If we consider that ## where 6 (t) is the delta-function, and if we compute 6 . 3 ) from (S.l) we get If we multiply both sides of (S.4) by t [ E (t);t] and integrate with respect to time from 0 to T , we have First, l e t us find the condition which is satisfied when the third term in (S.5) has a nonnegative value, i.e., For this, we shall use S. Bokhner's theorem [17], according to which. in order that for any y ( A ) and y(t) and any T > O where T(t) = cp (-t) is an even function, there is necessary and sufficient satisfaction of the inequality where 7 (ja) = Let us denote (t)= j ' -00 ( t ) e-jut dt. cP ( t ) ## +2(P (for t<O, for t) v where cp(t)=-0 cp(-t) EO t>O. Let us substitute (S.10) in (S.7), and then with regard to 6.11) and (S.1.2) we get Since the integrals in the square brackets are equal to each other, we get, 9 ( t ) = w,(t) ## 6(4 + -' k-r Y (A) =YtE(h); XI, tl, Y (t) = 'YIE(t) ; from (S.9) we get the necessary and sufficient condition for satisfying (S.6) where % ( j w ) = y w t r ( t ) e-jYf 0 dt ## is the frequency characteristic of the transformed linear part of the system. If w e now assume that condition (S.18) is satisfied and if we discard the third term in (S.5). we get the inequality Lct us take advanrage of Theorem 1 in V. M. Popov's paper [8], according to which [see Eq. (S.4)J for satisfying the condition (S.18) the value 1 E (t) (k - r)"l[k (t); r] f i ( t ) j is bounded, i.e.. ## If [ E (t); t] belongs to the sector [E ;k -r- I, i.e.. where E is an arbitrarily small positive number and t+" f 1(t) = 0, then from (S.20) i t follows that lim . bounded. I E (t) I is also h t if IC: (t)l is bounded, then i t is easy to show, by differentiating Eq. (S.1) with respect to , that ( ~ ' ( t 1 is I ) also bounded. From the boundedness of E (t)l , the boundedness of Y [C: (t); t] also follows, i.e., I* [ : ( t ) ; t ] l Cl. Therefore the right-hand side of inequality (S.19) may be evaluated in the following way: ## If we take (2.1) into account, we get C1 j I fi (t) Idt=Y,C1=C 0 m and consequently, where C > 0 is not dependent on T. If the nonstationary nonlinear characteristic [C:(t); t] belongs to the sector b; k r E ] , then the left-hand side of (S.24) will always be positive and will increase indefinitely as 1 6 (t)l increases indefinitely. If we l e t T go to infinity in (S.24) and if we use, as in C6. 7, and 131. Barbalat's lemma which also holds for this case, we conclude that for satisfying condition (S.18), the equality - - lim f ( t ) =O. t+OO occurs. If we apply Theorem I of paper [8] to the equation obtained from the process equation (1.7) by a transformation to a form similar to the one considered above, i.e., to the equation If =(t)l< ## MS the value I x(t)( is b0unded.t Thus, in order for the processes in a nonlinear system to be stable, besides inequality(S.18) being satisfied, the nonstationary characteristic [C(t); tl should belong to the sector [ E ; k r E 1. We shall express this condition by means of the stationary characteristic of the original nonlinear system. - - (t);t] ## may be represented in the form If is presupposed that l;(E(t), t)] <N, then tne tact that ( it V this case it is not necessary to include the V. M. P O ~ Otheorem. ?See the footnote on page 847 of paper [8]. ## B substituting [ y (t);t] in (S.21) add a d d i n g i to both sides of the inequality, we get I t is apparent, since this follows from the Lagrange theorem of the mean: that inequality (S.28) will Ibe satisfied for any x(t) if @(O)=O and rf e < dO(x) I d z < k--8, (S.30) i.e., if the derivative of the nonlinear element characteristic d@/dx belongs to the region (r + 6 ; k -& ). In addition, it is apparent that the characteristic @(x) itself will also belong to the sector (r + s ; k E ) . LITERATURE C I T E D A. I. LurWe, V. 1U. Postnikov, On the theory of control system stability [in Russian]. Prikl. matem. i and mekhan. 3 3 (1944). A. I. Lur'e, Some Nonlinear Problems in Automatic Control Theory [in Russian], Gostekhizdat (1951). A. M. Letov, Stability of Nonlinear Control Systems [in Russian]. Fizmatgiz (1962). I. G. Malkin, The Theory of Motion Stability [in Russian], Gostekhizdat (3952). E. N. Ro:senvasser, Some problems in the theory of nonlinear control systems [in Russian], Candidate dissertation, Gor'kovsk gos. univ. (1961). V. M. Popov, Criterii de stabilitate penuu sistemele nelineare de reglare automata bazate pe utilizarea transformatei Laplace, Stu~dii cercetari de energetica, Acad. R.P.R., arm. IX. 1 (1959). si V. M. Popov, Absolute stability of nonlinear automatic control systems [in Russian], Avtomatika i telemekhanika, 22. 8 (1961). V. M. Popov, Sur certains unbqalites intbgrales concernant la thgorie du reglage automatique, Compt. Rend., 256.17 l(1963). V. A. Yakubovich, ,4bsolute stability of nonlinear control systems in critical cases [in Russian], Avtomatika i telemekhanika, 24.3.6 (1963). V. A. ~ a k u b o i c x Absolute stability frequency conditions in nonlinear control systems with hysteresis nonlinearity [in Russian], Doklady Akad. Nauk SSSR. 149.2 (1963). E. N. Roz:envasser, Absolute stability of nonlinear systems [in Russianl, Avtomatika i telemekhanika.2, 3 (1 963). R. E. Kal~man,Liapunov functions for the problem of Lur'e in automatic control, Proc. Nat. Acad. Sci. USA, 49, 2 (1963). M. A. Aizerman and F, R. Gantmakher. Absolute Stability of Control Systems [in Russian], Izd. Akad. Nauk SSSR (1963). Ya. Z. Tsypkin. Absolute equilibrium state and process stability in nonlinear automatic sampled- data systems [in Russian]. Avtomatika i telemekhanika. 24. 1 2 (1963). V. A. Yakubovich, A matrix inequality method in the theory of nonlinear control systems. I. Absolute stability of forced vibrations [in Russian], Avtomatika i telemekhanika, 2,7 (1964). Ya. Z. Tsypkin, Basic theory of nonlinear sampled-data systems [in Russian], Reports on the Second International Congress, Internatiolial Federation on Automatic Control, Base1 (1962). C. Bokhn~er,Lectures, on Fourier Integrals [in Russianl, Fizmatgiz (1962). All abbreviations of periodicals in the above bibliography are letter-by-letter transliter ations a the abbreviations as given in the original Russian journal. Some or all of this periodical literature may well be available in English translation ## Frequency Domain Stability Criteria-Part R. W. BROCKETT, MEMBER , IEEE AND 5. L. WILLEMS Absfracf-The objective of this paper is to illustrate the limitations of the generalized Popov Theorem in establishing the stability of a loop containing a single nonlinearity, and to use the Liapunov Theory to give a new frequency domain stability criterion for such systems. The new criterion diifers from Popov's in that less restrictive assumptions are made on the linear part, and stronger assumptions are made on the nonlinearity. In this paper, it is assumed that the nonlinearity is monotone increasing. The approach used here is quite general, however, and in a companion paper various other restrictions are considered. ## If f(y) is a continuous function of y, and if 0 <yf(y) <ky2, then f will be said t o belong to the class Ak, hereafter written as f E A k . If Z(s) is any rational function of s which is real for all real s, then Z(s) is said to be positive real if (1) Re ZGw) 2 0 for all real w, (2) Z(s) has no right half-plane poles, and (3) all the imaginary axis poles of Z(s) are simple and the residue of Z(s) a t each one of them is real and positive. These two definitions are amalgamated in the following theorem. LTHOUGH A GREAT DEAL has been written about the stability of systems containing a single instantaneous nonlinearity in an otherwise linear loop, most of the results which do not depend in an essential way on the order of the equation are special cases of a rather simple theorem due, in essence, t o Popov. This result, stated in Theorem I, makes more restrictive assumptions on the linear part of the system than would seem t o be necessary on the basis of linear theory, i.e., it does not verify Aizerman's conjecture. On the other hand, the assumptions made on the nonlinearity are quite unrestrictive. The object of this paper is t o make a critical evaluation of Popov's Theorem and then t o establish a new criterion which enables one t o enlarge the class of systems which can be treated, provided it is possible t o constrain the slope of the nonlinearity. T h e system under consideration is shown in Fig. 1. Theorem 1 T h e null solution of the system shown in Fig. 1 is ASIL (asymptotically stable in the large) if f E A k and there exists an a! such that (1 +as)G(s) l / K is positive real. The original theorem of this type appearing in Popov [ I ] considered the special case where G(s) contains a pure integration and k = m . In that case a can be assumed positive without loss of generality. The idea of restricting f to belong t o the class AI,appeared, about a year later, in a paper by Popov and Halaney [2]. Rozenvasser [3], Narendra and Goldwyn [4], and Rekasius [s] also treated this case, getting essentially equivalent results. These authors made special assumptions about the nature of the imaginary axis poles and usually restricted a to be non-negative. Some work on relaxing the assumptions on the imaginary axis poles has been done by Yakubovich [6]. A complete survey of the Russian work done before 1964 can be formed in the recent monograph by Aizerman and Gantmacher [7]. These authors also remark1 that positive realness can be used in stating the conditions for stability. A proof of the theorem in the form given here appears in Willems [8]. In order t o use this theorem on a given system, one must determine if there exists an a such t h a t (l+as)G(s) + l / k is positive real. This is easily done when Popov's modified polar plot is used. I n the modified polar plot w I m G(jw) is plotted as a function of Re G(jw) instead of plotting Im GGo) vs. Re G&), a s is done in the ordinary polar (Nyquist) plot. From ## Fig. 1. Block diagram of the system under consideration. I t is assumed that p(s) and q(s) are polynomials without common factors and that the degree of p(s) exceeds t h a t of q(s). I t is also assumed that the nonlinearity f is sufficiently smooth t o insure the existence of a unique solution of the governing differential equation. Notice t h a t if x is defined by the differential equation p(D)x= y, then the differential equation governing the behavior of x is simply Thus the stability of the closed-loop system shown in Fig. 1 can be observed by studying this scalar equation. Manuscript received October 5, 1964; revised March 29, 1965. The work reported in this paper was supported in part by the National Aeronautics and Space Administration under Contract No. NsG-496 with the center for Space Research. The authors are with the Dept of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. it is seen that Re (1+oLjo)G(jw)fl/k>O for all w if, and only if, the modified polar plot lies to the right of a straight line passing through the - 1/k point having slope l / a (Fig. 2). If there is no straight line passing See [71, p 79. Reprinted from lEEE Trans. Automat. Contr., vol. AC-10, pp, 255-261, July 1965. 210 ZEEE TRANSACTIONS ON AUTOMAl T I C C O N T R O L through the - 1 / k point. such that the modified polar plot always lies to the right, then Theorem 1 cannot predict stability for all J'EAL, although it may predict stability for some smaller class of nonlinearities. Assuming that G(s) is stable, the only remaining thing t o be done, in connection with establishing the positive realness of (1 as)G(s) l / k , is to check the residues of the imaginary axis poles. I t is interesting to compare Popov's result with the usual Nyquist criterion. First, recall that the system G(s) will be stable for a n y linear feedback in Ah if, and only if, the Nyquist plot avoids the part of the negative real axis which lies t o the left of the - l / k point as shown in Fig. 3. For the special case where the Nyquist plot lies to the right of the vertical line passing through the - l/k point, i t is clear that by taking a to be zero, i t is possible t o make ( 1 +crs)G(s) l / k positive real, and in this case the generalized Popov criterion and the Nyquist criterion agree. I t is of some interest to examine the case where f E A , in more detail. The important question here is, "Under what circumstanc-es can one find an a such that (l+as)G(s) is positive real" Notice t h a t in this case a must be non-negative because positive real functions and their reciprocals are stable. T h e Nyquist plot of a positive real function lies entirely in the right half-plane. Since the effect of the factor ( l + a s ) is to rotate each point on the Nyquist plot of G ( s ) in the counterclockwise direction, i t is clear t h a t it will not be possible t o find an a such that ( 1 -kas)G(s) is positive real, if the Nyquist plot of G ( s ) enters the second quadrant. Even if the Nyquist plot of G ( s ) does not enter the second quadrant, it may be impossible to find an a such t h a t ( l + a s ) ( G ) ( s ) is positive real. If i t avoids both the second and the first quadrants, however, then one can always make Re (l+ajw)GCjw) 2 0 by taking a sufficiently large. What this means is that if 0 arg G(jw) for 0 I w a, then the null solution will be ASIL for all f E A , if, and only if, it is ASIL for all negative linear feedback. < > Before discussing ways of extending Theorem 1 it seems appropriate to examine some examples which illustrate its power and its limitations. One special form of (I), which has been examined repeatedly, is This corresponds t o a system obtained by applying nonlinear feedback around a system with no zeros. In particular, for systems with 2, 3, and 4 poles [ l l ] , y(4)+ay(3)+by(2)+cy(1)+f(y)=O; a, b, c, ab-c, >O. (6) Fig. 2. ## A modified polar plot. Fig. 3. The Nyquist plot of a system for which the Popov criterion is necessaly and sufficient. The modified polar plots of these systems are shown in Fig. 4-6. By examining the plot corresponding to (4), it is seen that it lies t o the right of a line passing through the origin having slope l / a . Therefore, from Theorem 1 it follows that the null solution of this system is ASIL for all f A,. This is a well known result. The modified polar plot of l/(s3+as2 +bs) intersects the negative rela axis a t - l / a b just as its Nyquist plot does, and thus it will certainly not be possible to take k to be greater than ab. Is it possible to use Theorem 1 to prove stability for f dab? T o show t h a t it is, it must be shown t h a t the modified polar plot lies to the right of a line passing through the - l / a b point. Clearly, if such a line exists it must have the same slope as the modified polar plot has a t that point. A short calculation shows that this slope is b / a ; with a little further calculation one can show that the modified polar plot actually lies t o the right of this line, and therefore, that the null solution of the third-order system is ASIL for all fAab. This result has also appeared before, but this explanation of why a should be b / a seems t o be new. For the fourth-order system the modified polar may take either of the shapes shown in Fig. 6. I t crosses the negative real axis a t -a2/(abc-c2) ; however, it is not always possible to find a straight line through the point such t h a t the modified polar plot is always on the right. T h e slope of the modified polar plot at the point where i t crosses the negative real axis is ac/(ab-26). The modified polar plot will lie t o the right of this line if, and only if, one imposes the additional constraint a3- (2ab - 2 6 ) > 0. Thus in the fourth-order case, unlike the second- and third-order cases, it is not always possible t o prove asymptotic stability in the large for as wide a class as would be predicted on the basis of linear theory. Brockett and Willems: Frequency Domain Stability Criteria I t is interesting to note that the conditions aa- (2ab - 2c) >0 can be given a simple interpretation in the root plane. If one replaces f ( x ) by the linear term x(abc-c2)/a2, then the system has a pair of imaginary axis poles a t w = rtjv'c/a. The remaining poles satisfy s2 An even worse situation is illustrated by the two equations ## + as + (ab - c)/a = 0. + z / b - a2/4-c/a (7) abc - c2 - a2d > 0. (10) Since the real and imaginary parts of the roots of this equation are R e s ~ = a/2; I m = (8) it follows that the inequality a3- 2(ab - c) >0 implies that the two roots with nonzero real parts must lie in a wedge bounded by two lines making a 45' angle with the real axis (Fig. 7). The null solutions of the linearized versions of both these equations are ASIL for all positive k, yet the generalized Popov cannot predict stability for f E A , , regardless of any additional assumptions placed on the coefficients. One of the objectives of this paper is to develop a method of dealing with such equations. These examples will be considered again. OF CONJECTURE IV. A REFORMULATIONTHE AIZERMAN In terms of scalar equations, the Aizerman conjecture suggests that if the null solution of the linearized equation p(D)x Fig. 4. The modified polar plot of a second-order system. + kq(D)x = 0 (11) ## Fig. 5. Modified polar plot of a third-order system. is ASIL for all kl<k <k2, then the null solution of the nonlinear equation (1) should also be ASIL if k~<f(y)/y <k2 for all y. Even though examples have been given which show that the Aizerman conjecture is not true, it remains as an upper limit which one strives for. The objective here is to reformulate this conjecture in such a way as to place in evidence the relationship between it and what the Popov Theorem proves; this will also provide the motivation for the work in later sections. I t can be seen that without loss of generality one can always assume that the linearized system is stable over the range 0 < k* <k, since if the original equation is stable over the range kl <k* <k2, then it can be trans formed into Fig. 6. Two possible forms of the modified polar plot of a fourth-order system. which is stable over the range 0 <k < kz - k ~ . Theorem 1 can be regarded as an attempt to verify Aizerman's conjecture. As such, it works in some cases and not in others; the relationship between the information it provides and the Aizerman conjecture is considerably clarified by the following theorem. Theorem 2 The null solution of (11) is stable for all k* in the range 0 <k* <k if, and only if, there exists a rational positive real function Z(s) such that Z(s) (G(s) l/k) is positive real. Proof: Assume such a Z(s) exists; write it as m(s)/n(s). Since the sum of two positive real functions is, itself, a real function, it follows that for all positive k* the quantity Z(s) (G(s) l/k) +kZ(s) is positive real. The numerator of this rational function is ## Fig. 7. Admissible pole location for Aizerrnan's conjecture to be verified. ZEEE T R A N S A C T I O N S O N AUTOMATIC: C O N T R O L Since the numerator and denominator polynomials of a In view of this, it is obvious that moving the I l k term positive real function must have their zeros in the half inside the parenthesis does not alter anything, and that plane Re s<O, we see that p(s)(l +kk) +kq(s) is a poly- taking the minus sign for an exponent of (l+as) is nomial of the stable type. By dividing this by (1 kk) , equivalent to permitting a to be negative. The main conclusion which these results lead to is it is seen that the null solution of (11) is indeed stable that Aizerman's conjecture asserts that a system should for the specified range of gains. The converse is a little more difficult. I t can be seen be stable with any f EAk if there exists any positive real that since the null solution of \$(D)x+kq(D)x=O is multiplier Z(s) such that Z(s) (G(s) l/k) is positive stable for all 0 <k* <k, it follows that the null solution of real. Theorem 1 asserts that the system will be stable with any fEAk if there exists a positive real multiplier p(D)x -t k(p(D) \$- kq(D))x = 0 (13) o the special form (1+as)'' such that f is stable for all 0 . k < a. In view of this, it follows ( that the phase plot of (p(r;)-tkq(s))/p(s) always lies between plus and minus 180"; if this were not the case, is positive rea.1. The problem is, then, one of seeking ways to generalize the class of admissable multipliers. there would be instability for some k* (Fig. 8). In Section V one such result is described. 180' arg Gtsl Fig. 8. The argument of ;I system which is stable for all positive linear feedback. In order to make further progress in establishing the stability of (I), it will be necessary to make additional assumptions on the form of the nonlinearity. In particular, i t is convenient to deal with the class of functions Mk having the properties (1)f (0) = 0, (2) 0 Idf(y)/dy <k, and (3) f-l(y) exist. Clearly, Mk is a subclass of Ah; every h member of M is monotone increasing, and M, is just the class of all invertible functions passing through the origin. The following theorem gives a stability criterion valid when f is thus restricted. Theorem 3 The null solution of the system shown in Fig. 1 is ASIL iff E Mk, and there exists a positive real multiplier of the form , X, a t which the phase Identify the points XI, Xz, . curve crosses the 0" line from below. Define to be - 1, if the first nonzero value of the phase curve is positive, and let it be + l if its first nonzero value is negative. Define Z(s) in terms of these quantities by Z(S) = s ~ T ~ ( s ~ Xi2)/~(s2 ui2) (14) such that (Z(s))+'(G(s) l/k) is positive real. (ai, ci, and zi real and positive.) Proof: Since the proof is somewhat involved, a number of preliminary results will be established first. Lemma 1 : Let n(D) be a polynomial in D and assume that the null solution of n(D)x=O is ASIL. If the null solution of Clearly, Z(s) is positive reid since its poles and zeros alternate and lie on the imaginary axis. Moreover, from the way Z(s) has been constructed, it follows that the phase angle of Z(s) (G(s) l/k) always lies between plus and minus 90". To show that .Z(s)(G(s) l/k) is positive real i t remains only to show that the residues a t the imaginary poles are real andl positive. The only poles which need to be checked are those of Z(s), since any imaginary axis poles of G(s) will be canceled by the zeros of of Z(s); however, since the a~rglument G(s) is zero a t all poles of Z(s), we see that: the residues are, indeed, real and positive. Now, consider the follovving restatement of the generalized Popov theorem. ## is ASIL, then so is the null solution of the lower-order equation Theorem 1 (Alternate form) The null solution of (1) is ASIL if there exists a positive real function of the form (l+as)" such that (1+as)"(G(s) l/k) is positive real. Proof: To establish the equivalence between this and Theorem 1, as it was originally stated, observe that Re (l+orjw)GCjw)+l/k=Re (l-t+)(G(jw)+l/k), and the argument of (I +a+) equals that of (1 -@)-I. Proof: If 4(t) is a solution of (16), then n(D)\$(t) is a solution of (17). If all solutions of'(16) are asymptotically stable, then n(D)\$(t) is bounded and tends to zero for any choice of +(t) satisfying (16). Since the zeros of n(D) lie in the left half-plane, it follows that if n(D)\$(t) Similar results have recently been obtained by Zames [9], [lo], using the methods of functional analysis, and a somewhat different definition of stability. Brockett and Willems: Frequency Domain Stability Criteria is bounded and tends to zero, then the same must be true of cp(t). Lemma 2: The null solution of (16) is ASIL for all f E M , if, and only if, the null solution of n(D>q(D>x f(n(D>~(D)x> 0 = nite, note that the fact that m(s)q(s)/n(s)p(s) is positive real, implies that (22) is asymptotically table.^ Let C\$(x) denote the solution of (22) which starts a t x. Along this particular path V(x) is given by (18) is ASIL for all f EM,. Proof: Iff E M,, then f has an inverse, f-l, which also belongs to M,. Since any solution of (16) is also a solution of and conversely; the result follows. Lemma 3: If the positive realness of Z(s)G(s) for some Z(s) of a given form implies that the null solution of (16) is ASIL for all f E M,, then the positive realness of (Z(s))'(G(s)+l/k) implies that the null solution of (16) is ASIL for all f EMk. Proof: Since the stability properties of (16) and (18) are the same, it follows that if the positive realness of Z(s)G(s) implies stability for fEM,, then the positive realness of Z(s) (G(s))-I implies stability for all f E M,. Since the reciprocal of a positive real function is positive is ASIL, if there exists a positive definite radially unreal, this latter condition can be expressed by saying bounded V(x), having continuous first partial deriva, tives, and a ~ ( x )which is nonpositive and not identi(Z(s))-IG(s) is positive real. cally zero along any solution of x = f(x), which does not, If f E M k , then f-I exist, and (16) is equivalent to itself, tend to the origin. Proof: LaSalle [13] has shown that under the given hypothesis all solutions tend to the largest invariant set where g is defined through its inverse by in the set of points for which Since the slope of f-' is always greater than l/k, it follows that gEM,. Therefore, if (20) is asymptotically stable in the large for gEM,, it follows that (16) is asymptotically stable in the large for f EMk. The transfer function associated with (20) is (G(s)+l/k), and so by our previous remarks the lemma is proven. Lemma 4: Let x be a state vector for Clearly, V(x) is positive semidefinite; to see that it is positive definite observe that if it is to be zero, then there must be a solution to (22) along which p(D)+ and (Evm(D)q(D)n(-D)p( -Dl)-+ vanish' identically. In view of (22) it follows that n(D)p(D)C\$can vanish only if m(D)q(D)C\$does also; hence, if m(D)q(D), n(D)p(D), and (Evm(D)q(D)n(-D)p( -D)) have no common factors, as assumed, V(x) is positive for all xEO. Lemma 5 : The null solutions of the autonomous differential equation ## which has the components x, x(l), etc., and let denote the right half-plane spectral factor of the even part of m(D)q(D)n(-D)p(-D). If m(s)q(s)/n(s)p(s) is positive real, and if m(s)q(s), n(s)p(s), and ## is independent of path and is a positive definite function of x. ) If ~ ( x only vanishes along solutions which tend to the origin, it follows by a continuity argument that x goes to the origin, and hence that the hypothesis insures asymptotic stability in the large. In view of Lemma 3, it follows that if it can be shown that the existence of a Z(s), of the given form which makes Z(s)G(s) positive real, implies the stability of (1) for all f E M,, then the theorem will be established. This last phase of the proof is largely calculations. Let x be a state vector for (16), having the components x , x('), etc. If the degree of m(D)q(D)+n(D)p(D) exceeds that of n(D)p(D) +n(D)q(D), then x* will be a vector of higher dimension than x; however, if Z(s)G(s), is positive real, it will have a t most one more component. By using (16) it will be possible to express this additional component in terms of the components of x, so that in any case it will be possible to write x* = T(x), provided only that Z(s)G(s) is positive real. Suppose there exists a Z(s) of the given form such that Z(s)G(s) is positive real. Write Z(s) as m(s)/n(s) with m(s) and n(s) being polynomials without common factors. Multiply (16) by m(D)p(D)x and subtract ((Evm(D)q(D)n( -D)p( -D))-x)~from each side to get See Weinberg and Slepian LIZ] for a pmof. Proof: That this integral is independent of path has heen shown earlier 11 11. To see that i t is positive defi- m(D)q(D)xn(D)P(D)x + m(D)q(D>xf(n(D>q(D)x) = - ((Em(D)q(23)\$rz(- D)p(- D))-x)~. (27) ## Integrating this from t(0) to r ( ~ * ) , and using T(x) for x* gives4 Because f is assumed to monotone, these terms are also never positive. T o complete the proof, it remains only to note that the derivatives of the proposed Liapunov function is identically zero only if n(D)x=O. Since n(D) has all its zeros in the left half plane, this and Lemma 5 imply that the null solution of (16) is ASIL. VI. INTERPRETATION AND EXAMPLES Some study of the class of functions which can be represented by expansions of the form given by (15) seems appropriate. Actually, this expansion defines the most general driving point impedance which can be constructed from linear inductors and resistors; such functions have been studied in considerable detail by circuit theorists. The reciprocal also has a circuit theoretic interpretation; it defines the most general driving point impedance which can be synthesized from capacitors and resistors. I t may be shown that the poles and zeros of any Z(s) of the form given by (15) lie on the negative real axis and interlace (Fig. 9). J,,, t(x) ## ( ( E w ( D ) ~ ( D ) I ~D)p(- D))-x)2 dt (28) (- where V is defined by (2 3). From (15) it follows that the zeros of n(s) lie a t -zi/ci. Introduce the notation ni(s) = n(s)/(cis+zi), and notice that m(s) can be expressed as m(s) = aosn(s) + al(s + z,)nl('s) a,(s + z,)n,(s). (29) Define G(x) as and observe that even though G(x) is expressed as a integral, it is only a function of x, since each term in it is of the form aiSf(y)dy. Clearly, G(x) = G(T(x)) is nonnegative. Inserting G into (28) gives, after some manipulation, ## Fig. 9. An RL driving point impedance. - f ( z i n i ( ~ ) q ( ~ ) x dl. )) Taking V(T(x)) -tG(T(x) to be a Liapunov function, it is clear that its derivative is simply the integrand on the right side of (31). The first term in the integrand is never positive; the second is of the form -ayf(y), with a positive, and is, likewise, never positive. Each term in the last sum is of the form -cz(~(z+w)-f(w)) = -c((z+w) -w)(~(z+w)- f ( ~ ) ) . The notation used here shoulcl not be interpreted as implying that the integration is to be done along a solution of the given equation. Equations (28)-(30) should be viewed as an explanation of the fact that the derivative of the left side of (31) evaluated along solutions of (16) is, in fact, the integralnd on the right side of (31). I t is clear that the assumption of monotonicity and Theorem 3 makes it possible to prove stability in some cases where Theorem 1 cannot be used. Unfortunately, it is still impossible to prove stability in all the cases where linear theory is predicted. These points will be illustrated further. The question of determining exactly how far Theorem 3 goes toward filling the gap between what Theorem 1 proves, and what the Aizerman conjecture suggests, however, remains largely unresolved. Before returning to the examples, it seems worth while to point out one special case in Theorem 3 which covers a number of interesting cases. Corollary: The null solution of the system shown in k Fig. 1 is ASIL iff E M and if is positive real. This is an obvious soecialization t o the case where v = 1. NOW consider the third-order system + bxC1) + +f(xf2))= 0. GX (32) ( to~ ## Brodett and Willems: Frequency Domain Stability Criteria terion which is considerably more restrictive that that which would be predicted on the basis of linear theory. The relationship between the information it provides b ~ ( ~cx(l) f(x) = 0 ) x(~) (33) and the Nyquist criterion is clarified by Theorem 2 . As for the full range predicted by linear theory, but the a first step in the development of a more general theory restrictions on a, b, and c are somewhat weaker than which will enable one to place further restrictions on the those implied by the Popov theorem. In fact, a necessary nonlinear part, and in this way relax the restrictions on and sufficient condition for Theorem 3 to predict stabil- the linear part, the authors introduced the assumption of monotonicity, and developed a corresponding freity for all f E M ( a b c - c 2 ) / a 2 is that either quency domain stability criterion. Additional stability a3 - 2(ab - c) 2 0 or ab - 2c 2 0. (34) criteria imposing different restrictions on the nonThis means that when f ( x ) =x(abc-cz)/az the roots linearity will be given in the second part of the paper. should lie within either a wedge or a semicircular disk ACKNOWLEDGMENT of radius 4 3 ,Fig. 10. The authors would like to thank Prof. G. Zames of M.I.T., Cambridge, Mass., for contributing numerous ideas and suggestions to their research. They would also like to thank Prof. R. E. Kalman of Stanford University, Calif., and K. Meyer of Brown University, Providence, R. I., for pointing out several errors in an earlier draft. Theorem 3 is not capable of establishing the asymptotic stability of REFERENCES [I] Popov, V. M., Absolute stability of nonlinear systems of automatic control, Automation and Remote Control, vol 22, no 8, Mar 1962, pp 857-875. (Russian original published in Aug 1961.) [2] Popov, V. M., and A. Halaney, On the stability of nonlinear automatic control systems with lagging arguments, Automation and Remote Control, vol 23, no 7, Feb 1963, pp 783-786. (Russian original published in Jul 1962.) 131 Rozenvasser, E. N., The absolute stability of nonlinear systems, Automation and Remote Control,vol 24, no 3, Oct 1963, pp 283-294. (Russian original published in Mar 1963.) Fig. 10. Admissible pole locations for the [4] Narendra, K. S., and R. M. Goldwyn, Existence of quadratic monotone theorem to prove stability. type Liapunov functions for a class of nonlinear systems, Tech. Rept. no. 415, Cruft Laboratory, Harvard University, Cambridge, Mass., Aug 1963. Additional examples could be given illustrating appli[5] Rekasius, 2. V., A stability criterion for feedback systems with one nonlinear element, IEEB Trans. on Automatic Control, V O ~ cation of Theorem 3, however, it seems just as important AC-9, Jan 1964, pp 46-50. to indicate clearly those cases where it is least effective. [6] Yakubovich, V. A., Absolute stability of nonlinear control systems in critical cases, Parts I and 11, Automation and Remote Consider the equation Control, vol24, nos 3 and 6, Oct 1963 and Jan 1964, pp 273-282 and 655-668. (Russian originals published in Mar and Jun y(') a y ( 3 ) b y ( 2 ) cy'l) d y f(yC2)) = 0; 1963.) ,171 Aizerman, M. A., and R. F. Gantmacher, On critical cases in the a, b, c, d , abc c2 - a2d > 0. (35) theory of absolute stability of controlled systems, Automation and Control, vol 24, no 6, Jan 1964, pp 669-674. (Russian origiT o see that there exists no Z ( s ) of the form given by nal published in Jun 1963). of systems single ( I S ) , such that (Z(s))+ls2/(s4+as3+bs2 +CS +d) is POSi- [8] Willems, J. L., The stability Cambridge,containing a 1964. nonlinearity, S.M. thesis, MIT, Mass., Jul tive real, observe that such a Z ( S ) would have to pave a [g] Zames, G., Nonlinear time-varying systems-Contracting transformations for iteration and stability, Electronic Systems Lab. pole at the origin and a pole at infinity, which is not Rept., MIT, Cambridge, Mass., 1964. possible for a Z ( s ) of the given form. I t is of some inter- [lo] -On the stability of nonlinear time-varying feedback sysNECv est to note that a detailed analog computer study of this [ l l ] tems 1M4 PrOc. On thepp 725-730. nonlinear feedback systems, Brockett, R. W., stability of equation, with f ( y ) = y3, indicates that for moderately IEEE Trans. Applications and Industry, vol AP-83, Nov 1964, large initial conditions, (35) actually has limit cycles. 1121 pp 443-449.L., and P. Slepian, Positive real matrices, J. Math. Weinberg, (See Fitts [14].) Mech., vol 9, no 1, 1960, pp 71-83. [13] LaSalle, J. P., Some extensions of Liapunov's second method, IRE Trans. on Circuit Theory, vol CT-7, Dec 1960, pp 520-527. VII. CONCLUSIONS [14] Fitts, D., Summary of results on the stability of a fourth-order differential equation with monotonic nonlinearity, Internal On the basis of the done On the generalized Memorandum, Electronic Systems Laboratory, MIT, CamPopov Theorem, it is clear that it gives a stability cribridge, Mass., 1964. + + - ## Frequency Domain Stability Criteria-Part R. W. BROCKETT, MEMBER, IEEE, AND I1 J. L. WILLEMS Abstract-In Part I of this paper zl detailed analysis was made of the type of information Popov's Theorem gives about the stability of a closed-loop system containing a single instantaneous nonlinearity, and a new stability theorem, useful when the nonlinearity is monotone, was given. I n this part a general approach for generating stability criteria is described and several specific results are obtained. In particular, an improved criterion valid when f is an odd function is given, and criteria valid when f is a power law nonlinearity are also developed. Several examples are inchided to illustrate the theory. Manuscript received October 5, 1964:; revised July 7, 1965. This work was supported in part by the National Aeronautics and Space Administration under Contract NsG-498 with the Center for Space Research. The authors are with the Dept. of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. T WAS SHOWN I N Part I' that a linear system having a transfer function G(s) is stable for all linear feedback in Ak if and only if there exists a positive real function Z(s) such that Z(s) (G(s) l/k) is positive real. (See Part I for notation.) I t was also shown that Popov's Theorem can be viewed as predicting stability for all f EAk if there exists a positive real multiplier of the special form Z(s) = (1 +as)" such that 1 R. W. Brockett and J. L. Willems, "Frequency domain stability criteria-Part I," IEEE Transactions on Automatic Control, vol. AC-10, pp. 255-261, July 1965. Repilnted from lEEE Trans. Automat. Contr., vol. AC-10, pp. 407-413, Oct. 1965. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL OCTOBER Z(.Y) ( G ( s ) + l / k ) is positive real, and a new stability theorem was given which states that if the nonlinearity f E M k then stability is assured provided there exists a Z ( s ) of the form of a RC or RL impedance such that Z ( s ) (G(s)+ l / k ) is positive real. In this paper the stability of the nonlinear feedback loop is examined from the point of view of determining what assumptions should be made on the nonlinearity in order to make the conditions for nonlinear stability approach those predicted by linear theory. The principal new results are given in Theorems 4, 5, and 6. As in Part I, the stability of the closed-loop system will be studied by studying the stability of the solutions of the scalar equation I t is assumed throughout that f(0) =0, that f is monotone increasing, and that f has a continuous inverse; in short, f EM,. Additional assumptions will be imposed on f as required. The fact that the assumptions on f are more restrictive than those used in the Popov Theoreni is partially compensated for by the fact that i t will no longer be necessary to assume that the degree of q(D) is less than that of p(D). Some use will be made of the lemmas used in proving Theorem 3. In particular, the stability of (1) will be established by proving the stability of the related equations Whether or not there exists a Zts) of a certain type which makes Z(s)G(s)positive real depends to a large extent on the phase angle of G(s) and the type of phase angles which Z ( s ) is capable of generating. The product Z(s)G(s)can not be positive real unless its phase angle, i.e., the sum of the phase angles of Z ( s ) and G(s), lies between 90" and -90". If the phase angle of G(s) lies outside these limits then the phase of Z ( s ) must be selected in such a way to compensate. Moreover, the rate a t which the phase of G(s) varies, and the extent to which i t varies, determines what characteristics the phase of Z ( s ) must have. For this reason a knowledge of the phase characteristics of the functions defined by (4) is useful. For Z ( s )GZO,the Popov multiplier, the phase angle is just that of a first-order lead term; it starts a t zero and increases monotonicalIy to 90. For Z ( s )E Z l , the class of multipliers occurring in Theorem 3, the phase angle is again restricted to lie between 0 and 90" b u t need not be monotone increasing. Notice that this means that if the phase plot of G(s) l / k both exceeds 90" and falls below -90" then Theorems 1 and 3 cannot predict stability. Otherwise stated, a necessary condition for the application of Theorems 1 and 3 is that the Nyquist plot of G(s)+ l / k be restricted to 3 quadrants. If c > 1 then the phase angle of functions in 2,can be negative. The minimum value which can be achieved is limited however, and only approaches -90" as c approaches infinity. In the special case where v = 1 and cl=c, the functions in Z, assume the form As was shown in Lemma 3, if the nonlinearity is invertible then it is possible to get information about the cases where f is restricted to lie in a sector by treating the unrestricted case and then transforming. The possible zero configurations for c = 2 and z = 1 are shown in Fig. 1. For lower values of c the possible zero configurations lie inside the circle shown. In what follows, considerable use will be made of a special group of positive real functions. Let 2,denote the class of rational functions which can be expressed in the form with all constants being non-negative and c i s c . For example, if c=O then this sum reduces t o a term of the form Z(s) = a s + b which is only a slight generalization of the multiplier appearing in the Popov Theorem. If c = 1 then this is simply the class of functions appearing in Theorem 3. T h e stability criteria t o be derived here are stated in Fig. 1. Possible zero locations for (5); @ = z / 2 . terms of the existence of a multiplier Z ( s ) which makes (Z(s))l(G(s) l / k ) a positive real product. By this terminology i t i meant that (Z(s))l(G(s) s +l/K) is posiAn analysis of the 20 defined by tive real in the ordinary sense and that z ( ~ ) and phase starts a t 0" assumes a minimum value of (G(s)+l/k) do not have common imaginary axis poles tan-l((l -c)/2v':) for c t 1, and then goes to go0 as u and zeros. approaches infinity. Since any Z(s) in Z can be ex. 1965 ## BROCKETT AND WILLEMS: FREQUENCY DOMAIN STABILITY CRITERIA pressed as a sum af terms of ithis type, and since the argument of a sum of complex numbers can not exceed the maximum argument of its individual terms, it follows that the minimum phase angle of any Z(s)EZ, is not less than tan-l((1- c)/21/;), c:> 1. 111. Two GENERAL RESULTS STABILITY ON In searching for sufficient conditions for the stability of equation (1) it is evident that there is some trade-off between the class of admissible multipliers [the admissible form of Z(s)] and the restrictions placed on the nonlinearity f . The following lemma isolates the basic problem. Lemma 6 : If f E A , then the null solution of (1) is ASIL if there exists a positive real multiplier Z(s) =m(s)/n(s) such that Z(s)G(s) is a positive real product and J:::)m(~)yf(n(~)r) dt = and thus by Lemma 5 asymptotic stability exists. Using Lemma 6 it is very easy to prove either Theorem 1 or 3 and other theorems of that form. (See Theorem 6 below.) If monotonicity is assumed, however, it is possible to establish an even more direct connection between the class of admissible multipliers and the form of the nonlinearity. Lemma 7: If f E M , then the null solution of (1) is ASIL if there exists a Z(s) EZ, and constants b 2 0 and d>O, depending only on c, such that Z(s)G(s) is a positive real product and the inequality holds for all y and 3. Proof: From Lemma 7 it follows that it is enough to show that G(X) + J:o(;~(x) dt ## (6) with G(x) 2 0 and g(x) its numerator m(s) as with G(x) 2 0 , and g(x) 2 0 . Here y =q(D)x and x is a vector whose components are x, x ( l ) ,. . , x(P-l), /3 being the degree of m(D)q(D) +n(D)p(D). Proof: In view of Lemma 1 it is enough to show that the null solution of (2) is ASIL. In order to construct a Liapunov function for (2), multiply by m(D)q(D)x, subtract ((Evm(D)q(D)n(- D)p(- D))-X) from both sides, and integrate the result to get ## >0. Assume Z(s)EZ,.Expand where n(s) is the denominator of Z(s) and ni(s) = n(s)/(cis+zi). In terms of this notation the integral on the right side of (11) can be expressed as ## + C ai(D + zi)fii(D)yf((ciD + z,)ni(D)y) dt. V 2=1 (13) In view of Lemma 4 and the assumptions made on G ( x ) , the quantity on the right side of (7) is a positive definite function of x. If the degree of n(D)p(D) +n(D)q(D) equals that of n(D)p(D) +m(D)q(D) then x is a state vector for (2). If, on the other hand, the degree n(D)q(D) is less than n(D)p(D) of n(D)p(D) +m(D)q(D) it can not be more than one less, because the denominator and numerator of a positive real function differ in degree by one or :zero. Thus (2) can be used to express x in terms of a state vector for (2). In either case the quantity on the left side of (7) is a positive definite function of the state vector associated with (2). Its derivative, the integrand of the term on the right, is obviously negative semidefinite; this shows that the null solution of (2) is a t least (.weakly) stable. T o show that it is actually asymptotically stable, note that the derivative, given by The first term on the right is of the form lf(z)dz and thus can be put in G(x). A typical term in the remaining sum can be written as + aibDni(D)yf(d~ini(D)y)dt. (14) ## vanishes identically only if The integrand of the first term on the right is positive by virtue of inequality (10); that of the second term is positive because f is monotone. The last term clearly integrates to givle a positive function. Thus, if gi(x) is taken to be the sum of the first two integrands and if IEEE ## TRANSACTIONS ON AUTOMATIC CONTROL OCTOBER Gi(x) is taken to be the last integral, then it is clear that the sum of the g;(x) makes a suitable choice for g(x) and a sum of the Gi(x) plus the first term on the right side of (13) makes a suitable choice for G. Since the requirements of Lemma 6 are met this completes the proof. In the Sections IV and V Lemma 6 is used to get more explicit stability criteria. same sign. This completes the proof. As an example of where Theorem 4 can be used to predict stability where the previous results fail, consider a fourth-order system having a transfer function G(s)= (10s+1)(2s+ l)/(s2+20s+400)(2s2+5s+4). (21) org G(s) I80-- f ## IV. ODD MONOTONE NONLINEARITIES If f ( y ) = -f ( - y ) then f is said to be an odd function. If f G M k is odd then f will be said to belong to the class O k . Theorem 4 gives a stability criterion which is somewhat stronger than that given by Theorem 3 , provided that f is odd. Theorem 4 If f GOk then the null solution of ( 1 ) is ASIL if there exists a Z ( s ) E Z z such that ( Z ( s ) ) * I ( G ( s ) + l / k ) is a positive real product. Proof: The case where f E Ok can be reduced to the case where fEO, in exactly the same way as it was in proving Theorem 3. If f G 0 k then define g by the equation Notice that g and its inverse belong to 0, for the inverse of an odd function is odd and the inverse of a function in Mkhas a slope everywhere greater than l / k . Notice that all solutions of ( 2 ) are solutions of the equations fl(D)P(D>x+g(n(D>q(o>x+n(D)P(D)x/k) 0 = (16) I 9 0 ' 0 ## Fig. 2. The phase angles for G and Z for a fourth-order example. Since the Nyquist diagram is not restricted to three quadrants it is clear that Theorems 1 and 3 will not n(D)a(D)x+n(~)p(')x/k+g-l(n(D)q(D)x) = 0. (17) work. Yet, if f E O , it is possible to prove stability; a If it can be shown that the null solutions of these equa- suitable choice for z ( ~ ) E Z ~ is tions are stable for all gGO, then the theorem will be Z(s) = (2s2 5s 4 ) / ( 2 s 1). (22) proven. + + From Lemma 7 i t follows that if it can be shown that there exists b > O and d>O such that inequality (10) holds with c = 2 then (16) will be stable if Z ( s ) ( G ( s ) 1 / k ) is a positive real product, and ( 1 7 ) will be stable if Z(s)(G(s) l/k)-1 is a positive real product. Since the reciprocal of a positive real function is a positive real function it follows that if inequality (10) can be established with c = 2 then the proof will be complete. Consider inequality (10) with c = 2 and b = d = 1. In this case i t becomes The phase curve of Z ( s ) is sketched in Fig. 2. Of course Theorem 4 will not predict stability for all four pole systems whose linearized equations are stable. The maximum phase lag associated with a Z 2 multiplier is approximately 19.5'. In view of this it is clear that if the phase approaches 180" a t high frequency then the maximum phase lead must be less than 109.5" if Theorem 4 is to be used. Even this is only a necessary condition on G ( s ) ; no really interesting sufficient conditions have yet been found. V. STRONGER ASSUMPTIONS ON THE NONLINEARITY From Lemma 6 i t follows that one way to get a stability theorem allows a more general class of multipliers is to make assumptions on f which will allow inequality (10) to be satisfied for larger values of c. One type of nonlinearity which immediately suggests itself is the The first makes it obvious that the desired result holds power law. For example, if f(y) = ky3, k > 0, what is the if y and z have different signs whereas the second makes best choice of b and d, and what is the corresponding i t obvious that the desired result holds if they have the value of c? In this case inequality (10) becomes 1965 ## BROCKETT AND WILLEMS: FREQUENCY DOMAIN STABILITY CRITERIA k(j + y)(cj + Y ) -- k b 9 ( d ~ )2~ 0. ~ (23) If this is multiplied by l/k3r4, it becomes, on substituting z for y/y, (Z+ ~ ) ( c z 1:13- bd3z 2 0. the value used for functions of class Ok. On the basis of Lemma 7 it follows that the equation 0 p(D)x \$- k q ( ~ ) 1 sgn(q(~)x) 0; k I (29) x " = isstable if there exists aZ(s) GZ+(,, such that (Z(s))+lG(s) is a positive real product. However, it is possible to prove a considerably more general result if the proper classification of nonlinear functions is used. The function f(y) will be said to belong to the class P,(u) if it is odd ,and if for all y ] ] z (24) I t may be shown that if bds is taken to be 64 then c may be taken to be 9 and that this is the highest value of c for which inequality (23) can be validated, regardless of the choice of bd3. The details of this argument may be found in Willems [8], Part I. For the general power law, where f(y) i ~ ' ~ i v e n by I(Y) = 121 ## 1 > 1 1 y/z 1 'Iu I I f (~)/f(z)1 5 I y/z l". 1% I (30) YI" sgn (Y) (25) it is not known what the best values of b and d are nor is it known what the corresponding value of c is. Other special cases which have been worked out are u = 2, for (jf y)+(u)yi-y sgn(4y+y) - b j by ) ~ s ~ n 0 ~ ) (31) 2( which the best value of c is 10+6.\/3, and u = 5 , for which the best value of c is 5. The best value o~fc associa.ted with the power u is the both hold for a suitable choice of b. Since a function same as that associated with the power l/u. T o prove f G P,(u) lies between these two extremes it follows that this, start by considering th~euth power equivalent of for f E Pm(u> inequality (25) which is (Y df ( ~ ( u > Y \$4 - b~f;f(by) 0. 2 (33) These are the functions which lie between a l / u power law and a u power law. I t has already been shown that the inequalities This inequality is assumed t:o hold for all z. By taking the uth root of both terms tlhis becomes (cz f 1) z f 1 I l l u sgn(z + 1) - dbl~uzllu 0. 2 -t1) - dbllY/z > 0. (27) (28) ## Multiply this by z--l-l/~to get (c 4- l/z) I 1 9l/z (ltu sgn ( z The functions in P,(u) are permitted to have infinite gains. In order to modify this theory in such a way as to treat systems for which the gain is limited, it is only necessary to re-examine the steps leading to (16). Define the set Rk(u) by saying that f E P k ( u ) if f-l(y) -y/k belongs to P,(u). In terms of this notation, Theorem 5 summarizes the results of this section. Theorem 5 If fEPk(u) then the null solution of (1) is ASIL if such there exists a .Z(s) EZ+(,, that (Z(s))"(G(s) +l/k) is a positive real product. Notice that this theorem actually includes Theorem 4 as a special case. VI. OTHER SUFFICIENT CONDITIONS While the assumptions made on the nonlinearity in the previous theorems have been natural enough, they are but a few of the many which could be used. For example, they say nothing about what improvement, if any, can be made on the Popov Theorem i f f is monotone, and in AI,but not in Mk so that Theorem 4 cannot be used. Likewise, if fEAk and Om,can Theorem 1 be improved upon? The following theorem treats the first case. Let w = 1/62 and divide inequality (28) by c. This shows that the value of c with a given u is the same as that associated with l / u , but it also shows that if b = d then the same values of b and d are appropriate for both u and l/u. Let +(u) denote the best value of c associated with a given u. A sketch of this function is shown in Fig. 3. / 01 02 . 03 . 0.5 ." Theorem 6, If f E M ## Fig. 3. A sketch of the function +(u). This sketch is only approximate; it is based on the data given above and some additional numerical work. Notice that as u approaches infinity +(u) approaches 2, and Aa then the null solution of (1) is ASIL if there exists a Z(s) EZI such that Z(s) (G(s) fZ(0) /kG(s)) is a positive real product. Proof: If (2) is multiplied by m(D)p(D)x +n(D)p(D)xZl(O)/k and then integrated, this result can be written as ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL provided the term appearing on the right is subtracted from both sides. The integral containing the nonlinear term can be rewritten a s linearization, although this would seem to be a worthwhile thing to do. Consider a transfer function G(s) which is stable for all linear feedback in A,. The results obtained here suggest that if the phase angle always lags then this system will be stable for any nonlinear feedback of the correct sign. If the phase approaches 180" a t high frequencies and is positive for some range of frequencies then it still may be possible to prove stability for all f E A , if the phase lead never exceeds 90'. If the phase angle of such a system exceeds 90" then it seems that linearization must be used with more caution. Roughly speaking, the larger and the more rapid the phase angle excursions, the less likely it is that linearization techniques will accurately predict stability behavior. The integrand of the first term on the right is clearly positive. If it can be shown that the second term can be expressed as a sum of the form G(x+g(x) as in Lemma 5, then the theorem will be proven. Notice however, -n(D)q(D)Z(O))/n(D)q(D) that the ratio (m(D)q(D) =m(D)/n(D) -Z(O) is, itself, a 21function; to prove this merely examine a typical term in (4). In the proof of Theorem 3 it was shown if m(s) n(s) E k then the integral In Parts I and I1 of this paper a detailed study of the stability of a class of nonlinear feedback systems has been made. Sufficient conditions for asymptotic stability in the large have been given in terms of the existence of a multiplier Z(s) of a certain class, which can make the product Z(s)G(s) positive real. The class of multipliers is defined by (4) and the stability criteria are listed in the following. The null solution of ( 1 ) will be ASIL if: 1 ) f E A k and there exists a positive real function Z ( s ) E z o such that (Z(s))"(G(s)+l/k) is a positive real product. 2) f E M k and there exists a positive real function Z(s)E Z l such that (Z(s))"(G(s) l / k ) is a positive real product. 3) -f E Oe and there exists a positive real function Z ( s ) E Z z such that ( Z ( S ) ) * ( G ( S ) + I / K ) is a positive real product. 4) fEPk(u) and there exists a positive real function Z ( s )EZ,,,, such that (Z(s))l(G(s) l / k ) is a + positive real product. 5) fE Ak and f E M , and there exists a positive real function Z(s)E Z l such that ( Z ( s ) ) (G(s)+Z(O) /kG(s)) is a positive real product. with G(x)>0 and g ( x ) 2 0 ; the remaining details are left to the reader. Corresponding results, valid when fEAk and 0, or P,(u) are given in reference [8] of Part I. Other possibilities, such as f EMk., and f GAk, k' > k, have not been worked out; however, the way to proceed seems clear. VII. LINEARIZATION Certainly one of the most interesting questions in stability theory is, "When it is possible to determine the stability properties of a nonlinear system by looking a t some linearized version of it?" T h e question is complicated because of various types of linearizations which can be used. In this paper, the results obtained from the Liapunov theory have compared with those predicted heuristicly by 1) total linearization, [the Aizerman problem] and 2) local linearization [linearization a t each point on the nonlinearity]. No comparison was made with the describing function On the basis of the criteria given, i t seems clear that the cases in which Aizerman type linearization can give wrong answers are those for which the phase angle exceeds 90" or those for which the phase angle changes rapidly from a positive value to a large (-180") negative value. I t seems that any type of linearization can give a wrong answer if the phase angle approaches 180" as w approaches zero or if the phase changes rapidly from near 180" to near - 180". A great deal more work is required to translate the given stability criteria into engineering terms. This, and an investigation of the necessary conditions for stability, seem to be two worthwhile areas for future study. ## IEEE TRANSACTIONS ON A I J T O M A ~ C CONT~XOL VOL. AC-10, NO. 4 OCTOBER, 1965 The object of this Appendix: is to describe in more detail the procedure used for calculating v in the various proofs in Parts I and I1 of this paper. Basically what is done is to first define a scalar function V(x) by a line integral which is independent of path; e.g., v(x) = t + t~ P(D)xq(D)x - ((E~Y(D)P(D))-xI2 dt (39) provided that now d t ) , the path of integration, is a solution of (38) and t is identified with time. By taking the limit, v(x, 2) = p(D)xq(D) - ((Ewq(D)p(- D)x)-x)2 dt and, using (38), . V(X,x) = J::;P(D)X~(D)X - ((:E~!?(D)P(-D))-x)~ (37) (40) Of course, t need not be identified with time but it should be understood that in evaluating V(x) a path is traversed which starts a t x = 0 and ends a t x and that t(x) depends on the particular path chosen. Now let x and XO be points On the of the equation whose solutions V ( X ) is to be calculated; e.g., - ((Evq(D)p(- D))-x)~. (41) similar reasoning applies in the more complicated cases. ~h~ reason for showing a dependency on in is that in the event that p(D) is of the same degree as p(D), the nth derivative of x will appear in V . This p ( D ) x = 0 or x = Ax. (38) Suppose that the state a t time t is x and the state a t time t+At is xo. Then clearly can, of course, always be eliminated by using the equation of motion but it often simplifies things not to do so. This is particularly true in proving the Popov Theorem (see Brockett I:11], Part I). On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems Part I: Conditions Derived Using Concepts of Loop Gain, Conicity, and Positivity G. ZAMES, Abstract-The object of this paper is to outline a stability theory for input-output problems using functional methods. More particularly, the aim is to derive open loop conditions for the boundedness and continuity of feedback systems, without, at the beginning, placing restrictions on linearity or time invariance. It will be recalled that, in the special case of a linear time invariant feedback system, stability can be assessed using Nyquist's criterion; roughly speaking, stability depends on the amounts by which signals are amplified and delayed in flowing around the loop. An attempt is made here to show that similar considerations govern the behavior of feedback systems in general-that stability of nonlinear time-varying feedback systems can often be assessed from certain gross features of input-output behavior, which are related to amplification and delay. This paper is divided into two parts: Part I contains general theorems, free o restrictions on linearity or time invariance; Part f 11, which will appear in a later issue, contains applications to a loop with one nonlinear element. There are three main results in Part I, which follow the introduction of concepts o gain, conicity, positivity, f and strong positivity: THEOREM 1 :If THEOREM 2: MEMBER, IEEE tems we might ask: What are the kinds of feedback that are stabilizing? What kinds lead to a stable system? Can some of the effects of feedback on stability be described without assuming a very specific system representation? Part I of this paper is devoted t o the system of Fig. 1, which consists of two elements in a feedback loop.' This simple configuration is a model for many controllers, amplifiers, and modulators; its range of application will be extended t o include multi-element and distributed systems, by allowing the system variables t o be multidimensional or infinite-dimensional. the open loop gain is less than one, then the closed loop is bounded. ## Fig. 1. A feedback loop with two elements. If the open loop can be factored into two, suitably proportioned, conic relations, then the closed loop is bounded. the open loop can be factored into two positive relations, one of which is strongly positive and has finite gain, then the closed loop is bounded. rHEoREM 3: If Results analogous to Theorems 1-3, but with boundedness replaced by continuity, are also obtained. The traditional approach to stability involves Lyapunov's method; here it is proposed to take a different course, and t o stress the relation between input-output behavior and stability. An input-output system is one in which a function of time, called the output, is required t o track another function of time, called the input; more generally the output might be required to track some function of the input. In order t o behave properly an input-output system must usually have two properties: 1) Bounded inputs must produce bounded outputs-- ## EEDBACK, broadly speaking, affects a system in one of two opposing ways: depending on circumstances it is either degenerative or regenerativeither stabilizing or destabilizing. In trying t o gain some perspective on the qualitative behavior of feedback sysY- ## i.e., the system must be nonexplosive. 2) Outputs must not be critically sensitive to small changes in inputs-changes such as those caused by noise. The system of Fig. 1 has a single input x , multiplied by constants al and az, and added in a t two points. This arrangement has been chosen because it is symmetrical and thus convenient for analysis; it also remains invariant under some of the transformations that will be needed. Of course, a single input loop can be obtained by setting al or a2 t o zero. The terms w and w are fixed bias functions, which l z will be used t o account for the effects of initial conditions. The variables el, ez,yl, and y . are outputs. ~ Manuscript received December 29, 1964; revised October 1, 1965; February 2, 1966. This work was carried out a t the M.I.T. Electronic Systems Laboratory in part under support extended b NASA under Parts of this Contract NsG-496 with the Center for Space ~esearcK. paper were presented a t the 1964 National Electronics Conference, Chicago, Ill., and a t the 1964 International Conference on Microwaves, Circuit Theory, and Information Theory, Tokyo, Japan. The author is with the Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Reprinted from IEEE Trans. Automat. Contr., vol. AC-11, pp. 228-238, Apr. 1966. ## ZAMES: STABILITY OF NONLINEAR FEEDBACK. SYSTEMS These two properties will form the basis of the definition of stability presented in this paper. I t is desired t o find z conditions on the elements HIand H (in Fig. 1) which will ensure t h a t the overall loop will remain stable after H Iand Hzare interconnected. I t is customary t o refer and prior to interconnection as the "open-loop" to HI H2 elements, and t o the interconnected structure as the "closed loop." The problem t o be considered here can therefore be described as seeking open-loop conditions for closed-loop stability. Although the problem a t hand is posed as a feedback problem, it can equally well be interpreted as a problem in networks; i t will be found, for example, t h a t the equations of the system of Fig. 1 have the same form as those of the circuit of Fig. 2, which consists of two elements in series with a voltage source, and in parallel with a current s o u r ~ e . ~ of these results are paralleled in the work of Brockett and Willems [4], who use Lyapunov based methods. Several others have obtained similar or related results by functional methods: Sandberg [5a] extended the nonlinear distortion theory mentioned above; later [5b] he obtained a stability theorem similar to Theorem 1 of this paper. Kudrewicz [6] has obtained circle conditions by fixed point methods. Contraction methods for incrementally positi~ve operators have been developed by Zarantonello 1[7], Kolodner [8], Minty [9], and Browder [lo]. A stability condition for linear timevarying systems has been described by Bongiorno [I 1]. There are several preliminaries t o settle, namely, to specify a system model, t o define stability, and to write feedback equations. What is a suitable mathematical model of a feedback element? A "black box" point of view towards defining a model will be taken. T h a t is t o say, only input-output behavior, which is a purely external property, will be considered ; details of internal structure which underlie this behavior will be omitted. Accordingly, throughout Part I , a feedback element will be represented by an abstract relation, which can be interpreted as a mapping from a space of input functions into a space of output functions. More concrete representations, involving convolution integrals, characteristic graphs, etc., will be considered in Part 11. Some of the elementary notions of functional analysis will be used, though limitations of space prevent an Fig. 2. A circuit equivalelit to the loop of Fig. 1. introduction to this subject.3 Among the concepts which will be needed and used freely are those of an abstract relation, a nor~ned linear space, an inner product space, 1.1 Historical Note and the L, spaces. T h e problem of Ryapunov stability has a substantial T h e practice of omitting the quantifier "for all" shall history with which the names of Lur'e, Malkin, Yaku- be utilized. For example, the statement bowitch, Kalman, and many others, are associated. On "-x2 I x2(x E X)" the other hand, functional methods for stability received less attention until relatively recently, although is t o be read: some versions of the well-known Popov [I] theorem "for a11 x E X, -x2 x2." might be considered as fitting: into this category. . T h e present paper has its origin in studies [2a, b ] of CONVENTIOXI: A n y expression containing a condition of nonlinear distortion in bandlimited feedback loops, in the type "xEX," free of quantifiers, holds for all x E X . which contraction methods were used t o prove the existence and stability of a n inversion scheme. The 2.1 The Extended Normed Linear Sbace X, author's application of contraction methods t o more In order to specify what is meant by a system, a suitgeneral stability problems was inspired in part by conable space of input and output functions will first be versations with Narendra during 1962-1963; using defined.4 Since unstable systems will be involved, this Lyapunov's method, Narendra and Goldwyn [3] later space must contain functions which ('explode," i.e., obtained a result similar t o the circle condition of Part which grow without bound as time increases [for examI1 of this paper. ple, the exponential exp ( t ) ] .Such functions are not conThe key results of this paper, in a somewhat different tained in the spaces commonly used in analysis, for exformulation, were first presented in 1964 [2d, el. Many ample, in the L, spaces. Therefore it is necessary to < It is assumed that the source voltage v and the source current i are inputs, with v=u~x-twl i = u ~ x f P ; currents andvoltages and W the in the two elements are outputs. A good reference is Kolmogorov and Fornin [12]. The space of input functions will equal the space of output functions. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL APRIL construct a special space, which will be called X,. X,, will contain both "well-behaved" and "explodingn functions, which will be distinguished from each other by assigning finite norms to the former and infinite norms t o the latter. X, will be an extension, or enlargement, of an associated normed linear space X in the following sense. Each finite-time truncation of every function in X, will lie in X ; in other words, the restriction of x E X , t o a finite time interval, say t o [0, t], will have a finite norm-but this norm may grow without limit as t-+ m . First a time interval T and a range of input or output values V will be fixed. DEFINITION: T is a given subinterval o the reals, o the f f type [to, m) or (- m , m). V is a given linear space. The point of assumptions (2)-(3) on X can now be appreciated; these assumptions make it possible to determine whether or not an element x E X , has a finite norm, by observing whether or not limt,, llxtll exists. For example : EXAMPLE 1: Let L2[0, m) be the norrned linear space consisting of those real-valued functions x on [0, m) for which the integral j,"x2(t)dt exists, and let this integral equal Let X = L2[0, a ) , and let L2.=X,; that is, Lzc is the extension of L 2[0, a ) . Let x be the function on [0, a ) given by x(t) = exp (t). Is l\xllcfinite, that is, is x in X ? No, because llxtll grows without limit as t+m, or in other words, IIxIIe= m . 11~11~. DEFINITION: A relation H on X , is any subset o the f Let x be any function mapping T into V, product space X, X X,. I (x, y) is any pair belonging to H f that is, x: T-+ V; let t be any point in T; then the symbol then y will be said to be H-related to x; y will also be xr denotes the truncated function, xt: T+ V, which as- said to be an image of x under H.5 sumes the values xt(7) =x(r) for T <t and x~(T) 0 else= In other words, a relation is a set of pairs of functions in where. X,. I t will be convenient to refer to the first element (A truncated function is shown in Fig. 3.) Next, the in any pair as an input, and to the second element as an space X is defined. output, even though the reverse interpretation is sometimes more appropriate. A relation can also be thought of as a mapping, which maps some (not necessarily all) inputs into outputs. In general, a relation is multivalued; i.e., a single input can be mapped into many outputs. The concept of state, which is essential to Lyapunov's method, will not be used here. This does not mean that initial conditions cannot be considered. One way of accounting for various initial conditions is to represent a system by a multi-valued relation, in which each input Fig. 3. A truncated function. is paired with many outputs, one output per initial condition. Another possibility is t o introduce a separate DEFINITION: X a space consisting o ,functions o the relation for each initial condition. f f type x: T-+V; the following assumptions are made conNote that the restrictions placed on X , tend to limit, a cerning X: priori, the class of allowable systems. In particular, the (1) X is a normed linear space; the norm o x E X is requirement that truncated outputs have finite norms f means, roughly speaking, that only systems having infidenoted by Ilxll. nite "escape times," i.e., systems which do not blow up (2) V x E X then x t E X for all t E T . in finite time, shall be considered. (3) I x:T+V, and i x t E X for all t E T , then: f f Some additional nomenclature follows: f (a) llxtll is a nondecreasing function o t E T. (b) I limt,.. vxtlI exists, then x E X and the limit f DEFINITION: I H is a relation on X,, then the domain o f f equals H denoted Do(H), and the range of H denoted Ra(H), are (For example, it can be verified that assumptions (1)- the sets, (3) are satisfied by the L, spaces.) Next, X, is defined. Do(H) = ( x xEX,, and there exists y E X , such that DEFINITION: The extension o X , denoted by X,, i s the f (x, Y) s#ace consisting of those functions whose truncations lie i n X , that is, X,= x:T+V, and x t x for all ~ E T ] . lia(H)= { y 1 yEX,, and there exists x E X d such that (x, Y) (NOTE: is a linear space.) An extended norm, denoted X, [IxII., is assigned to each x E X , as follows: llxll.=llx/l if 6 In general x can have many images. x E X , and IIxII,= 00 i f x g X . example, in the analysis of multielement (or distributed) networks, V is multidimensional (or infinite-dimensional).] Second, the notion of a truncated function is introduced. NOTATION: o or IIx I . HI {XI CHI 1966 ## LAMES: STABILITY OF NONLINEAR FEEDBACK SYSTEMS DEFINITION: A relation H on X, is continuous i;f H has the ,following property: Given any x e X (that is, Ilxll e < a),and any A >0,there exists 6 >0 such that, for all y E X , i f <6 then HX- ~ ~ <A. 1 1 DEFINITION: A relation H on X, is input-output stable i H is bounded and continuous. f NOTATION: I H is a relation on Xel and i x is a given f f f element o X,, then the symbol Hx denotes a n image o x f under H.6 The idea here is to use a special symbol for an element instead of indicating that the element belongs to a certain set. For example, the st;atement, "there exists HX having property R" is shorthand for "there exists yERa(H), such that y is an irniage of x, and y has property P."6 Observing that Hx is, according to the definitions used here, a function on T, the following symbol for the value of Hx a t time t is adopted : NOTATION: The symbol Hx~(t) denotes the value assumed by the fundion Hx at time t ET. Occasionally a special type of relation, called an operator, will be used: DEFINITION: An operator H is a relation on X. which satisfies two conditions: 1) Do(H) = X,. 2) H is singlevalued; that is, i x , y, and z are elements o X,, and y f f and z are images o x under AT, then y =z. f 2.3 The Class ( 3 DEFINITION: (R is the class o those relations H on X. f having the property that the zero element, denoted o, lies i n Do(H) , and Ho = 0. jfI~-~ll (1 2.5 Feedback Equations Although negative feedback loops will be of interest, the positive feedback configuration of Fig. 1 has been The chosen because it is ~~mmetrica1.l equations describing this system, t o be known as the FEEDBACK EQUATIONS, are : el ez = = wl + alx + yz w + a2x + yl 2 (2b) yz = H2ez yl = Hlel ## in which it is assumed that: HI and H are relations in (R z a1 and a z are real constants wl and wt are fixed biases in X in Xe is an input and e2 in Xe are (error) Outputs yl and y2 in X, are outputs. (The biases are used t o compensate for nonzero zeroinput responses and, in particular, for the effects of initial conditions.) The closed-loop relations El, Ez, F1, and F2,are now defined as follows. DEFINITION : El is the relation that relates el to x or, more precisely, El = { (x, e 3 (x, el) EX,XX,, and there exist e2,y I, y ?, H1el, and H2ez,all in X,, such that (1) and (2) are satisJied.) Similarly E2 relates ez to x; F1relates y, to x; F~ relates yz to x. The that H maps zero into zero many derivations; if this condition is not met a t the outset, it can be obtained by ad,dilnga compensating bias to the feedback equations. If H and K are relations in and C is a real constant, then the sum (HS-K), the Product cH, and the composition product K H of foll@wing Hl are defined in the usual way17and are relations in (R. The inverse o H in f (R, denoted by H-l, always exists. The identity operator on X, is denoted by I. 2.4 Input-Output Stability All the prerecluisites are now assembled for defining the problem of interest which is: Find conditions on H and 1 H which ensure that El, E2, F1 and FZ 2 1 are bounded or The term astable" has been used in a variety of ways, to indicate that a system is; somehow well behaved. A stable. In general it will be enough to be concerned with and F1 and Fz, since every system shall be called stable: if it is well behaved in two El and E2 is respects: (1) I t is bounded, i.e., not explosive. (2) it is F ~ x related to some EIX by the equation F2x = EIX -alx-w1, so that F 2 is bounded (or stable) whenever continuous, i.e., not critically sensitive t o noise. E l is, and similarly for Fl vs. E2. DEFINITION: A subset Y oj"X. is bounded i there exists f It should be noted that by posing the feedback probA > o such for Y EY?jfIyII.<'. A On lem in terms of relations (rather than in terms of X, is boundedEV 'he image under H of every bounded sub- operators) all questions of existence and uniqueness of ' set o X. is a bounded subset if X,. f solutions are avoided. For the results to be practically significant, it must usually be known from some other 6 In keeping with the usual convention used here, any statement sourceg that solutions exist and are unique (and have containing Hx free of quantifiers homldsfor all x in Ra(H). For example, U H x > l (xX.)" means that "for all x in X. and for all Hx in infinite "escape times"). . Ra(H), %>l.' In particular, Do(H+K) = Do(H)nDo(K). Note that (Ris nota linear space; for example, ~f Do(;H) f Do(K) then Do[(H+K) -K] f Do(H). . - ,--, This definition irnplies that inputs of finite norm produce outputs of finite norm. More than that, it implies that the sort of situation is avoided in which a bounded sequence of inputs, say IIx.ll< 1 where n= 1, 2, , produces a sequence of outputs having norms that are finite Ibut increasing without limit, say I]Hx,,ll =n. - - .. 9 Existence and stability can frequently be deduced from entirely separate assumptions. For example, existence can often be deduced, by iteration methods, solely from the fact that (loosely speaking) the open loop delays signals; stability can not. (The connection between existence and generalized delay is discussed in G. Zames, "Realizability conditions for nonlinear feedback systems," IEEE Trans. on Circuit Theory, vol. CF-11, pp. 186-194, June 1964.) ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL APRIL 3. SMALL LOOP GAINCONDITIONS T o secure a foothold on this problem a simple situation is sought in which it seems likely, on intuitive grounds, that the feedback system will be stable. Such a situation occurs when the open loop attenuates all signals. This intuitive idea will be formalized in Theorem 1; in later sections, a more comprehensive theory will be derived from Theorem 1. T o express this idea, a measure of attenuation, i.e., a notion of gain, is needed. 3.1 Gains I~HX - HYII~ < HI .IIx - 1 9 YEDO(H)]x (9) In the Feedback Equations (1)-(2), the product g(H1) .g(Hz) will be called the open-loop gain-product, and similarly, g(H1) .g(H2) will be called the incremental open-loop gain-product. 3.2 A Stability Theorem ## Consider the Feedback Equations (1)-(2). THEOREM 1:lo a) I g(H1) .g(H2)<1, then the closed loop f relations EI and E2 are bounded. b) I g(H1) g(H2) < 1, f then El and E2 are input-output stable. Gain will be measured in terms of the ratio of the norm of a truncated output to the norm of the related, truncated input. DEFINITION: ## The gain o a relation H in f (R, denoted by is Theorem 1 is inspired by the well known Contraction Principle.ll PROOF OF THEOREM 1: (a) Since eqs. (1)-(2) are symmetrical in the subscripts 1 and 1, it is enough to consider El. This proof will consist of showing that there are positive constants a, b, and c, with the property that any pair (x, el) belonging to E l [and so being a solution of eqs. (1)-(2)], satisfies the inequality where the supremum is taken over all x in Do(H), all Hx in Ra(H), and all t in T for which xt# 0 . In other words, the supremum is taken over all possible input-output pairs, and over all possible truncations. The reason for using truncated (rather than whole) functions is that the norms of truncated functions are known t o be finite a priori. I t can be verified that gains have all the properties of norms. In addition, if H a n d K belong to (R then g(KH) <g(K)g(H). Gains also satisfy the following inequalities: I t will follow that if x is confined to a bounded region, say llxll <A, then el will also be confined to a bounded +bll w211+cA. Thus El region, in this case Ilelll 5all will be bounded. PROOF OF INEQUALITY (10): If (x, el) belongs to El then, after truncating eqs. (la) and (Ib), and using the triangle inequality to bound their norms, the following inequalities are obtained : will where (4) is implied by (3), and (5) is derived from (4) by taking the limit as t+ w . If g(H) < w then (5) implies that H is bounded. In fact, conditions for boundedness will be derived using the notion of gain and inequalities such as (5). In a similar way, conditions for continuity will be derived using the notion of incremental gain, which is defined as follows : DEFINITION: The incremental gain o any H in (R, def noted by g(H), is I l ( ~ x ) - (~Y)tll t g(H) = sup (6) IIxt - ytll where the supremum is taken over all x and y in Do(H), all Hx and Hy in Ra(H), and all t in T for which xt #yt. Furthermore, applying Inequality (4) to eqs. (2), the following is obtained, for each t in T: Letting g(H1) 4 a and g(H2) LP, and applying (1la) to (10a) and ( l l b ) to (lob), the following inequalities are obtained : ## + I al 1 -llxtll+ /311eztll lleztll < 11w2tll + I a2 I .IIxtll + alleltll lleltll .-< llwltll (t E T) (128) (t E T). (12b) Applying (12b) t o //eztl/ (12a), and rearranging, in (1 - d)l)eit(l 5 llwitll ~IIw2tll (t TI, (13) + ( I a1 l + /3 l a2 I >IIxtll Incremental gains have all the properties of norms, and satisfy the inequalities g(KH) 5 g(K) .g(H) (7) l l ( ~ x ) t (HY)~[I g ( H ) xt - ~ t l l [x, yEDo(H) ; t~ T ] (8) < -11 10 A variation of Theorem 1 was originally presented in [2d]. A related continuity theorem was used in [2c]. An independent, related result is Sandberg's [Sb]. 1 If X is a complete space, if all relations are in fact operators, 1 and if the hypothesis of Theorem l b holds, then the Contraction Principle implies existence and uniqueness of solutions-a matter that has been disr,egardedhere. 1966 ## Z.AMES: STABILITY OF NONLINEAR FEEDBACK SYSTEMS Since (1-4) (as (llS < 1, by hypothesis), Inequality >0 (13) can be divided by (1 -a@) ; after dividing and taking the limit of both sides as t-+ m , the Inequality (10) remains. Q.E.D. (b) Let (x', el') and (xf', elt1) be any two pairs belonging to El. Proceeding as in Part (a) an inequality --xlII is obtained, which of the form Ile~"-el'll implies that El is continuous;. Moreover, since the hypothesis of Part (b] implies the hypothesis of Part (a), El is bounded too. Therefore El is input-output stable. EXAMPLE 2: In eqs. (1)-(2) (and in Fig. 1) let one of the two relations, say HI, be the identity on Lze. (Lz, is z defined in Example 1.) Let the other relation, H on Lz,, be given by the equation Hg:(t) = k ~ [ x ( t ) ] , where k>O is a constant, and N is a function whose graph is shown f in Fig. 4. For what sralues o k are the closed loop relations (a) bounded? (b) stable? (a) First the gain is calcullated. ICI X" unaffected; however, HI is changed into a new relation HI', as in effect - c I appears in feedback around HI. Under what conditions does this transformation give a gain product less than one? I t will appear that a sufficient condition is that the input-output relations of the open loop elements be confined to certain "conic" regions in the product space X , 2,. Fig. 5. A transformation. ~ ( H z = SUP ) {Som / ## I V ~ [ X (at : ) I Jomx2(t) at) 'Iz ~ RESTRICTION: I n the remainder o this paper, assume f that X is a n imner-product space, that (x, y) denotes the inner product on X, and that (x, x) = 11~11~. This restriction is made with the intention of working mainly in the extended L ~ [ o ,a) norm,12 with (x, y) =J,"x(t)y(t)dt. where the first sup is over [~:DO(H); HxERa(H); t E T , xt#O]. That is, g(H) is X: times the supremum of the absolute slopes of lines drawn from the origin to points on the graph of N. Here g(H) = k, so Theorem 1 implies boundednes:; for k < l . This example is trivial in a t least one respect, namely, in that H has no memory; examples with mennory will be given in Part 11. (b) g(H) can be worked out to be k times the supremum of the absolute Lipschitzian slopes of N, that is, g(H) = k sup,, , N(x) - ~ ( y ) / -y = 2k. The closed ,,,I x loop is therefore stable for k < 1/2. ## 4.1 Definitions: o Conic and Positive Relations f DEFINITION: A relation H i n is interior conic if there are real constants r 2 0 and c for which the inequality is satisfied. H is exterior conic if the inequality sign i n f (14) is reversed. H i s conic i it is exterior conic or interior conic. The constant c will be called the center parameter of H , and r will be called the radius parameter. of The truncated output ( H x ) ~ a conic relation lies either inside or outside a sphere in X , with center proportional to the truncated input xt and radius proportional to IIxtll. The region thus determined in X,XX, will be called a "cone," a term suggested by the following special case: EXAMPLE 3: Let H be a relation on Lz, (see Example 1); let Hx(t) be a function of x(t), say Hx(t) = ~ [ x ( t ) ] , where N has a graph in the plane; then, as shown in Fig. 6, the graph lies inside or outside a conic sector of the plane, with a center line of slope c and boundaries of slopes c -r and c+r. More generally, for H to be conic [without Hx(t) necessarily being a function of x(t) , that is, if H has memory], i t is enough for the point [x(t), Hx(t)] to be confined to a sector of the plane. In this case, it will be said that H is instantaneously confined to a sector o the plane. f Inequality (14) can be expressed in the form (HX) - cxtl12-rllxtl12<0. If norms are expressed in ddenotes slope ## Fig. 4. Graph of the rlelation in Example 2. The usefulness of Theorem 11 is limited by the condition that the open-loop gain-product be less than o n e a condition seldom met in practice. However, a reduced gain product can often be obtained by transforming the feedback equations. For example, if c I is added to and subtracted from &, as shown in Fig. 5, then e2 remains 11 l2 However, in engineering applications it is often more interesting to prove stability in the L, norm. The present theory has been extended in that direction in the author's [2f]. The idea is IZf] is to transform LZfunctions into L, functions by means of exponential weighting factors. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL CASE APRIL la: I a > 0 then H-I is inside { l/b, f l/a CASE l b : I a < O then H-l is outside { l / a , f l/b CASE 2: If a = 0 then (H-l- (l/b) I) is positive. 1 1 Interior of sector (v) Properties (ii), (iii), and (iv) remain valid with the terms "inside ( ] " and "outside ( interchanged throughout. (vi) g ( H ) s m a x ( l a [ , Ibl). Henceif H i s i n { -r, r ] then g(H) <r. The proofs are in Appendix A. One consequence of these properties is that it is relatively easy to estimate conic bounds for simple interconnections, where it might be more difficult, say, t o find Lyapunov functions. 4.3 A Theorem on Boundedness ## Fig. 6. A conic sector in the plane. terms of inner products then, after factoring, there is obtained the equivalent inequality ((Hx), - ax,, (Hx)t - bx,) Consider the feedback system of Fig. 1, and suppose find a condition on HI which will ensure the boundedness of the closed loop. A condition will be found, which places Hl inside or outside a sector depending on a and b, and which requires either Hl or H to be bounded away z from the edge of its sector by an arbitrarily small amount, A or 6. THEOREM 2a: [In eqs. (1)-(z)] Let HI and H be conic z f relations. Let A and 6 be constants, o which one is strictly positive and one is zero. Suppose that (I) - H2 is inside the sector ( a + A, b-A} where b>O, and, (11) Hl satis-es one o the following conditions. f CASE 50 [x ## E Do(H); t E T] (15) that HZis confined to a sector {a, b]. I t is desirable to where a = c - r and b = c+r. I t will often be desirable to such as (15)1 and a inspired by Fig. 6 is introduced: NOTATION: A conic relation H i s said to be inside the sector {a, b ) , if a 5b and if Inequality (15) holds. H is outside the sector { a , b ) if a < b and if (15) holds with the inequality sign reversed. The following relationship will frequently be used: If H is interior (exterior) conic with center c and radius r Conthen H is inside (outside) the sector ( c -r, c+r versely, if H is inside (outside) the sector ( a , b ] , then H i s interior (exterior) conic, with center (b+a)/2 and radius (b -a)/2. DEFINITION: A relation H in 6l is positive13if 1. ## l a : If a>O then HI i s outside (xt,(Hx)t)>O [xDo(H);tE~l. (16) CASE A positive relation can be regarded as degenerately conic, with a sector from 0 t o co. [Compare (15) and (16).] For example, the relation H on L ,is positive if it z is instantaneously confined (see Example 3) to the first and third quadrants of the plane. ## 1b : If a <0 then HI is inside CASE 2 : If a = 0 then ## 4.2 Some Properties o Conic Relations f Some simple properties will be listed. I t will be assumed, in these properties, that H and HI are conic relations; that H is inside the sector {a, b], with b>O; that HI is inside {al, b l ) with b1>0; and that k>O is a constant. (i) I is inside { 1, 1 ] (ii) k H i s inside {ka, kbj; - H i s inside {-b, -a]. (iii) SUM RULE: (H+ HI) is inside (a+al, b f bl]. (iv) INVERSE RULE l3 Short for "positive semidefinite." The terms "passiven and "nondissipative" have also been used. ## is positive; in addition, if A = 0 then g(H1) < 00. Then El and Ez are bounded. The proof of Theorem 2a is in Appendix B. Note that the minus sign in front of Hzreflects an interest in negative feedback. EXAMPLE 4: If HI and H are relations on Lz, instanz taneously confined t o sectors of the plane (as in Example 3), then the closed Ioop will be bounded if the sectors are related as in Fig. 7. (More realistic examples will be discussed in Part 11.) ## ZA,MES: STABILITY OF NONLINEAR FEEDBACK. SYSTEMS b CASE THEOREM 2b: Let HI and Hz be incrementally conic relations. Let A and 6 be constants, of which one i s strictly positive and one i s zero. Suppose that, 'v-4," \-m," (I) -Hz i s incrementally inside the sector { a + ~ , b -A where b > 0, and, (I I) HI satisjies one of the following conditions: 1, CASE 2: a::O CASE CASE I ## CASE lb: u<O CASE 2 : I a = 0 then f NOTE: I N ALL CASES. A>O,S=O. AND b>O. ADMISSIBLE REGlOFlS ARE SHADED Fig. 7. ## Mutually admissible sectors for Hzand H I i s incrementally positive; in addition, if A = 0 then g(H1) < 0 3 . Then El and Ez are input-output stable. The proof is similar t o that of Theorem la, and is omitted. ## 4.4 Incrementally Conic and Positive Relations Next, it is desired to find a s,tahility result similar to the preceding theorem on boundedness. T o this end the recent steps are repeated with all definitions replaced by their "incrernental" counterparts. DEFINITION: A relation H in (R i s incrementally interior (exterior) conic if there are real constants r > 0 and c for which the inequality A special case of Theorem 2, of interest in the theory of passive networks, is obtained by, in effect, letting a = 0 and b-+ a. Both relations then become positive; also, one of the two relations becomes strongly positive, 1.e. : DEFINITION: A relation H in (R i s strongly (incrementally) positive i f , for some o > 0, the relation (H- oI) i s (incrementally) positive. i s satisfied (with inequality sign reversed). A n incremenThe theorenn, whose proof is in Appendix C, is: tally conic relation H i s incrementally inside (outside) THEOREM 3 : a) " ' [1n eqs. (1)-(2)] If HI and -Hz are the sector ( a , b i j f a b and i f the inequality positive, and if -Hz is strongly positive and has finite gain, then El and E2 are bounded. (b) If HI and -Hz are incrementally positive, and if -Hz is strongly incrementally positive and has finite incremental gain, then i s satisfied (with inequality sign reversed). A relation H El and E2 are input-output stable. z For example, if H on L2, is instantaneously confined in @ i s incrementally positivle14 if to a sector of the plane, then, under the provisions of Theorem 3, the sector of Hzlies in the first and third quadrants, and is bounded away from both axes. EXAMPLE 5: Consider the relation H on Lz,, with Hx(t) = N [x(t) 1, where N is a function having a graph in the plane. If N is incremeatally inside { a , b } then N 5.1 Positivity and Passivity in Networks satisfies the Lipscl~itz conditions, a ( x - y) N ( x ) - N ( y ) A passive element is one that always absorbs energy. b ( x -y ) . Thus N lies in a slector of the plane, as in the I s a network consisting of passive elements necessarily nonincremental case (see Fig. 6), and in addition has stable An attempt will be made to answer this question upper and lower bounds to its slope. for the special case of the circuit of Fig. 2. Incrementally conic relatiions have properties similar First, an elaboration is given on what is meant by a to those of conic relations (see Section 4.2). 1, < < < The terms "monotone" and "'incrementally passive" have also been used. l4 l6 A variation of this result was originally presented in [2d]. Kolodner [S] has obtained related results, with a restriction of linearity on one of the elements. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL APRIL passive element. Consider an element having a current i and a voltage v; the absorbed energy is the integral j,"i(t)v(t)dt, and the condition for passivity is that this integral be non-negative. Now, let 2 be a relation mapping i into v ; by analogy with the linear theory, i t is natural to think of Z as an impedance relation; suppose Z is defined on Lz,, where the energy integral equals the inner product (i, v); then passivity of the element is equivalent to positivity of Z. Similarly, if Yon L2,is an admittance relation, which maps v into i, then passivity is equivalent to positivity of Y. Now consider the circuit of Fig. 2. This circuit consists of an element characterized by an impedance relation Zz, an element characterized by an admittance relation Yll a voltage source v, and a current source i. The equations of this circuit are, positive constant." In fact, the conic sectors defined here resemble the disk-shaped regions on a Nyquist chart. However, Theorem 2 differs from Nyquist's Criterion in two important respects: (1) Unlike Nyquist's Criterion, Theorem 2 is not necessary, which is hardly surprising, since bounds on HI and Hz are assumed in place of a more detailed characterization. (2) Nyquist's criterion assesses stability from observation of only the eigenfunctions exp (jwt), where Theorem 2 involves all inputs in X,. There is also a resemblance between the use of the notions of gain and inner product as discussed here, and the use of attenuation and phaseshift in the linear theory. A further discussion of this topic is postponed to Part 11, where linear systems will be examined in some' detail. One of the broader implications of the theory developed here concerns the use of functional analysis for the study of poorly defined systems. I t seems possible, from only coarse information about a system, and perhaps even without knowing details of internal structure, to make useful assessments of qualitative behavior. It is observed that these equations have the same form as the Feedback Equations, provided that the sources i and v are constrained by the equations v=alx+wl, and i = azx+wz. (By letting a1 = 0 the familiar "parallel circuit" is obtained. Similarly, by letting az=O the "series circuit" is obtained.) Thus there is a correspondence between the feedback system and the network considered here. Corresponding to the closed loop relation El there is a voltage transfer relation mapping v into al. Similarly, corresponding to Ez there is a current transfer relation mapping i into in.If Theorem 3 is now f applied to eqs. (20)-(21) it may be concluded that: I both elements are passive, and if, in addition, the relation o one o the elements is strongly positive and has finite f f gain, then the network transfer relations are bounded. ## A . Proofs o Proferties (i-wi) f Properties (i, ii). These two properties are immediately implied by the inequalities ( ( I x ) ~ 1-xt, ( I x ) ~ 1.xt) = = ((cHx)~ tax,, (cHx)~ cbxt) c ~ ( ( H x-~axt, ( H x ) ~ bxt) 5 0 ) in which c is a (positive or negative) real constant. Property (iii). I t is enough to show that (H+Hl) has center \$(b+bl+a+al) and radius \$(b+bl-a -al) ; the following inequalities establish this: ## 11 [(H + H~)x]t \$(b + 61 + a + al)xtll - The main result here is Theorem 2. This theorem provides sufficient conditions for continuity and boundedness of the closed loop, without restricting the open loop to be linear or time invariant. Theorem 2 includes Theorems 1 and 3 as special cases. However, all three theorems are equivalent, in the sense that each can be derived from any of the others by a suitable transformation. There are resemblances between Theorem 2 and Nyquist's Criterion. For example, consider the following, easily derived, limiting form of Theorem 2: "If H 2 = k I then a sufficient condition for boundedness of 1 the closed loop is that H be bounded away from the critical value - (1/k) I, in the sense that + 11 (HIX)~ + ( b ~ al)ll (Triangle Ineq.) + 5 +(b - a)llxtll + t(bl - a1)llxtIl +(b + b l - a - al)llxtll = ## < j l ( ~ x ) t- \$(b + a)xtl( (Ala) (Alb) where eq. (Alb) follows from eq. (Ala) since H has center +(b+a) and radius +(b-a), and since HI has center +(bl+al) and radius + ( b ~ - a d . Property (iv). CASES la AND 1966 ## ZAMES: STABILITY OF NONLINEAR FEEDBACK SYSTEMS where H-lx =y and x = Hy. Since, by hypothesis, H is inside {a, b } and b > 0, the sign of the last expression is opposite to that of a. Thus the Inverse Rule is obtained. CASE 2: Here a=O. The property is implied by the inequality, 1 (xt, (B1x)t - - xt) b = (111) (Using Fig. 5 as a guide,) define two new elements of X,, yz' = yz el' = el + cyl. + cez (A64 (A6b) ## - ((Hy)t, byt - (Hy)t) > 0. b Property (v). Simply reverse all the inequality signs. Property (vi). II<~x>tll I l ( ~ x ) - +(b -t a)xtll t < I t shall now be sbown that there are elements Hltel' and Hzlez' in X, that satisfy eqs. (A3)-(A4) simultaneously with the elements defined in (I)-(111). Taking eqs. (A3)-(A4) one a t a time: Equation (A3a). Substituting eq. (la) for el in eq. (A6b), and eq. (lb) for yl, ## + 11\$(b + a)xtli (Triangle Ineq.) (A2a) (A2b) where eq (A2b) follows from e:q (A2a) since, from the hypothesis, H has center \$(b+a) and radius +(b-a). I t follows that g(H) S m a x (I a ] , (bl). Q.E.D. B. Proof o Theorem 2a f The proof is divided into three: parts: (1) The transformation of Fig. 5 is carried out, giving a new relation Ezl; Ez' is shown to contain Ez. (2) The new gain product is shown t o be less than one. (3) Ez' is shown to be bounded, by Theorem 1; the boundedness of E2 and El follows. Let c=+(b+a) and r=+(b-a). If w1' =wl-cwz and al' =al-caz, then, with the aid of eq. (A6a), eq. (AL7) reduces to eq. (A3a). Equation (A3b): This is merely eq. (lb), repeated. Equation (A4a): Recalling that H2'=Hz+cI, it follows, from eqs. (A6a) and (2a), that there is an Hzte2 in X, for which eq. (A4a) holds. Equation (A4b): If eq. (A6b) is substituted for el in eq. (2b), it is found that there exists Hl(elt-cyl) in X, such that y l = Hl(el' - cyl). Therefore, (after inversion) (after inversion) Hl-lyl = e - cy1 : (after rearrangement) (HI-' cI)yl = el' yl (HI-' + cI)-'el'. B.1 Transformation o Eqs. (1)f (i?) The proof will be worked biackwards from the end; the equations of the transformed system of Fig. 5 are, el' ez y2' yl where := That is, there exists Hllel' in X, for which eq. (A4b) holds. Since eqs. (A3)-(A4) are all satisfied, (x, ez) is in Ezl. Since (x, ez) is an arbitrary element of Ez, Ez' contains Ez. w,' = = + al:x + y; .= H,'e,' ## Hz' = (Hz GI) I\$< = (HIw1 c1)-1. +. (A54 (A5b) B.2 Boundedness o E: f I t will be shown that g(H1') .g(Hzt)< 1. The Case A >0, 6 = 0: g(Hzl) will be bounded first. Since H z i sin ( -b+A, -a-A} by hypothesis, (Hz+cI) is in (-b+A+c, -a-A+C} by the Sum Rule of Section 4.2. Observing that (Hz+cI) =Hz', that (- b+c) = -7, and that (-a+c) = r, it is concluded that H ' is 2 in ( -r+A, r - A } . Therefore g(Hsl) < r -A. Next, g(Hll) will be bounded. In Case l a , where a>O and HI is outside ( I t may be observed that these equations have the same form as eqs. (1)-(2), but HI is replaced by HI' and Hz is replaced by HZ'.) Let Ez' be the closed-loop relation that consists of all pairs (x, ez) satisfying eqs. (A3)(A4). I t shall now be: shown that Ez' contains Ez, that is, that any solution of eqs. (1)-('2) is also a solution of - .. .. eqs. ( ~ 3 ) - ( ~ 4 )thus boundedness of E2' will imply ; boundedness of Ez. In greater detail the Inverse Rule of Section 4.2 implies that H1-' is outside { -b, - a } ; the same result is obtained in Cases l b and 2. In all cases, therefore, the Sum Rule implies that (HI-l+cI) is outside ( -r, r }. By the Inverse Rule again, (HI-'+GI)-' is in {-;'+1}. Therefore g(Hll) 5 l/r. Finally, (I) let (x, ez) be any given1 element of Ez. (11) Let el, yl, - Hlel, anld Hzez be fixed elements - - YZ, of X, that satisfy eqs. (1)-(2) simultaneously with x and er. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL The Case A = 0, 6 >0: I t shall be shown that this is a special case of the case A>0, 6 =O. In other words, it will be shown that there are real constants a, b, and A* for which the conditions of the case A >0, 6 = 0 are fulfilled, but with a replaced by a, b by b, and A by A. Consider Case l a , in which a >0. (Cases l b and 2 have similar proofs, which will be omitted.) I t must be shown , that: (1) -H2 is in { a + ~ b-A]. (2) HI is outside ## for any x in X , and for any t in T. Hence, for any r>O, Without loss of generality it can be assumed that 6 is smaller than either l/a or l/b. Choose a and b* in the ranges a b < a* < a and b < b* < --1 as 1 - bs Equation (All) was obtained by expanding the square on its l.h.s., and applying eqs. (A9) and (A10). Constants X, r , and A, are selected so that X>a, r=X2/u, and A = r [1 -dl - (U/X)~].Now it can be verified that, for this choice of constants, the term (A2-2ru+r3 in eq. (A11) equals (r -A)2; also, 0 <A <r since ( a h ) < 1) ; therefore eq. (All) implies that H is conic with center z - r a n d radiusr-A. Q.E.D. The author thanks Dr. P. Falb for carefully reading the draft of this paper, and for making a number of valuable suggestions concerning its arrangement and concerning the mathematical formulation. [I] V. M. Popov, "Absolute stability of nonlinear systems of automatic control," Automation and Remote Control, pp. 857-875, March 1962. (Russian original in August 1961.) [2] (9) G;, Zames, "Conservation of bandwidth in nonlinear operations, M.I.T. Res. Lab. of Electronics, Cambridge, Mass., Quarterly Progress Rept. 55, pp. 98-109, October 15, 1959. (b) , "Nonlinear operators for system analysis," M.I.T. Res. Lab. of Electronics, Tech. Rept. 370, September 1960. (c) ----:, "Functional analysis applied to nonlinear feedback systems, IEEE Trans. on Circuit Theory, vol. CT-10, pp. 392404, September 1963. (d) , "On the stability of nonlinear, time-varying feedback svstems." Proc. NEC. vol. 20. DD. 725-730. October 1964. (6) --, "Contracting transf&mations-A theory of stability and iteration for nonlinear, time-varying systems," Summaries, 1964 Internat'l Conf. on Microwaves, Circuit Theory, and Information Theory, pp. 121-122. (f), ''Nonlinear time varying feedback systerns-conditions for L m -boundedness derived using conic operators on exponentially weighted spaces," Proc. 196.5 Allerton Conf., pp. 460-471. [3] K. S. Narendra and R. M. Goldwyn, "A geometrical criterion for thestability of certain nonlinearnonautonomous systems," IEEE Trans. on Circuit Theory (Correspondence),vol. CT-11, pp. 406408, September 1964. [4] R. W. Brockett and J. W. Willems, "Frequency domain stability criteria," pts. I and 11, 1965 Proc. Joint Automatic Control Conf., pp. 735-747. [S] (a) I. W. Sandberg, "On the properties of some systems that distort signals," Bell Sys. Tech. J., vol. 42, p. 2033, September 1963, and vol. 43, pp. 91-112, January 1964. (b) -,"On the Lz-boundedness of solutions of nonlinear functional equations," Bell. Sys. Tech. J., vol. 43, pt. 11, pp. 1581-1599, July 1964. 1 1 1 Kudrewicz. "Stabilitv of nonlinear feedback svstems." Auto6 . matika i ~e1emechunika;vol. no. 8, 1964 (and-other papers). 25, [7] E. H. Zarantonello, "Solving functional equations by contractive averaging," U. S. Army Math. Res. Ctr., Madison, Wis. Tech. Summary y p t . 160, 1960. [8] I . I. Kolodner, Contractive methods for the Hammerstein equation in Hilbert spaces," University of New Mexico, Albuquerque, Tech. Rept. 35, July 1963. [9] G. J. Minty, "On nonlinear integral equations of the Hammerstein type," survey appearing in Nonlinear Integral Equations, P. M. Anselone, Ed. Madison, Wis.: University Press, 1964, pp. 99-154. [lo] F. E. Browder, "The solvability of nonlinear functional equations," Duke Math. J., vol. 30, pp. 557-566, 1963. [I 11 J . J.. Bongiorno, Jr., "An extension of the Nyquist-Barkhausen stability criterion to linear lumped-parameter systems with timevarying elements," IEEE Trans. on Automatic Control (Correspondence), vol. AC-8, p 166-170, April 1963. [12] A. N. Kolmogorov and V. Fomin, Funcfional Analysis, vols. I and 11. New York: Graylock Press, 1957. ~ Since -Hz is in (a, b ) by hypothesis, and since a* < a and b* > b by construction, there must be a A* > O.such that Hzsatisfies condition (1). Since HI is outside ## by hypothesis, and since by construction condition (2) is satisfied. Hence this is, indeed, a special case of the previous one. ## B.3 Conclusion o the Proof f Since g(Hll) .g(H2') < 1, Ez' is bounded by Theorem 1, and so is E2, which is contained in Ezl. Moreover, the boundedness of Ez implies the boundedness of El; for, if (x, el) is in El and ( x , ez) is in Ez, then <const. Thus, if Ilxl1 Iconst. and Ilezl[Iconst., then /[ell\ (Inequality (A8) was obtained by applying the Triangle Inequality and Inequality (4) to eq. (la), and taking the limit as t-+ a,. I t may be noted that g(Hz) < a ,since -Hz is in ( a , b ) by hypothesis.) Q.E.D. C. Proof of Theorem 2 This shall be reduced to a special case of Theorem 2 [case 2 with 6 = 01. In particular, it shall be shown that there are constants b > 0 and A> 0 for which (I) -Hz is inside {A, b - A ) , and, (11) the relation [HI+ (l/b)i] is positive. [HI+ (l/b)i] is clearly positive for any b> 0, since by hypothesis Hz is positive; the second condition is therefore satisfied. T o prove the first condition it is enough to show that Hz is conic with center -r and radius r-A, where r=b/2. This is shown as follows: The hypothesis implies that, for some constant o>O and for any constant X >g(Hz), the following inequalities are true > " 8.. On the Input-Output Stability of Time-Varying Nonlinear Feedback Systems-Pa,rt 11: Conditions 1nvolvin.g Circles in the Frequency Plane and Sector Nonlinearities G. ZAMES, MEMBER, I E E E x--Abstract-The object of this papter is to outline a stability theory LINEAR TIME-INVARIANT E l based on functional methods. Part 1 of the paper was devoted to a \ general feedback configuration. Part I1 is devoted to a feedback system consisting of two elements, one of which is linear time-invariant, and the other nonliiear. An attempt is made to unify several stability conditions, including Popov's condition, into a single principle. This principle is based on the concepts of conicity and positivity, and provides a link with the notions of gain and phase shift of the linear theory. Part I1 draws on the (generalized) notion of a L'sector non. ' linearity." A nonlinearity N i s said to be INSIDE THE SECTOR (a,P ) if it IE2-. X NONLINEAR f satisfies an inequality of the type ((Nx-ax),, (Nx-px)\$? 5 0 . I N NO MEMORY 1 is memoryless and is characterized by a graph in the plane, then this simply means that the graph lies inside a sector of the plane. Fig. 1. A feedback system. However, the preceding definition extends the concept to include nonlinearities with memory. THEOREM, There are two main results. The first result, the CIRCLE asserts in part that: If the nonlinearity is inside a sector {a,P } , and if the frequency response of the linear element avoids a ('critical region" in the complex plane, then the closed loop is bounded; if CX>O then the critical region is a disk whose center is halfway between the points -l/a and -I/& and whose diameter is greater than the distance between these points. The second result is a method for taking into account the detailed properties of the nonlinearity to get improved stability conditions. This method involves the removal of a "multiplier" from the linear element. The frequency response of the linear element is modified by the removal, and, in effect, the size of the critical region is reduced. Several conditions, including Popov's condition, are derived Fig. 2. If Nx(t) vs. x ( t ) and H ( j w ) lie in the shaded regions, and if the Nyquist diagram of H(jw) does not encircle the critical disk, by this method, under various restric.tions on the nonlinearity N; the then the closed loop is bounded. following cases are treated : (i) N i s instantaneously inside a sector (a,p} (ii) N satisfies (i) and is memoryless and time-invariant. (iii) N satisfies (ii) and has a restricted slope. supposed, for the moment, that N has no memory. These assumptions are among, the simplest which ensure that the system is both (i) general enough to have many applications (ii) complicated enough t o exhibit such characteristic nonlinear phenomena as jump resonances, subharmonics, etc. T h e object hlere is to find stability conditions for the closed-loop system. For practical reasons, it is desirable to express these conditions in terms of quantities that can be measured experimentally, such as frequency responses, transfer characteristics, etc. I n particular, the following question is of interest: Imagine that the graph of N lies inside a sector of the plane, as shown in Fig. 2(a), and t h a t the frequency response of H is plotted in the complex plane; can the complex plane be divided HE feedback system of Fig. 1 consists1 of a linear time-invariant element H and a (not necessarily linear or time-invariant) element N. I t will be Manuscript received December 29, 1964; revised October 1, 1965, and May 10, 1966. Parts of this paper were presented a t the 1964 National Electronics Conference [la], Chicago, Ill. This work was carried out in part under support extended by the National Aeronautics and Space Administration under Contract NsG-496 with the M.I.T. Center for Space Research. The author is with the Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Mass. A single input x , multiplied by real constants a1 and az, is added in a t two points. By setting al or a2 tto zero, it is possible to obtain a single-input system, in which the element closest to the input is either the linear element or the nonlinearity. The terms wl and wz are fixed bias functions, which will be used to account for the effects of initial conditions. The variables el anld ez are outputs. Reprinted from IEEE Trans. Automat. Contr., vol. AC-11. pp. 465-476, Jul. 1966. 235 IEEE ## TRANSACTIONS ON AUTOMATIC CONTROL JULY into regions that are "safe" or "unsafen as far as stabil- cases, the LZconditions imply L,-boundedness or continuity. For physical applications the most appropriate ity is concerned? I t will be shown that, with certain qualifications, such definitions of boundedness and continuity are, of course, a division is possible. In fact it has already been shown obtained in the L, norm. in Part I that such regions, called "conic sectors," exist f DEFINITION: (Ro be the class o relations on L2, havLet in a quite general sense. Here these general results will be ing the property that the zero element, denoted o, is in applied to some concrete situations, involving frequency Do(H), and Ho = o. An operator H on Lzeis any function responses, etc. (Fig. 2, which illustrates the simplest of the type H: Lz,-+L2,. DEFINITION: operator H on Lzeis TIME-INVARIANT An of the results to be obtained here, gives some idea of i it commutes with all delays. That is, for t 2 0 let T, be f what is being sought.) the operator on L2, given by: T,x(t) =x(t-T) for t > ~ , and T,x(t)=O for t < ~ Then HT,=T,Hfor all 7 2 0 . . H is MEMORYLESS if Hx(t) is a function o x(t) (i.e., f The purpose of this section is to define H a n d N, and f to write feedback equations. H and N will be repre- only o x(t)) for all x i n L2, and for all t 2 0 . sented by input-output relations or by operators, in 2.1. The Operator Classes 3t and 2 keeping with the theory outlined in Part I. DEFINITIONR [o, m) is the space o real-valued func: f DEFINITION X is the class o operators on L2, having : f f the following property: I N is in 3t then there s a function, tions on the interval [0, w ). L,, where p = 1, 2, . . - , is the space consisting o those N: Reals-+Reals, satisfying2 f x in R [0, a ) for which the integral f," x(t) ( pdt is finite. I n addition, for the case fi = 2, it is assumed that L2 is an and having the following properties: (i) N(0) = 0, (ii) inner-product space, with inner-product N(x) <const. XI, and (iii) for any real x, ~ ~ ~ ( x ' ) d x ' is jinite. An operator in 3t is memoryless, time-invariant, not necessarily linear, and can be characterized by a graph and norm IIxll2. The symbol IIxIl, without subscript, will in the plane. The letter N will indicate the graph of N. often be used instead o IIxllz. f 6: f DEFINITION. is the class o those operators H on Lz, L, is the space consisting o those functions x i n f satisfying a n equation of the type2 R[O, m) that are measurable and essentially bounded. L, is assumed to be a normed linear space, with norm in which h, is a real constant, and the impulse response h is a function i n LI with the property that, for some cro<0, h(t) exp (-gat) is also i n h. Operators in 6: are linear and time-invariant. 2.2. Feedback Equations Consider the feedback system of Fig. 1, but with two modifications: (i) N is not necessarily memoryless; (ii) a1 and a2 are operators on Lze,multiplying x. (This L,,= ( X ~ X E R [ O , and x t E L p f o r a l l t > O } . m) amount of generality will be needed for some of the An extended norm [[xllpe defined on L,,, where (lxlIp, intermediate results; ultimately, the interesting case is is if x E L , and 1Ixllpe=m if XGL,. The symbol that in which N has no memory, and al and a2 are real x ze will usually be abbreviated to IIxIl,. constants.) The equations of this system are The concept of a relation H on L,,, with domain Do(H) and range Ra(H) was introduced in Part I. A relation H on L,, is L,-bounded if H maps bounded subsets of L,, into bounded subsets of L,,. H is L,-continuous if, given any x in Do(H) and any A>0, there is a in which i t is assumed that: 6>0 such that, for any y in Do(H), if l l ~ - ~ l l , , < 8 then H is a n operator in 6: N is a relation in (Ro Part I1 will be devoted entirely to finding L2 conditions (for boundedness and continuity), since these are easier to derive than the other L, conditions. However, V t can be verified that every mapping of !he type N:L,+R[O, m ) is Lzc. Slm~larly, every most of the results of this paper have been extended to satisfying (1)a) in fact an operator onoperator on L2. [see mapping H:L2+R[0, satisfying ( 2 ) is an (B1) of the L, norm, in [lb]. I t has been found that, in most Appendix B]. No distinction will be made between functions difering over sets o zero measure. f Those definitions which were introduced in Part I will only be summarized here. Following the convention of Part I, the subscripted symbol xt denotes a function in R[o, m) truncated after [0, t). The space L,,, where p = 1 , 2 , . - ., m , is the extension of L,, i.e., r1Ix" , IIHX-HY~~, ~ <A. 1966 ## ZA.MES: STABILITY OF TIME-VARYING FEEDBACK SYSTEMS (a) N IS INSTANTANEOUSLY INSIDE A SECTOR x in L2, i s a n input el and en in Lz, are outputs w , and wz in L are fixed bizses 2 either a l and a2 are real constants, or, more generallyJ3 al i s a relation on L2, having the property that [lalxl[,<const. IIxII,, and similarly for az. REMARK: to begin with, the linear element satisIf, fies a state equation, then Hel is set equal to the "zeroinitial-condition response" of the state equation, and wz is set equal to the "zero-input response." The closed-loop relations which map x into el and ez will be denoted El and Ez. The objective here is: Find conditions on N and H which ensure that El and Et are L2-bounded and L2-continuous. 3. CONDITIONS CONICITY POSITIVITY FOR AND (b) N IS POSITIVE jd.~) Fig. 3. ## Permissible regions (shaded) for instantaneously confined nonlinearities. This section has some preliminary results, which will be needed later in the analysis of stability. The following questions are fundamental to this analysis. Under what conditions is an operatolr conic or positive? Under what conditions is the composition product of two operators conic or positive? The definitions of conicity and positivity were introduced in Part I. They are repeated here, for the special case of relations on Lzc. DEFINITION: H be a reliztion in 6to. H i s INTERIOR Let CONIC if G and r 2 0 are real constants and the inequality II(flx)t-cxtl\I~xtl\ Fig. 4. ## Permissible regions (shaded) for the frequency response HCjo). (xEDo(H);tLo) (4) holds. H i s exterior conic if (4) holds with the inequality sign reversed. c will be called the center parameter of H, and r will be called the radius parameter of H. H i s INSIDE (OUTSIDE) SECTOR { a , 0 j if a la and THE if the inequality inside { a,0 if, in addition, N satisfies the slope restric-X tions a 5 [ ~ ( x )N ( ~ ) ] / ( y ) I P . N is positive if its graph lies in fhe first and third quadrants; N is incrementally positive if, in addition, N is nondecreasing. 3.2. Linear Time-Invariant Operators holds (with the inequality sign reversed). H i s POSITIVE if it satisjies the inequality (xt, ( H x )t) 2 0 for all x in Do(.H) and ah! t>O. In Part I , the concepts of conicity and positivity were subdivided into categories such as "instantaneous" conicity, "incrementaln conicity, etc. The definitions of these terms are listed in Appendix A. REMARK: The following conditions are equivalent: (i) H is interior conic with parameters c and r. (ii) H is inside {c-r, c+r f . Consider the operator class S ; i t will be shown, roughly speaking, that a conic sector has a counterpart in the frequency plane, in the form of a circular disk (see Fig. 4). This disk degenerates into a half-plane in the case of a positive operator. : Let DEFINITION s = u jw denote a point in the complex plane. The LAFLACE TRANSFORM H ( s ) of H in d i s : ~ ( s = h. ) ## + Jomh(t) exp (-st)dt (0 > 0) (6) (The integral on the right-hand side of ( 6 ) exists and is analytic for u 3 0 [See ( B l ) of Appendix B ] . ) DEFINITION: The NYQUIST DIAGRAM of H ( s ) i s a curve in the complex plane consisting of: (i) the image of the 3.1. Memoryless, Time-Invariant Nonlinearities jw-axis under the mapping H ( s ) , and (ii) the point h,. Consider the operator class X ;the conditions for N LEMMA Let H be a n operator in 2, and let c and r 0 1. in X to be conic, positive, etc., are simply the "instan- be real constants. taneous" conditions of Appendix A. Some of these con(a) If H ( s ) satisfies the inequality ditions are illustrated in Fig. 3. In particular, N is inside the sector {a, 3 if its graph lies in a sector of the plane bounded by lines of slopes a! and 0; N is incrementally > ## The more general assumption vvill be needed in Section 5 only. then H i s incrementally interior conic with center parameter c and radius parameter r. JULY ## (b) I H(s) satisfies the inequality f and if the Nyquist diagram o H(s) does not encircle f the point (c, 0), then H is incrementally exterior conic with center parameter c and radius parameter r. (c) I ~ e { ~ C j w ) ) for w E ( - w , w ) then H is f 20 incrementally positive. The Proof of Lemma 1 is in Appendix B. REMARK. The gains g(H) and g(H) were defined in Part I. I t follows from Lemma 1(a) that if H(jw) I c , then g(H) =g(H) I c . Fig. 5. Products of intervals. ## 3.3. Composition Products and Sector Products The composition product of two positive operators need not be positive. Those special cases in which the product is positive are of interest because they give a tighter bound on the composite behavior than would be obtained in general. (They form the basis of the factorization method of Section 5.2.) Similarly, those special cases in which the product of two sector operators lies inside the "product sector" are Fig. 6. at3 operators. (a) Typical pole-zero pattern. (b) Circle of confinement of Kern). of interest. DEFINITION: PRODUCT SECTOR (al, PI ] X t a z , Pz) The 3.5. A Memoryless Nonlinearity and a n at3 Multiplier is the sector (a, ) , where [a, /3] is the interval o the p f A situation resembling Lemma 2, but with N more reals defined by [a,PI = { X Y x E [ a , PI] and Y E[az, Pz] ] . In other words, product sectors behave like point- restricted and K more general, is considered next. K is wise products of the corresponding real intervals (see taken to be a sum of first-order terms, of a type that can be realized as the driving-point impedance of an Fig. 5). RC network (see Guillemin [2], ch. 4). I t is easy to show that if both operators are memoryDEFINITION: ae be the class o those operators K Let f less, say if both operators are in %, then their product in 8 having Laplace transforms o the form f has the above mentioned properties. [This can be shown by expressing the ratio Nl(Nz(x))/x as the product (Nl(y)/y) X (Nz(x)/x) , where & ~ z ( x ) . ]More difficult are cases in which one operator is in % and the other is in 8 , as in Lemmas 2 and 3. where ki 2 0 , Xi > 0, and K ( w ) 0 are real constants. An operator K i n at3 has poles and zeros alternating 3.4. A Memoryless Nonlinearity and a First-Order on the negative-real axis, with a pole nearest but not a t Multiplier the origin. The frequency response of K lies inside a The following lemma is the basis for Popov's condi- circle in the right-half plane, located as shown in Fig. 6(b); it follows that K is positive and inside the sector tion (Section 5.1). LEMMA Let N be an operator in %, K be a n operator { K( a, ), K(0) ) . (Observe that 2. f in C ,and let the Laplace transform o K be K(s) = kX/s+X where k > 0 and X >0. (a) I N is positive4 then NK is positive. f (b) If N is inside5 a sector ( a , 0) then NK is inside LEMMA Let N be an operator in Z, 3. and K be a n the product sector { a , p ) X (0, k ) . operator in m . e > The proof of Lemma 2 is in Appendix C. (Note that K itself is positive and inside ( 0 , k ) , since K(jw) lies entirely in the right-half plane and since K(jw) -\$kl = +k.) (a) If N is incrementally positive then NK is positive. f (b) I N is incrementally inside the sector (or, fl]then NK is inside the product sector (a, ) X { K ( w ) , P K(O) ) . In other words, multiplication by K affects the composite sector as if K had no memory. The proof of Lemma 3 is in Appendix D. 1966 ## ZA.MES: STABILITY OF TIME-VARYING FEEDBACK SYSTEMS Consider now the niain problem of this paper, namely, the problem of stability for the loop of Fig. 1. Suppose that N is a relation (which may or may not be memoryless) inside a sector { a , p ] . What conditions on the frequency response H ( j w ) are suficient to ensure boundedness of the closed loop? I t will appear that the following "circle conditions" are sufficient: DEFINITION: H(joj will be said to SATISFY THE CIRCLE CONDITIONS FOR THE SECTOR f a , P ) , WITH OFFSET 8, where a 5 P, 6 > 0, and 6 >_ 0 are real constants, if the following conditions hold: CASE1 ~If. a>O, then CONTINUITY D1SK FOR (b) Fig. 7. Critical disks for Example 1. (Broken curve indicates edges of jump region in H ( j o ) plane.) and the Nyquist diagram o HCjw) does not encircle the f point -+(l/or+l/P). CASE1 ~I . a < O , then f CASE 2. I a=O, then Re{HCjw)f -(I/P)+6 for f wE(- Q), ). a In other words, the complex plane is divided into two regions, shaped either like a circuliar disk and its complement, or like two half-planes. (The case a > O is illustrated in Fig. 2") One of the regions will be called "permissible" and the other will be called "critical." If H(jw) does not enter or encircle the critical region, then the closed loop is bounded. I f , in addition, N is incrementally inside for, /3), then the closed loop is continuous. These results are formalized in the following theo1 > The Circle Theorem can be viewed as a generalization of Nyquist's ~ r i t e r i o n , ~ which a critical region in replaces the critical point. For a given N there are two critical regions, one for boundedness and one for continuity. I t can be shown that the continuity region always contains the boundedness region (see Example l and Fig. 7). The Circle Theorem will serve as the generating theorem for the rest of this paper; i.e., the remaining results will be obtained as corollaries to the Circle Theorem by variously constraining the form of N. In particular, the following corollary is obvious. ## 4.1. A Circle Conditionfor Instantaneous Nonlinearities7 COROLLARYI (I) N in 6iois instantaneously (incre1. f mentally) inside the sector f a + A , & A ) where 8 > 0, and i conditions (11) and (111) o the Circle Theorem hold, f f then El and E2are Lz-bounded (L2-continuous). EXAMPLE (a) Let No be the relation shown in Fig. 1. ,-om. c111. 7(a), and N be the relation in 6iodefined by the equation Nx(t) = [l+sin2 ( t ) ] . No(x(t)). Find the critical regions A CIRCLE THEOREM. Suppose that (I) N is a relation in (Ro, (i~zcrementally)inside the for boundedness and continuity. (b) Repeat for the function shown in Fig. 7(b). sector f a + A , @-A], n~here p>0. (a) Observe tlhat No is inside the sector ( 113, 1 ) and (11) N is an oper.ator i n C ,zohich satisJies the circle that the time-varying gain [1+sin2 ( t ) ] is inside the conditions for the sector { a ,@ ) with offset 6. f (I 11) 6 and A are %on-negativeconstants, at least one o sector { 1, 2 ) . I t follows that N is inside the product sector, (113, 1 ) X (1, 2 ) = f1/3, 2 ) . Corollary 1 therewhich is greater than zero. fore implies that the critical region for boundedness is a Then the closed-loop operators , land E2 are L2-bounded disk, as shown in Fig. 7 . However, since N is multiE valued, and therefore not incrementally in any sector, (Lz-continuous). The Circle Theorem is based on Theorem 2 of Part I. Corollary 1 provides no information about continuity. (b) The same results as in (a) are obtained for boundI t was assumed in Theorem 2 that al and az were real edness. In addition, N is incrementally inside f 1/6, 4 ) , constants. However, with only minor changes in the proof, it can be shown that Theorem 2 holds more gen- and a continuity disk is obtained, as shown in Fig. 7. erally if a1 and az are relations on Lz,, provided al and a2 satisfy inequalities of the type lla,x\l,<const. IIxlle. The More accurately, of the sufficient part of Nyquist's criterion. Similar or closely related circle conditions were found indeCircle Theorem then follows i~mmediatelywith the aid pendently by the author [la], Sandberg [S], Narendra and Goldwyn of Lemma 1 of Part 11. [6], arid Kudrewicz (71. ## IEEE TRANSACTIONS O N AUTOMATIC CONTROL, JULY Observe that the nonlinearity N in Corollary 1 can be time-varying and can have memory. In fact, very little has been assumed about the detailed character of N. The price paid for this is that Corollary 1 is often conservative, i.e., the critical region is too large. This is especially true of the boundedness condition (see Example 2). T h e continuity condition probably gives a quite fair estimate of what to expect. In fact, an approximate analysis, based on the harmonic balance method (cf. Fig. 8 A transformation. . Hatanaka [3]), suggests that continuity breaks down in the following way: There is a zone, inside the critical Letting wz' = wzfX-lwz, and recalling that a2= 0, concontinuity disk, in which jump-resonance phenomena sider the equations of the transformed system of Fig. 8, occur. The zone is not much smaller than the continuity disk. Furthermore, the magnitudes of jump resonances depend on the Nyquist diagram behavior inside the continuity disk. Let E' and Ez' be the closed-loop relations for (12a)I (12b). STEP 1. El1 and &I are L2-bounded. This follows from T h e next two corollaries can be viewed as attempts to the Circle Theorem whose hypotheses are satisfied bereduce the size of the critical region, a t the cost of added restrictions on N. In certain cases, i t will be possible to cause: wz' is in L2 by assumption IV; N K is in the prodremove a "multiplier" K from the linear element, be- uct sector ( a , p} X ( 0, 1) by Lemma 2; and N K satisfore applying the Circle Theorem. The removal of K fies the appropriate circle conditions. STEP I t will be shown (below) that 2. will shift the frequency response of the remainder, HI&), away from the critical region. Thus the effective size of the critical region will be reduced. ## 5.1. Popov's Condition Consider the feedback system of Fig. 1, under the same conditions as in Corollary 1, but with the added constraint that N is a memoryless, time-invariant operator. The following condition for boundedness (not continuity) involves the removal of a first-order multiplier from H . COROLLARY I 2. f (I) N is a n operator i n %, inside the sector ( a , p ] where j3 >0. (11) H i s a n operator i n d: that can be factored into a product H = KHl, where HI and K are i n C , and K(s) =X/(s+X) where X > 0. (111) HI satisfies the circle conditions for the product p sector (a, ] X (0, 1 } with ofset 6, where 6 >0. 2 (IV) a2 = 0 and wz is i n Lz, where w denotes the derivative on [0, a). Then El and E are L2-bounded. z REMARKS: For a > 0, Condition I11 simply means (i) (ii) Condition IV that Re { Cjw+X)HCjw) ) 2 -X/P+6. limits the result to that configuration in which the directions of flow is from the input to H to N. PROOFOF COROLLARY The feedback equations will 2. be transformed, as illustrated in Fig. 8; i.e., H will be split into a product, H = K H l , and the multiplier K will be transferred into a composition with N. I t will then be shown, in Step 1, that the transformed equations are bounded, and, in Step 2, that they are equivalent to the original equations as far as stability is concerned. Since &' and Ez' have been proved L2-bounded, and since K is certainly Lz-bounded, i t follows that El and E are Lz-bounded. z T o prove (13b), recall that Ez and Ezf are subsets (of a product space), so that it is enough to establish that each contains the other. Suppose that ( x , ez) is an element of Ez; by definition of Ez, there is an el in L2, satisfying (3a)-(3b) ; let el1=el and ez' = wz' f Hlel. Direct substitution shows that (x, el1, ezf)satisfies (12a)-(12b), so that (x,-ezl) belongs to Ezf.Substitution also shows that e2= Kez', so that (x, ez) is in KEz'. Since ( x , e2) is an arbitrary element of Ez, i t follows that KEZ1 contains Ez. I t can similarly be shown that Ez contains KE;, so that (13b) holds. The proof of (13a) is similar. Q.E.D. EXAMPLE Let N be the operator in % whose graph 2. is shown in Fig. 9(a), and Iet H(s) = k/(s+X)(s+p). For what values of k is the closed-loop L2-bounded? Compare Corollaries 1 and 2. Here N is inside (0, I } , so the critical region is a half. plane, Re ( . 1 - 1 6, in both corollaries. In Corollary 2 let K(s) =X/(s+X) and Hl(s) = k/X(s +p) ; the following estimates are obtained: Corollary 1: - ~ p < k < ( ~ + p ) ( ~ - t - , u + 2 d ~ Corollary 2: -Xp <k Corollary 2 is less conservative than Corollary 1, as i t shows the closed loop to be bounded for all positive k. Figure 10 also illustrates the following point: For a I O , both corollaries predict the same critical region; however, in many cases of interest, H l o is further from the critical region than HCjw). ## 5.3. A Slope-Restricted Nonlinearity and an plier (Fit? Multi- Consider the feedback system of Fig. 1, under the conditions of Corollary 2, but with an added slope restriction on IV, and a more general type of multiplier K. COROLLARY If: 3.8 Fig. 9. An example of Popov's Condition. Arrows indicate shift away from critical region. (I) N i s a n operator i n X , incrementally insideg {a, where fl> 0. 13) (11) H is a n operator i n 2,which can be factored into a product H = KHl, where K is i n (Fit? and HI i s i n 2. (111) HI satisjies. the circle conditions for the product , with ofset 6, sector (or, fl) x ( ~ ( w ) ~ ( 0 ) ) where 6 > 0. (IV) There is1O a wz' such that K(wz') = wz. (V) Eith'er K ( w ) > O or az=O. Then EI and E2 are Lz-bounded. Corollary 3 is obtained immediately by the factorization method, with the help of Lemma 3. ## 5.2. The Factorization Method T h e proof of C:orollary 2 suggests a method for generating a class of Popov-like conditions. The method consists of a factorization of H into H = KHl, followed by the transformation of (3a)-(3b) into (12a)-(12b) followed by an application of the Circle Theorem. Various stability conditions are produced by variously choosing the multiplier K . The method has two preca'nditions: (Ia) Either K--l exists or a2= 0. (Ib) There is a wz' in LZsuch that K(w4) = wz. These preconditilons ensure that (3) are transformable into (12). Note that if K-' exists, then a2 need not be zero; however, in that case, (12b) must be modified by the addition of a term az'x where a z f x =K-lazx (that is, a2' is a relation on L2.). The method is worthwhile only if i t gives a smaller effective critical region than Corollary 1. This happens if: REMARK: For suitably restricted N, Corollary 3 has several advantages over Popov's method: (i) The shift away from the critical region, which depends on KGw), can be controlled more flexibly as a function of w . This is likely to be useful where a negative magnitude slope d/dwl H(ju) is followed by a positive slope a t a larger w. (ii) a2 need not be zero if K ( w ) > 0. (iii) If a>>O,the critical region predicted by Corollary 3 ( a disk) is sometimes smaller than by Popov's method (which always gives a half-plane for a>O). EXAMPLE Let N be the operator in 32. whose graph 3. is shown in Fig. 9(a) (the same as in Example 2), and let H have the Laplace transform where r>>l. For what values of k is the closed loop bounded? Compare Corollaries 1, 2, and 3. Figure 10(a) illustrates the significant features of the (IIa) N K lies in a sector not greater than the product Nyquist diagram of H(jw) (not drawn to scale). Obof the sectors of N and K. (IIb) K H l lies in a sector greater than the product of serve that FI@) has two "pass bands," one for r-2<w <r-l and the other for r < u < r 2 ; these ('pass bands" the sectors of K and HI. produce the two loops in Fig. 10(a). Note that the If Requirements (1Ia)-(Il[b) are satisfied, then i t is critical region is the same half-plane in all corollaries, advantageous to transfer K: firom a composition with HI namely, ~ e . ) 2 - 1 + 6 . { into a composition with N. Requirement (IIa) usually means that the multiplier K has a very special form, 8 Corollary 3 and the factorization method, in a functional setting, and the difficulty in finding suitable multipliers is the in Reference main problem in applying this method. Once K is fixed, were introduced by the author exploited by [la!. A related method in a Liapunov setting has been Brockett, Willems, and Requirement (IIS) defines a (limited) class of operators Forys [4a]-[4b]. N.x)-.N(y) H for which this method is, useful. i.e., a < As an illustration of this method, a condition rex-Y<P lo This condition is satisfied automatically if K ( m ) > O . If sembling Popov's is derived next. K( m ) =0, then it is satisfied if w , is in Lz. ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL JULY Corollary 1 predicts boundedness for - 1<k < 8 of frequency response, have no strict meaning in nonapproximately. linear or time-varying systems. However, stability does Popov's method is useless here. A comparison of Figs. seem to depend on certain measures of signal amplifica10(a) and 10(b) shows the effect of removing the multi- tion and signal shift. Thus the norm ratio plier X/s+X: Hl(jw) is moved away from the critical plays a role similar to the role of gain. Furthermore, the region in the lower left half-plane (in the decaying edge inner product (x, Hx), a measure of input-output crossof the lower pass band, r-l<w < I ) ; however, this im- correlation, is closely related to the notion of phase shift. provement is more than offset by the bulge introduced For example, for linear time-invariant operators in d: in the upper left half-plane (in the rising edge of the the condition of positivity, (x, Hx)?O, is equivalent upper pass band, 1 < w <r). (by Lemma 1) to the phase condition, What is obviously needed here is a multiplier that acts like Popov's in the lower pass band, but has no effect in the upper pass band. The multiplier K(s) I t may be worthwhile to see what the theorems of r-l(s+ 1) (s+r-l) accomplishes just this. Its removal Part I mean in terms of gain and phase shift. This can shifts HlCjw) entirely into the right half-plane [Fig. be done with the help of Lemma 1. Theorem 1 of Part 10(c)]. Corollary 3 therefore implies that the closed loop I can be viewed as a generalization to nonlinear timeis bounded for all positive k, in fact, for k > - 1. varying systems of the rule that, "if the open-loop gain . - . is less than one, then the closed loop is stable." Theorem 6 . COMMENTS CONCLUSIONS AND 3 can be viewed as the generalization of, "if the open6.1. Circle Conditions loop absolute phase shift is less than 180' then" the The main result here is the Circle Theorem. The Cir- closed loop is stable." Theorem 2 places gain and phase cle Theorem is a sufficient condition for closed-loop shift in competition, permitting large gains a t small stability, which requires the nonlinearity N to lie in- phase shifts, etc. side a sector, but which leaves N free otherwise. The other conditions are all corollaries of the Circle Theo- 6.4. Conclusions Some of the salient features of the functional theory rem. are : Corollary 1 is probably the most useful result, since (i) I t provides an alternative to the method of Liai t roughs out the region of stability, with a minimum of punov, an alternative resembling the classical NyquistIlowever, i t is often conservative. restrictions on N. Corollaries 2 and 3 provide a tradeoff between limita- Bode theory. (ii) I t is well suited to input-output problems. tions on N a n d limitations on H(jw). Probably more sig(iii) I t is free of state-space restrictions, and is therenificant than the actual conditions is the fact that there fore useful for distributed systems, hysteritic systems, is a method of generating them, namely, the factorizaetc. I t also lends itself well to multivariable systems. tion method. (iv) I t unifies several results in stability theory. In T h e results derived in Part I1 hold for nonzero initial particular, i t is noteworthy t h a t Popov's condition, the conditions in the linear element, provided the "zeroslope-restricted-N result, etc., can all be derived from input response" wz satisfies the indicated restrictions. the Circle Theorem. (v) I t has led to some new results, notably Corollary 6.2. Extensions of the Theory 3 and [lb]. T h e theory has been extended in, several directions The theory outlined here is probably still far from its (see [lb]), notably, definitive form. Nevertheless, i t provides enough in1) to L,, sight to make possible a reasonably systematic design 2) to systems with a limited rate of time variation. of stabilizers. The extension to L, involves the use of exponential A APPENDIX weighting factors, which transform L, functions into Lz DEFINITIONS CONICITY OF AND POSITIVITY functions. The extension to time-varying systems involves the use of a shifted Nyquist diagram, H(u+jw), I t will be assumed that H is a relation in 6to and that in which u depends on the rate of time variation. c, r 2 0 , and a </3 are real constants. IIHx]I/~IxII ## H is incrementally interior conic if II(Hx - Hy)t - 4% y)tji I - ~ ) t l l ; rij(x l1 There are two positive elements in the open loop; each contributes a n absolute phase shift of less than 90'; the open-loop absolute phase shift is therefore less than 180'. The stability of a linear time-invariant feedback system depends on the amounts of gain and phase shift introduced by the open loop. Are similar considerations involved in nonlinear, time-varying problems? ofcourse the classical definitions of gain and phase shift, in terms JULY that is, ## For this purpose, let yA(Hx-cx) - [HX]t. Hence and ~ % H ( x ~ ) Now xt is in Lz. Hence H(xt) is in Lz by Property D of ) t Appendix B.1. Since [ ~ ( x ~ =](HX)~,i t follows that ( H x ) ~ in L2. Thus 6 is in Lz, all terms in (B5) are in is L2, and by Property E, Fig. 11. Contours. T o prove that H(rR) has Property N observe that H(rR) = H(Fw) +H(rR -I?,) ; since H(I',) has Property N by hypothesis, it is enough now t o show that H(rB-rw) has Property N. This can be accomplished I, lies in a circle centered a t by showing that H(rR -?) hm and not including the origin. The last assertion is a consequence of two facts. (i) There is an Ro>O for which, for R 2 R o and s in ( F R - ~ ~ ) , ( s ) - h ~5 1 hml. l~ l4 (ii) h, # 0. (i) is obtained from Property C of Appendix B.l for sl Ro, and therefore certainly holds for s in ( r R-?) I,. (ii) holds since where Yt(s), Xt(s), and A(s) are the 1.i.m. transforms of yt, xi, and 6. Hence Now the braced terms in (B7) are l.i.m. transforms of functions in Lz; for Xt(s) this is true by definition; for the remaining terms, this can be proved by the reasoning given below in Assertion 1. Suppose that A(s)/H(s) -c is the 1.i.m. transform of a function ~ ( 7 )i;t follows t h a t I > h, = lim H(jw), w-t m (Parseval's Theorem) and since H h ) #O by Property N. (b) This is a special case of the Maximum Modulus Theorem of Phragmgn-Lindelof. The theorem implies that a function analytic in a half-plane, and bounded on the boundary, is bounded throughout the half-plane. =- ## A complex-valued function W(s) will be said to satisfy the Paley-Wiener conditions if (PW1) W(s) is analytic for a (PWZ) 11 Yt1(2 (Parseval's Theorem) (B8) > 0, and (o I t will be shown, in Assertion 1, that q(7) = 0 for almost all 7 <t. Therefore ## J I W(U + jo) lzdw _< const. -m > 0). llxt - q1I2 = ll~t11~ + 11q1I2 2 ll~t11~. 039) The following lemma is a modification of Theorem 5 of Paley-Wiener [lo], and is stated without proof. LEMMA (a) If w is in Lz, wt=O, and W(s) is the 5. 1.i.m. transform of ~ ( r ) then W(s) exp (st) satisfies the , Paley-Wiener conditions. (b) Conversely, if t>O and W(s) exp (st) satisfies the Paley-Wiener conditions, then there is a function w in Lz having the properties that wt= 0 and that W(s) is the l.i.m. transform of w. B.4. Proof o Lemma 1(b) f Let x in Lz, and t 2 0 be given. Since H is linear, it is enough to show that H is conic with parameters c and r ; 17 ## Titchmarsh [9b], sec. 5.61. (B8) and (B9) imply (B4). ASSERTION The expression A(s)/ [ ~ ( s- c] is the 1. ) 1.i.m. transform of a function q in L z ; furthermore g(7) = O for almost all T <t. 1. PROOFOF ASSERTION By Lemma 5b, i t is enough to show that [A(s) exp (st)]/[H(s)-G] satisfies the Paley-Wiener conditions. T o prove (PWl), observe that the following three terms are analytic for a > O : A(s), because i t is the 1.i.m. transform of an Lz function; exp (st), because i t is analytic throughout the plane; [H(s)--GI-', by Lemma 4 and the hypothesis on the Nyquist diagram. The product of these terms must therefore also be analytic for u>o. T o prove (PW2), observe that 1966 (a < const. 2 11 (NY) tl12 equals the > 0) ## [by inequality (8)] (Since 6EL2 and a t = O by construction, the last inequality is implied by Lemma 5(a).) Q.E.D. APPENDLX C LEM:MA 2 where y = Kx. Now recalling that (xt, left-hand side of (CS), we get A preliminary assertion will be proved first. 2. ASSERTION If N in X is positive, K is in S, and y = Kx, then ## Observing that, for a 2 0, N satisfies the inequality by(?) N ( Y ( ~ ) ) [ ~ ((7)) we get 2 y I=, Since K is in 2, y(t) is given by a convolution integral, whose kernel is fixed for a fixed x, and whose limits of integration are 0 and t ; therefore y(0) = 0. Furthermore, since N is positive, its graph lies in the first and third quadrants. I t follows that the right-hand side of (C2) is non-negative. Q.E.D. which implies (C7). Q.E.D. CASE B. Suppose a < O . Decompose N(x) into two parts, N(x) =:N+(x)+N-(x); let N+(x) = N(x) for N(x) 2 0 and N+(x) = O elsewhere, and let N-(x) be similarly defined. Since N+ is clearly inside (0, p ) , Case A implies that N+K is inside (0, k/3 Similarly N-K is inside { ka, 0 ) . On summing the sectors of N+K and N-K (by the Sum Rule of Part I) it is found that NK is inside {ka, k a ] ; that is, inside f a , p ) X {o, k ) . Q.E.D. 1. ## C.1. Proof o Lemma 2 f Part a) I t is required to show that, for any given x in Lz, and any givenl t 2 0 , the inequality Before proving Lemma 3, a few related assertions will be introduced. ASSERTION Let K be an operator in 6ie, x a fixed 3. element of Lze,and A Kx. Then x has a "Foster expansion" in y;18 that is, x can be expressed as a finite sum, holds. For this purpose, make the following substitutions: Write y 5 KX, and observe that, since K(s) = kX/(s+X), y is differentiable and in which F, are operators mapping the image under K of L2, into R[O, w ), as follows: CASE1. Fay = K-'(0) .y. CASE^. If i = l , 2, - . . , (m-1), t h e n F i i s i n S a n d has a Laplace Transform, Fi(s) = ## (C3) is therefore equivalent to his/(s + Oi), (hi > 0, ei > 0). CASE 3. Fmy= hmyif K ( a,) = 0 and Fmy= 0 otherwise, where k 2 0 . ASSERTION If N is incrementally positive, and 4. (xt, [ ~ ~ l t ) then (xt, [ ~ ( x + y ) l i ) 2 0 . 20 PROOF ASSERTION I t is enough to show that OF 4. Now k and X are positive by hypothesis; the first integral in (C5) is non-negative by Assertion 2 ; the second integral is non-negative, since .N is a positive operator; therefore (C5) is true. Q.E.D. Part b) I t will be assumed, for simplicity, that P > 0. CASE A. Suppose a 2 0 . I t must be shown that N K is inside ( a , 8) X (0, k ) . This is equivalent to saying that N K is inside {o, kp}, or that ((NKx) t), (NKx - kPx):)t) 0. ## But the left-hand side of (Dl) can be expressed as which has the form (xlt -xzt, Nxlt - Nxzt), and is nonnegative, since N is an incrementally positive operator. Therefore (Dl) holds. Q.E.D. l8 < (C6) ## IEEE TRANSACTIONS ON AUTOMATIC CONTROL ASSERTION If N is an operator in %, incrementally 5. inside a sector {a,/3 f where a <0 and /3 > 0, then N can be decomposed into N = N++N-, where N- is inside {a,01, and N+ is inside 10, PROOF ASSERTION Since N is incrementally inOF 5. side a sector, its graph N is continuous and has bounded variation on every finite interval. Consequently N can be expressed as an integral, N(x) =loxn(x')dx'. Let n+(x) =n(x) if n(x) 2 0 , and n+(x) = O if n(x) < O ; let N+(x) =Jozn+(x')dx'. Clearly N+ has the desired property. N- is constructed similarly. Q.E.D. of. Hence N K is inside { 0, bK(0) f , which equals (0, 0 f X (0, K(O) ). CASE B. If a > O then N K is decomposed into three parts D.1. Proof o Lemma 3 f Part a ) Let x be any given element of L2,, and t any given point in [0, a ) ; it is required to show that Letting y & Kx, and recalling that x can be expressed by the Foster expansion Now the three parts lie in the sectors {o, (P-a)K(O) f 0 , ~ [ K ( o ) - ~ ( m ) ] ) , { ~ K ( w ) , ~ ( a ) frespecand a , tively. (The first two of these sectors are determined by the rule formed in Case A, after observing that [N-a11 is inside ( 0 , P-a f , and that [K-K( oo) . I ] is inside (0, [K(o) -K ( oo) ] ) ; the third sector is simply the sector of a constant times the identity.) On summing the three sectors (by the Sum Rule of Part I), it is found a that N K is inside ( a ~ ( ) , PK(0) ) ; that is, inside o j x { ~ ( m )~, ( 0 ) CASEC. If a <0, N is decomposed into N+ and N-, as in Assertion 5. This case then follows by the reasoning used in Case B of Lemma 2. Q.E.D. (ff9 1, 1. ## (see Assertion 3), (D2) is equivalent to I t will be shown that each component on the left-hand side of (D3) is non-negative. CASE 1. Here Foy=K-l(O).y. Hence ((Fay),, (Ny),) = [K-~(o)].(yt, (NY)~); this is non-negative since N is a positive operator, and since K(0) is necessarily positive. CASE2. Here Fi(s) = his/(s+Oi). Let The author thanks Dr. P. Falb for correcting the manuscript, and for offering many valuable suggestions. He also thanks Dr. G. Kovatch and NASA's Electronic Research Center, Cambridge, Mass., for supporting the completion of the paper, and Mrs. Iris McDonald for typing it. [la] G. Zames, "On the stability of nonlinear, time-varying feedback systems," Proc. 1964 NEC, vol. 20, pp. 725-730. [lb] -, "Nonlinear, time-varying feedback systems-Conditions for L,-boundedness derived using conic operators on exponentially weighted spaces," Proc. 1965 Allerton Conf., pp. 460-471. [lc] , "On the input-output stability of time-varying nonlinear feedback systems-Part I. Conditions derived using concepts of loop gain, conicity, and positivit	{"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": 0.9182112216949463, "perplexity": 1341.6813321393684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314852.37/warc/CC-MAIN-20190819160107-20190819182107-00038.warc.gz"}
https://www.physicsforums.com/threads/left-right-inverses.463173/	Left, Right Inverses • Start date • #1 136 0 Suppose we have a linear transformation/matrix A, which has multiple left inverses B1, B2, etc., such that, e,g,: $$B_1 \cdot A = I$$ Can we conclude from this (i.e., from the fact that A has multiple left inverses) that A has no right inverse? If so, why is this? • #2 HallsofIvy Homework Helper 41,847 969 Suppose there were a right inverse, say, R. Multiplying the equation $B_1A= I$ on the right by R gives $(B_1A)R= IR$ so that $B_1(AR)= B_1= R$ But then doing the same with $B_2A$ leads to $B_2= R$. In other words, if a matrix has both right and left inverses, then it is invertible and both right and left inverses are equal to its (unique) inverse. • Last Post Replies 1 Views 5K • Last Post Replies 3 Views 4K • Last Post Replies 2 Views 3K • Last Post Replies 3 Views 4K • Last Post Replies 3 Views 8K • Last Post Replies 2 Views 979 • Last Post Replies 2 Views 2K • Last Post Replies 9 Views 3K • Last Post Replies 2 Views 2K • Last Post Replies 1 Views 4K	{"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": 0.9628791213035583, "perplexity": 1778.6070872783368}, "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-2022-21/segments/1652662539131.21/warc/CC-MAIN-20220521143241-20220521173241-00175.warc.gz"}
https://www.physicsforums.com/threads/electrostatics-help.170204/	# Electrostatics help 1. May 14, 2007 ### miamiheat5 Again I am not ssure what to do. An electron(mass 9.11x10^-31kg) is accelerated in a uniform E field (E=3.0x10^4 N/C) between two parallel charged plates. The separation of the plates is 1.6cm. The electron is accelrated from rest near the negative plate and passes through a tiny hole in the positive plate. With what speed does the electron emerge from the hole? I was thinking you would use qE=ma to find the acceleration, then use kinematics to find the velocity but i am not sure. I would really be happy if anyone can help. 2. May 14, 2007 ### malawi_glenn Have you tried to solve this based on those assumptions? ;) 3. May 14, 2007 ### miamiheat5 yes and i got v as being 1.3x10^-24m/s 4. May 14, 2007 ### malawi_glenn did you remember to take account for the electrons charge?.. 5. May 14, 2007 ### malawi_glenn because i got 1.310^7 m/s 6. May 14, 2007 ### miamiheat5 what do u mean take account for the electrons charge? 7. May 14, 2007 ### malawi_glenn Some books do not give the electron charge in SI units. Why didnt you post your work done? So we can see were you have done wrong?.. How can one help you if you dont post your work?.. Post your work and I will tell you were it went wrong :) And I also post how I solve this problem. 8. May 14, 2007 ### miamiheat5 ok this what i did: qE=ma 1.602x10^-19(3.0x10^4)=(9.11x10^-31)a 4.806x10^-15=9.11x10^-31a a=5.27x10^-47 Then: vf^2=0^2+2(5.27x10^-4709.016) sqroot vf^2=sqroot 1.686x10^-48 vf=1.3x10^-24m/s 9. May 14, 2007 ### malawi_glenn you shold pratice more on how to calculate with exponents.. 4.806x10^-15=9.11x10^-31a dividing both left and right side with 9.11x10^-31 does not give a=5.27x10^-47 .. 4.806x10^-15 / 9.11x10^-31 = 0.52810^(-15+31) 10. May 14, 2007 ### miamiheat5 oh ya maybe I should thank you for catching my mistake. Do you think you can help me with one other problem. With this one I don't have any work because I don't know where to start. A negative charge -q is fixed to one corner of a rectangle. What postitive charge must be fixed to corner A and what positive charge must be fixed to corner B, so that the total electric field at the remaining corner is zero? Express your answer in terms of q. Here is the drawing, i will simulate a rectangle without side lines 2d----------------------------->(should be in middle) A --------------------------------------------------------- d---------------------------------------------->(should be on other side of rectangle) -q O-----------------------------------------------------B Last edited: May 14, 2007 11. May 14, 2007 ### malawi_glenn Look at composants in x- and y direction. Use coloumbs law and some trigonemtry. So if the Eletric field should be zero, then the force should be zero too. E = F/q Try from here =) good luck 12. May 14, 2007 ### miamiheat5 Ok this I promise will be the last question I need help with again i am not sure how to start the question off. So this question uses information from: question (7) "An electron(mass 9.11x10^-31kg) is accelerated in a uniform E field (E=3.0x10^4 N/C) between two parallel charged plates. The separation of the plates is 1.6cm. The electron is accelrated from rest near the negative plate and passes through a tiny hole in the positive plate. With what speed does the electron emerge from the hole?" The resulting electron beam from question (7) then passes into an evacuated region outside the plates where the E field drops to zero. It then passes through two horizontally oriented defelction plates, the beam travels another 20cm horizontally and strikes a phosphor coated screen. What vertical distance(measured ffrom the path taken if no deflection filed were present) would be measured for the beams defelction?) Is this another question that will use kinematics to be solved? 13. May 14, 2007 ### malawi_glenn yes its a free fall thing. 14. May 14, 2007 ### miamiheat5 what formulas will i need to use? 15. May 14, 2007 ### malawi_glenn projectile motion.. i.e free fall combined with transversal motion. The deflection plates will prevent the particle from avoid its original path, it will not be accelerated by an electric field, and gravity is much much weaker than the effect of the deflection plates. So if you remoce the deflection plates, and dont have any electric field, the electron will just fall beacuse of its mass.	{"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": 0.8513711094856262, "perplexity": 1851.2051802130952}, "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/1474738660996.20/warc/CC-MAIN-20160924173740-00066-ip-10-143-35-109.ec2.internal.warc.gz"}
https://www.groundai.com/project/effects-of-site-asymmetry-and-valley-mixing-on-hofstadter-type-spectra-of-bilayer-graphene-in-a-square-scatter-array-potential/	Effects of site asymmetry and valley mixing on Hofstadter-type spectra of bilayer graphene in a square-scatter array potential # Effects of site asymmetry and valley mixing on Hofstadter-type spectra of bilayer graphene in a square-scatter array potential Danhong Huang, Andrii Iurov, Godfrey Gumbs and Liubov Zhemchuzhna Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, NM 87117, USA Department of Electrical & Computer Engineering, University of New Mexico, Albuquerque, NM 87131, USA Center for High Technology Materials, University of New Mexico, Albuquerque, NM 87106, USA Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10065, USA Donostia International Physics Center (DIPC), P de Manuel Lardizabal, 4, 20018, San Sebastian, Basque Country, Spain July 19, 2019 ###### Abstract Under a magnetic field perpendicular to an monolayer graphene, the existence of a two-dimensional periodic scatter array can not only mix Landau levels of the same valley for displaying split electron-hole Hofstadter-type energy spectra, but also couple two sets of Landau subbands from different valleys in a bilayer graphene. Such a valley mixing effect with a strong scattering strength has been found observable and studied thoroughly in this paper by using a Bloch-wave expansion approach and a projected effective Hamiltonian including interlayer effective mass, interlayer coupling and asymmetrical on-site energies due to a vertically-applied electric field. For bilayer graphene, we find two important characteristics, i.e., mixing and interference of intervalley scatterings in the presence of a scatter array, as well as a perpendicular-field induced site-energy asymmetry which deforms severely or even destroy completely the Hofstadter-type band structures due to the dependence of Bloch-wave expansion coefficients on the applied electric field. PACS: ## I Introduction Shortly after its discovery and fabrication in 2004, graphene has captured tremendous attention and generated an enormous wave of research activities due to its unique Dirac-cone-type electronic band-structures and properties. Novoselov et al. (2005); Geim and Novoselov (2007); Neto et al. (2009) This wave also includes a huge amount of research works concerned with magnetic-field behavior, electronic properties, Landau levels (LLs) Goerbig (2011); Zhang et al. (2006) and quantum Hall effect Zhang et al. (2005); Novoselov et al. (2007); Kane and Mele (2005). Nearly at the same time, bilayer graphene (BLG), which consists of two closely-located graphene sheets, was also fabricated and tested experimentally. Novoselov et al. (2004); Oostinga et al. (2008); Ohta et al. (2006) The BLG electronic properties are found significantly different depending on details of an A-B stacking process, or Bernal-stacked form, with relatively shifted carbon-atom positions in two layers. Yan et al. (2011a) Bilayer graphene revealed some highly unusual properties, e.g., unconventional quantum Hall effect Novoselov et al. (2006) and cyclotron resonance Henriksen et al. (2008). A comprehensive theoretical study of the LL degeneracy and quantum Hall effect for BLG in Bernal stacking was reported in Ref. [McCann and Fal’ko, 2006]. Based on an effective two-dimensional Hamiltonian, it was concluded that the low-energy spectrum of BLG can be characterized as parabolic dispersion of chiral quasi-particles with a Barry phase . Meanwhile, its magnetic-filed dependent energy spectrum is found consisting of a set of nearly equidistant four-fold degenerateLLs. In this paper, we will employ such an effective-Hamiltonian approach to establish theoretical formalism for modulated LLs in the presence of a square-scatter array potential in Sec. II. One of the most unusual and fascinating phenomena related to the electronic spectrum under a perpendicular quantizing magnetic field is the so-called Hofstadter butterflyHofstadter (1976); Azbel (1964), theoretically predicted in 1976. Here, a recursive fractal electron spectrum was obtained as a function of prime ratio of the magnetic flux passing through a lattice unit cell to a fundamental flux quanta, and these degenerate electronic subbands split and clustered themselves into different patterns corresponding to the value of a given magnetic-flux ratio. By performing first-principles calculations for hexagonal two-dimensional graphene-type lattice, tight-binding approximation resulted in a Hofstadter-butterfly-like clustering pattern, except for an asymmetry with respect to zero wave vector Gumbs and Fekete (1997). Such types of Hofstadter band-structure were also predicted to exist in carbon nanotubes as pseudo-fractal magneto-electronic spectrum Nemec and Cuniberti (2006) and also in bilayer graphene Nemec and Cuniberti (2007). In a recent experiment, Hofstadter’s butterfly and fractal quantum Hall effect have been extended to Moire superlattices, which are formed as BLG or flakes are coupled to a rotationally aligned hexagonal boron nitride layer Dean et al. (2013); Ponomarenko et al. (2013); Woods et al. (2014) inside a van der Waals heterostructure sample. Hunt et al. (2013) The main idea involved in such experiments is that an elementary lattice-unit cell through which the magnetic flux was measured Hofstadter (1976) will be replaced by a much bigger supercell of the Moire lattice, Schmidt et al. (2014); Yang et al. (2016); Wang et al. (2015) so that the butterfly is expected to be seen at a much lower magnetic field. Additionally, the theory for such butterfly structures in twisted BLG was proposed in Ref. [Bistritzer and MacDonald, 2011], in which long-period spatial patterns can be created precisely at small twist angles. Later, the coexistence of both fractional-quantum-Hall and integral-quantum-Hall states associated with fractal Hofstadter spectrum was confirmed experimentally within such twisted-bilayer structures. Wang et al. (2012) Moreover, specific subband gaps of a Hofstadter’s butterfly were also found for interacting Dirac fermions in graphene. Apalkov and Chakraborty (2014) On the other hand, in the absence of a magnetic field, a periodic electrostatic field gives rise to new zero-energy states with minigaps and chirality Brey and Fertig (2009), and their composite wave functions can still satisfy the required Bloch periodic condition. Apart from this, new massless Dirac fermions with strong anisotropic properties Park et al. (2008a) are realized in graphene subjected to a slowly-varying periodic potential. Park et al. (2008b) In contrast, a spatially-uniform interaction of Dirac electron with an off-resonant optical field can lead to the formation of either gapped Kibis (2010); Iurov et al. (2011, 2013) or anisotropic dressed Kibis et al. (2017) states depending on polarizations of an imposed irradiation. Very interestingly, two unique features associated with BLG system have been found. The first property is the intervalley mixing and the quantum interference effect coming from two valleys in the presence of a two-dimensional scattering-lattice potential, while the second property results from a site-energy asymmetry induced by a perpendicular electric field. Here, the latter factor is able to destroy the Hofstadter-type fractal band structures established by an in-plane scattering-lattice potential and an out-of-plane quantizing magnetic field, resulting in strongly deformed self-repeated patterns. Such a phenomena is attributed to the dependence of Bloch-wave expansion coefficients on an applied electric field, leading to an electro-modulation of the Hofstadter-type subband splittings. The rest of the paper is organized as follows. In Sec. II we present theoretical formalism and acquire a set of characteristic equations, describing electron energy spectrum and corresponding eigenstates for BLG in the presence of both a perpendicular quantizing magnetic field and a two-dimensional periodic electrostatic modulation potential. These results expand the previously studies for a two-dimensional electron gas Kühn et al. (1993) and for a monolayer graphene Gumbs et al. (2014a, b). In Sec. III, we display and discuss our numerical results demonstrating fractal Hofstadter band-structures in different ranges of magnetic field of interest and with various modulation strengths in a close up view for separate LLs and self-repeated superstructures as well. Finally, a brief summary with remarks is given in Sec. IV. ## Ii Model and Theory By considering and valleys, where , and Å, and including sublattices and as well as bilayer structure, the four-component wave functions for each valley can be formally written as McCann and Fal’ko (2006) ΨK=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ϕAKϕ~BKϕ~AKϕBK⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ , Ψ~K=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ϕ~B~KϕA~KϕB~Kϕ~A~K⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦ , (1) where and label the bonds in the bottom layer and and label the bonds in the top layer. For each valley, the graphite tight-binding Hamiltonian matrix within the -plane for Bernal-stacking Yan et al. (2011b) bilayer takes the form ^HTBξ=vF⎡⎢ ⎢ ⎢ ⎢ ⎢⎣V+ξu/2ξv3(^px+i^py)0ξv(^px−i^py)ξv3(^px−i^py)V−ξu/2ξv(^px+i^py)00ξv(^px−i^py)V−ξu/2γ1ξv(^px+i^py)0γ1V+ξu/2⎤⎥ ⎥ ⎥ ⎥ ⎥⎦ , (2) where represents the () or () valley, cm/s is the intralayer (monolayer) Fermi velocity, characterizes the effective mass of electrons in the parabolic band, measures the strength of the interlayer coupling, represents the bias-induced on-site energies of bilayer, with electric field and bilayer separation , and corresponds to a symmetrical bilayer. In addition, we have introduced canonical momentum operators and , where the Landau gauge is chosen for a uniform magnetic field along the vertical direction. The potential of a two-dimensional (2D) scatter array in Eq. (2) is assumed as V≡V(x,y)=V0[cos(πxdx)cos(πydy)]2N , (3) where is an integer, stands for the scattering-potential strength, and are the two array periods in the and directions, respectively. Even in the absence of the scatter potential (i.e., ), the eigen-energies and eigen-states correspond to the Hamiltonian in Eq. (2) can only be calculated numerically. For low-energy states of electrons (with kinetic energy less than ), however, the Hamiltonian in Eq. (2) can be projected onto a one. For such a situation, the wave functions in Eq. (1) for each valley also reduce to a two-component form (4) and the projected effective Hamiltonian matrix becomes ^Heffξ=−12m∗[0(^px−i^py)2(^px+i^py)20]+ξv3[0^px+i^py^px−i^py0] +ξu2[1001]−ξuv2γ21[(^px−i^py)(^px+i^py)00−(^px+i^py)(^px−i^py)]+V(x,y)^I0 , (5) where in the last term is the identity matrix, the first, second and the rest two terms represents the intralayer, interlayer and bias effects, respectively. By taking in Eq. (5) as a start, in the strong-field limit, i.e., with a cyclotron frequency , we can formally set in Eq. (5). Based on this simplification, we obtain the analytical form of the eigen-energy levels for each valley () Eξ±,n=⎧⎪ ⎪⎨⎪ ⎪⎩±ℏωc√n(n−1)−ξδ/2 , for n≥2ξu/2−ξδ , % for n=1ξu/2 , for n=0 , (6) where , and correspond to electron and hole energy levels at each valley, respectively, each energy level is spin degenerate, and the lowest two energy levels are four-fold degenerate with respect to both spins and electron-hole pseudospins. If , we get from Eq. (6), which becomes eight-fold degenerate now. The corresponding eigen-states to these electron (hole) energy levels () are calculated as ΨKξ±,n,kx(x,y)=C±n(ξ)ei(kx+Kξ)x√Lx[ϕn,kx+Kξ(y)D±n(ξ)ϕn−2,kx+Kξ(y)] , (7) where is the sample length in the direction, () for (), is the harmonic-oscillator wave functions with a guiding center , the magnetic length, and two coefficients D±n(ξ)=Eξ±,n−ξu/2+ξnδℏωc√n(n−1) , C±n(ξ)=1√1+∣∣D±n(ξ)∣∣2 . (8) Assuming , we have and for , which becomes independent of and . On the other hand, for and we obtain ΨKξ±,0,kx(x,y)=ei(kx+Kξ)x√Lx[ϕ0,kx+Kξ(y)0] , (9) After the scatter array has been included in the strong-field limit, the wave function of the system can be expanded as Φξℓ;α,n,k∥(x,y)=1√Ny∞∑s=−∞{eikyℓ2B(sp+ℓ)K1ΨKξα,n,kx−(sp+ℓ)K1(x,y)} , (10) where , corresponds to electron and hole states, , and for the first magnetic Brillouin zone, is the number of unit cells spanned by and in the direction, () is the sample length in the direction, is the reciprocal lattice vector in the direction, and is a new quantum number for labeling split subbands from a -degenerated LL in the absence of scatters. Importantly, the above constructed wave function satisfies the usual Bloch condition, i.e., Φξℓ;α,n,k∥(x+dx,y+qdy)=eikxdxeikyqdyΦξℓ;α,n,k∥(x,y) . (11) Substituting the expression for wave function at each valley in Eq. (7) into Eq. (9), we find Φξℓ;α,n,k∥(x,y)=1√NyLx∞∑s=−∞{eikyℓ2B(sp+ℓ)K1Cαn(ξ) ×ei[kx+Kξ−(sp+ℓ)K1]x[ϕn,kx+Kξ−(sp+ℓ)K1(y)Dαn(ξ)ϕn−2,kx+Kξ−(sp+ℓ)K1(y)]} , (12) where and . Now, by taking into account of the term in Eq. (5), a tedious but straightforward calculation leads to an explicit expression for the matrix elements of the potential , yielding Vℓ′,n′,α′ℓ,n,α(\boldmathk∥,ξ)≡∑ξ′,k′∥∫∫dxdy[Φξ′ℓ′;α′,n′,k′∥(x,y)]†V(x,y)Φξℓ;α,n,k∥(x,y) +δℓ,ℓ′δn,n′δξ,ξ′[1+Dα′n(ξ)Dαn(ξ)][(2N)!(N!)2]2⎫⎬⎭ , (13) where correspond to electron and hole levels, respectively. From Eq. (13) we find two valleys for bilayer graphene can be coupled to each other, which is different from the monolayer graphene Gumbs et al. (2014b). Here, the terms with come from the intravalley contribution, whereas the terms with stand for the intervalley coupling which presents an interference effect. Moreover, we have defined in Eq. (13) two intervalley () coupling factors F(B,A)ij(ξ′,ξ)=(2Ni)(2NN)A(B,A)1(0,N−i∣∣ξ′,ξ) +(2Nj)(2NN)A(B,A)2(N−j,0)+2(2Ni)(2Nj)A(B,A)3(N−j,N−i∣∣ξ′,ξ) , (14) where the binomial expansion coefficient for is (mn)≡m!n!(m−n)! . (15) Finally, we have introduced in Eq. (14) the following three self-defined functions A(B,A)1(r,s∣∣ξ′,ξ)=Drs(B,A)n′,nTsℓ(ξ′,ξ)δℓ,ℓ′ , (16) A(B,A)2(r,s)=Drs(B,A)n′,n{δℓ−ℓ′,r[sgn(n′−n)]β+δℓ′−ℓ,r[sgn(n−n′)]β} , (17) A(B,A)3(r,s∣∣ξ′,ξ)=Drs(B,A)n′,n{δℓ−ℓ′,r[sgn(n′−n)]βcos[Θℓ′rs(n′,n∣∣ξ′,ξ)] +δℓ′−ℓ,r[sgn(n−n′)]βcos[Θℓrs(n,n′∣∣ξ′,ξ)]} , (18) where , Drs(B)n′,n=√n1!n2!e−Wrs/(2ϕ)(Wrsϕ)β/2L(β)n1(Wrsϕ) , (19) with and being the integers prime to each other, is the magnetic flux per unit cell, is the flux quanta, , , is the associated Laguerre polynomial, , , , Tsℓ(ξ′,ξ) =⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩±2cos[s(~kx(ξ′,ξ)dx−ℓ2π)ϕ] ,(+) for β=4N% and (−) for β=4N+2±2sin[s(~kx(ξ′,ξ)dx−ℓ2π)ϕ] ,(+) for β=4N+1 and (−) for β=4N+3% , (20) Θℓrs(n′,n∣∣ξ′,ξ)=s[~kx(ξ′,ξ)dx−2π(ℓ+r/2)]ϕ−sgn(n′−n)βtan−1(sdxrdy) , (21) and characterizing the intervalley coupling for and the interference effect as well. Here, the range of extends to all magnetic Brillouin zones in this direction for Umklapp scatterings. The energy dispersion of the th magnetic band around each valley for this modulated system is a solution of the eigenvector problem with elements of the coefficient matrix given by {\tensor\boldmathM(\boldmathk∥,ξ)}j,j′=[Eξα,n−ε(\boldmathk∥,ξ)]δn,n′δℓ,ℓ′δ(n)α,α′+Vℓ′,n′,α′ℓ,n,α(\boldmathk∥,ξ) , (22) where for (i.e., degenerate electron-hole levels) and for , is a composite index, and is an orthonormal eigenvector. Furthermore, the eigenvalues of the system are determined by roots of the characteristic equation . ## Iii Numerical Results and Discussions ### iii.1 Two-Dimensional Electron Gas and Monolayer Graphene As a starting point, we first briefly discuss the effect of a two-dimensional (2D) periodically-modulated scattering-lattice potential in Eq. (3) on a 2D electron gas (EG) under a perpendicular quantizing magnetic field . In the absence of this scattering-lattice potential, 2DEG will be quantized into a series of discrete LLs: with , as the cyclotron frequency, and as the effective mass of electrons. These uncoupled LLs are highly degenerate with respect to their guiding centers (or with different cyclotron orbits), where is the magnetic length. In the presence of the scattering-lattice potential, however, these degenerate LLs are strongly coupled to each other and expand into a set of split Landau bands, as shown in Fig. 1. Furthermore, a close-up view in Fig. 1 reveals that a self-similar pattern occurs within the fourth () Landau band at low , just as predicted early by Hofstadter in his seminal work Hofstadter (1976). If the 2DEG is replaced by a monolayer graphene, a different set of LLs with appears in the absence of a scattering-lattice potential, where , is the Fermi velocity of graphene, and () corresponds to electrons (holes), respectively. In this case, we find that the LL sits at the zero-energy Dirac point instead of for 2DEG, and but not proportional to for 2DEG. After the scattering-lattice potential in Eq. (3) has been employed, these guiding-center degenerated energy levels also expand into a Landau band through mutual couplings, as seen in Fig. 2. However, the mirror symmetry with respect to the band center is lost in Fig. 2 for monolayer graphene, as discussed in details recently by us Gumbs et al. (2014b). Here, one crucial difference between 2DEG and monolayer graphene is the LL separation for graphene, in contrast with a uniform one, , for 2DEG. Consequently, the graphene energy-level separation will decrease with increasing , and therefore, overlaps of many Hofstadter butterflies will show up for higher values as in Fig. 2. ### iii.2 Bilayer Graphene Now, Let us turn our attention to discussions on development of Landau bands in a bilayer graphene. For bilayer graphene subjected to a scattering-lattice potential given by Eq. (3) and under a perpendicular quantizing magnetic field at the same time, our numerical solutions for the eigenvalue equation in Eq. (22) are presented in Figs. 3 - 5 with various scattering strengths . As a whole, we find that degenerate LLs with different guiding centers tend to couple to each other and lead to band-center asymmetric Landau bands within which a fractal Hostadter structure is seen for high magnetic fields . Furthermore, the developed Landau bands for two valleys () are coupled to each other in a bilayer graphene through an Umklapp scattering process across whole magnetic Brillouin zones, which is in contrast with the case for a monolayer graphene where the Landau bands are found independent of a valley. As indicated in Section II, the valley mixing and interference effect contained in the modulation potential in Eq. (13) are described explicitly by the wave number , where , and nm. The integer power , which measures the peak sharpness of the scattering potential in Eq. (3), is selected as . In the absence of the 2D scattering-lattice potential, each LL under the magnetic flux ratio has a -fold degeneracy for magnetic subbands. We have taken , and , respectively, in Figs. 35. For all three graphs, we only show the lowest four Landau bands for both electrons and holes. All the numerical results which display self-repeated Hofstadter butterfly structures are presented as a function of . Here, all energy levels, except for and , are shifted upwards by a fixed energy offset for . Therefore, we have to made an adjustment to our plots in Figs. 35 so that the electron-hole symmetry can be restored with respect to the zero-energy point. With fixed lattice period , a magnetic-flux ratio can be uniquely related to a magnetic-field strength . The upper bound of in Figs. 35 for observing Hofstadter spectra is found within the range of . The unperturbed LL spectrum is shown in Eq. (6). In our numerical calculations, we have set and so that the LL structure consists of a few pairs of extremely closely-located levels, corresponding to for two valley indexes. This on-site energy-level separation () depends on or . Additionally, two groups of LaLs associated with and are nearly degenerate due to their very small separations , as found from the inset of Fig. 3. Furthermore, the spin degeneracy in these LLs is kept since none of them depends on spin index. All scaled higher levels staring for have the same dependence which becomes nearly equidistant as and in contrast with the monolayer graphene. Here, the pseudospin index hints a complete electron/hole symmetry for these LLs. After the scattering-lattice potential given by Eq. (3) has been introduced to bilayer graphene, the previously uncoupled and highly-degenerate LLs expand into many magnetic bands with self-similar structures, as can be verified directly from Fig. 3. Since the higher LLs become almost equally separated in bilayer graphene, we expect similar self-repeated structures within a magnetic band for large values. Because the mixing of LaLs depends on , we present comparisons in Figs. 4 and 5 for strong and intermediate scattering strengths . When the strong scattering strength is , the mixing of and Landau bands is severe, as seen in Fig. 4. In addition, the band mixing is found to increase with magnetic field in this case. If the scattering strength, , is weak, on the other hand, no band mixing appears, as can be verified from Fig. 4. For intermediate scattering strength in Fig. 5, we find the band mixing still happen, but it occurs at a higher magnetic field. For lower values of , on the other hand, such band mixing is completely negligible, as found from Fig. 5. Therefore, in order to observe Hofstadter butterflies and band mixing effects simultaneously, a stronger scattering strength is preferred. More importantly, the large value of also brings down the required magnetic field for observation to an experimentally accessible level. ### iii.3 Effect of Breaking Down of Inversion Symmetry For a monolayer graphene, the group of wavevector associated with the or point within the crystal first Brillouin zone is found isomorphic to the point group Malard et al. (2009) . For a bilayer graphene with a Bernal stacking, on the other hand, this point group is downgraded to with a lower symmetry. Furthermore, in the presence of a vertical bias field, these two point groups Malard et al. (2009) become and , respectively, for a gated monolayer graphene and a biased bilayer graphene. The loss of an inversion symmetry for a bilayer graphene under a vertical electric field has a profound effect on the formation of fractal Landau subbands in the presence of a square-scatter array potential, as can be seen from Eqs. (8) and (13) where both LL-coupling coefficients and are dependent and the intervalley coupling also becomes possible. Compared with a monolayer graphene, a bilayer graphene can bring into additional valley mixing and site asymmetry after a perpendicular electric field has been applied. Such intervalley interference and electro-modulation effects can be seen clearly from Eq. (13) for the matrix elements of the scattering potential, i.e., the summation over for fixed and changing coefficients and with and for . In Figs. 3 - 5, only a negligible electric field is employed (), and therefore, no visible distortions of the Hofstadter butterfly, which results from a square 2D periodically-modulated scattering-lattice potential, can be resolved. However, as is slightly increased from to in Fig. 6 for a very weak modulation with and , we find from Fig. 6 that the previously found self-similar patterns within the third Landau band under is very strongly distorted, and therefore disappears. Moreover, as is further increased from to in Fig. 7 for but , we find from a direct comparison between Fig. 6 and Fig. 7 that the previously observed self-repeated patterns within the fourth Landau band under is destroyed completely. Meanwhile, the mixing of the third and fourth Landau bands is seen clearly even for such a small modulation amplitude in contrast with the result in Fig. 4 for . ## Iv Brief Summary In conclusion, we have developed a theoretical formalism to demonstrate the Hofstadter-type fractal band structure for bilayer graphene in the presence of a two-dimensional periodic electrostatic modulation. The current work can be viewed as a generalization of the previous reported results based on Bloch-wave expansion approach applied to both a two-dimensional electron gas Kühn et al. (1993) and a monolayer graphene Gumbs et al. (2014b). As in previous studies Kühn et al. (1993); Gumbs et al. (2014b), this work includes explicitly deriving a non-perturbative eigenvalue equation, finding numerical solutions which display self-repeated split Landau subbands as a function magnetic flux and periodic subband dispersions as a function of electron wave number in a full magnetic Brillouin zone. Both Hofstadter butterflies and band mixing effects can be displayed simultaneously for a strong scattering strength which further reduces a required magnetic field for such observations to an accessible level. Interestingly, we find two unique features for the bilayer-graphene system in this study. The first one is related to a bias-modulated mixing of and an interference from two valleys (i.e., non-vanishing intervalley scattering with ) in the presence of a scattering-lattice potential. 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The feedback must be of minimum 40 characters and the title a minimum of 5 characters	{"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": 0.8567662835121155, "perplexity": 2351.183015948376}, "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-2020-24/segments/1590348509264.96/warc/CC-MAIN-20200606000537-20200606030537-00047.warc.gz"}
https://mathoverflow.net/questions/158940/immersed-seifert-surfaces-of-minimal-genus	# Immersed Seifert surfaces of minimal genus Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$ is an immersion $f\colon S\to X$ such that $f(\partial S)$ is a longitude of $K$. We refer to the genus of $S$ as the genus of the immersed Seifert surface. Gabai showed that the minimal genus of an immersed Seifert surface is the same as the minimal genus of an embedded Seifert surface. I was wondering whether a stronger statement holds: every immersed Seifert surface of minimal genus is an embedded Seifert surface in disguise', i.e. homotopic (through immersions) to an embedded Seifert surface. For higher genus I can think of immersed Seifert surfaces that certainly do not from Seifert surfaces, e.g. there exists an immersed Seifert surface $f\colon S\to X_U$ of genus one for the unknot $U$ such that there are curves on $f(S)$ which are non-trivial in $H_1(X_U)$. • Maybe I'm missing something but why can't you just take a minimal genus embedded SS and then just re-immerse it by inserting some self-intersections? You could have a self-intersection curve parallel to the knot, which bounds an annulus in the surface, whose only double points are on the boundary. This immersed annulus gives a torus in the knot complement, and the torus bounds a solid torus in the knot complement. Initially I do not think there's a path in the space of immersions to an embedding. – Ryan Budney Feb 28 '14 at 1:14 I think this should be false for fibered knots. Consider a fibered knot, which is the mapping torus of a mapping class $\phi: S\to S$. Suppose there is a non-separating simple curve $c\subset S$ such that $\phi(c)\cap c=\emptyset$. Then one can form a "crossjoin" surface, by removing annulus neighborhoods $\mathcal{N}(c)$ and $\mathcal{N}(\phi(c))$, and inserting two crossing annuli which connect one boundary of $\mathcal{N}(c)$ to the opposite boundary of $\mathcal{N}(\phi(c))$. This was used by Cooper-Long-Reid to construct immersed surfaces in fibered manifolds. This immersed crossjoin surface will have a curve on it which is homologically non-trivial, and therefore the surface cannot be homotoped to an embedding. Addendum: Actually, I realized that there is a simple way to form counterexamples in torus knots. The monodromy of a torus knot is finite-order, say $k$. Take a non-separating simple closed curve, and remove an annulus neighborhood. Then insert an annulus that winds $k$ times around the mapping torus direction, and connect up with the other boundary component of the annular neighborhood. The cross-cut is not needed, since this produces one component which is an immersed torus which is homologically trivial. So this works even for the trefoil knot.	{"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": 0.86739182472229, "perplexity": 252.11494755135536}, "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/1550247496080.57/warc/CC-MAIN-20190220190527-20190220212527-00339.warc.gz"}
http://www.maa.org/publications/periodicals/convergence/eulers-analysis-of-the-genoese-lottery-probabilities-in-the-genoese-lottery?device=mobile	Euler's Analysis of the Genoese Lottery - Probabilities in the Genoese Lottery Author(s): That the Genoese lottery captured Euler's mathematical imagination is evidenced by the entries he made in his notebook, known today as H5, which he probably filled between 1748 and 1750 [6, p. xxiv]. His first published article on the lottery did not appear until 1767. However, the Genoese lottery was only established in Berlin in 1763, and on March 10 of the same year, Euler delivered an address entitled "Reflections on a Singular Type of Lottery called the Genoese Lottery" [11] to the Berlin Academy. The text of the address was finally published in 1862, in Euler's Opera Posthuma I. Leonard Euler's publications were numbered from E1 through E856, in the early 20th century by Gustav Eneström, in the order of their publication. The fact that this paper was published posthumously is reflected in its high number in Eneström's index: E812. Unlike Euler's later papers on the Genoese lottery, which are works of pure mathematics that set out to answer questions of little or no practical value inspired by the lottery, E812 is a work of applied mathematics. Euler proves little new mathematics, but instead applies elements of probability theory, interspersed with dashes of common sense and vaguely justified rules of thumb, in describing how one might go about designing a Genoese style lottery from scratch, determining, in particular, fair prize levels. This is a different undertaking from his analysis of Roccolini's proposal, where the prize amounts were given, and Euler determined the fair price of a ticket, comparing these to Roccolini's prices. Euler's goal in the first portion of the paper is to calculate the probability pk,i that a player who bets on k numbers will in fact match i of them (Euler did not use the notation pk,i; we are adopting it here to make the discussion easier to follow). The values depend on the parameters n and t, where the lottery consists of choosing t tokens at random from a collection numbered 1, 2, 3, ¼, n. A modern probability text would say that i has hypergeometric distribution, with parameters n, t and k, and thus pk,i = æ ç ç è t i ö ÷ ÷ ø æ ç ç è n-t k-i ö ÷ ÷ ø æ ç ç è n k ö ÷ ÷ ø . However, the hypergeometric distribution, the distribution of sampling without replacement, did not yet have the status of a standard random variable in Euler's time. Euler's analysis was indeed one of the first treatments of this distibution in print, although Jacob Bernoulli and Abraham de Moivre had considered the hypergeometric distribution earlier in the 18th century [12, pp. 201-202]. Like Bernoulli and de Moivre, Euler did not give the pk,i's in the above form; in fact, he had not yet even developed a convenient notation for binomial coefficients, although he would develop a notation similar to the modern one later in his career. Instead, Euler gave the pk,i in a form that was better suited to efficient recursive calculation. Euler gives a complete derivation of the desired probabilities. His method of proof - an implicit or `socratic' induction - is a common one in Euler's writings: he solves the simplest cases in order, clearly and persuasively argued, until the pattern is clear to the reader. Paper E812 is divided into sections which Euler calls Problems, some with corollaries or scholia. He considers the distribution of i in the case k=1 in Problem 1, and then proceeds through the next three values of k in Problems 2-4. We summarize these in Table 2, where the kth column represents the results of the corresponding Problem. pk,i k = 1 k = 2 k = 3 k = 4 i=0 n-t n (n-t)(n-t-1) n(n-1) (n-t)(n-t-1)(n-t-2) n(n-1)(n-2) (n-t)(n-t-1)(n-t-2)(n-t-3) n(n-1)(n-2)(n-3) i=1 t n 2t(n-t) n(n-1) 3t(n-t)(n-t-1) n(n-1)(n-2) 4t(n-t)(n-t-1)(n-t-2) n(n-1)(n-2)(n-3) i=2 t(t-1) n(n-1) 3t(t-1)(n-t) n(n-1)(n-2) 6t(t-1)(n-t)(n-t-1) n(n-1)(n-2)(n-3) i=3 t(t-1)(t-2) n(n-1)(n-2) 4t(t-1)(t-2)(n-t) n(n-1)(n-2)(n-3) i=4 t(t-1)(t-2)(t-3) n(n-1)(n-2)(n-3) Table 2: Summary of Problems 1-4	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9204738140106201, "perplexity": 1485.0839075102833}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246651471.95/warc/CC-MAIN-20150417045731-00071-ip-10-235-10-82.ec2.internal.warc.gz"}
http://star-www.rl.ac.uk/docs/sc6.htx/node10.html	Next: Colour indices Up: Background Material Previous: Magnitudes # Photometric Systems The intensity of the light emitted by stars and other astronomical objects varies strongly with wavelength. Thus, the apparent magnitude, , observed for a given star by a detector depends on the range of wavelengths to which the detector is sensitive; a detector sensitive to red light will usually record a different brightness than one sensitive to blue light. The first estimates of stellar magnitudes were made either using the unaided eye or later by direct observation through a telescope. Magnitudes estimated in this way are referred to as visual magnitudes, 4. The sensitivity of the human eye peaks at a wavelength of around 5500Å. The bolometric magnitude, , is the notional magnitude measured across all wavelengths. Clearly the bolometric magnitude cannot be measured directly, because of absorption in the terrestrial atmosphere (see Section ) and the practical difficulties of constructing a detector which will respond to a sufficiently wide range of wavelengths. The bolometric correction is the difference between and : (10) Note, however, that sometimes the opposite sign is given to . The concept of a bolometric magnitude is only really applicable to stars, which to a first approximation emit thermal radiation as black bodies. The bolometric correction is used to derive an approximation to the bolometric magnitude from the observed one. It would clearly be absurd to try to apply a bolometric correction to the observed visual magnitude of some exotic object which was emitting most of its energy non-thermally in the X-ray or radio regions of the spectrum. Schmidt-Kaler[65] gives tables of stellar bolometric corrections. Another type of magnitude which is sometimes encountered is the photographic magnitude, . Photographic magnitudes were determined from the brightness of star images recorded on photographic plates and thus are determined by the wavelength sensitivity of the photographic plate. Early photographic plates were relatively more sensitive to blue than to red light and the effective wavelength of photographic magnitudes is about 4200Å. Note that photographic magnitudes refer to early plates exposed without a filter. Using a combination of more modern emulsions and filters it is, of course, possible to expose plates which are sensitive to different wave-bands. However, modern photometric systems are defined for photoelectric, or latterly, CCD detectors. In modern usage a photometric system comprises a set of discrete wave-bands, each with a known sensitivity to incident radiation. The sensitivity is defined by the detectors and filters used. Additionally a set of primary standard stars are provided for the system which define its magnitude scale. Photometric systems are usually categorised according to the widths of their passbands: wide band systems have bands at least 300Å wide, intermediate band systems have bands between 300 and 100Å, narrow band systems have bands no more than a few tens of Å wide. The optical region of the spectrum is only wide enough to accommodate three or four non-overlapping wide bands. A plethora of photometric systems have been devised and a large number remain in regular use. The criteria for designing photometric systems and descriptions of the more common systems are given by Sterken and Manfroid[67], Straizys[70], Lamla[49], Golay[32] and Jaschek and Jaschek[40]. Some of the more common ones are summarised below. Johnson-Morgan UBV System The Johnson-Morgan UBV wide-band system[43,44] is easily the most widely used photometric system. It was originally introduced in the early 1950s. The longer wavelength R and I bands were added later[42]. Table includes the basic details of the Johnson-Morgan system and Figure shows the general form of the filter transmission curves. Tabulations of these curves are given by Jaschek and Jaschek[40]. The Johnson-Morgan and bands should not be confused with the similar, and similarly named, bands in the Cousins VRI system[13,14]. The Cousins band (complemented by and ) is identical to the Johnson-Morgan system. However, the Cousins and bands respectively have wavelengths of 6700Å and 8100Å and thus both are bluer than the corresponding Johnson-Morgan bands. They are usually indicated by , where C' stands for Cape'. For further details see Straizys[70], pp294-296 and pp309-312. The zero points of the UBV system are chosen so that for a star of spectral type A0 which is unaffected by interstellar reddening (see Appendix ) . Despite its ubiquity the UBV system has some disadvantages. In particular, the short wavelength cutoff of the filter is partly defined by the terrestrial atmosphere rather than the detector or filter. Thus, the cutoff (and hence the observed magnitudes) can vary with altitude, geographic location and atmospheric conditions. Table: Details of common photometric systems. The values are taken from Astrophysical Quantities[1], except those for the system which are taken from Sterken and Manfroid[67] and those for the band which are from the UKIRT on-line documentation System Band Effective Bandwidth Wavelength (FWHM) Å Å visual - photographic - Johnson-Morgan 3650 680 4400 980 5500 890 7000 2200 9000 2400 Strömgren 3500 340 4100 200 4670 160 5470 240 m m 1.25 0.38 1.65 0.48 2.2 0.70 3.5 1.20 3.8 0.6 4.8 5.70 Strömgren System Another widely used system is the Strömgren intermediate-band uvby system[71,72]. The details of this system are included in Table . Filter transmission curves for the Strömgren system are given by Jaschek and Jaschek[40]. Strömgren magnitudes are well-correlated with Johnson-Morgan magnitudes. System The system is an extension of the system to longer wavelengths. It was originally introduced by Johnson and his collaborators though modern versions of it derive from the work of Glass[29]. Details of the system are included in Table . The bands are matched to, and share the same names as, the windows in which the terrestrial atmosphere is transparent at infrared wavelengths (see Section ). The band is a later addition. It is better matched to the corresponding atmospheric window than the the original band. The system is less well-standardised than other systems and each observatory will often define its own system which differs slightly from the others. These differences arise because the atmospheric windows which are transparent at infrared wavelengths are themselves different at different observatories and, in particular, vary with altitude. Consequently, great care must be exercised in inter-comparing observations made at different observatories. Table summarises some of the more common systems. For further details see Bersanelli et al.[4], Bouchetet al.[5] and Straizys[70], pp292-307. Leggett[54] gives details of the transformations between the various infrared systems. Simons and Tokunaga[66] have recently reported an attempt to standardise infrared photometric systems. Table: Common JHKLM systems. Adapted from Bersanelli et al.[4] System Institute Bands Reference Arizona Lunar and Planetary Lab. Johnson[41] SAAO South African Astron. Obs. Glass[29] Carter[6,7] ESO European Southern Obs. Wamsteker[76] Engels et al.[26] AAO Anglo-Australian Obs. Allen and Cragg[3] MSO Mount Stromlo Obs. Jones and Hyland[45] CIT California Inst. Technol. Frogel et al.[28] Elias et al.[25,24] UNAM Univ. Autonoma de Mexico Tapia et al.[73] UKIRT Joint Astron. Centre, Hawaii Casali and Hawarden[8] As for the original Johnson-Morgan system, the zero point of the JHKLM system is defined so that an unreddened A0 star has the same magnitude in all colours: . The standard star used is Vega ( Lyræ). Observing programmes which use a given photometric system need not necessarily observe in all the bands of that system. Often only some, or perhaps even only one, of the bands will be used. The choice of bands will be dictated by the aims of the programme and the observing time available. Subsections Next: Colour indices Up: Background Material Previous: Magnitudes The CCD Photometric Calibration Cookbook	{"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": 0.8741645216941833, "perplexity": 2033.2366252719878}, "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/1416400380866.29/warc/CC-MAIN-20141119123300-00229-ip-10-235-23-156.ec2.internal.warc.gz"}
https://hal.archives-ouvertes.fr/hal-01612071	# Stein characterizations for linear combinations of gamma random variables Abstract : In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as linear combinations of (non necessarily independent) gamma distributed random variables. The connection with Malliavin calculus for random variables in the second Wiener chaos is detailed. An application to McKay Type I random variables is also outlined. Keywords : Type de document : Pré-publication, Document de travail This corresponds to the second section of https://arxiv.org/abs/1601.03301 New results added and .. 2017 Domaine : Liste complète des métadonnées https://hal.archives-ouvertes.fr/hal-01612071 Contributeur : Marie-Annick Guillemer <> Soumis le : vendredi 6 octobre 2017 - 14:20:59 Dernière modification le : mardi 11 décembre 2018 - 01:25:32 ### Identifiants • HAL Id : hal-01612071, version 1 • ARXIV : 1709.01161 ### Citation Benjamin Arras, Ehsan Azmoodeh, Guillaume Poly, Yvik Swan. Stein characterizations for linear combinations of gamma random variables. This corresponds to the second section of https://arxiv.org/abs/1601.03301 New results added and .. 2017. 〈hal-01612071〉 ### Métriques Consultations de la notice	{"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": 0.8105725646018982, "perplexity": 4268.940635875923}, "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-2018-51/segments/1544376823702.46/warc/CC-MAIN-20181211194359-20181211215859-00505.warc.gz"}
http://www.docoja.com/kan/jkantxtg11-9.html	## p\|󎫏ETFEρEEEՁEEEoEDEK t[CgE\tgEGA[JpΉ̃ICaT pꂩ{A͑̋t̖\|̈ӖȒPɌ܂B ̎̃ANZXpaTXgB Ђ[Wp AbvEACtH[ AbvEACpbh O[OEAhChql̃oC@ɃCXg[ΎE̊OŃC^[lbgĂ̎g܂B p\ ̎: p󊿘aT() 󊿘aT() Ɩ󊿘aT() 󊿘aT() ɖ󊿘aT() p󍑌ꎫT() 󍑌ꎫT() Ɩ󍑌ꎫT() 󍑌ꎫT() ɖ󍑌ꎫT() pOꎫT() OꎫT() ƖOꎫT() OꎫT() ɖOꎫT() {̖ƃQ[(p) {̖ƃQ[() {̒nƗs(p) {̒nƗs() {̗jƓ`(p) {̗jƓ`() {̐ƃfBA(p) {̐ƃfBA() AtKjX^푈(p) CN푈(p) CN푈() tXs(p) tXs() tX̐(p) tX̐() hCcs(p) hCcs() L[[h A ANZT[ AWA w ʒu ߕ F EC^[X\|[c C f q GlM[ y I[fBI L v w Ȋw Ƌ َq Ƒ ` J_[ y wZ @Bw D Z s z ʕ RK o ϕi N z \|p Q[ w H L q y ЊQ ٔ d R Љ @ j A V[ j w X\|[c _ ZbNX 푈 f Hו P 䏊pi _X nw n ʐM eNmW[ Nw VC V dC ` O { {j zn _ ঒ ƍ s@ aC sY D w @ ybg ۈ @ { X fBA 싅 [bp s j => => : 1 2 2 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 19 19 19 19 19 19 20 20 20 21 21 22 22 23 23 24 ڃANZXF , , , , , , , o , D , K JeS[F F L[[hF @Bw `E : ͂: stretch, extend, spread, full, string, stick out, throw out (one's chest), square, slap, give a slap, be expensive, be dear : ͂: tension, unit to count sheets of paper グ: ͂肠: raise [lift up] (one's voice) <<< ؂: ͂肫: be in high spirits, be full of pep, be enthusiastic [zealous] <<< 􂯂: ͂肳: split [burst] open, rend <<< o: ͂肾: project (v.), jut [stand] out, overhang <<< o \|: ͂肽: knock [strike] (a person) down, slap [a person in the face], punch (a person's head) <<< \| : : net <<< nFg , , ْ , , o , , , c , , ~ FR𒣂 , X , , 𒣂 , I𒣂 , 𒣂 , X𒣂 , , 𒣂 , 𒣂 , , l , l̒Ȃ , , ܂𒣂 , 𒣂 , , Ԃ𒣂 , j𒣂 , 𒣂 , j , Ӓn , Ӓn𒣂 , 𒣂 , V , V , V , V炸 , , 𒣂 , lp , j𒣂 , ͂𒣂 , V𒣂 , li , {l , h𒣂 , Aei𒣂 , V[𒣂 , ^C , ^C𒣂 , eg𒣂 , lbg𒣂 , uL , \|X^[𒣂 `FbNF \ JeS[F F \|Fcross, end (ext.), save TC, ZC ς: 킽: cross (vi.) <<< n ς: 킽: crossing point, ferry ς܂: ܂: end (vt.), finish, terminate, get through, conclude, settle, make do with ς: : end (vi.), finish, come to an end, be finished, be over, be cleared off, be settled <<< I ς܂Ȃ: ܂Ȃ: I am very sorry (for what I have done), I really did you wrong, I regret ς݂܂: ݂܂: I am very sorry (to trouble you), Excuse me ς: : save <<< ~ nF~ , , o , Fς܂ , тς܂ , ւɍς܂ , ς܂ , { , , , x , oLς , o^ς , zBς , , ōς , pς܂ , , ̎ JeS[F F L[[hF , \|Fnest, beehive, cobweb, lair \E : : nest (n.), beehive, cobweb, lair : : nest (v.) : : build a nest <<< ɒ: ɂ: sit (on eggs), brood <<< \|: : weave [spin] a web <<< \| nF , F̑ , , ̑ , , w偂̑ , w偂̑炯 , , JeS[F F L[[hF { \|Fmeaning, reason, cause, ground, translate N, GL : 킯: meaning, reason, cause, ground <<< R : 킯Ȃ: without reason <<< 󂪗L: 킯: for a certain reason [some reason] <<< L q˂: 킯˂: ask [inquire into] the reason [cause] <<< q ̖: 킯̂Ȃ: easy, simple <<< , ȒP ̕Ȃ: 킯̂킩Ȃ: not know why, absurd, incomprehensible <<< : ₭: translate nF , Ζ , , ʖ , \| , \ , , a F , JeS[F F L[[hF j \|Ffestival, fete, fest, gala TC Ղ: ܂: deify, dedicate a shrine (to), enshrine Ղ: ܂: festival, fete, fest, gala Ղ: ܂: celebrate a festival, hold a fete, hold a memorial service (for) Ղグ: ܂肠: set (a person) up, put (a person) on a pedestal, exalt <<< nFJ , Վi , Փ , Ւd , ՓT , ՗ , i , F , ̍ , f , y , , Ӎ , w , LO , ] , ܌ , n , \N , ԓ , O , , Ɨ , ՚q , SN , , `FbNF J JeS[F F L[[hF \|Ffish, fishing, me (bor.) M, S : : fish (n.) : : : me, I <<< ǂ: Ȃǂ: fish (v.), angle for ނ: Ȃ <<< Hׂ: Ȃׂ: eat fish <<< H ߂܂: Ȃ܂: catch a fish <<< ̍: Ȃ̂ق: fish bone <<< nF , , , , H , , , ϋ , l , ċ , FC , ͊ , Ϗ܋ , R , s , [C , VNȋ , , ̝g , , W , X , ̓Vw , , , , ̃tC JeS[F F L[[hF X\|[c , @Bw \|Fstrong, robust, solid LE, SE : 悢: strong, powerful, mighty, robust, vigorous, solid : 悢: strong [hard] drink <<< : 悢: strong [violent] wind <<< : 悢Ђ: strong light <<< : 悭: strongly, powerfully, firmly, hard, severely, violently ߂: ߂: intensify (vt.) ܂: ܂: become strong, intensify (vi.) : 悭Ȃ <<< : 悳: strength, power, intensity, vigor : 悪: bluff (n.), show of courage : 悪, 悪䂤: bluff (v.) <<< ߂: Ƃ߂: strive, endeavor <<< w : : compel, force : 킢: tough, stiff nF , , , s , d , , , s , , , x , , , , , , , ŋ , , ׋ , F͂ , ̋ , , , S苭 , ӂ , ̋ , ӎű , ݒ , ̋ , , Η͂̋ , , , DS̋ , ̋ , , ͂ , S , h , h , M , i , Ŏ , , , , ӌ: ## o JeS[F F L[[hF \|Fwarp, pass (ext.) PC, LE o: 傤: Buddhist scriptures, sutra oǂ: 傤: chant [recite] a sutra, read a service <<< o: 傤ƂȂ <<< o: ւ: pass away, elapse, go by, pass [go] through o: o: Ă: warp (n.) nFo , oT , o , oc , o , o , o , ox , o , o , o , oH , o , _o , o , o Fo , No , o , oϊw , bo , o , Ԃo ## D JeS[F F L[[hF D \|Fship, vessel Z D: ӂ D: ӂ D: ӂ˂: on board a ship, by ship, by sea D~: ӂ˂: leave [get off] a ship, disembark <<< ~ Do: ӂ˂ł: sail, leave port <<< o D: ӂ˂: arrive in port, enter port <<< Dɏ: ӂ˂ɂ̂: go [get] aboard [on board] a ship, embark (vi.) <<< Dɏ悹: ӂ˂ɂ̂: embark (vt.) <<< Dɋ: ӂ˂ɂ悢: be a good sailor <<< DɎア: ӂ˂ɂ킢: be a bad sailor <<< D𑆂: ӂ˂: row (a boat), nod (in a theater) <<< Dɐ: ӂ˂ɂ悤: get [become] seasick <<< nF͑D , DD , qD , D , D , D , D , Dc , D , D , D , D , D , D , Do , D , D FnɑD , ږD , FD , ^D , ݍqHD , D , CD , ݕD , όD , ĎD , OD , CD , ~D , D , D , ߊCqHD , q͑D , HD , ؑD , oD , qD , CD , D , ʑD , BD , qD , vD , D , D , zD , ؑD , jD , sD , YD , a@D , ⋋D , ߌ~D , AD , D , VD , HD , AD , KD , AD , I_D , N[D , ReiD , Tx[WD `FbNF M ## K JeS[F F L[[hF \|Flearn, study, lesson VE K: Ȃ炤: learn, take lessons (in, on), be taught, study, practice K: Ȃ炢: learning, study, lesson, habit, custom KƐ: Ȃ炢ƂȂ: become a custom, fall into the habit of (doing) <<< K킹: Ȃ킹: make (a person) learn [practice, exercise] K芵: Ȃ炤Ȃ: Practice makes perfect <<< nFK , K , wK , uK , K , K , K , K , K , K , K , K , K , K , \K , K FԂK , ̂K , sVK , @K , K , oGK , sAmK ӌ: w ӌ: ̃y[WɗLLF1380 - 1389ASŁF2742. \|Pbgdq http://www.docoja.com/kan/jkantxtg11-9.html	{"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": 0.9370432496070862, "perplexity": 2247.862218691725}, "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/1508187823229.49/warc/CC-MAIN-20171019050401-20171019070401-00747.warc.gz"}
https://math.stackexchange.com/questions/545794/for-sets-a-b-c-a-setminus-b-subset-a-setminus-c-cup-c-setminus-b	# For sets $A,B,C$, $(A\setminus B)\subset (A\setminus C)\cup (C\setminus B)$ First of all, I am sorry for my bad english, I am from Brazil :-) I have problem with proof for some set theory task. Here it is: $A,B,C$ are three sets. Show that: $$(A\setminus B) \subset (A\setminus C) \cup (C\setminus B)$$ It is clear by looking at diagrams but I do not know how I show! It is not too hard I think... Thank you for the help, Igor Suppose $\;x\in A\setminus B\;$ . We have only two possibilites: \begin{align}(1)&\;\;x\in C : \;\;\implies x\in C\setminus B\implies x\in (A\setminus C)\cup (C\setminus B)\\ (2)&\;\;x\notin C :\;\;\implies x\in A\setminus C\implies x\in (A\setminus C)\cup (C\setminus B)\end{align} Hint: Take any $x\in A\setminus B,$ which means that $x\in A$ and $x\notin B$. Now consider two cases (and note that there are no other cases to consider): • $x\in C$ • $x\notin C$ Show that, in either case, $x\in (A\setminus C)\cup(C\setminus B).$ If you take out from $A$ things that are in $B$, what is left is certainly things in $A$ not in $C$, except for things in $C$ that were not in $B$, so if you add the latter you are all set. The formula is nothing more than this.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9432592391967773, "perplexity": 510.5090587470892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319082.81/warc/CC-MAIN-20190823214536-20190824000536-00157.warc.gz"}
http://mathhelpforum.com/calculus/138355-differential-using-chain-rule-print.html	# Differential using chain Rule • April 10th 2010, 01:34 PM charikaar Differential using chain Rule suppose that f:R-->R is continuous. using the chain rule, or otherwise, show that the function F defined by $F(x) = \exp(\int_0^xf)$ is differentiable and find its derivative. Fundamental Theorem of calculus say that the derivative of $\int_a^xf(t)dt = f(x)$ I tried to use this to solve the above but no vain. thanks for any help. • April 10th 2010, 01:45 PM You have the FTC wrong: it is $\int_a^b f'(x)\, \mathrm{d}x = f(b) - f(a)$, or equivalently, $\frac{\mathrm{d}}{\mathrm{d}x}\int_a^xf(s)\,\mathr m{d}s = f(x)$. Does this help? • April 10th 2010, 01:55 PM charikaar sorry still can't solve it with exponential! • April 10th 2010, 02:02 PM Let $h(x) = \int_0^x f$. Then $F(x) = \exp(h(x))$. Applying the chain rule gives $F'(x) = \exp(h(x))\cdot h'(x)$. The exponential is now out of the way and you only have to differentiate h. Use the FTC for this. Define the function $g(t)$ to be the antiderivative of $f(t)$ Then, $\int_{0}^{x} f(t) dt = g(x) - g(0)$ So, $F(x) = e^{g(x) - g(0)}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9607023596763611, "perplexity": 680.6280370973686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464051299749.12/warc/CC-MAIN-20160524005459-00201-ip-10-185-217-139.ec2.internal.warc.gz"}
https://www.maplesoft.com/support/help/maple/view.aspx?path=odeadvisor%2FAbel	Solving Abel's ODEs of the First Kind - Maple Programming Help Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : First Order : odeadvisor/Abel Solving Abel's ODEs of the First Kind Description • The general form of Abel's equation of the first kind is given by: > Abel_ode := diff(y(x),x)=f3(x)y(x)^3+f2(x)y(x)^2+f1(x)y(x)+f0(x); ${\mathrm{Abel_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{\mathrm{f3}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{3}}{+}{\mathrm{f2}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{2}}{+}{\mathrm{f1}}{}\left({x}\right){}{y}{}\left({x}\right){+}{\mathrm{f0}}{}\left({x}\right)$ (1) where f3(x), f2(x), f1(x) and f0(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 24. There is as yet no general solution for this ODE. For Abel's equation of the second kind, see Abel2A and Abel2C. • The most general method available at the moment to solve Abel ODEs seems to be the method of "Abel's invariant", described in E. Kamke, p. 26, as sub-method (g) due to M. Chini. The invariant of an Abel equation with f2=0 is the following quantity: > Abel_invariant := -1/27/f3(x)^4(-diff(f0(x),x)f3(x)+f0(x)diff(f3(x),x)+ 3f0(x)f3(x)f1(x))^3/f0(x)^5; ${\mathrm{Abel_invariant}}{≔}{-}\frac{{\left({-}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f0}}{}\left({x}\right)\right){}{\mathrm{f3}}{}\left({x}\right){+}{\mathrm{f0}}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f3}}{}\left({x}\right)\right){+}{3}{}{\mathrm{f0}}{}\left({x}\right){}{\mathrm{f3}}{}\left({x}\right){}{\mathrm{f1}}{}\left({x}\right)\right)}^{{3}}}{{27}{}{{\mathrm{f3}}{}\left({x}\right)}^{{4}}{}{{\mathrm{f0}}{}\left({x}\right)}^{{5}}}$ (2) If the invariant does not depend on x, then the equation can be solved directly. For an Abel equation with f2<>0, the f2 term can be removed by using the following transformation: > y(x)=u(x)-f2(x)/3/f3(x); ${y}{}\left({x}\right){=}{u}{}\left({x}\right){-}\frac{{\mathrm{f2}}{}\left({x}\right)}{{3}{}{\mathrm{f3}}{}\left({x}\right)}$ (3) The invariant can then be calculated as in the previous case. Note that if an Abel ODE has a constant invariant, then any other Abel ODE obtained from it by a transformation of the form > {y(x)=G(t)u(t)+H(t), x=F(t)}; $\left\{{x}{=}{F}{}\left({t}\right){,}{y}{}\left({x}\right){=}{G}{}\left({t}\right){}{u}{}\left({t}\right){+}{H}{}\left({t}\right)\right\}$ (4) will also have a constant invariant (that is, is also solvable by this method). The method "Chini" (see ?odeadvisor,Chini), also due to Chini, generalizes this method of the constant invariant for Abel ODEs. Examples > $\mathrm{Abel_ode}≔\frac{ⅆ}{ⅆx}y\left(x\right)=\mathrm{f3}\left(x\right){y\left(x\right)}^{3}+\mathrm{f2}\left(x\right){y\left(x\right)}^{2}+\mathrm{f1}\left(x\right)y\left(x\right)+\mathrm{f0}\left(x\right)$ ${\mathrm{Abel_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{\mathrm{f3}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{3}}{+}{\mathrm{f2}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{2}}{+}{\mathrm{f1}}{}\left({x}\right){}{y}{}\left({x}\right){+}{\mathrm{f0}}{}\left({x}\right)$ (5) > $\mathrm{Abel_invariant}≔-\frac{1{\left(-\left(\frac{ⅆ}{ⅆx}\mathrm{f0}\left(x\right)\right)\mathrm{f3}\left(x\right)+\mathrm{f0}\left(x\right)\left(\frac{ⅆ}{ⅆx}\mathrm{f3}\left(x\right)\right)+3\mathrm{f0}\left(x\right)\mathrm{f3}\left(x\right)\mathrm{f1}\left(x\right)\right)}^{3}}{27{\mathrm{f3}\left(x\right)}^{4}{\mathrm{f0}\left(x\right)}^{5}}$ ${\mathrm{Abel_invariant}}{≔}{-}\frac{{\left({-}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f0}}{}\left({x}\right)\right){}{\mathrm{f3}}{}\left({x}\right){+}{\mathrm{f0}}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f3}}{}\left({x}\right)\right){+}{3}{}{\mathrm{f0}}{}\left({x}\right){}{\mathrm{f3}}{}\left({x}\right){}{\mathrm{f1}}{}\left({x}\right)\right)}^{{3}}}{{27}{}{{\mathrm{f3}}{}\left({x}\right)}^{{4}}{}{{\mathrm{f0}}{}\left({x}\right)}^{{5}}}$ (6) > $y\left(x\right)=u\left(x\right)-\frac{\mathrm{f2}\left(x\right)}{3\mathrm{f3}\left(x\right)}$ ${y}{}\left({x}\right){=}{u}{}\left({x}\right){-}\frac{{\mathrm{f2}}{}\left({x}\right)}{{3}{}{\mathrm{f3}}{}\left({x}\right)}$ (7) > $\left\{y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right),x=F\left(t\right)\right\}$ $\left\{{x}{=}{F}{}\left({t}\right){,}{y}{}\left({x}\right){=}{G}{}\left({t}\right){}{u}{}\left({t}\right){+}{H}{}\left({t}\right)\right\}$ (8) > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$ $\left[{\mathrm{odeadvisor}}\right]$ (9) > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{dchange}\right)$ $\left[{\mathrm{dchange}}\right]$ (10) > $\mathrm{odeadvisor}\left(\mathrm{Abel_ode}\right)$ $\left[{\mathrm{_Abel}}\right]$ (11) 1) An example of an Abel ODE having a constant invariant solved using the related scheme: > $\mathrm{ODE}≔\frac{ⅆ}{ⅆx}y\left(x\right)=\frac{1\left(12x+27{x}^{3}+27{x}^{3}{y\left(x\right)}^{2}+18{x}^{2}y\left(x\right)+27{y\left(x\right)}^{3}{x}^{3}+27{x}^{2}{y\left(x\right)}^{2}+9xy\left(x\right)+1\right)}{27{x}^{3}}:$ > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ODE}\right)$ ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}\frac{{29}{}{\mathrm{RootOf}}{}\left({-}{81}{}\left({{\int }}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{841}{}{{\mathrm{_a}}}^{{3}}{-}{27}{}{\mathrm{_a}}{+}{27}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){+}{x}{+}{3}{}{\mathrm{_C1}}\right){}{x}{-}{3}{}{x}{-}{3}}{{9}{}{x}}$ (12) Any "linear transformation" of ODE will also be solved by the same method. For example: > $\mathrm{TR_LIN}≔\left\{y\left(x\right)=G\left(t\right)u\left(t\right)+H\left(t\right),x=F\left(t\right)\right\}$ ${\mathrm{TR_LIN}}{≔}\left\{{x}{=}{F}{}\left({t}\right){,}{y}{}\left({x}\right){=}{G}{}\left({t}\right){}{u}{}\left({t}\right){+}{H}{}\left({t}\right)\right\}$ (13) > $\mathrm{ODE_p}≔\mathrm{dchange}\left(\mathrm{TR_LIN},\mathrm{ODE},\left[u,t\right]\right):$ > $\mathrm{ans_p}≔\mathrm{dsolve}\left(\mathrm{ODE_p},u\left(t\right)\right)$ ${\mathrm{ans_p}}{≔}{u}{}\left({t}\right){=}{-}\frac{{9}{}{H}{}\left({t}\right){}{F}{}\left({t}\right){-}{29}{}{\mathrm{RootOf}}{}\left({-}{81}{}\left({{\int }}_{{}}^{{\mathrm{_Z}}}\frac{{1}}{{841}{}{{\mathrm{_a}}}^{{3}}{-}{27}{}{\mathrm{_a}}{+}{27}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){+}{F}{}\left({t}\right){+}{3}{}{\mathrm{_C1}}\right){}{F}{}\left({t}\right){+}{3}{}{F}{}\left({t}\right){+}{3}}{{9}{}{G}{}\left({t}\right){}{F}{}\left({t}\right)}$ (14) 2) A case for which the solving method is known: f0(x) = f1(x) = 0, and diff(f3(x)/f2(x),x)=a*f2(x). In this case, one can proceed as follows: > $\mathrm{ode}≔\mathrm{subs}\left(\left\{\mathrm{f1}\left(x\right)=0,\mathrm{f0}\left(x\right)=0\right\},\mathrm{Abel_ode}\right)$ ${\mathrm{ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{\mathrm{f3}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{3}}{+}{\mathrm{f2}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{2}}$ (15) First introduce r(t) and t as new variables using: > $\mathrm{ITR}≔\left\{y\left(x\right)=\frac{\mathrm{f2}\left(t\right)r\left(t\right)}{\mathrm{f3}\left(t\right)},x=t\right\}$ ${\mathrm{ITR}}{≔}\left\{{x}{=}{t}{,}{y}{}\left({x}\right){=}\frac{{\mathrm{f2}}{}\left({t}\right){}{r}{}\left({t}\right)}{{\mathrm{f3}}{}\left({t}\right)}\right\}$ (16) > $\mathrm{new_ode}≔\mathrm{dchange}\left(\mathrm{ITR},\mathrm{ode},\left[r\left(t\right),t\right]\right):$ Now, introduce the condition on the derivative of f3(t)/f2(t): > $\mathrm{constraint}≔\frac{ⅆ}{ⅆt}\left(\frac{\mathrm{f3}\left(t\right)}{\mathrm{f2}\left(t\right)}\right)-a\mathrm{f2}\left(t\right)=0$ ${\mathrm{constraint}}{≔}\frac{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f3}}{}\left({t}\right)}{{\mathrm{f2}}{}\left({t}\right)}{-}\frac{{\mathrm{f3}}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{f2}}{}\left({t}\right)\right)}{{{\mathrm{f2}}{}\left({t}\right)}^{{2}}}{-}{a}{}{\mathrm{f2}}{}\left({t}\right){=}{0}$ (17) and simplify new_ode with regard to this relation: > $\mathrm{new_ode2}≔\mathrm{simplify}\left(\mathrm{new_ode},\left\{\mathrm{constraint}\right\},\left\{\frac{ⅆ}{ⅆt}\mathrm{f3}\left(t\right)\right\}\right)$ ${\mathrm{new_ode2}}{≔}\frac{{-}{r}{}\left({t}\right){}{{\mathrm{f2}}{}\left({t}\right)}^{{3}}{}{a}{+}{\mathrm{f2}}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{r}{}\left({t}\right)\right){}{\mathrm{f3}}{}\left({t}\right)}{{{\mathrm{f3}}{}\left({t}\right)}^{{2}}}{=}\frac{{{\mathrm{f2}}{}\left({t}\right)}^{{3}}{}{{r}{}\left({t}\right)}^{{2}}{}\left({r}{}\left({t}\right){+}{1}\right)}{{{\mathrm{f3}}{}\left({t}\right)}^{{2}}}$ (18) This ODE is separable. > $\mathrm{odeadvisor}\left(\mathrm{new_ode2}\right)$ $\left[{\mathrm{_separable}}\right]$ (19) 3) Rewrite in "normal form" (no square term in the RHS) when: f3(x)=1/x, f2(x)=1/x, f1(x)=0, f0(x)=4 > $\mathrm{ode}≔\mathrm{eval}\left(\mathrm{subs}\left(\left\{\mathrm{f3}\left(x\right)=\frac{1}{x},\mathrm{f2}\left(x\right)=\frac{1}{x},\mathrm{f1}\left(x\right)=0,\mathrm{f0}\left(x\right)=4\right\},\mathrm{Abel_ode}\right)\right)$ ${\mathrm{ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{{y}{}\left({x}\right)}^{{3}}}{{x}}{+}\frac{{{y}{}\left({x}\right)}^{{2}}}{{x}}{+}{4}$ (20) First of all, Abel's ODEs of the first kind can be rewritten in normal form (which is sometimes useful) by making the appropriate change of variables. The transformation is of a general type. After introducing > $w\left(x\right)≔{ⅇ}^{∫\left(\mathrm{f1}\left(x\right)-\frac{{\mathrm{f2}\left(x\right)}^{2}}{3\mathrm{f3}\left(x\right)}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx}$ ${w}{}\left({x}\right){≔}{{ⅇ}}^{{\int }\left({\mathrm{f1}}{}\left({x}\right){-}\frac{{{\mathrm{f2}}{}\left({x}\right)}^{{2}}}{{3}{}{\mathrm{f3}}{}\left({x}\right)}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}$ (21) the following transformations (where {x,y(x)} = old vars; {t,r(t)} = new vars) will yield the desired normal form: > $\mathrm{tr}≔\left\{t=∫\mathrm{f3}\left(x\right){'w'\left(x\right)}^{2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx,r\left(t\right)=\frac{1\left(3y\left(x\right)\mathrm{f3}\left(x\right)+\mathrm{f2}\left(x\right)\right)}{3{ⅇ}^{∫\left(\mathrm{f1}\left(x\right)-\frac{1{\mathrm{f2}\left(x\right)}^{2}}{3\mathrm{f3}\left(x\right)}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx}\mathrm{f3}\left(x\right)}\right\}$ ${\mathrm{tr}}{≔}\left\{{t}{=}{\int }{\mathrm{f3}}{}\left({x}\right){}{{w}{}\left({x}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{,}{r}{}\left({t}\right){=}\frac{{3}{}{y}{}\left({x}\right){}{\mathrm{f3}}{}\left({x}\right){+}{\mathrm{f2}}{}\left({x}\right)}{{3}{}{{ⅇ}}^{{\int }\left({\mathrm{f1}}{}\left({x}\right){-}\frac{{{\mathrm{f2}}{}\left({x}\right)}^{{2}}}{{3}{}{\mathrm{f3}}{}\left({x}\right)}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}}{}{\mathrm{f3}}{}\left({x}\right)}\right\}$ (22) The transformation equations required for this case are obtained from the general transformation tr (above) as follows: > $\mathrm{TR}≔\mathrm{eval}\left(\mathrm{subs}\left(\left\{\mathrm{f3}\left(x\right)=\frac{1}{x},\mathrm{f2}\left(x\right)=\frac{1}{x},\mathrm{f1}\left(x\right)=0,\mathrm{f0}\left(x\right)=4\right\},\mathrm{tr}\right)\right)$ ${\mathrm{TR}}{≔}\left\{{t}{=}{-}\frac{{3}}{{2}{}{{x}}^{{2}}{{3}}}}{,}{r}{}\left({t}\right){=}\frac{{{x}}^{{4}}{{3}}}{}\left(\frac{{3}{}{y}{}\left({x}\right)}{{x}}{+}\frac{{1}}{{x}}\right)}{{3}}\right\}$ (23) > $\mathrm{ITR}≔\mathrm{solve}\left(\mathrm{TR},\left\{y\left(x\right),x\right\}\right)$ ${\mathrm{ITR}}{≔}\left\{{x}{=}{-}\frac{{3}{}{\mathrm{RootOf}}{}\left({2}{}{t}{}{{\mathrm{_Z}}}^{{2}}{+}{3}\right)}{{2}{}{t}}{,}{y}{}\left({x}\right){=}{-}\frac{{\mathrm{RootOf}}{}\left({2}{}{t}{}{{\mathrm{_Z}}}^{{2}}{+}{3}\right){-}{3}{}{r}{}\left({t}\right)}{{3}{}{\mathrm{RootOf}}{}\left({2}{}{t}{}{{\mathrm{_Z}}}^{{2}}{+}{3}\right)}\right\}$ (24) and the change of variables is implemented as follows: > $\mathrm{new_ode}≔\mathrm{dchange}\left(\mathrm{ITR},\mathrm{ode},\left[t,r\left(t\right)\right],'\mathrm{known}'=\mathrm{indets}\left(\mathrm{ode},'\mathrm{unknown}'\right)\right):$ > $\mathrm{new_ode2}≔\mathrm{simplify}\left(\mathrm{op}\left(1,\mathrm{map}\left(\mathrm{allvalues},\left[\mathrm{solve}\left(\mathrm{new_ode},\left\{\frac{ⅆ}{ⅆt}r\left(t\right)\right\}\right)\right]\right)\right)\right)$ ${\mathrm{new_ode2}}{≔}\left\{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{r}{}\left({t}\right){=}\frac{{18}{}{{t}}^{{3}}{}{{r}{}\left({t}\right)}^{{3}}{-}{{t}}^{{2}}{}\sqrt{{6}}{}\sqrt{{-}\frac{{1}}{{t}}}{-}{243}}{{18}{}{{t}}^{{3}}}\right\}$ (25) Finally, the normal form can be made explicit as follows: > $\mathrm{collect}\left(\mathrm{new_ode2},r\left(t\right),\mathrm{factor}\right)$ $\left\{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{r}{}\left({t}\right){=}{{r}{}\left({t}\right)}^{{3}}{-}\frac{{{t}}^{{2}}{}\sqrt{{6}}{}\sqrt{{-}\frac{{1}}{{t}}}{+}{243}}{{18}{}{{t}}^{{3}}}\right\}$ (26)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 52, "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": 0.9542539715766907, "perplexity": 1090.5174836120416}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668772.53/warc/CC-MAIN-20191116231644-20191117015644-00329.warc.gz"}
https://pure.pucv.cl/en/publications/interior-solutions-of-relativistic-stars-in-the-scale-dependent-s	# Interior solutions of relativistic stars in the scale-dependent scenario Grigoris Panotopoulos, ANGEL RINCON RIVERO, Ilídio Lopes Research output: Contribution to journalArticlepeer-review 15 Scopus citations ## Abstract We study relativistic stars in the scale-dependent scenario, which is one of the approaches to quantum gravity, and where Newton’s constant is promoted to a scale-dependent quantity. First, the generalized structure equations are derived here for the first time. Then they are integrated numerically assuming a linear equation-of-state in the simplest MIT bag model for quark matter. We compute the radius, the mass and the compactness of strange quarks stars, and we show that the energy conditions are fulfilled. Original language English 318 European Physical Journal C 80 4 https://doi.org/10.1140/epjc/s10052-020-7900-3 Published - 1 Apr 2020 Yes ## Fingerprint Dive into the research topics of 'Interior solutions of relativistic stars in the scale-dependent scenario'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9658163189888, "perplexity": 2889.531088109932}, "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/1664030335448.34/warc/CC-MAIN-20220930082656-20220930112656-00654.warc.gz"}
https://infoscience.epfl.ch/record/161377	Field-induced decay dynamics in square-lattice antiferromagnets Dynamical properties of the square-lattice Heisenberg antiferromagnet in applied magnetic field are studied for arbitrary value S of the spin. Above the threshold field for two-particle decays, the standard spin-wave theory yields singular corrections to the excitation spectrum with logarithmic divergences for certain momenta. We develop a self-consistent approximation applicable for S≥1, which avoids such singularities and provides regularized magnon decay rates. Results for the dynamical structure factor obtained in this approach are presented for S=1 and S=5/2. Published in: Physical Review B Condensed Matter, 82, 14, 144402 Year: 2010 ISSN: 0163-1829 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": 0.8908607959747314, "perplexity": 2502.6992254828056}, "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-2018-13/segments/1521257649683.30/warc/CC-MAIN-20180324034649-20180324054649-00560.warc.gz"}
http://www.vixra.org/anal/	# Functions and Analysis Previous months: 2007 - 0703(1) 2009 - 0903(1) 2010 - 1003(1) - 1004(2) - 1005(2) - 1008(1) 2011 - 1106(3) - 1110(1) - 1112(1) 2012 - 1202(4) - 1203(8) - 1204(1) - 1206(3) - 1207(2) - 1210(3) - 1211(1) - 1212(2) 2013 - 1301(2) - 1302(1) - 1303(2) - 1304(4) - 1305(7) - 1306(16) - 1307(5) - 1308(3) - 1309(1) - 1310(5) - 1311(2) 2014 - 1402(1) - 1403(5) - 1404(2) - 1405(1) - 1406(1) - 1407(3) - 1408(3) - 1409(1) - 1410(1) - 1411(3) - 1412(1) 2015 - 1502(5) - 1503(3) - 1505(1) - 1506(1) - 1507(4) - 1508(5) - 1509(2) - 1510(10) - 1511(5) - 1512(1) 2016 - 1601(5) - 1602(3) - 1603(3) - 1604(7) - 1605(2) - 1606(4) - 1608(5) - 1609(4) - 1610(2) - 1611(5) - 1612(4) 2017 - 1701(8) - 1702(2) - 1703(12) - 1704(2) - 1705(7) - 1707(4) - 1708(3) - 1709(9) - 1710(5) - 1711(4) - 1712(7) 2018 - 1801(1) - 1802(6) - 1803(2) - 1804(2) - 1806(8) - 1807(4) ## Recent submissions Any replacements are listed farther down [276] viXra:1807.0233 [pdf] submitted on 2018-07-12 19:46:47 ### The Isotropic Constant Depends Only on the Dimension of the Domain Authors: Johan Aspegren In this preprint we will prove that the isotropic constant depends only on the dimension of the domain. Thus we prove a result that implies the sharp isotropic constant conjecture. Category: Functions and Analysis [275] viXra:1807.0228 [pdf] submitted on 2018-07-11 05:35:49 ### News Limit Formulas for Exponential of the Digamma Function, K-Power and Exponential Function Authors: Edigles Guedes We derive some identities for limit of the exponential for digamma function, k-power and exponential function, involving gamma functions and Pochhammer symbols. Category: Functions and Analysis [274] viXra:1807.0227 [pdf] submitted on 2018-07-11 05:38:23 ### News Limit Formulas for the Exponential of Pi 8, Involving Pochhammer Symbols and Secant Function Authors: Edigles Guedes I derive some news identities for limit of the exponential of Pi/8, involving Pochhammer symbols and secant function. Category: Functions and Analysis [273] viXra:1807.0135 [pdf] submitted on 2018-07-07 01:53:55 ### Dynamic Equations for 2nd-Order Curves Authors: Viktor Strohm The motion of a point along an ellipse under the action of a generalized force is investigated. Result: differential equation of second-order curves with respect to the focus, differential equation of curves of the second order with respect to the center, general differential equation of second order curves. Several examples of the application of these equations are proposed. Category: Functions and Analysis [272] viXra:1806.0464 [pdf] submitted on 2018-06-30 13:06:43 ### Np?=exp Authors: Thinh D. Nguyen We only point out that the work of algorithmic algebra community is not enough, at least so far. Category: Functions and Analysis [271] viXra:1806.0444 [pdf] submitted on 2018-06-28 10:42:13 ### The Optimization Principle for the Riemann Hypothesis Authors: Hassine Saidane Based on the observation that several physical, biological and social proceesses seem to be optimizing an objective function such as an action or a utility, the Central Principle of Science was deemed to be Optimization. Indeed, optimization proved to be an efficient tool for uncovering several scientific laws and proving some scientific theories. In this paper, we use this paradigm to identify the location of the nontrivial zeros of the Riemann Zeta function (RZF).This approach enabled the formulation of this problem as a constrained optimization problem where a simple objective function referred to here as the “Push-Pull Action” is maximized. The solution of the resulting constrained nonlinear optimization problem proved that nontrivial zeros of RZF are located on the critical line. In addition to proving the Riemann Hypothesis, this approach unveiled a plausible law of “Maximum Action of Push-Pull” that seems to be driving RZF to its equilibrium states at the different heights where it reaches its nontrivial zeros. We also show that this law applies to functions exhibiting the same properties as RZF. Category: Functions and Analysis [270] viXra:1806.0360 [pdf] submitted on 2018-06-24 12:50:58 ### Computing Multi-Homogeneous Bezout Numbers is Hard Authors: Thinh Nguyen The multi-homogeneous B´ezout number is a bound for the number of solutions of a system of multi-homogeneous polynomial equations, in a suitable product of projective spaces. Given an arbitrary, not necessarily multi-homogeneous system, one can ask for the optimal multi-homogenization that would minimize the B´ezout number. In this paper, it is proved that the problem of computing, or even estimating the optimal multi-homogeneous B´ezout number is actually NP-hard. In terms of approximation theory for combinatorial optimization, the problem of computing the best multi-homogeneous structure does not belong to APX, unless P = NP. Moreover, polynomial time algorithms for estimating the minimal multihomogeneous B´ezout number up to a fixed factor cannot exist even in a randomized setting, unless BPP⊇NP. Category: Functions and Analysis [269] viXra:1806.0326 [pdf] submitted on 2018-06-22 12:30:06 ### On the New Method for Finding Sum of an Infinite Series in Which 1/n (N∈N) is Common from Every Term Such that N→∞ Authors: Tejas Chandrakant Thakare Using method of integration as the limit of sum we can easily evaluate sum of an infinite series in which 1/n is common from every term such that n→∞ (n∈N). However in this method we do some rigorous calculations before integration. In this paper, in order to minimize the labor involved in this process I propose an alternative new method for finding the sum of an infinite series in which 1/n is common from every term such that n→∞. Category: Functions and Analysis [268] viXra:1806.0239 [pdf] submitted on 2018-06-17 23:43:19 ### The Similarity Between Rules for Essentially Adequate Quaternionic and Complex Differentiation Authors: Michael Parfenov This paper is the third paper of the cycle devoted to the theory of essentially adequate quaternionic differentiability. It is established that the quaternionic holomorphic (ℍ -holomorphic) functions, satisfying the essentially adequate generalization of Cauchy-Riemann’s equations, make up a very remarkable class: generally non-commutative quaternionic multiplication behaves as commutative in the case of multiplication of ℍ -holomorphic functions. Everyone can construct such ℍ-holomorphic functions by replacing a complex variable as a single whole by a quaternionic one in expressions for complex holomorphic functions, and thereafter verify their commutativity. This property, which is confirmed by a lot of ℍ-holomorphic functions, gives conclusive evidence that the developed theory is true. The rules for quaternionic differentiation of combinations of ℍ-holomorphic functions find themselves similar to those from complex analysis: the formulae for differentiation of sums, products, ratios, and compositions of H-holomorphic functions as well as quaternionic power series, are fully identical to their complex analogs. The example of using the deduced rules is considered and it is shown that they reduce essentially the volume of calculations. The base notions of complex Maclaurin series expansions are adapted to the quaternion case. Category: Functions and Analysis [267] viXra:1806.0067 [pdf] submitted on 2018-06-07 04:22:20 ### Exactly Solving Arbitrary Order Linear Ordinary Differential Equations Authors: Claude Michael Cassano Theorems establishing exact solution for any linear ordinary differential equation of arbitrary order (homogeneous and inhomogeneous) are presented and proven. Category: Functions and Analysis [266] viXra:1806.0047 [pdf] submitted on 2018-06-06 04:42:10 ### Exactly Solving Second Order Linear Ordinary Differential Equations Authors: Claude Michael Cassano Further development of exactly solving second order linear ordinary differential equations, and related non-linear ordinary differential equations. Category: Functions and Analysis [265] viXra:1804.0405 [pdf] submitted on 2018-04-26 11:14:24 ### Mixed Generalized Multifractal Densities for Vector Valued Quasi-Ahlfors Measures Authors: Adel Farhat, Anouar Ben Mabrouk In the present work we are concerned with some density estimations of vector valued measures in the framework of the so-called mixed multifractal analysis. We precisely consider some Borel probability measures satisfying a weak quasi-Alfors regularity. Mixed multifractal generalizations of densities are then introduced and studied in a framework of relative mixed multifractal analysis. Category: Functions and Analysis [264] viXra:1804.0264 [pdf] submitted on 2018-04-20 06:18:07 ### On Expanding a Function Into Raw Moment Series Authors: Andrej Liptaj Abstract I focus in this text on the construction of functions f_{j} with the delta property C_{i}\left(f_{j}\right)=\delta_{i,j}, where C_{i} are operators which associate to a function its i-th raw moment. A formal method for their construction is found, however results are divergent, from what a non-existence of such functions is conjectured. This also prevents an elegant series expansion with order-by-order moment matching. For a finite interval some partial results are presented: a method of expansion into raw moment series for finite number of moments and a “non-delta” method based on computing Legendre-expansion coefficients from moments (an already known method [1]). As by-product some coefficients formulas are found: coefficients for expanding a Hermite function into the Taylor series and coefficients for expanding into the Taylor series an element of a Fourier series (i.e. common formula for sine and cosine) thus formally merging the two (sine and cosine) Fourier sub-series into one. Category: Functions and Analysis [263] viXra:1803.0498 [pdf] submitted on 2018-03-22 20:24:32 ### On the Non-Trivial Zeros of the Riemann Zeta Function Authors: John Herapath, Quincy Howard Xavier, Carl Wigert In this document, we present several important insights concerning the Riemann Zeta function and the locations of its zeros. More importantly, we prove that we should be awarded the $1 000 000 prize for proving or disproving the Riemann hypothesis Category: Functions and Analysis [262] viXra:1803.0001 [pdf] submitted on 2018-03-01 03:59:30 ### Interesting Expansion Based on Matching Definite Integrals of Derivatives: Simple, Elegant, But Unexplored Authors: Andrej Liptaj Comments: 9 Pages. A novel method of function expansion is presented. It is based on matching the definite integrals of the derivatives of the function to be approximated by appropriate polynomials. The method is fully integral-based, it is easy to construct and it presumably slightly outperforms Taylor series in the convergence rate. Category: Functions and Analysis [261] viXra:1802.0267 [pdf] submitted on 2018-02-19 17:56:17 ### Abel's Lemma and Dirichlet's Test Incorrectly Determine that a Trigonometric Version of the Dirichlet Series$\zeta(s)=\sum N^{-S}$is Convergent Throughout the Critical Strip at$t\ne0$Authors: Ayal Sharon Comments: 18 Pages. Euler's formula is used to derive a trigonometric version of the Dirichlet series$\zeta(s)=\sum n^{-s}$, which is divergent in the half-plane$\sigma \le 1$, wherein$s \in \mathbb{C}$and$s=\sigma +it$. Abel's lemma and Dirichlet's test incorrectly hold that trigonometric$\zeta(s)$is convergent in the critical strip$0<\sigma \le 1$at$t\ne0$, because they fail to consider a divergent monotonically decreasing series (e.g. the harmonic series) in combination with a bounded oscillating function having an increasing period duration (e.g.$f(t, n) = \sin(t \cdot \ln(n))$). Category: Functions and Analysis [260] viXra:1802.0191 [pdf] submitted on 2018-02-14 18:58:19 ### New Special Function and It's Application Authors: Zeraoulia Rafik Comments: 06 Pages. Thank's to the special function In this note we present a new special function such that behaves more like error function and since we arn’t able to express it as elementary function using previous standard functions ,we only give it’s simple expression in some range values using numerical approximation . and we will show how it helps to get values of complicated integral which they arn’t available at wolfram alpha and in the same time we will show it’s relationship with error function and cumulative distribution function . Category: Functions and Analysis [259] viXra:1802.0126 [pdf] submitted on 2018-02-10 07:24:37 ### A Note on the Possibility of Incomplete Theory Authors: Han Geurdes, Koji Nagata, Tadao Nakamura, Ahmed Farouk Comments: 11 Pages. None In the paper it is demonstrated that Bells theorem is an unprovable theorem. This inconsistency is similar to concrete mathematical incompleteness. The inconsistency is purely mathematical. Nevertheless the basic physics requirements of a local model are fulfilled. Category: Functions and Analysis [258] viXra:1802.0120 [pdf] submitted on 2018-02-10 14:44:28 ### Analyticity and Function Satisfying :$\displaystyle \ F'=e^{{f}^{-1}}$Authors: Zeraoulia Rafik Comments: 23 Pages. I wish my results w'd be considerable for any futur refeered journal In this note we present some new results about the analyticity of the functional-differential equation$ f'=e^{{f}^{-1}}$at$ 0$with$f^{-1}$is a compositional inverse of$f$, and the growth rate of$f_-(x)$and$f_+(x)$as$x\to \infty$, and we will check the analyticity of some functional equations which they were studied before and had a relashionship with the titled functional-differential and we will conclude our work with a conjecture related to Borel- summability and some interesting applications of some divergents generating function with radius of convergent equal$0$in number theory Category: Functions and Analysis [257] viXra:1802.0094 [pdf] submitted on 2018-02-08 07:08:19 ### Upper Bound for the Product of the Sum of the Reciprocals of N Real Numbers Greater Than or Equal to 1 by the Product of These Incremented by 1. Authors: Jesús Álvarez Lobo Comments: 2 Pages. Revista Escolar de la Olimpiada Iberoamericana de Matemática. Volume 22. Spanish. Upper bound for the product of the sum of the reciprocals of n real numbers greater than or equal to 1 by the product of those increased by 1, and some variants. Se establece una cota superior para el producto del sumatorio de los recíprocos de n números reales mayores o iguales que 1 por el producto de éstos incrementados en 1, y para algunas variantes. Category: Functions and Analysis [256] viXra:1802.0021 [pdf] submitted on 2018-02-02 16:57:10 ### The Signum Function of the Second Derivative and Its Application to the Determination of Relative Extremes of Fractional Functions (SF2D). Authors: Jesús Álvarez Lobo Comments: 10 Pages. Usually, the complexity of a fractional function increases significantly in its second derivative, so the calculation of the second derivative can be tedious and difficult to simplify and evaluate its value at a point, especially if the abscise isn't an integer. However, to determine whether a point at which cancels the first derivative of a function is a relative extremum (maximum or minimum) of it, is not necessary to know the value of the second derivative at the point but only its sign. Motivated by these facts, we define a signum function for the second derivative of fractional functions in the domain of the roots of the first derivative of the function. The method can dramatically simplify the search for maximum and minimum points in fractional functions and can be implemented by means of a simple algorithm. Category: Functions and Analysis [255] viXra:1801.0096 [pdf] submitted on 2018-01-08 07:56:30 ### Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers Authors: Martin Nicholson Comments: 6 Pages. In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. Category: Functions and Analysis [254] viXra:1712.0539 [pdf] submitted on 2017-12-20 06:47:39 ### Integrals Containing the Infinite Product$\prod_{n=0}^\infty\left[1+\left(\frac{x}{b+n}\right)^3\right]$Authors: Martin Nicholson Comments: 8 Pages. We study several integrals that contain the infinite product${\displaystyle\prod_{n=0}^\infty}\left[1+\left(\frac{x}{b+n}\right)^3\right]$in the denominator of their integrand. These considerations lead to closed form evaluation$\displaystyle\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$and to some other formulas. Category: Functions and Analysis [253] viXra:1712.0519 [pdf] submitted on 2017-12-19 19:49:52 ### The Bilateral Laplace Transform of the Positive Even Functions and a Proof of Riemann Hypothesis Authors: Seong Won Cha Comments: 11 Pages. We show that some interesting properties of the bilateral Laplace transform of even and positive functions both on the line z=x+iy0 and on a circle. We also show the Riemann hypothesis is true using these properties. Category: Functions and Analysis [252] viXra:1712.0478 [pdf] submitted on 2017-12-15 08:30:49 ### Two-Dimensional Fourier Transformations and Mordell Integrals Authors: Martin Nicholson Comments: 10 Pages. Several Fourier transformations of functions of one and two variables are evaluated and then used to derive some integral and series identities. It is shown that certain two- dimensional Mordell integrals factorize into product of two integrals and that the square of the absolute value of the Mordell integral can be reduced to a single one-dimensional integral. Some connections to elliptic functions and lattice sums are discussed. Category: Functions and Analysis [251] viXra:1712.0463 [pdf] submitted on 2017-12-16 01:01:52 ### Proof that a Derivative is a Fraction, and the Chain Rule is the Product of Such Fractions Authors: Carl Wigert, Quincy-Howard Xavier Comments: 1 Page. In this paper, we define very small numbers and very very small numbers and use them to construct derivatives as ratios of real numbers. We then use that result to rigorously prove that the chain rule treats derivatives as fractions being multiplied. Category: Functions and Analysis [250] viXra:1712.0355 [pdf] submitted on 2017-12-08 19:58:12 ### On_the_bilateral_laplace_transform_of_the_positive_even_functions_and_proof_of_the_riemann_hypothesis Authors: Seong Won Cha Comments: 9 Pages. This is a brief report before writing a full paper. We proved the Riemann hypothesis using the properties of the bilateral Laplace transform. Category: Functions and Analysis [249] viXra:1712.0113 [pdf] submitted on 2017-12-04 21:50:14 ### Multiplicative Versions of Infinitesimal Calculus Authors: D Williams Comments: 8 Pages. An overview of some types of multiplicative infinitesimal calculi is given. Analogs of standard results ("Simpson's" Product, "Maclurin's" Product, fundamental theorems, etc) are shown. An area that deserves more attention. Category: Functions and Analysis [248] viXra:1712.0019 [pdf] submitted on 2017-12-02 12:52:22 ### The Troncated Integral Authors: Antoine Balan Comments: 2 pages, written in french It is showed that a large class of functions defined by integrals verify the Riemann Hypothesis. Category: Functions and Analysis [247] viXra:1711.0356 [pdf] submitted on 2017-11-18 15:54:02 ### On the Attempt to Use a Stochastic Interpretation to Compute the Trace of a Regular Representation U on X = Ak/k* Authors: Matanari Shimoinuda Comments: 12 Pages. The group X, which is proposed by A.Connes, is an interesting thing for number theory. Let's think of the trace of a regular representation U on X of the idele class. However it is hard to compute it since X is non-compact. In this article, we try to show that the trace is computable. Category: Functions and Analysis [246] viXra:1711.0298 [pdf] submitted on 2017-11-13 20:01:20 ### Some Infinite "Continuous" Products Over the Interval (0,1) Authors: D Williams Comments: 3 Pages. Some "continuous" (that is, over real numbers in the interval (0,1)) infinite products are given with their finite product approximations. THESE PRODUCTS DESERVE MORE STUDY. Category: Functions and Analysis [245] viXra:1711.0297 [pdf] submitted on 2017-11-13 20:05:52 ### Dx-less Integrals Authors: D Williams Comments: 3 Pages. Some examples of dx-less integrals are given with their finite sum approximations. They appear to have use in estimating long-term values of certain stochastic recursive functions. A request is made for determining convergence of such integrals. Category: Functions and Analysis [244] viXra:1711.0257 [pdf] submitted on 2017-11-09 15:57:32 ### Improvement on Stirling's Formula for n! Using Product Integrals Authors: D Williams Comments: 10 Pages. An improved version of Stirling's Formula (which I call Neylon's Approximation) for n! is constructed using Product Integrals. Category: Functions and Analysis [243] viXra:1710.0246 [pdf] submitted on 2017-10-22 16:35:58 ### Prime Enumerability. Authors: Paris Samuel Miles-Brenden Comments: 2 Pages. Riemann-Zeta Note. None. Category: Functions and Analysis [242] viXra:1710.0140 [pdf] submitted on 2017-10-12 11:04:15 ### Chur-Type Theorems for K-Triangular Lattice Group-Valued Set Functions with Respect to Filter Convergence Authors: Antonio Boccuto, Xenofon Dimitriou Comments: 4 Pages. We prove some Schur and limit theorems for lattice group-valued k-triangular set functions with respect to filter convergence, by means of sliding hump-type techniques. As consequences, we deduce some Vitali-Hahn-Saks and Nikodym-type theorems. Category: Functions and Analysis [241] viXra:1710.0126 [pdf] submitted on 2017-10-11 21:03:23 ### Matematical Certainty Authors: Paris Samuel Miles-Brenden Comments: 2 Pages. Mathematical certainty often does not translate; but here the stringent analytical means of it's establishment are presented. Mathematical certainty is defined in terms of sets and deterministic variables; in terms of the error root mean squared deviation. Category: Functions and Analysis [240] viXra:1710.0083 [pdf] submitted on 2017-10-08 03:04:30 ### Non-Standard General Numerical Methods for the Direct Solution of Differential Equations not Cleared in Canonical Forms Authors: Carlos Oscar Rodríguez Leal Comments: 16 Pages. Paper writting in spanish. Paper presented at the VII International Congress of Numerical Methods, CUCEI, Universidad de Guadalajara, Guadalajara, Jalisco, Mexico. In this work I develop numerical algorithms that can be applied directly to differential equations of the general form f (t, x, x ) = 0, without the need to cleared x . My methods are hybrid algorithms between standard methods of solving differential equations and methods of solving algebraic equations, with which the variable x is numerically cleared. The application of these methods ranges from the ordinary differential equations of order one, to the more general case of systems of m equations of order n. These algorithms are applied to the solution of different physical-mathematical equations. Finally, the corresponding numerical analysis of existence, uniqueness, stability, consistency and convergence is made, mainly for the simplest case of a single ordinary differential equation of the first order. Category: Functions and Analysis [239] viXra:1710.0036 [pdf] submitted on 2017-10-03 21:20:37 ### Laws of General Solutions of Partial Differential Equations Authors: Hong Lai Zhu Comments: 18 Pages. In this paper, four kinds of Z Transformations are proposed to get many laws of general solutions of mth-order linear and nonlinear partial differential equations with n variables. Some general solutions of first-order linear partial differential equations, which cannot be obtained by using the characteristic equation method, can be solved by the Z Transformations. By comparing, we find that the general solutions of some first-order partial differential equations got by the characteristic equation method are not complete. Category: Functions and Analysis [238] viXra:1709.0442 [pdf] submitted on 2017-09-30 11:38:21 ### Causality Authors: Antoine Warnery Comments: 11 Pages. French The purpose of this study is to explore the mathematical principle of causality. Category: Functions and Analysis [237] viXra:1709.0393 [pdf] submitted on 2017-09-26 07:41:30 ### Neumann Series Seen as Expansion Into Bessel Functions J_{n} Based on Derivative Matching. Authors: Andrej Liptaj Comments: 8 Pages. Text presents known results. Multiplicative coefficients of a series of Bessel functions of the first kind can be adjusted so as to match desired values corresponding to a derivatives of a function to be expanded. In this way Neumann series of Bessel functions is constructed. Text presents known results. Category: Functions and Analysis [236] viXra:1709.0357 [pdf] submitted on 2017-09-23 12:37:20 ### The Kakeya Tube Conjecture Implies the Kakeya Conjecture Authors: Johan Aspegren Comments: 3 Pages. In this article we will give a proof that the Kakeya tube conjecture implies the Kakeya conjecture. Category: Functions and Analysis [235] viXra:1709.0310 [pdf] submitted on 2017-09-20 13:49:06 ### Calculation of the Sum for Sequence of Square Natural Numbers. Russian Authors: Misha Mikhaylov Comments: 2 Pages. This is the Russian version of my previous publication. Category: Functions and Analysis [234] viXra:1709.0305 [pdf] submitted on 2017-09-20 13:07:26 ### Calculation of the Sum for Sequence of Square Natural Numbers Authors: Misha Mikhaylov Comments: 2 Pages. This sum for natural values is, of course, already calculated by Bernoulli himself – at least modern or relatively recent authors that deal with it usually refer to take into account Bernoulli numbers. But, apparently, this method is rather cumbersome. Therefore, there can be suggested another, easier way to do this, but without claiming of its superfluous rigidity. Category: Functions and Analysis [233] viXra:1709.0304 [pdf] submitted on 2017-09-20 07:24:33 ### Boys' Function Computed by Gauss-Jacobi Quadrature Authors: Richard J. Mathar Comments: 48 Pages. Most of the content is the source code listing Boys' Function F_m(z) that appears in the quantum mechanics of Gaussian Type Orbitals is a special case of Kummer's confluent hypergeometric function. We evaluate its integral representation of a product of a power and an exponential function over the unit interval with the numerical Gauss-Jacobi quadrature. We provide an implementation in C for real values of the argument z which basically employs a table of the weights and abscissae of the quadrature rule for integer quantum numbers m <= 129. Category: Functions and Analysis [232] viXra:1709.0281 [pdf] submitted on 2017-09-18 20:00:56 ### On the Kakeya Conjecture Authors: Johan Aspegren Comments: 3 Pages. In this paper we will prove a theorem that implies the Kakeya conjecture. Category: Functions and Analysis [231] viXra:1709.0113 [pdf] submitted on 2017-09-10 07:44:48 ### On Kakeya Set Conjecture Authors: Johan Aspegren Comments: 5 Pages. In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained. Category: Functions and Analysis [230] viXra:1709.0047 [pdf] submitted on 2017-09-05 05:45:21 ### Describing a Fluid Motion with 3-D Rectangular Coordinates Authors: Valdir Monteiro dos Santos Godoi Comments: 9 Pages. We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure. Category: Functions and Analysis [229] viXra:1708.0123 [pdf] submitted on 2017-08-11 10:14:24 ### Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations. Category: Functions and Analysis [228] viXra:1708.0006 [pdf] submitted on 2017-08-02 03:18:34 ### Two Compartmental Mathematical Analysis of the Diffusion of a Therapeutic Agent in Human Tissues Authors: E. U. Agom, M. S. Atureta Comments: 4 Pages. ijsr.net publication In this paper, we present a unified minimal compartmental model to estimate mathematically the concentration of a Therapeutic Agent injected intravenously in a steady state into Human tissues divided into two compartments; the blood and tissues. The model takes into consideration most, if not all physiological factors of the Human system in conformity with the physical realities vis-a-vis the Therapeutic Agent concentration before uptake by the compartments. The models were a system of first order non-homogeneous ordinary differential equations. And, the result from the models gives a zero concentration in both the blood and the tissues before the advent of the Therapeutic agent. Category: Functions and Analysis [227] viXra:1708.0005 [pdf] submitted on 2017-08-02 03:37:25 ### Application of Adomian Decomposition Method in Solving Second Order Nonlinear Ordinary Differential Equations Authors: E. U. Agom, A. M. Badmus Comments: 6 Pages. ijesi.org paper In this paper, we use Adomian Decomposition Method to numerically analyse second order nonlinear ordinary differential equations and implement the continuous algorithm in a discrete domain. This is facilitated by Maple package. And, the results from the two test problems used shows that the Adomian Decomposition Method is almost as the classical solutions. Category: Functions and Analysis [226] viXra:1707.0246 [pdf] submitted on 2017-07-18 08:02:22 ### Neutrosophic Crisp Mathematical Morphology Authors: Eman.M.El-Nakeeb, Hewayda ElGhawalby, A.A.Salama, S.A.El-Hafeez Comments: 13 Pages. In this paper, we aim to apply the concepts of the neutrosophic crisp sets and its operations to the classical mathematical morphological operations, introducing what we call "Neutrosophic Crisp Mathematical Morphology". Several operators are to be developed, including the neutrosophic crisp dilation, the neutrosophic crisp erosion, the neutrosophic crisp opening and the neutrosophic crisp closing.Moreover, we extend the definition of some morphological filters using the neutrosophic crisp sets concept. For instance, we introduce the neutrosophic crisp boundary extraction, the neutrosophic crisp Top-hat and the neutrosophic crisp Bottom- hat filters.The idea behind the new introduced operators and filters is to act on the image in the neutrosophic crisp domain instead of the spatial domain. Category: Functions and Analysis [225] viXra:1707.0155 [pdf] submitted on 2017-07-11 08:27:57 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 27 Pages. A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement. Category: Functions and Analysis [224] viXra:1707.0131 [pdf] submitted on 2017-07-09 17:37:53 ### New Infinite Product Representation for Cosine Function and Power Series for Tangent Function Authors: Edigles Guedes Comments: 3 Pages. In this paper, I demonstrate one new infinite product representation for cosine function, one new power series representation for tangent function and amazing identities involving radical. Category: Functions and Analysis [223] viXra:1707.0130 [pdf] submitted on 2017-07-09 17:41:03 ### New Proof of the Infinite Product Representation for Gamma Function and Pochhammer's Symbol and New Infinite Product Representation for Binomial Coefficient Authors: Edigles Guedes, Cícera Guedes Comments: 6 Pages. In this paper, we demonstrate some limit's formulae for gamma function and binomial coefficient among other things. Category: Functions and Analysis [222] viXra:1705.0410 [pdf] submitted on 2017-05-29 06:39:46 ### New Principles of Differential Equations Ⅰ Authors: Hong Lai Zhu Comments: 71 Pages. This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct. Category: Functions and Analysis [221] viXra:1705.0399 [pdf] submitted on 2017-05-28 00:50:44 ### Variable Axis Angles in Coordinate Systems Part. 1 (Ziennokątowe Układy Współrzędnych) Authors: Andrzej Peczkowski Comments: 15 Pages. This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OY axis that crosses the OX axis at any angle. Category: Functions and Analysis [220] viXra:1705.0398 [pdf] submitted on 2017-05-28 00:55:17 ### Variable Axis Angles in Coordinate Systems Part.2 (Zmiennokątowe Układy Współrzęnych) Authors: Andrzej Peczkowski Comments: 14 Pages. This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Hi 1 is the OX axis that crosses the OY axis at any angle. Category: Functions and Analysis [219] viXra:1705.0397 [pdf] submitted on 2017-05-28 01:04:24 ### Variable Axis Angles in Coordinate Systems Part. 3 (Zmiennokątowe Układy Współrzędnych ) Authors: Andrzej Peczkowski Comments: 17 Pages. This is mathematics where the axes of the OX and OY coordinate systems do not intersect at right angles. Part 3. Axes OX and OY intersect at any angle Category: Functions and Analysis [218] viXra:1705.0249 [pdf] submitted on 2017-05-16 08:26:20 ### Decomposition of Exponential Function Into Derivative Ring. Derivative Ring: Suitable Basis for Derivative-Matching Approximations. Authors: Andrej Liptaj Comments: 6 Pages. A set of functions which allows easy derivative-matching is proposed. Several examples of approximations are shown. Category: Functions and Analysis [217] viXra:1705.0165 [pdf] submitted on 2017-05-09 17:00:33 ### One Word: Navier-Stokes Authors: Nicholas R. Wright Comments: 6 Pages. We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards. Category: Functions and Analysis [216] viXra:1705.0028 [pdf] submitted on 2017-05-02 15:33:07 ### An Efficient Computational Method for Handling Singular Second-Order, Three Points Volterra Integrodifferenital Equations Authors: Morad Ahmad, Shaher Momani, Omar Abu Arqub, Mohammed Al-Smadi, Ahmed Alsaedi Comments: 13 Pages. In this paper, a powerful computational algorithm is developed for the solution of classes of singular second-order, three-point Volterra integrodifferential equations in favorable reproducing kernel Hilbert spaces. The solutions is represented in the form of series in the Hilbert space W₂³[0,1] with easily computable components. In finding the computational solutions, we use generating the orthogonal basis from the obtained kernel functions such that the orthonormal basis is constructing in order to formulate and utilize the solutions. Numerical experiments are carried where two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values of the unknown variables. Error estimates are proven that it converge to zero in the sense of the space norm. Several computational simulation experiments are given to show the good performance of the proposed procedure. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to multipoint singular boundary value problems restricted by Volterra operator. Category: Functions and Analysis [215] viXra:1704.0282 [pdf] submitted on 2017-04-21 20:56:48 ### The Asymptotic Behavior of Defocusing Nonlinear Schrödinger Equations Authors: En-Lin Liu Comments: 6 Pages. Quite trivial research XD This article is concerned with the scattering problem for the defocusing nonlinear Schrödinger equations (NLS) with a power nonlinear \|u\|^p u where 2/n < p < 4/n. We show that for any initial data in H^{0,1} x the solution will eventually scatter, i.e. U(-t)u(t) tends to some function u+ as t tends to innity. Category: Functions and Analysis [214] viXra:1704.0030 [pdf] submitted on 2017-04-04 03:53:37 ### Examples of Approximations by Series Based on Derivatives Matching Authors: Andrej Liptaj Comments: 11 Pages. Inspired by Taylor polynomials, several other approximations based on derivative-matching are proposed. Category: Functions and Analysis [213] viXra:1703.0295 [pdf] submitted on 2017-03-31 06:45:36 ### Higher Order Derivatives of the Inverse Function Authors: Andrej Liptaj Comments: 7 Pages. A general recursive and limit formula for higher order derivatives of the inverse function is presented. The formula is next used in couple of mathematical applications: expansion of the inverse function into Taylor series, solving equations, constructing random numbers with a given distribution from uniformly distributed randomnumbers and expanding a function in the neighborhood of a given point in an alternative way to the Taylor expansion. Category: Functions and Analysis [212] viXra:1703.0261 [pdf] submitted on 2017-03-28 03:24:35 ### Field Equations Authors: J.A.J. van Leunen Comments: 4 Pages. Field equations occur in many physical theories. Most dynamic fields share a set of first and second order partial differential equations and differ in the kinds of artifacts that cause discontinuities. The paper restricts to first and second order partial differential equations. These equations can describe the interaction between the field and pointlike artifacts. The paper treats periodic and one-shot triggers in maximally three spatial dimensions. The paper applies quaternionic differential calculus. It uses the quaternionic nabla operator. This configuration implements the storage of dynamic geometric data as a combination of a proper timestamp and a three-dimensional spatial location in a quaternionic storage container. The storage format is Euclidean. The paper introduces warps and clamps as new types of super-tiny objects that constitute higher order objects. Category: Functions and Analysis [211] viXra:1703.0253 [pdf] submitted on 2017-03-27 00:05:14 ### Universal Evolution Model Authors: Ramesh Chandra Bagadi Comments: 9 Pages. In this research investigation, the author has prevented a novel scheme of Universal Evolution Model. Category: Functions and Analysis [210] viXra:1703.0184 [pdf] submitted on 2017-03-19 09:23:11 ### Density Problem of Périat Authors: F.L.B. Périat Comments: 2 Pages. This could help science going faster. Category: Functions and Analysis [209] viXra:1703.0135 [pdf] submitted on 2017-03-13 17:20:52 ### Global in Time Solvability of Incompressive NSIVP in the Whole Space Authors: Naoya Isobe Comments: 13 Pages. Global in time solvability of incompressive Navier-Stokes initial value problem in the whole space is proved using time transformation analysys. Category: Functions and Analysis [208] viXra:1703.0134 [pdf] submitted on 2017-03-13 17:22:16 ### Key point of the proof (Global in Time Solvability of Incompressive NSIVP in the Whole Space) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Global in Time Solvability of Incompressive NSIVP in the Whole Space). Category: Functions and Analysis [207] viXra:1703.0133 [pdf] submitted on 2017-03-13 17:59:21 ### Global in Time Solvability of Incompressive Nsivp in Periodic Space Authors: Naoya Isobe Comments: 15 Pages. Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys. Category: Functions and Analysis [206] viXra:1703.0132 [pdf] submitted on 2017-03-13 18:00:32 ### Key Point of the Proof (Global in Time Solvability of Incompressive Nsivp in Periodic Space) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Global in Time Solvability of Incompressive NSIVP in periodic Space). Category: Functions and Analysis [205] viXra:1703.0121 [pdf] submitted on 2017-03-13 06:11:06 ### Global in Time Solvability of Incompressive Nsivp in the Whole Space (Japanese Version) Authors: Naoya Isobe Comments: 12 Pages. Global in time solvability of incompressive Navier-Stokes initial value problem in the whole space is proved using time transformation analysys. Category: Functions and Analysis [204] viXra:1703.0120 [pdf] submitted on 2017-03-13 06:17:44 ### Key Point of the Proof (Global in Time Solvability of Incompressive Nsivp in the Whole Space) (Japanese Version) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Global in Time Solvability of Incompressive NSIVP in the Whole Space). Category: Functions and Analysis [203] viXra:1703.0119 [pdf] submitted on 2017-03-13 06:19:44 ### Global in Time Solvability of Incompressive Nsivp in Periodic Space (Japanese Version) Authors: Naoya Isobe Comments: 14 Pages. Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys. Category: Functions and Analysis [202] viXra:1703.0118 [pdf] submitted on 2017-03-13 06:21:47 ### Key Point of the Proof (Global in Time Solvability of Incompressive Nsivp in Periodic Space) (Japanese Version) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Global in Time Solvability of Incompressive NSIVP in periodic Space). Category: Functions and Analysis [201] viXra:1703.0073 [pdf] submitted on 2017-03-07 21:27:59 ### On the Riemann Zeta Function Authors: Jonathan Tooker Comments: 5 Pages. Five figures. Uploading rough draft for posterity. We discuss the Riemann zeta function and make an argument against the Riemann hypothesis. While making the argument in the classical formalism, we discuss the argument as it relates to the theory of infinite complexity. We extend Riemann's own (planar) analytic continuation$\mathbb{R}\to\mathbb{C}^2$into (bulk) hypercomplexity with$\mathbb{C}^2\to\,^\star\mathbb{C}$. Category: Functions and Analysis [200] viXra:1702.0119 [pdf] submitted on 2017-02-09 07:58:42 ### Lagrangian Solid Modeling Authors: Matthew Marko Comments: 114 pages including supplementary code The author demonstrates a stable Lagrangian solid modeling method, tracking the interactions of solid mass particles, rather than using a meshed grid. This numerical method avoids the problem of tensile instability often seen with Smooth Particle Applied Mechanics by having the solid particles apply stresses expected with Hooke's law, as opposed to using a smoothing function for neighboring solid particles. This method has been tested successfully with a bar in tension, compression, and shear, as well as a disk compressed into a flat plate, and the numerical model consistently matched the analytical Hooke's law as well as Hertz contact theory for all examples. The solid modeling numerical method was then built into a 2-D model of a pressure vessel, which was tested with liquid water particles under pressure and simulated with Smoothed Particle Hydrodynamics. This simulation was stable, and demonstrated the feasibility of Lagrangian specification modeling for Fluid Solid Interactions. Category: Functions and Analysis [199] viXra:1702.0039 [pdf] submitted on 2017-02-03 03:27:16 ### Proof to Euler's Formula Authors: Carl-Gustav Hedenby Comments: 1 Page. - The author proves Euler's formula for the imaginary exponential without reverting to series expansions. Category: Functions and Analysis [198] viXra:1701.0617 [pdf] submitted on 2017-01-26 01:27:43 ### Non-Trivial Extension of Real Numbers Authors: Ilya Chernykh Comments: 8 Pages. In Russian language We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size. Category: Functions and Analysis [197] viXra:1701.0511 [pdf] submitted on 2017-01-15 16:19:44 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 5/6. Authors: Stephen C. Pearson. Comments: 42 Pages. This particular submission contains a copy [PART 5/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 4/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'. Category: Functions and Analysis [196] viXra:1701.0510 [pdf] submitted on 2017-01-15 17:01:34 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 6/6. Authors: Stephen C. Pearson. Comments: 24 Pages. This particular submission contains a copy [PART 6/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 5/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'. Category: Functions and Analysis [195] viXra:1701.0505 [pdf] submitted on 2017-01-15 14:34:25 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 3/6. Authors: Stephen C. Pearson. Comments: 42 Pages. This particular submission contains a copy [PART 3/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 2/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'. Category: Functions and Analysis [194] viXra:1701.0504 [pdf] submitted on 2017-01-15 15:44:13 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 4/6. Authors: Stephen C. Pearson. Comments: 42 Pages. This particular submission contains a copy [PART 4/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 3/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'. Category: Functions and Analysis [193] viXra:1701.0502 [pdf] submitted on 2017-01-15 11:15:18 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 1/6. Authors: Stephen C. Pearson. Comments: 42 Pages. This particular submission contains (inter alia) a copy [PART 1/6] of the author's original paper, which was completed on 31st March 1984 and thus comprises a total of 161 handwritten foolscap pages. Subsequently, its purpose is to enunciate various definitions and theorems, which pertain to the following topics, i.e. (a) the algebra of quaternion hypercomplex numbers; (b) functions of a single quaternion hypercomplex variable; (c) the concepts of limit and continuity applied to such functions; (d) the elementary principles of differentiation and integration applied to quaternion hypercomplex functions. Many of the concepts presented therein are analogous to well established notions from real and complex variable analysis with any divergent results being due to the non-commutativity of quaternion products. Category: Functions and Analysis [192] viXra:1701.0501 [pdf] submitted on 2017-01-15 12:58:04 ### An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 2/6. Authors: Stephen C. Pearson. Comments: 42 Pages. This particular submission contains a copy [PART 2/6] of the author's original paper and is therefore a continuation of his previous submission, namely - "An Introduction to Functions of a Quaternion Hypercomplex Variable - PART 1/6", which has been published under the 'VIXRA' Mathematics subheading:- 'Functions and Analysis'. Category: Functions and Analysis [191] viXra:1701.0324 [pdf] submitted on 2017-01-08 04:19:38 ### Existence of Traveling Waves in the Fractional Burgers Equation Authors: Adam Chmaj Comments: 6 Pages. Original 2014 version of the result is posted here. Some minor corrections are left to the reader. The existence of traveling waves for the fractional Burgers equation is established, using an operator splitting trick. This solves a 1998 open problem. Category: Functions and Analysis [190] viXra:1612.0413 [pdf] submitted on 2016-12-30 16:56:35 ### Area of Torricelli's Trumpet or Gabriel's Horn, Sum of the Reciprocals of the Primes, Factorials of Negative Integers Authors: Sinisa Bubonja Comments: 9 Pages. In our previous work [1], we defined the method for computing general limits of functions at their singular points and showed that it is useful for calculating divergent integrals, the sum of divergent series and values of functions in their singular points. In this paper, we have described that method and we will use it to calculate the area of Torricelli's trumpet or Gabriel's horn, the sum of the reciprocals of the primes and factorials of negative integers. Category: Functions and Analysis [189] viXra:1612.0394 [pdf] submitted on 2016-12-29 16:05:22 ### Perceptive or P-Calculus: Ordinale & Residuale Noesis Authors: Arthur Shevenyonov Comments: 10 Pages. novel foundations An early formal glimpse at a survey of results yet to be revealed bridging topics as diverse as, the extensions of Cauchy functional equation, Taylor expansion, ABC conjecture, and Fermat LP to name but a few. Category: Functions and Analysis [188] viXra:1612.0245 [pdf] submitted on 2016-12-14 08:14:24 ### New Expansions in Series for Tangent and Secant Functions Authors: Edigles Guedes Comments: 3 Pages. In this paper, the author proved new expansions in series for tangent and secant functions. Category: Functions and Analysis [187] viXra:1612.0238 [pdf] submitted on 2016-12-14 01:05:36 ### On the Navier-Stokes Equations Authors: Daniel Thomas Hayes Comments: 7 Pages. The problem on the existence and smoothness of the Navier--Stokes equations is considered. Category: Functions and Analysis [186] viXra:1611.0368 [pdf] submitted on 2016-11-26 20:05:27 ### Infinite Product Representations for Gamma Function and Binomial Coefficient Authors: Edigles Guedes Comments: 7 Pages. In this paper, I demonstrate one new infinite product for binomial coefficient and news Euler's and Weierstrass's infinite product for Gamma function among other things. Category: Functions and Analysis [185] viXra:1611.0073 [pdf] submitted on 2016-11-05 14:49:21 ### Coefficient-of-determination Fourier Transform CFT Authors: Matthew Marko Comments: 13 Pages, English This algorithm is designed to perform Discrete Fourier Transforms (DFT) to convert temporal data into spectral data. What is unique about this DFT algorithm is that it can produce spectral data at any user-defined resolution; existing DFT methods such as FFT are limited in resolution proportional to the temporal resolution. This algorithm obtains the Fourier Transforms by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate. Category: Functions and Analysis [184] viXra:1611.0056 [pdf] submitted on 2016-11-04 07:11:54 ### How to Project Onto Extended Second Order Cones Authors: O. P. Ferreira, S. Z. Németh Comments: 10 Pages. The extended second order cones were introduced by G. Zhang and S. Z. N\'emeth for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. G. Zhang and S. Z. Németh found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods. Category: Functions and Analysis [183] viXra:1611.0049 [pdf] submitted on 2016-11-03 23:41:33 ### In nite Product Representations for Binomial Coefcient, Pochhammer's Symbol, Newton's Binomial and Exponential Function Authors: Edigles Guedes Comments: 15 Pages. In this paper, I demonstrate one infinite product for binomial coefficient, Euler's and Weierstrass's infinite product for Pochhammer's symbol, limit formula for Pochhammer's symbol, limit formula for exponential function, Euler's and Weierstrass's infinite product for Newton's binomial and exponential function, among other things. Category: Functions and Analysis [182] viXra:1611.0002 [pdf] submitted on 2016-11-01 01:50:33 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 13 Pages. This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for approximation orders 2, 4, 6, and 8, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [181] viXra:1610.0278 [pdf] submitted on 2016-10-23 16:35:25 ### Applying the Second Order Two-Scale Approximation to a Dispersive Wave Equation Authors: Yigal Gurevich Comments: 8 Pages. The method of multiple scales is applied and the second order two-scale approximation is calculated for a linear dispersive wave equation with a small perturbation proportional to the amplitude cubed. Category: Functions and Analysis [180] viXra:1610.0244 [pdf] submitted on 2016-10-21 04:28:14 ### Operator Exponentials for the Clifford Fourier Transform on Multivector Fields in Detail Authors: David Eelbode, Eckhard Hitzer Comments: 16 Pages. published in Advances in Applied Clifford Algebras, 26(3), pp. 953-968 (2016), Online First: 22 Oct. 2015, DOI: 10.1007/s00006-015-0600-7. In this paper we study Clifford Fourier transforms (CFT) of multivector functions taking values in Clifford’s geometric algebra, hereby using techniques coming from Clifford analysis (the multivariate function theory for the Dirac operator). In these CFTs on multivector signals, the complex unit i∈C is replaced by a multivector square root of −1, which may be a pseudoscalar in the simplest case. For these integral transforms we derive an operator representation expressed as the Hamilton operator of a harmonic oscillator. Category: Functions and Analysis [179] viXra:1609.0349 [pdf] submitted on 2016-09-25 01:25:44 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 17 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. Category: Functions and Analysis [178] viXra:1609.0141 [pdf] submitted on 2016-09-11 08:38:51 ### Solutions for Euler and Navier-Stokes Equations in Powers of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 7 Pages. We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1. Category: Functions and Analysis [177] viXra:1609.0074 [pdf] submitted on 2016-09-06 15:08:57 ### General Solution For Navier-Stokes Equations With Any Smooth Initial Data Authors: Valdir Monteiro dos Santos Godoi Comments: 6 Pages. We present a solution for the Navier-Stokes equations for incompressible case with any smooth (C^∞) initial velocity given a pressure and external force in n = 3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, n≥1. Category: Functions and Analysis [176] viXra:1609.0006 [pdf] submitted on 2016-09-01 04:47:12 ### Adequate Quaternionic Generalization of Complex Differentiability Authors: Michael Parfenov Comments: 46 Pages. Abstract: The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory. Category: Functions and Analysis [175] viXra:1608.0432 [pdf] submitted on 2016-08-31 08:23:50 ### General Solution for Navier-Stokes Equations with Conservative External Force Authors: Valdir Monteiro dos Santos Godoi Comments: 10 Pages. We present two proofs of theorems on solutions of the Navier-Stokes equations for incompressible case with a conservative external force in n = 3 spatial dimensions. Without major difficulties, it can be adapted to any spatial dimension, n>=1. Category: Functions and Analysis [174] viXra:1608.0394 [pdf] submitted on 2016-08-29 06:04:21 ### Limit Theorems for Lattice Group-Valued K-Triangular Set Functions Authors: Antonio Boccuto, Xenofon Dimitriou Comments: 2 Pages. Using sliding hump-type techniques, we prove some Schur, Vitali-Hahn-Saks and Nikodým-type theorems for lattice group-valued k-triangular set functions. Category: Functions and Analysis [173] viXra:1608.0229 [pdf] submitted on 2016-08-21 11:27:59 ### One Dimensional Method for Clifford Analysis Authors: Fu Yuhua Comments: 12 Pages. Unlike Dirac operator method, this paper discusses one dimensional method for Clifford analysis, namely the n-dimensional problem is simplified into n problems of one dimension, even reduced to only one problem of one dimension. For example, the typhoon track is a two-dimensional problem related to latitude and longitude, but as forecasting typhoon track, it can be simplified into two problems of one dimension: forecasting longitude and forecasting latitude respectively. Again, the stock index is effected by various factors, however as forecasting stock index we may assume that it is only a function of time. In order to improve the effect of one dimensional method, we can change finding the solution suitable for whole space or a domain, into finding the solution suitable for one point only with single point method. As applying one dimensional method, the fractal model is very effective. Category: Functions and Analysis [172] viXra:1608.0155 [pdf] submitted on 2016-08-16 03:44:17 ### Exact Solutions for Sine-Gordon Equations and F-Expansion Method Authors: L. A. N. de Paula Comments: 5 Pages. A large number of methods have been proposed for solving nonlinear differential equations. The Jacobi elliptic function method and the f-expansion methods are generalizations from a few of them. These methods produce not only single-solitons but also multi-soliton solutions. In this work we applied the f -expansion method and found novel solutions besides those known for three main equations of the kind sine-Gordon: Triple Sine-Gordon (TSG), Double Sine-Gordon (DSG) and Simple Sine-Gordon (SSG). Category: Functions and Analysis [171] viXra:1608.0065 [pdf] submitted on 2016-08-05 21:34:41 ### Carotid-Kundalini Functions and Chaos Authors: Sai Venkatesh Balasubramanian Comments: 4 Pages. This article explores a case of signal based chaos generation, using the Carotid-Kundalini function, shown in literature to possess fractal artifacts. Specifically, we set the input to a two tone signal, with the frequency ratio between the sinusoids acting as the control parameter. We explore the iterative map using the time derivatives, and upon plotting the bifurcation plot, observe the chaotic nature of the generated signal. Phase portraits are plotted for different orders, and presence of rich patterns are observed. True to the nonlinear nature, the frequency spectrum shows a horde of new frequency components generated at the output. Lyapunov Exponents also quantitatively confirm the presence of generated chaos in the Carotid-Kundalini signal. Category: Functions and Analysis [170] viXra:1606.0344 [pdf] submitted on 2016-06-30 08:34:29 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 2 Pages. Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Category: Functions and Analysis [169] viXra:1606.0324 [pdf] submitted on 2016-06-28 16:56:02 ### The Theory of$n$-scales Authors: Furkan Semih Dundar Comments: 5 Pages. We give a theory of$n$-scale previously called as$n$dimensional time scale. In previous approaches to the theory of time scales, multi dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. Here we define an$n$-scale as an arbitrary closed subset of$\mathbb R^n$. Modified forward and backward jump operators,$\Delta$-derivatives and multiple integrals on$n$-scales are defined. Category: Functions and Analysis [168] viXra:1606.0180 [pdf] submitted on 2016-06-17 22:42:03 ### 1. Universal Relative Metric That Generates A Field Super-Set To The Fields Generated By Some Number Of Distinct Relative Metrics 2. Universal Function Generation (FAA) Authors: Ramesh Chandra Bagadi Comments: 18 Pages. In this research investigation, the author has presented a theory of ‘Universal Relative Metric That Generates A Field Super-Set To The Fields Generated By Various Distinct Relative Metrics’. Category: Functions and Analysis [167] viXra:1606.0141 [pdf] submitted on 2016-06-14 08:42:21 ### Some Definite Integrals Over a Power Multiplied by Four Modified Bessel Functions Authors: Richard J. Mathar Comments: 16 Pages. The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x) K_1^v(x) dx are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are Modified Bessel Functions. There are essentially 15 types of the 4-fold product of which 10 are discussed here. Partial integration of each of these types leads to correlations between these integrals. The main result are (forward) recurrences of the integrals with respect to the exponent j. Category: Functions and Analysis [166] viXra:1605.0290 [pdf] submitted on 2016-05-28 22:52:24 ### A Formalized Implied Integral Authors: Eric Stucky Comments: 6 Pages. In this document, we define a fully formalized notion of the Implied Integral'' which works for a broad variety of symbolic expressions. We show that it is equal to the ordinary integral up to a generalized constant term, and we show that this generalized constant is in some sense optimal. Category: Functions and Analysis [165] viXra:1605.0236 [pdf] submitted on 2016-05-22 15:47:43 ### Two Theorems on Solutions in Eulerian Description Authors: Valdir Monteiro dos Santos Godoi Comments: 8 Pages. We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions. Category: Functions and Analysis [164] viXra:1604.0376 [pdf] submitted on 2016-04-29 06:12:02 ### Abstract Theorems on Exchange of Limits and Preservation of (Semi)continuity of Functions and Measures in the Filter Convergence Setting Authors: Antonio Boccuto, Xenofon Dimitriou Comments: 13 Pages. We give necessary and sufficient conditions for exchange of limits of double-indexed families, taking values in sets endowed with an abstract structure of convergence, and for preservation of continuity or semicontinuity of the limit family, with respect to filter convergence. As consequences, we give some filter limit theorems and some characterization of continuity and semicontinuity of the limit of a pointwise convergent family of set functions. Category: Functions and Analysis [163] viXra:1604.0369 [pdf] submitted on 2016-04-28 08:05:07 ### Homogeneous Reduction of Order Homogeneous and Inhomogeneous Second Order Linear Ordinary Differential Equations Solution in One Process Authors: Claude Michael Cassano Comments: 4 Pages. The Homogeneous , Reduction of Order Homogeneous , and Inhomogeneous Second Order Linear Ordinary Differential Equations Solution may be established in one analytical proof-process. Category: Functions and Analysis [162] viXra:1604.0244 [pdf] submitted on 2016-04-15 22:16:18 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. In english. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [161] viXra:1604.0235 [pdf] submitted on 2016-04-15 04:23:52 ### On Some Series Related to Möbius Function and Lambert W-Function Authors: Danil Krotkov Comments: 12 Pages. We derive some new formulas, connecting some series with Möbius function with Sine Integral and Cosine Integral functions, give the formal proof for full version of Stirling's formula; investigate the values of new Dirichlet series function at natural numbers >1 and it's behavior at the pole s=1, connecting it with elementary constants. Category: Functions and Analysis [160] viXra:1604.0204 [pdf] submitted on 2016-04-13 02:17:14 ### On Zeros of Some Entire Functions Authors: Bing He Comments: 16 Pages. Let \begin{equation} A_{q}^{(\alpha)}(a;z)=\sum_{k=0}^{\infty}\frac{(a;q)_{k}q^{\alpha k^2} z^k}{(q;q)_{k}}, \end{equation} where$\alpha >0,~0<q<1.$In a paper of Ruiming Zhang, he asked under what conditions the zeros of the entire function$A_{q}^{(\alpha)}(a;z)$are all real and established some results on the zeros of$A_{q}^{(\alpha)}(a;z)$which present a partial answer to that question. In the present paper, we will set up some results on certain entire functions which includes that$A_{q}^{(\alpha)}(q^l;z),~l\geq 2$has only infinitely many negative zeros that gives a partial answer to Zhang's question. In addition, we establish some results on zeros of certain entire functions involving the Rogers-Szeg\H{o} polynomials and the Stieltjes-Wigert polynomials. Category: Functions and Analysis [159] viXra:1604.0107 [pdf] submitted on 2016-04-05 12:31:45 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 14 Pages. We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis ## Replacements of recent Submissions [240] viXra:1806.0444 [pdf] replaced on 2018-07-01 07:28:05 ### The Optimization Principle for the Riemann Hypothesis Authors: Hassine Saidane Comments: 8 Pages. Abstract. Based on the observation that several physical, biological and social processes seem to be optimizing an objective function such as an action or a utility, the Central Principle of Science was deemed to be Optimization. Indeed, optimization proved to be an efficient tool for uncovering several scientific laws and proving some scientific theories. In this paper, we use this paradigm to identify the location of the nontrivial zeros of the Riemann Zeta function (RZF). This approach enabled the formulation of this problem as a constrained optimization problem where a simple objective function referred to here as the “Push-Pull Action” is maximized. The solution of the resulting constrained nonlinear optimization problem proved that nontrivial zeros of RZF are located on the critical line. In addition to proving the Riemann Hypothesis, this approach unveiled a plausible law of “Maximum Action of Push-Pull” that seems to be driving RZF to its equilibrium states at the different heights where it reaches its nontrivial zeros. We also show that this law applies to functions exhibiting the same properties as RZF. Keywords: Zeta function, Riemann Hypothesis, Constrained Optimization Category: Functions and Analysis [239] viXra:1806.0082 [pdf] replaced on 2018-06-09 05:42:18 ### Derivation of the Limits of Sine and Cosine at Infinity Authors: Jonathan W. Tooker Comments: 4 Pages. two figures This paper examines some familiar results from complex analysis in the framework of hypercomplex analysis. It is usually taught that the oscillatory behavior of sine waves means that they have no limit at infinity but here we derive definite limits. Where a central element in the foundations of complex analysis is that the complex conjugate of a$\mathbb{C}$-number is not analytic at the origin, we introduce the tools of hypercomplex analysis to show that the complex conjugate of a$^\star\mathbb{C}$-number is analytic at the origin. Category: Functions and Analysis [238] viXra:1804.0264 [pdf] replaced on 2018-04-23 04:22:49 ### On Expanding a Function Into Raw Moment Series Authors: Andrej Liptaj Comments: 10 Pages. I focus in this text on the construction of functions f_{j} with the delta property C_{i}\left(f_{j}\right)=\delta_{i,j}, where C_{i} are operators which associate to a function its i-th raw moment. A formal method for their construction is found, however results are divergent, from what a non-existence of such functions is conjectured. This also prevents an elegant series expansion with order-by-order moment matching. For a finite interval some partial results are presented: a method of expansion into raw moment series for finite number of moments and a “non-delta” method based on computing Legendre-expansion coefficients from moments (an already known method [1]). As by-product some coefficients formulas are found: coefficients for expanding a Hermite function into the Taylor series and coefficients for expanding into the Taylor series an element of a Fourier series (i.e. common formula for sine and cosine) thus formally merging the two (sine and cosine) Fourier sub-series into one. Category: Functions and Analysis [237] viXra:1803.0001 [pdf] replaced on 2018-03-05 04:58:02 ### Expansion Into Bernoulli Polynomials Based on Matching Definite Integrals of Derivatives Authors: Andrej Liptaj Comments: 8 Pages. A method of function expansion is presented. It is based on matching the definite integrals of the derivatives of the function to be approximated by a series of (scaled) Bernoulli polynomials. The method is fully integral-based, easy to construct and presumably slightly outperforms Taylor series in the convergence rate. Text presents already known results. Category: Functions and Analysis [236] viXra:1803.0001 [pdf] replaced on 2018-03-01 15:48:19 ### Expansion Into Bernoulli Polynomials Based on Matching Definite Integrals of Derivatives Authors: Andrej Liptaj Comments: 7 Pages. A novel method of function expansion is presented. It is based on matching the definite integrals of the derivatives of the function to be approximated by a series of (scaled) Bernoulli polynomials. The method is fully integral-based, easy to construct and presumably slightly outperforms Taylor series in the convergence rate. Category: Functions and Analysis [235] viXra:1802.0126 [pdf] replaced on 2018-02-12 23:41:44 ### A Note on the Possibility of Icomplete Theory Authors: Han Geurdes, Koji Nagata, Tadao Nakamura, Ahmed Farouk Comments: 12 Pages. In the paper it is demonstrated that Bells theorem is an unprovable theorem. This inconsistency is similar to concrete mathematical incompleteness. The inconsistency is purely mathematical. Nevertheless the basic physics requirements of a local model are fulfilled. Category: Functions and Analysis [234] viXra:1801.0096 [pdf] replaced on 2018-01-16 06:00:50 ### Quadratic Transformations of Hypergeometric Function and Series with Harmonic Numbers Authors: Martin Nicholson Comments: 8 Pages. Presentation is improved, a theorem, a corollary and some references are added In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. Category: Functions and Analysis [233] viXra:1709.0357 [pdf] replaced on 2018-01-22 03:40:51 ### The Kakeya Tube Conjecture Implies the Kakeya Conjecture Authors: Johan Aspegren Comments: 3 Pages. In this article we will give a proof that the Kakeya tube conjecture implies the Kakeya conjecture. Category: Functions and Analysis [232] viXra:1709.0113 [pdf] replaced on 2017-09-15 04:26:30 ### On the Kakeya Set Conjecture Authors: Johan Aspegren Comments: 5 Pages. In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained. Category: Functions and Analysis [231] viXra:1709.0113 [pdf] replaced on 2017-09-13 00:01:34 ### On the Kakeya Set Conjecture Authors: Johan Aspegren Comments: 5 Pages. In this article we will prove the Kakeya set conjecture. In addition we will prove that in the usual approach to the Kakeya maximal function conjecture we can assume that the tube-sets are maximal. Moreover, we will construct a tube- set were the well known L2 bound for the Kakeya maximal function is attained. Category: Functions and Analysis [230] viXra:1709.0047 [pdf] replaced on 2017-09-19 07:48:27 ### Describing a Fluid Motion with 3-D Rectangular Coordinates Authors: Valdir Monteiro dos Santos Godoi Comments: 9 Pages. We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure. Category: Functions and Analysis [229] viXra:1709.0047 [pdf] replaced on 2017-09-11 14:27:48 ### Describing a Fluid Motion with 3-D Rectangular Coordinates Authors: Valdir Monteiro dos Santos Godoi Comments: 9 Pages. We describe a fluid in three-dimensional motion with at most one spatial variable by rectangular coordinate, beyond time, and conclude on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure. Category: Functions and Analysis [228] viXra:1708.0123 [pdf] replaced on 2017-09-07 19:37:37 ### Describing a Fluid in Three-Dimensional Circular Motion with at Most One Spatial Variable by Rectangular Coordinate Authors: Valdir Monteiro dos Santos Godoi Comments: 5 Pages. Describe a fluid in three-dimensional circular motion with at most one spatial variable by rectangular coordinate, beyond time, and concludes on the breakdown of Euler and Navier-Stokes solutions and the necessity of use of vector pressure. Category: Functions and Analysis [227] viXra:1708.0123 [pdf] replaced on 2017-08-15 06:33:11 ### Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate Authors: Valdir Monteiro dos Santos Godoi Comments: 5 Pages. Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations. Category: Functions and Analysis [226] viXra:1708.0123 [pdf] replaced on 2017-08-12 11:42:48 ### Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. Describe a fluid in three-dimensional circular motion with one independent variable by rectangular coordinate and concludes on the breakdown of Euler and Navier-Stokes equations. Category: Functions and Analysis [225] viXra:1707.0155 [pdf] replaced on 2017-09-01 11:06:05 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 31 Pages. Published at WISE Journal, Volume 7, No. 1 (Spring, 2018), pp. 110-140. A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid in movement. Category: Functions and Analysis [224] viXra:1707.0155 [pdf] replaced on 2017-08-27 13:57:42 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 31 Pages. A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid in movement. Category: Functions and Analysis [223] viXra:1707.0155 [pdf] replaced on 2017-08-22 12:28:29 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 30 Pages. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement. Category: Functions and Analysis [222] viXra:1707.0155 [pdf] replaced on 2017-08-21 07:31:25 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 30 Pages. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement. Category: Functions and Analysis [221] viXra:1707.0155 [pdf] replaced on 2017-07-24 06:41:52 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 28 Pages. This paper need some change. See also viXra:1708.0123, "Describing a Fluid in Three-Dimensional Circular Motion with One Independent Variable by Rectangular Coordinate", by Valdir M.S. Godoi A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement. Category: Functions and Analysis [220] viXra:1707.0155 [pdf] replaced on 2017-07-20 09:28:40 ### On a Problem in Euler and Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 27 Pages. A study respect to a problem found in the equations of Euler and Navier-Stokes, whose adequate treatment solves a centennial problem about the solution of these equations and a most correct modeling of fluid movement. Category: Functions and Analysis [219] viXra:1705.0410 [pdf] replaced on 2017-05-30 20:59:45 ### New Principles of Differential Equations Ⅰ Authors: Hong Lai Zhu Comments: 71 Pages. This is the first part of the total paper. Since the theory of partial differential equations (PDEs) has been established nearly 300 years, there are many important problems have not been resolved, such as what are the general solutions of Laplace equation, acoustic wave equation, Helmholtz equation, heat conduction equation, Schrodinger equation and other important equations? How to solve the problems of definite solutions which have universal significance for these equations? What are the laws of general solution of the mth-order linear PDEs with n variables (n,m≥2)? Is there any general rule for the solution of a PDE in arbitrary orthogonal coordinate systems? Can we obtain the general solution of vector PDEs? Are there very simple methods to quickly and efficiently solve the exact solutions of nonlinear PDEs? And even general solution? Etc. These problems are all effectively solved in this paper. Substituting the results into the original equations, we have verified that they are all correct. Category: Functions and Analysis [218] viXra:1705.0165 [pdf] replaced on 2018-02-14 22:45:03 ### One Word: Navier-Stokes Authors: Nicholas R. Wright Comments: 6 Pages. Replaced "Pareto" with "Pareto Improvement" We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto Improvement exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards. Category: Functions and Analysis [217] viXra:1705.0165 [pdf] replaced on 2017-10-11 04:08:40 ### One Word: Navier-Stokes Authors: Nicholas R. Wright Comments: 6 Pages. Replaced "irrationality of square" with "geometric rate" and Cambria Math font. We prove the Navier-Stokes equations, by means of the Metabolic Theory of Ecology and the Rule of 72. Macroecological theories are proof to the Navier-Stokes equations. A solution could be found using Kleiber’s Law. Measurement is possible through the heat calorie. A Pareto exists within the Navier-Stokes equations. This is done by superposing dust solutions onto fluid solutions. In summary, the Navier-Stokes equations require a theoretical solution. The Metabolic Theory of Ecology, along with Kleiber’s Law, form a theory by such standards. Category: Functions and Analysis [216] viXra:1703.0135 [pdf] replaced on 2017-03-29 06:22:52 ### Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis Authors: Naoya Isobe Comments: 13 Pages. Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis. Category: Functions and Analysis [215] viXra:1703.0134 [pdf] replaced on 2017-03-29 06:21:16 ### Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis). Category: Functions and Analysis [214] viXra:1703.0133 [pdf] replaced on 2017-03-29 06:19:34 ### Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis Authors: Naoya Isobe Comments: 15 Pages. Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis. Category: Functions and Analysis [213] viXra:1703.0132 [pdf] replaced on 2017-03-29 06:17:44 ### Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis). Category: Functions and Analysis [212] viXra:1703.0121 [pdf] replaced on 2017-03-29 06:11:05 ### Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis (Japanese Version) Authors: Naoya Isobe Comments: 12 Pages. Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis. Category: Functions and Analysis [211] viXra:1703.0120 [pdf] replaced on 2017-03-29 06:09:07 ### Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis) (Japanese Version) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in the Whole Space Using Time Transformation Analysis). Category: Functions and Analysis [210] viXra:1703.0119 [pdf] replaced on 2017-03-29 06:06:14 ### Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis (Japanese Version) Authors: Naoya Isobe Comments: 14 Pages. Global in time solvability of incompressive Navier-Stokes initial value problem in periodic space is proved using time transformation analysys. Category: Functions and Analysis [209] viXra:1703.0118 [pdf] replaced on 2017-03-29 06:02:51 ### Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis) (Japanese Version) Authors: Naoya Isobe Comments: 1 Page. Key point of the proof (Proof of Global in Time Solvability of Incompressive NSIVP in Periodic Space Using Time Transformation Analysis) Category: Functions and Analysis [208] viXra:1701.0617 [pdf] replaced on 2017-05-18 05:57:40 ### Non-Trivial Extension of Real Numbers Authors: Ilya Chernykh Comments: 11 Pages. Now in English We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used as a cardinality measure for fine comparison between infinite countable sets in metric space which respects the intuitive notion of the set's size. Category: Functions and Analysis [207] viXra:1701.0617 [pdf] replaced on 2017-05-12 07:46:53 ### Non-Trivial Extension of Real Numbers Authors: Ilya Chernykh Comments: 10 Pages. Now in Russian We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used as a cardinality measure for fine comparison between infinite countable sets in metric space which respects the intuitive notion of the set's size. Category: Functions and Analysis [206] viXra:1701.0617 [pdf] replaced on 2017-04-27 23:34:08 ### Non-Trivial Extension of Real Numbers Authors: Ilya Chernykh Comments: 9 Pages. We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size. Category: Functions and Analysis [205] viXra:1701.0617 [pdf] replaced on 2017-02-01 10:10:22 ### Non-Trivial Extension of Real Numbers Authors: Ilya Chernykh Comments: 8 Pages. In Russian language We propose an extension of real numbers which reveals a surprising algebraic role of Bernoulli numbers, Hurwitz Zeta function, Euler-Mascheroni constant as well as generalized summations of divergent series and integrals. We extend elementary functions to the proposed numerical system and analyze some symmetries of the special elements. This reveals intriguing closed-form relations between trigonometric and inverse trigonometric functions. Besides this we show that the proposed system can be naturally used for fine comparison between countable sets in metric space which respects the intuitive notion of the set's size. Category: Functions and Analysis [204] viXra:1612.0238 [pdf] replaced on 2018-04-26 17:21:49 ### On the Navier-Stokes Equations Authors: Daniel Thomas Hayes Comments: 9 Pages. The problem on the existence and smoothness of the Navier-Stokes equations is resolved. Category: Functions and Analysis [203] viXra:1612.0238 [pdf] replaced on 2018-01-29 19:07:23 ### On the Navier-Stokes Equations Authors: Daniel Thomas Hayes Comments: 11 Pages. The problem on the existence and smoothness of the Navier-Stokes equations is solved. Category: Functions and Analysis [202] viXra:1611.0056 [pdf] replaced on 2016-11-10 10:15:26 ### How to Project Onto Extended Second Order Cones Authors: O. P. Ferreira, S. Z. Németh Comments: 12 Pages. The extended second order cones were introduced by S. Z. Németh and G. Zhang in [S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. Németh and G. Zhang in [S.Z. Németh and G. Zhang. Positive operators of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods. Category: Functions and Analysis [201] viXra:1611.0002 [pdf] replaced on 2017-01-15 23:03:42 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 22 Pages. [v8] Revised Appendix B This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [200] viXra:1611.0002 [pdf] replaced on 2016-12-21 02:06:40 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 22 Pages. [v7] Algorithm runtime is improved. This version should now be stable. This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [199] viXra:1611.0002 [pdf] replaced on 2016-12-19 03:02:40 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 22 Pages. [v6] update: Further improvments to error/tolerance analysis This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [198] viXra:1611.0002 [pdf] replaced on 2016-12-16 18:53:00 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 23 Pages. [v5] update: Improved error/tolerance analysis This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [197] viXra:1611.0002 [pdf] replaced on 2016-12-13 20:10:43 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 22 Pages. [v4] update: robust error/tolerance analysis This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [196] viXra:1611.0002 [pdf] replaced on 2016-11-30 17:18:29 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 16 Pages. v3 includes links to associated MATLAB and Mathematica code This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [195] viXra:1611.0002 [pdf] replaced on 2016-11-29 23:00:31 ### Numerical Solution of Linear, Nonhomogeneous Differential Equation Systems via Padé Approximation Authors: Kenneth C. Johnson Comments: 16 Pages. This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed. Category: Functions and Analysis [194] viXra:1609.0349 [pdf] replaced on 2016-12-20 11:35:40 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 22 Pages. Published at WISE Journal, Volume 6, No. 2 (Summer, 2017), pp. 60-81. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and then, in special case, solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force, the Bernoulli’s law. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [193] viXra:1609.0349 [pdf] replaced on 2016-11-03 19:59:17 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 22 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [192] viXra:1609.0349 [pdf] replaced on 2016-11-03 11:39:04 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 22 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [191] viXra:1609.0349 [pdf] replaced on 2016-10-27 18:38:53 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 20 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [190] viXra:1609.0349 [pdf] replaced on 2016-10-25 00:35:09 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 20 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [189] viXra:1609.0349 [pdf] replaced on 2016-10-20 20:17:39 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 20 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. A little description of the Lamb’s solution for Euler equations is done. Category: Functions and Analysis [188] viXra:1609.0349 [pdf] replaced on 2016-10-07 11:19:51 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 18 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. Category: Functions and Analysis [187] viXra:1609.0349 [pdf] replaced on 2016-09-29 09:09:56 ### Solutions for Euler and Navier-Stokes Equations in Finite and Infinite Series of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 17 Pages. We present solutions for the Euler and Navier-Stokes equations in finite and infinite series of time, in spatial dimension N=3, firstly based on expansion in Taylor’s series of time and at end concluding on necessity of solutions for velocity given by irrotational vectors, for incompressible flows and conservative external force. Category: Functions and Analysis [186] viXra:1609.0141 [pdf] replaced on 2016-12-27 13:03:54 ### Solutions for Euler and Navier-Stokes Equations in Powers of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 10 Pages. We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1. Category: Functions and Analysis [185] viXra:1609.0141 [pdf] replaced on 2016-12-16 16:27:55 ### Solutions for Euler and Navier-Stokes Equations in Powers of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 10 Pages. We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1. Category: Functions and Analysis [184] viXra:1609.0141 [pdf] replaced on 2016-09-27 17:45:06 ### Solutions for Euler and Navier-Stokes Equations in Powers of Time Authors: Valdir Monteiro dos Santos Godoi Comments: 8 Pages. We present a solution for the Euler and Navier-Stokes equations for incompressible case given any smooth (C^∞) initial velocity, pressure and external force in N=3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, N≥1. Category: Functions and Analysis [183] viXra:1609.0074 [pdf] replaced on 2016-09-27 17:12:07 ### General Solution For Navier-Stokes Equations With Any Smooth Initial Data Authors: Valdir Monteiro dos Santos Godoi Comments: 6 Pages. We present a solution for the Navier-Stokes equations for incompressible case with any smooth (C^∞) initial velocity given a pressure and external force in n = 3 spatial dimensions, based on expansion in Taylor’s series of time. Without major difficulties, it can be adapted to any spatial dimension, n≥1. Category: Functions and Analysis [182] viXra:1609.0006 [pdf] replaced on 2017-01-19 09:35:22 ### Adequate Quaternionic Generalization of Complex Differentiability Authors: Michael Parfenov Comments: 46 Pages The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory. Category: Functions and Analysis [181] viXra:1609.0006 [pdf] replaced on 2016-09-22 07:34:07 ### Adequate Quaternionic Generalization of Complex Differentiability Authors: Michael Parfenov Comments: 46 Pages Abstract: The high efficiency of complex analysis is attributable mainly to the ability to represent adequately the Euclidean physical plane essential properties, which have no counterparts on the real axis. In order to provide the similar ability in higher dimensions of space we introduce the general concept of essentially adequate differentiability, which generalizes the key features of the transition from real to complex differentiability. In view of this concept the known Cauchy-Riemann-Fueter equations can be characterized as inessentially adequate. Based on this concept, in addition to the usual complex definition, the quaternionic derivative has to be independent of the method of quaternion division: on the left or on the right. Then we deduce the generalized quaternionic Cauchy-Riemann equations as necessary and sufficient conditions for quaternionic functions to be H-holomorphic. We prove that each H-holomorphic function can be constructed from the C-holomorphic function of the same kind by replacing a complex variable by a quaternionic in an expression for the C-holomorphic function. It follows that the derivatives of all orders of H- holomorphic functions are also H-holomorphic and can be analogously constructed from the corresponding derivatives of C-holomorphic functions. The examples of Liouvillian elementary functions demonstrate the efficiency of the developed theory. Category: Functions and Analysis [180] viXra:1608.0432 [pdf] replaced on 2016-09-01 12:40:33 ### General Solution for Navier-Stokes Equations with Conservative External Force Authors: Valdir Monteiro dos Santos Godoi Comments: 10 Pages. We present two proofs of theorems on solutions of the Navier-Stokes equations for incompressible case with a conservative external force in n = 3 spatial dimensions. Without major difficulties, it can be adapted to any spatial dimension, n>=1. Category: Functions and Analysis [179] viXra:1606.0344 [pdf] replaced on 2016-09-07 12:56:55 ### Notes on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 7 Pages. § 1: remembering the need of imposed the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. This section is historical only. § 2: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. § 3: non-uniqueness in time for incompressible and potential flows, if u≠0. § 4: a more general solution of Euler and Navier-Stokes equations for incompressible and irrotational (potential) flows, given the initial velocity. § 5: Solution for Euler and Navier-Stokes equations using Taylor’s series of powers of t around t=0. Category: Functions and Analysis [178] viXra:1606.0344 [pdf] replaced on 2016-08-18 14:56:52 ### Notes on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 5 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. This date is historical only. § 2: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. § 3: non-uniqueness in time for incompressible and potential flows, if u≠0. § 4: a more general solution of Euler and Navier-Stokes equations for incompressible and irrotational (potential) flows, given the initial velocity. Category: Functions and Analysis [177] viXra:1606.0344 [pdf] replaced on 2016-08-15 15:48:39 ### Notes on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Two last dates: non-uniqueness in time for incompressible and potential flows, if u≠0. Category: Functions and Analysis [176] viXra:1606.0344 [pdf] replaced on 2016-07-30 08:27:18 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Third date: non-uniqueness in time for incompressible and potential flows, if u≠0. Category: Functions and Analysis [175] viXra:1606.0344 [pdf] replaced on 2016-07-29 20:50:51 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Third date: non-uniqueness in time for incompressible and potential flows. Category: Functions and Analysis [174] viXra:1606.0344 [pdf] replaced on 2016-07-19 18:59:31 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 3 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential velocity and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. The first date is historical only. Category: Functions and Analysis [173] viXra:1606.0344 [pdf] replaced on 2016-07-19 14:28:34 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 3 Pages. First date: remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Second date: verifying that for potential velocity and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. The first date is historical only. Category: Functions and Analysis [172] viXra:1606.0344 [pdf] replaced on 2016-07-03 20:13:18 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 2 Pages. Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Category: Functions and Analysis [171] viXra:1606.0344 [pdf] replaced on 2016-06-30 18:11:07 ### Note on Uniqueness Solutions of Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 2 Pages. Remembering the need of impose the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. Category: Functions and Analysis [170] viXra:1606.0324 [pdf] replaced on 2017-01-20 16:47:57 ### The Theory of N-Scales Authors: Furkan Semih Dundar Comments: 6 Pages. v3. Clarifications have been made. We provide a theory of$n$-scales previously called as$n$dimensional time scales. In previous approaches to the theory of time scales, multi-dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}.$n$-scales make the mathematical structure more flexible and appropriate to real world applications in physics and related fields. Here we define an$n$-scale as an arbitrary closed subset of$\mathbb R^n$. Modified forward and backward jump operators,$\Delta$-derivatives and multiple integrals on$n$-scales are defined. Category: Functions and Analysis [169] viXra:1606.0324 [pdf] replaced on 2016-07-02 18:45:26 ### The Theory of N-Scales Authors: Furkan Semih Dundar Comments: 5 Pages. v2 typos are fixed We give a theory of$n$-scale previously called as$n$dimensional time scale. In previous approaches to the theory of time scales, multi dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. Here we define an$n$-scale as an arbitrary closed subset of$\mathbb R^n$. Modified forward and backward jump operators,$\Delta$-derivatives and multiple integrals on$n\$-scales are defined. Category: Functions and Analysis [168] viXra:1606.0141 [pdf] replaced on 2016-06-21 11:42:55 ### Some Definite Integrals Over a Power Multiplied by Four Modified Bessel Functions Authors: Richard J. Mathar The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x)K_1^v(x) dx are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are Modified Bessel Functions. There are essentially 15 types of the 4-fold product. Partial integration of each of these types leads correlations between these integrals. The main result are (forward) recurrences of the integrals with respect to the exponent j of the power. Category: Functions and Analysis [167] viXra:1605.0236 [pdf] replaced on 2016-08-28 12:57:14 ### Two Theorems on Solutions in Eulerian Description Authors: Valdir Monteiro dos Santos Godoi We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions. Category: Functions and Analysis [166] viXra:1605.0236 [pdf] replaced on 2016-05-27 07:05:31 ### Two Theorems on Solutions in Eulerian Description Authors: Valdir Monteiro dos Santos Godoi We present two proofs of theorems needed to the major work we are doing on existence and breakdown solutions of the Navier-Stokes equations for incompressible case in n = 3 spatial dimensions. Category: Functions and Analysis [165] viXra:1604.0244 [pdf] replaced on 2017-07-26 09:02:29 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [164] viXra:1604.0244 [pdf] replaced on 2017-04-03 08:56:40 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [163] viXra:1604.0244 [pdf] replaced on 2017-03-28 21:39:58 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [162] viXra:1604.0244 [pdf] replaced on 2017-03-27 09:28:38 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [161] viXra:1604.0244 [pdf] replaced on 2017-03-17 22:05:04 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [160] viXra:1604.0244 [pdf] replaced on 2016-08-27 06:08:44 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 12 Pages. The equations 19 and 20 are not correct for n > 1 We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [159] viXra:1604.0244 [pdf] replaced on 2016-07-14 12:31:22 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, supposing that there is some solution, following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, it is possible that there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [158] viXra:1604.0244 [pdf] replaced on 2016-07-07 14:39:22 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [157] viXra:1604.0244 [pdf] replaced on 2016-06-24 10:51:48 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [156] viXra:1604.0244 [pdf] replaced on 2016-06-23 12:10:31 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [155] viXra:1604.0244 [pdf] replaced on 2016-06-20 07:01:30 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations following the Eulerian and Lagrangian descriptions, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [154] viXra:1604.0244 [pdf] replaced on 2016-06-12 22:01:12 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 8 Pages. Sorry, this paper yet is not good. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [153] viXra:1604.0244 [pdf] replaced on 2016-06-09 12:53:33 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: ERRATA: Page 4, between equations (12) and (13) include the word "square" before the word "module": square module. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [152] viXra:1604.0244 [pdf] replaced on 2016-06-05 18:00:27 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [151] viXra:1604.0244 [pdf] replaced on 2016-06-02 09:33:17 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [150] viXra:1604.0244 [pdf] replaced on 2016-05-31 09:37:15 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [149] viXra:1604.0244 [pdf] replaced on 2016-05-12 11:26:02 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 6 Pages. ERRATA: x,y,z in equation (7) are in Lagrangian description We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [148] viXra:1604.0244 [pdf] replaced on 2016-05-09 13:52:01 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [147] viXra:1604.0244 [pdf] replaced on 2016-05-06 20:29:57 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [146] viXra:1604.0244 [pdf] replaced on 2016-04-27 08:05:51 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 5 Pages. It´s necessay enter the factor 1/3 multiplying the second ni in the equations 1, 5, 6 and 7. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [145] viXra:1604.0244 [pdf] replaced on 2016-04-23 21:37:32 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [144] viXra:1604.0244 [pdf] replaced on 2016-04-23 04:30:18 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. Sorry. Unfortunately, x, y, z are not so free. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [143] viXra:1604.0244 [pdf] replaced on 2016-04-19 11:14:23 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [142] viXra:1604.0244 [pdf] replaced on 2016-04-18 08:03:49 ### Solution for Navier-Stokes Equations – Lagrangian and Eulerian Descriptions Authors: Valdir Monteiro dos Santos Godoi Comments: 4 Pages. Equations (6) and (7) are wrongs, both versions 1 and 2. Nothing like they is possible. We find an exact solution for the system of Navier-Stokes equations, following the description of the Lagrangian movement of an element of fluid, for spatial dimension n = 3. As we had seen in other previous articles, there are infinite solutions for pressure and velocity, given only the condition of initial velocity. Category: Functions and Analysis [141] viXra:1604.0235 [pdf] replaced on 2016-06-12 04:36:26 ### On Some Series Related to Möbius Function and Lambert W-Function Authors: Danil Krotkov We derive some new formulas, connecting some series with Möbius function with Sine Integral and Cosine Integral functions, give the formal proof for full version of Stirling's formula with remainder term in form of definite integral of elementary function; investigate the values of new Dirichlet series function at natural numbers >1 and its behavior at the pole at s=1, connecting it with elementary constants. Category: Functions and Analysis [140] viXra:1604.0107 [pdf] replaced on 2016-11-18 06:21:50 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [139] viXra:1604.0107 [pdf] replaced on 2016-04-15 07:02:55 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [138] viXra:1604.0107 [pdf] replaced on 2016-04-14 23:45:15 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi We seek some attempt solutions for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. These solutions have the principal objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [137] viXra:1604.0107 [pdf] replaced on 2016-04-11 08:05:39 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [136] viXra:1604.0107 [pdf] replaced on 2016-04-10 07:28:48 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [135] viXra:1604.0107 [pdf] replaced on 2016-04-08 21:17:12 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: Pages. Sorry, the case (B) is damaged. (2.34) and related are wrong. For now, it is safe the case (C). We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [134] viXra:1604.0107 [pdf] replaced on 2016-04-07 11:27:47 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi Comments: 16 Pages. Surprisingly, we also solved the case (B) of the Millenium Problem on Navier-Stokes equations. We seek a solution attempt for the system of Navier-Stokes equations for spatial dimensions n = 2 and n = 3. This solution has the most objective to provide a better numerical evaluation of the exact analytical solution, thus contributing to the solution not only from a theoretical mathematical problem, but from a practical problem worldwide. Category: Functions and Analysis [133] viXra:1604.0107 [pdf] replaced on 2016-04-06 11:51:43 ### A Naive Solution for Navier-Stokes Equations Authors: Valdir Monteiro dos Santos Godoi	{"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": 0.9222962856292725, "perplexity": 1497.115171478309}, "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/1531676588972.37/warc/CC-MAIN-20180715203335-20180715223335-00234.warc.gz"}
https://mca2017.org/prog/session/cp	# Contributed Papers All abstracts Size: 125 kb Organizers: • Gantumur Tsogtgerel (McGill University, Canada) • Abdullahi Adem North-West University Multiple wave solutions and conservation laws of the Date-Jimbo-Kashiwara-Miwa (DJKM) equation via symbolic computation PDF abstract Size: 39 kb In this talk, we present soliton solutions and conservation laws for the DJKM equation with the aid of symbolic computation. The soliton solutions of the DJKM equation are constructed by using the multiple exp-function method, which is a generalization of Hirota's perturbation scheme. The solu- tions obtained involve generic phase shifts and wave frequencies. Furthermore, in nitely many conservation laws are derived by using the multiplier method which is an indicator of the integrability of the underlying equation. • Karim Samei Bu Ali Sina University, Hamedan, Iran. Singleton Bounds for R-additive Codes PDF abstract Size: 64 kb Shiromoto (Linear algebra Applic 295 (1999) 191-200) obtained the basic exact sequence for the Lee and Euclidean weights of linear codes over $\mathbb{Z}_{\ell}$ and as an application, he found the Singleton Bounds for linear codes over $\mathbb{Z}_{\ell}$ with respect to Lee and Euclidean weights. Huffman (Adv. Math. Commun 7 (3) (2013) 349-378) obtained the Singleton Bound for $\mathbb{F}_{q}$-linear $\mathbb{F}_{q^{t}}$-codes with respect to Hamming weight?. ?Recently the theory of $\mathbb{F}_{q}$-linear $\mathbb{F}_{q^{t}}$-codes were generalized to $R$-additive codes over $R$-algebras by Samei and Mahmoudi. In this paper, we generalize Shiromoto's results for linear codes over $\mathbb{Z}_{\ell}$ to $R$-additive codes. As an application, when $R$ is a chain ring, we obtain the Singleton Bounds for $R$-additive codes over free $R$-algebras. Among other results, the Singleton Bounds for additive codes over Galois rings are given. • Lahcen Laayouni Al Akhawayn University On the efficiency of the Algebraic Optimized Schwarz Methods PDF abstract Size: 39 kb In this study we investigate the efficiency of the Algebraic Optimized Schwarz Methods (AOSM) in solving large-scale linear systems. The AOSM used as preconditionners in solving linear systems converge in two iterations for a decomposition with two sub-domains using optimal transmission blocks. These blocks require the inverse of large sub-matrices of the original matrix of the linear system. In this paper we are interested in approximating the transmission blocks with adequate approximations. Numerical comparisons will be presented for different types of problems. This is joint work with M. Gander and D. Szyld. • Pedro Pablo CARDENAS ALZATE Universidad Tecnológica de Pereira The Zhou’s method for solving delay differential equations applied to biological models PDF abstract Size: 39 kb In this work, we apply the Zhou’s method or Differential Transfor- mation Method (DTM) for solving some models that arises in biological sciences, which are nonlinear delay differential equations. The efficiency of DTM is illustrated by investigating the convergence results on numerical models that show the reliability and accuracy of this method. • Domingo Tarzia CONICET and Universidad Austral The one-phase Stefan problem with a latent heat of fusion depending of the position of the free boundary and its velocity PDF abstract Size: 41 kb From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat depends inversely with the rate of change of the seepage front (e.g. Zhou-Bu-Lu, Int. J. Numerical and Analytical Methods in Geomechanics, 37 (2013), 2825-2832). A one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. The aim of this analysis is to extend prior results, finding an analytical solution that recovers, by specifying some parameters, the solutions already examined in the literature regarding Stefan problems with variable latent heat. Moreover, we also consider different boundary conditions at the fixed face. This is a joint paper with Julieta Bollati (CONICET and Universidad Austral). • Nazish Iftikhar National University of Computer and Emerging Sciences, Lahore Campus, Pakistan. Classifying Robertson-Walker scale factor using Noether’s approach PDF abstract Size: 40 kb The universe can be depicted in the best way by using Friedmann-Robertson-Walker (FRW) models. FRW models of the universe are considered to have properties like homogeneity and isotropy. The universe is continuously expanding which can be represented by considering Robertson-Walker scale factor. Robertson-Walker scale factor is the function of time 't'. The scale factor is useful to define red shift and the Hubble parameter. The Hubble parameter gives information about the evolution of the universe and is also useful in calculating the age of the universe. In present research work, Noether’s approach was applied to classify FRW spacetime. The spacetime was considered for three types of universe i.e. closed, open, and flat. For closed, open and flat universe, curvature parameter 'k' was -1, 1, and 0 respectively. Different values of Robertson-Walker scale factor were considered which gave the nontrivial symmetries. By using Noether equation and Perturbed Lagrangian an over-determined system of partial differential equations were obtained. For the closed, open and flat universe, maximal and minimal set of Noether operators were acquired. For every Noether operator, the corresponding energy type first integral of motion was calculated. • Imran Naeem Lahore University of Management sciences (LUMS), Pakistan A new approach to construct first integrals and closed-form solutions of dynamical systems for epidemics PDF abstract Size: 39 kb A new class of non-standard Hamiltonian known as the "artificial Hamiltonian" is introduced which results in an artificial Hamiltonian system of first-order ordinary differential equations (ODEs). The notion of an artificial Hamiltonian is developed for the systems of dynamical systems of ODEs. Also, it is shown that every system of second-order ODEs can be expressed as an artificial Hamiltonian system of first-order ODEs. The newly developed notion of an artificial Hamiltonian system gives a new way to solve the dynamical systems of first-order ODEs or systems of second-order ODEs which can be expressed as an artificial Hamiltonian system by utilizing the known techniques applicable to the non-standard Hamiltonian systems. We employed this proposed notion to solve dynamical systems of first-order ODEs arising in epidemics. • Rehana Naz Lahore School of Economics, Pakistan The first integrals and closed-form solutions of optimal control problems PDF abstract Size: 64 kb The Pontrygin's maximum principle (Pontryagin, 1987) provides the necessary conditions for the optimum in the optimal control problems in terms of variables time $t$, state variables $q^i$, costate variables $p_i$ and control variables $u_i$. One can eliminate the control variables in terms of state and co-state variables which reduces the conditions of Pontrygin's maximum principle to following non-standard Hamiltonian system: $$\dot q^i=\frac{\partial H}{\partial p_i}, \dot p^i=-\frac{\partial H}{\partial q_i}+\Omega^i(t,q^i,p_i.)$$ This type of non-standard Hamiltonian system arises widely in optimal control problems in different fields of the applied mathematics. A mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces results in a non-standard Hamiltonian system. In optimal control problems of economic growth theory involving a non-zero discount factor these type of system arise and are known as a current value Hamiltonian systems. It is proposed how to modify the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al 2014 in order to apply it to the epidemics, mechanics and other areas as well. To show the effective of the approach developed here, it is utilized to construct the first integrals and closed form solutions of some models from real world. Moreover, the essential aspects of infectious diseases spread are uncovered and polices are provided to public health decision makers to compare and implement different control programs. For the Economic growth model some policies are provided to the government in order to have a sustainable growth. • Vladislav Bukshtynov Florida Institute of Technology Optimal Reconstruction of Constitutive Relations for Porous Media Flows PDF abstract Size: 55 kb Comprehensive full-physics models for flow in porous media typically involve convection-diffusion partial differential equations whose parameters are unknown and have to be reconstructed from experimental data. Quite often these unknown parameters are coefficients represented by space-dependent, sometimes correlated, functions, e.g. porosity, permeability, transmissibility, etc. However, special complexity is seen when the reconstructed properties are considered as state-dependent parameters, e.g. the relative permeability coefficients $k_{rp}$. Modern petroleum reservoir simulators still use simplified approximations of $k_{rp}$ as single variable functions of $p$-phase saturation $s_p$ given in the form of tables or simple analytical expressions. This form is hardly reliable in modern engineering applications used, e.g., for enhanced oil recovery, carbon storage, modeling thermal and capillary pressure relations. Thus, the main focus of our research is on developing a novel mathematical concept for building new models where $k_{rp}$ are approximated by multi-variable functions of fluid parameters, namely phase saturations $s_p$ and temperature $T$. Reconstruction of such complicated dependencies requires advanced mathematical and optimization tools to enhance the efficiency of existing engineering procedures with a new computational framework generalized for use in various earth science applications. • Buthinah Bin Dehaish King Abdullaziz University Fixed Point Theorem for monotone Lipschitzian mappings PDF abstract Size: 37 kb Among this talk we will consider a new class of Lipschitzian mappings which are monotone and then we will discuss some fixed point theorems for these mappings. • Chuang Xu University of Alberta Best finite constrained approximations of one-dimensional probabilities PDF abstract Size: 52 kb This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. Special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The elementary approach is based on best approximations of (monotone) $L^r$-functions by step functions, which is different from, and naturally complementary to, the classical Voronoi partition approach. This is a joint work with Dr. Arno Berger. • Eric Jose Avila Universidad Autonoma de Yucatan Global dynamics of a periodic SEIRS model with general incidence rate PDF abstract Size: 38 kb We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate, we will show that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence. We estimate the basic reproduction number and we provide numerical simulations to illustrate our findings. • Eugen Mandrescu Holon Institute of Technology, Israel Shedding vertices and well-covered graphs PDF abstract Size: 80 kb A set $S$ of vertices in a graph $G$ is independent if no two vertices from $S$ are adjacent. If all maximal independent sets are of the same cardinality, then $G$ is well-covered (or unmixed) (Plummer, 1970). $G$ belongs to class $\mathbf{W}_{2}$ if every $2$ disjoint independent sets are included in $2$ disjoint maximum independent sets (Staples, 1975). There are deep interactions between shellability, vertex decomposability and well-coveredness (Castrill\'{o}n, Cruz, Reyes, 2016). Let $v\in V\left( G\right)$ and $N\left( v\right)$ be its neighborhood. If for every independent set $S$ of $G-\left( N\left( v\right) \cup\{v\}\right)$, there is some $u\in N\left( v\right)$ such that $S\cup\left\{ u\right\}$ is independent, then $v$ is a shedding vertex of $G$ (Woodroofe, 2009). Let $Shed\left( G\right)$ denote the set of all shedding vertices. Clearly, no isolated vertex is shedding, and no graph in $\mathbf{W}_{2}$ has isolated vertices. In this talk, we show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given independent set. Specifically, for well-covered graphs, it means that a non-isolated vertex $v\in Shed\left( G\right)$ if and only if $G-v$ is well-covered. Thus $G$ belongs to class $\mathbf{W}_{\mathbf{2}}$ if and only if $Shed\left( G\right) =V\left( G\right)$. There exist well-covered graphs without shedding vertices; e.g., $C_{7}$. On the other hand, there are non-well-covered graphs with $Shed\left( G\right) =V\left( G\right)$. Problem 1. Find all well-covered graphs having no shedding vertices. Problem 2. Find all graphs having $Shed\left(G\right) =V\left( G\right)$. • Yuan-Jen Chiang University of Mary Washington Leaf-wise Harmonic Maps of Manifolds with 2-dimensional Foliations PDF abstract Size: 47 kb In the 1980s, A. Connes [Proc. of Symp. in Pure Math, AMS, 1982] and E. Ghys [J. Func. Anal., 1988] proved the Gauss-Bonnet type theorem for compact manifolds with 2-dimensional foliations. In this paper, we derive the expressions of harmonic non $\pm$ holomorphic maps of Riemann surfaces. We study the relationship between leaf-wise harmonic maps and harmonic maps. We investigate the Gauss-Bonnet type theorem for leaf-wise harmonic maps between manifolds with 2-dimensional foliations, which generalize the results of Connes and Ghys. This paper has recently appeared in the Bulletin of the Institute of Mathematics, Academia, Sinica. • Stefan Veldsman Nelson Mandela University Generalized complex numbers over near-fields PDF abstract Size: 41 kb In the early 20th century, Dickson (1905) investigated the redundancy or not of the field axioms. By a clever disturbance of the multiplication of a field, he demonstrated the existence of an algebraic structure fulfilling all the requirements of a field except one of the distributive axioms. These structures are known as Dickson near-fields, but there are many near-fields not of this type. Almost immediately near-fields were shown to be not just an algebraic curiosity. Veblen and Wedderburn (1907) showed that near-fields are exactly the algebraic structures required for coordinization of geometries that lead to non-Desarguesian planes. In a monumental paper, Zassenhaus (1935/6) showed that all finite near-fields are Dickson near-fields except for 7 strays. There are many other applications of near-fields and the more general near-rings became an important and useful area of investigation with its own concerns and problems catering for non-linear algebraic systems. The construction of the complex numbers over the reals has been generalized in many ways leading to the 2-dimensional elliptical complex numbers (= complex numbers) and the parabolic and hyperbolic complex numbers. These can be extended to higher dimensions and using an arbitrary ring as the base ring. It is possible to define matrices and polynomials over near-rings. Using these, one can construct generalized complex numbers over a near-field. In this talk, this construction will be formalized. We also report on properties of this algebraic structure and highlight similarities and differences with its motivating example; the usual complex numbers over the real field. • Ryad Ghanam Virginia Commonwealth University in Qatar Non-Solvable subalgebras of gl(4,R) PDF abstract Size: 37 kb In this talk, I will present all the simple, then semi-simple, subalgebras of gl(4, R). Each such semi-simple subalgebra acts by commutator on gl(4, R). In each case the invariant subspaces are found and the results used to determine all possible subalgebras of gl(4, R) that are not solvable	{"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": 0.8512537479400635, "perplexity": 875.7234508759243}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2020-50/segments/1606141740670.93/warc/CC-MAIN-20201204162500-20201204192500-00337.warc.gz"}
https://quant.stackexchange.com/questions/22041/link-between-two-it%C3%B4s-lemma-written-in-different-ways	# Link between two Itô's Lemma written in different ways I have been told that these two expressions of Itô's Lemma are the same, but written in different ways : $$f(t,X_t) = f(0, X_0) + \int_{0}^{t} \frac{\partial f}{\partial s} ds + \int_{0}^{t} \frac{\partial f}{\partial X_s} dX_s + \frac{1}{2}\int_{0}^{t} \frac{\partial^2 f}{\partial {X_s}^2}{\sigma_s}^2 ds$$ and $$df(X_t, Y_t) = \frac{\partial f}{\partial X_t}dX_t + \frac{\partial f}{\partial Y_t}dY_t + \frac{1}{2} \frac{\partial^2 f}{\partial {X_t}^2}d<X_t>_t + \frac{1}{2} \frac{\partial^2 f}{\partial {Y_t}^2}d<Y>_t + \frac{\partial^2 f}{\partial X_t \partial Y_t}d<X, Y>_t$$ where $<X, Y>_t$ is the quadratic variation operator (and $<X>_t = <X,X>_t$). I cannot figure out why they are similar. In my eyes, they seem pretty different. How can I pass from the first one to the second one? Actually I don't really understand how the second one can be interpreted and applied. • The second one is more general than the first one. Replace $X_{t}$ by $t$ then you have the first formula because $<.>_{t}=0$. Then take he integrated form – glork Dec 1 '15 at 20:12 • I think the two expressions refer to the integral and differential forms. The two you wrote are not for the same thing, that is, one is for $f(t, X_t)$ and the other is for $f(X_t, Y_t)$, then they are of course not the same. Try to write the second one for $f(t, X_t)$. – Gordon Dec 1 '15 at 20:14 • Thank you. But I don't understand why $<.>_t = 0$. And why $<X>_t = \sigma_t^2$(by identification). And why $<X, Y>_t = 0$. As you can see, there are many things unclear... Could you please develop your answer a bit? – MarinD Dec 1 '15 at 22:01 I assume you're confused between the integral and SDE writings of Ito's lemma, since the two equations you have are indeed different. Let $X_t$ be an Ito process defined by $$X_t = X_0 + \int_0^t \alpha_s \, ds + \int_0^t \sigma_s \, dW_s$$ for adapted processes $\alpha_s$ and $\sigma_s$ (and assuming some technical boundedness condition on the integrals). This equation may be written in shorthand as an SDE as $$dX_t = \alpha_t dt + \sigma_t dW_t.$$ The SDE is not rigorous - it is simply a shortcut way of writing down the integrals above, and provides a bit of intuition behind the evolution of $X$ over "infinitesimally small" time intervals. Now consider a measurable function $f: [0,T] \times \mathbb{R} \to \mathbb{R}$ such that $$f(\cdot,x)\in C^1([0,T]) \quad \forall x \in \mathbb{R},$$ and $$f(t,\cdot) \in C^2(\mathbb{R}) \quad \forall t \in [0,T].$$ I would argue the correct (mathematically rigorous) way of stating Ito's lemma is $$f(T,X_T) = f(0,X_0) + \int_0^T \frac{\partial}{\partial t}f(s,X_s) \, ds + \int_0^T \frac{\partial}{\partial x}f(s,X_s) \, dX_s \\ \qquad + \frac{1}{2}\int_0^T \frac{\partial^2}{\partial x^2}f(s,X_s) \, d<X,X>_s$$ The quantity $<X,X>_s$ is the quadratic variation accumulated by the Ito process $X$ up until time $s$. You can show (Shreve II, page 143-144, e.g.) that this is given by $$<X,X>_s = \int_0^s \sigma^2_u \, du,$$ or, in differential (shorthand) form as $$d<X,X>_s = \sigma^2_s \, ds.$$ Plugging this into Ito's lemma gives your first equation. Now, just like the Ito process $X$ was written in shorthand as an SDE, so may $f$, since it, too, is an Ito process. That is, we also have $$df(t,X_t) = \frac{\partial}{\partial t}f(t,X_t)dt + \frac{\partial}{\partial x}f(t,X_t) dX_t + \frac{1}{2}\frac{\partial^2}{\partial x^2}f(t,X_t) d<X,X>_t.$$ The first boxed equation had precise mathematical meaning. The second boxed equation is just shorthand for the first. Update: Your second equation is often called Ito's product rule. Ito's lemma is "usually" stated for functions of one Ito process as it was for my answer above. If you have a function of two Ito processes then both processes' quadratic variation and cross variation appear in Ito's lemma, aka Ito's product rule. See Shreve II, page 168, e.g. for a decent explanation • Thanks, it helps a lot! But what about my second equation? In which way should it be used? Or is it false? – MarinD Dec 2 '15 at 8:58 • @MarinD See update, does that help? – bcf Dec 2 '15 at 15:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9642100930213928, "perplexity": 288.12059458630466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317817.76/warc/CC-MAIN-20190823020039-20190823042039-00517.warc.gz"}
https://www.physicsforums.com/threads/tough-flux-question.216707/	# Tough Flux question 1. Feb 20, 2008 ### dgoudie [SOLVED] Tough Flux question 1. The problem statement, all variables and given/known data A cubic cardboard box of side a = 0.250 m is placed so that its edges are parallel to the coordinate axes, as shown in the figure. There is NO net electric charge inside the box, but the space in and around the box is filled with a nonuniform electric field of the following form: E(x,y,z) = Kz j + Ky k, where K= 3.20 N/C.m is a constant. [Broken] What is the electric flux through the top face of the box? (The top face of the box is the face where z=a. Remember that we define positive flux pointing out of the box.) Hint:Since E is not constant, you will need to do an area integral for this problem. Remember the definition of electric flux. Only one component of E contributes to the flux out of the top of the box. In doing the integral to find the flux, it might help to imagine breaking the top surface into strips along which the z-component of the electric field is constant. If all else fails, ask for help. 2. Relevant equations Flux=$$\int$$EdA 3. The attempt at a solution The Nonuniform part is what's tripping me up. I know how to do it with uniform electric field. I'll call the top face faceA, Flux=I. So I$$_{A}$$=$$\int$$EdA= $$\int$$(3.20 N/C z $$\widehat{j}$$ + 3.20 N/C y$$\widehat{k}$$)* (dA $$\widehat{k}$$) So it says that for this face Z=the side length, but what do I do with y? and with the top face wouldnt only the K-hat vectors apply anyway? and once I get an asnwer will it be negative? anyway if someone could give me some insight, let me know if ive set it up right it would help a lot. Thanks in advance! Last edited by a moderator: Apr 23, 2017 at 10:58 AM 2. Feb 20, 2008 ### dgoudie I'm also really confused by the z being with the j and the y being with the k. 3. Feb 20, 2008 ### awvvu Flux is the surface integral of E dotted with the normal vector. So, find E dot n first. Then, I think, you have to do a simple double integral to integrate over all the points on the surface. (edit: or instead, since the integrand only depends on y, you only need to integrate over y and then just multiply by a. It's the same thing really.) Also, you should write entire expressions in latex tags, not switch to it only to put symbols in. Last edited: Feb 20, 2008 4. Feb 20, 2008 ### dgoudie yeah i don't really know how to use the tags yet. And I though flux was E dot A, where does the normal vector come into play? :S thanks 5. Feb 20, 2008 ### awvvu Well, the dA in the integral you use is really a vector quantity. It points in the direction perpendicular to the surface. So, when you take the dot product of E and dA, the normal vector I was talking about comes into play. 6. Feb 20, 2008 ### dgoudie so the dot product would be: (0.8 j + 3.20 y k)(.25^2) times cos90 am I starting it right? 7. Feb 20, 2008 ### awvvu Nope, because that quantity is 0. Remember that the theta you use is between E and dA (which is perpendicular to the surface). They are parallel. Also, keep in mind that the dot product is a scalar quantity, so there is no more vector stuff in it. You need to multiply together the magnitudes of the two vectors, but I don't think that will work for this expression. The best way to find the dot product is to use the other formula for it, where you sum up the products of their components. 8. Feb 20, 2008 ### dgoudie do I sub anything in for z and y yet? i got 3.20z +0.0625y as my dot product without subbing anything in. Sorry i'm being a noob :S 9. Feb 20, 2008 ### awvvu I don't know where you got those numbers from. It's an easy dot product: $$\vec{E} \cdot d \vec{A} = <0, Kz, Ky> \cdot <0, 0, dA>$$ Note that the dA's are different. Then to integrate, have you done double integrals yet? 10. Feb 20, 2008 ### dgoudie yeah thats what I did. K=3.20 so Kz +Kyda=3.20z + .2y yeah crap forgot to multiply K by da. so now that I have the dot, I just integrate? 11. Feb 20, 2008 ### awvvu The Kz term shouldn't be there. What's Kz 0? :p You can't just plug in for dA, or just make the integral E * A, because the electric field is non uniform. Also, you should keep everything in terms of variables until the very end. Yup, you just integrate. But do you know how to do an area integral? 12. Feb 20, 2008 ### dgoudie i'm not sure. I know z times y will be the area with z constant and y changing. edit is it: $$\int_{1} EdA + \int_{2} EdA$$ Last edited: Feb 20, 2008 13. Feb 20, 2008 ### awvvu The integrand only depends on y. So integrate over that. But keep in mind that that's only one side of the square you're taking into account. For your edit: That looks like how to calculate the net flux through a closed surface. But we're only doing this for one face. What you need to do is set up the integral. What are the limits of integration and what variable are you integrating over? Last edited: Feb 20, 2008 14. Feb 20, 2008 ### dgoudie 0 and .250? over y? 15. Feb 20, 2008 ### awvvu Yeah but you should really keep .250 in terms of a. Then your final answer will be an expression that you can see the validity of, not just a number. 16. Feb 20, 2008 ### dgoudie So does this integrand look right or is there no dA involved? sorry for bothering so much. $$\int^{a}_{0} 0.2y dA$$ 17. Feb 20, 2008 ### awvvu I don't know where you're getting 0.2 from. The dot product of E and dA was K y dA, which I was trying to lead you into realizing before. I don't think I can explain this too well, but: you can't integrate over dA. However, dA = dx*dy. so, what you really need is a double integral: $$\iint_{square} K y dA = \int_0^a \int_0^a K y dx dy$$ I dunno if you know how to do this (it's not too hard though; do one integral at a time, starting from the inside one), but that's how I would set up the integral. Since x is constant in the integrand, instead of needing a double integral, you should realize integral is simply this (or this can be figured out by just doing the double integral): $$a \int_0^a K y dy$$ Perhaps someone else should explain this more clearly. Last edited: Feb 20, 2008 18. Feb 20, 2008 ### dgoudie .2 was k times A. its not your explaining thats the problem, we did no questions like this and there are none in the text. I got the right answer now thanks for all your help! know any place I can do some supplimentry reading on this subject? my text book sucks Last edited: Feb 20, 2008 19. Feb 20, 2008 ### awvvu You can't use A like that, since the electric field is non-uniform. A really good book for this stuff is Div, Grad, Curl and All That. It's a lot more advanced that what you need to know for an introductory physics class though but it's going to be really useful later on. Last edited by a moderator: Apr 23, 2017 at 10:58 AM 20. Feb 20, 2008 ### dgoudie yeah I think i'm doing vector calc next year. Im doing Calculas III now. Series and sums and stuff. we do double integrals in a month or so I thin, ill probbaly understand better than. thanks for you help Last edited by a moderator: Apr 23, 2017 at 10:58 AM Similar Discussions: Tough Flux question	{"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": 0.9421432018280029, "perplexity": 885.1927854921672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120694.49/warc/CC-MAIN-20170423031200-00549-ip-10-145-167-34.ec2.internal.warc.gz"}
http://mathoverflow.net/revisions/72683/list	Return to Question 3 Added the formulas I am struggling to link; added 7 characters in body EDIT:I am trying to link together the martingale representation theorem as in the answer to this question (Cont Tankov use the analogous notation): http://mathoverflow.net/questions/70981/martingale-representation-theorem-for-levy-processes Under some integrability conditions, there exist $\phi$ and $\psi$ st: $$M_t=M_0+\int^t_0 \phi(s)dW_s+\int^t_0 \int_{\textbf{R}}\psi(s,x)\tilde{N}(ds,dx)$$Where $\tilde{N}(ds,dx)$ is the compensated measure of the Lévy process $X$. and the Ito's formula for Levy processes (taken from Cont Tankov) is: $$f(X_t) - f(X_0) = \int_0^t \frac{\sigma^2}{2} f''(X_s) ds + \int_0^t f'(X_{s-}) dX_s$$$$+ \sum_{0\leq s \leq t} \textbf{1}_{(\Delta X_s \neq 0)}[f(X_{s}) - f(X_{s-}) - \Delta X_s f'(X_{s-})]$$ Please note that the formulas above are for scalar processes. My question is how does the term $\int^t_0\int_{\textbf{R}}\psi(s,x)\tilde{N}(ds,dx)$ relate to $\sum_{0\leq s \leq t} \textbf{1}_{(\Delta X_s \neq 0)}[f(X_{s}) - f(X_{s-}) - \Delta X_s f'(X_{s-})]$ and to the Levy measure of the process $X$. Also Proposition 8.16 from Cont-Tankov suggests that a funtion of a Levy process is a martinale iff we have: $$f(X_t) - f(X_0) = \int_0^t f'(X_s)\sigma dWs + \int_{[0,t]\times R} \tilde{N}(ds,dx) [f(X_{s-} + y) - f(X_{s-})]$$ And again I am not sure how to relate the formula above to the Ito's formula. 2 added 2 characters in body This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone clarify what is the relationship between the jump measure, levy measure and if it is possible to express these quantities as a $$\sum_{\Delta X X_t \neq 0} sth$$ in the context of Ito's lemma? In particular I am looking at the application to the exponential function at the moment, if it simplifies anything. 1 Levy jump measure vs. Levy measure vs. sum of jumps This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone clarify what is the relationship between the jump measure, levy measure and if it is possible to express these quantities as a $$\sum_{\Delta X \neq 0} sth$$ in the context of Ito's lemma? In particular I am looking at the application to the exponential function at the moment, if it simplifies anything.	{"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": 0.9746757745742798, "perplexity": 270.85573082396263}, "config": {"markdown_headings": false, "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-2013-20/segments/1368706009988/warc/CC-MAIN-20130516120649-00055-ip-10-60-113-184.ec2.internal.warc.gz"}
http://drops.dagstuhl.de/opus/frontdoor.php?source_opus=1807	When quoting this document, please refer to the following DOI: 10.4230/LIPIcs.STACS.2009.1807 URN: urn:nbn:de:0030-drops-18079 URL: http://drops.dagstuhl.de/opus/volltexte/2009/1807/ Go to the corresponding LIPIcs Volume Portal ### Tractable Structures for Constraint Satisfaction with Truth Tables pdf-format: ### Abstract The way the graph structure of the constraints influences the complexity of constraint satisfaction problems (CSP) is well understood for bounded-arity constraints. The situation is less clear if there is no bound on the arities. In this case the answer depends also on how the constraints are represented in the input. We study this question for the truth table representation of constraints. We introduce a new hypergraph measure {\em adaptive width} and show that CSP with truth tables is polynomial-time solvable if restricted to a class of hypergraphs with bounded adaptive width. Conversely, assuming a conjecture on the complexity of binary CSP, there is no other polynomial-time solvable case. ### BibTeX - Entry @InProceedings{marx:LIPIcs:2009:1807, author = {Daniel Marx}, title = {{Tractable Structures for Constraint Satisfaction with Truth Tables}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {649--660}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Susanne Albers and Jean-Yves Marion}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},	{"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": 0.9471526145935059, "perplexity": 2530.2918866052155}, "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-2014-41/segments/1412037663417.12/warc/CC-MAIN-20140930004103-00406-ip-10-234-18-248.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/79842/is-an-ellipse-a-circle-transformed-by-a-simple-formula?answertab=votes	# Is an ellipse a circle transformed by a simple formula? Does any ellipse $E$ have a circle $C$ such that you can obtain $E$ by transforming $C$ by a simple formula $F$? In details , both $E$ and $C$ have the same center and the axes of $E$ are the XY axes. And F moves $(x,y)$ to $( m*x , y)$ . Where m is a real number. - Yes, but the formula is of the specific kind you mention only after rotating the ellipse and centering it at the origin. – ShreevatsaR Nov 7 '11 at 12:58 Sure, if you scale a circle differently in the horizontal and vertical directions, you'll certainly get an ellipse. – Ｊ. Ｍ. Nov 7 '11 at 12:59 @J.M.: Of course, it's enough to scale (a different circle) only horizontally. :-) – ShreevatsaR Nov 9 '11 at 3:08 I think you'll see it from $${\rm ellipse:\ \ }{x^2\over a^2}+{y^2\over b^2}=1.$$ $${\rm circle:\ \ } {x^2\over b^2}+{y^2\over b^2}=1.$$ That formula for an ellipse is only valid for axis-parallel ellipses centered at the origin. Anyway, to complete it into the form the OP wants: $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ means $\displaystyle \frac{b^2}{a^2}x^2 + y^2 = b^2$, which is the circle $x'^2 + y^2 = b^2$ with $(x',y)$ sent to $(\frac{a}{b}x', y)$. – ShreevatsaR Nov 7 '11 at 13:22	{"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": 0.925848126411438, "perplexity": 650.358931414329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999661726/warc/CC-MAIN-20140305060741-00033-ip-10-183-142-35.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/35238/how-thick-does-steel-have-to-be-to-be-able-to-withstand-300-bar-sphere	# How thick does steel have to be to be able to withstand 300 bar (sphere) How thick does the material in a sphere have to be to withstand the (inner)pressure of 300 bar if the material is steel? (With an inner-radius of 2cm) Atmosphere pressure = same as 0 meter above sea level Is there some "golden rule" like sqrt(bar)cm thickness times stress-constant? - The usual simplification in structural engineering is to consider the sphere as infinitely thin. That is what is called a 'shell' as opposed to thicker 'plates'. Shells react to pressure by undergoing traction, but without bending. That makes things easier. The book by Timoshenko is the place to go for a detailed explanation of either case. The easiest way to turn your problem into numbers is to consider only half of the sphere. The force exerted on that half sphere by the inner pressure will be $F=p \pi r^2$, and what keeps the sphere in place is the tension in the rim, exerted by the other half sphere. For a thickness $t$ of the shell means an area of $A = 2 \pi r t$, so that the tension is $\sigma = p r / (2 t)$. So if you know the tensile strength of steel, you can calculate the thickness from $t = p r / (2 \sigma)$. This is another tricky question... steel has a lower yield limit, at which it starts to deform permanently, and a tensile limit, at which it snaps. These limits are also highly dependent on how the steel has been manufactured. You have some values here. Being conservative and choosing $\sigma = 150\;\mathrm{MPa}$, and your pressure of $p= 300\;\mathrm{bar} = 30\; \mathrm{MPa}$, for a radius of $r=2\;\mathrm{cm}=0.02\;\mathrm{m}$ the resulting minimal thickness is $t=0.002\;\mathrm{m}=2\;\mathrm{mm}$. Depending on the actual material you are using, and how much risk of your sphere blowing to pieces on your face you are willing to take, you may be able to get away with less, or want to avoid risks and go with even more. Normally using the thin shell approximation is deemed safe when $r/t > 10$, so for a much thicker plate things would be different. If you plan to use this information in any real life project, be extremely careful with what you do: if your sphere suddenly breaks in half due to material failure, each half will be initially propelled by a force close to $40\;\mathrm{kN}$... -	{"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": 0.8545377850532532, "perplexity": 556.5239975123126}, "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/1464053209501.44/warc/CC-MAIN-20160524012649-00186-ip-10-185-217-139.ec2.internal.warc.gz"}
http://mathhelpforum.com/discrete-math/119741-solved-need-help-there-equation-solve-easily.html	# Math Help - [SOLVED] Need help...is there an equation to solve this easily? 1. ## [SOLVED] Need help...is there an equation to solve this easily? I'm new to this site, and I apologize if I've put this question in the wrong forum, but I really wasn't sure where it should go. This is not for a class. ... So, right now I'm stuck with guess & check for this, unless someone knows a formula I can use. I have the following equation: A1+2B+C4+D8+E16+F32 = X X is given. It is one value out of the set {1,2,3...63) A,B,C,D,E,F can only be 0 or 1. Is there a way to solve for A through F? There is only one right answer per value of X. For example, If X=1, then A=1 and B through F=0. Or, If X=3 then A=1, B=1, and C through F=0. Having an equation to solve this will make this a lot simpler for me than creating a table of all 63 possibilities (64 if you include X=0). Thanks! 2. Originally Posted by elleg I'm new to this site, and I apologize if I've put this question in the wrong forum, but I really wasn't sure where it should go. This is not for a class. ... So, right now I'm stuck with guess & check for this, unless someone knows a formula I can use. I have the following equation: A1+2B+C4+D8+E16+F32 = X X is given. It is one value out of the set {1,2,3...63) A,B,C,D,E,F can only be 0 or 1. Is there a way to solve for A through F? There is only one right answer per value of X. For example, If X=1, then A=1 and B through F=0. Or, If X=3 then A=1, B=1, and C through F=0. Having an equation to solve this will make this a lot simpler for me than creating a table of all 63 possibilities (64 if you include X=0). Thanks! If I am understanding this correctly You want to know how many solutions there are to the eqution $A+2B+4C+8D+16E+32F=X$ where $X=1...63$ This equation is just like binary numbers $2^0A+2^2B+2^2C+2^3D+2^4E+2^5F=X$ The expression of numbers is Binary is unique so there is only one solution for each X. I hope this helps 3. You're right. I was thinking of A through F as booleans, but rather than true-false they are 1 or 0, so yes, binary numbers. My problem is that I'm not sure how to solve that equation. It's been years since I had to deal with binary numbers. I'm currently googling binary/boolean equations, binary/boolean algebra, etc. to see if I can find an answer. 4. Any number from 0..63, when it is converted into binary, can be written using 6 binary digits. In fact, there is a bijection between the set X={0,...,63} and the set Y={(A,B,C,D,E,F) \| each of A,...,F is 0 or 1}. This means that there are two mutually inverse functions f: X -> Y and g: Y -> X. The function g(A,B,C,D,E,F) is in fact given by the formula A+2B+4C+8D+16E+32F. The function f is the one that converts decimal into binary. So, given n in X, to find the solution to the equation A+2B+4C+8D+16E+32F = n you calculate f(n). This is the only solution because f and g constitute a 1-1 correspondence. I may have described it a little too heavily; the intuitive idea is simpler than this. 5. Awesome...that just clicked for me. So, basically, my variables A,B,C,D,E,F actually make up the positions of the base 2 version of my base 10 variable X. Thus If X=1 Then: FEDCBA 000001 (A=1, All others=0) If X=2 Then: FEDCBA 000010 (B=1, All others=0) If X=3 Then: FEDCBA 000011 (A=1,B=1, All others=0) So, all I have to do is convert X to base 2, and then use each position of the base 2 number. 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": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8306184411048889, "perplexity": 462.3515812409132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678683543/warc/CC-MAIN-20140313024443-00075-ip-10-183-142-35.ec2.internal.warc.gz"}
https://terrytao.wordpress.com/2015/11/23/a-cheap-version-of-halaszs-inequality/	A basic estimate in multiplicative number theory (particularly if one is using the Granville-Soundararajan “pretentious” approach to this subject) is the following inequality of Halasz (formulated here in a quantitative form introduced by Montgomery and Tenenbaum). Theorem 1 (Halasz inequality) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function bounded in magnitude by ${1}$, and suppose that ${x \geq 3}$, ${T \geq 1}$, and ${ M \geq 0}$ are such that $\displaystyle \sum_{p \leq x} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} \geq M \ \ \ \ \ (1)$ for all real numbers ${t}$ with ${\|t\| \leq T}$. Then $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{\sqrt{T}}.$ As a qualitative corollary, we conclude (by standard compactness arguments) that if $\displaystyle \sum_{p} \frac{1 - \hbox{Re}(f(p) p^{-it})}{p} = +\infty$ for all real ${t}$, then $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) = o(1) \ \ \ \ \ (2)$ as ${x \rightarrow \infty}$. In the more recent work of this paper of Granville and Soundararajan, the sharper bound $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M} + \frac{1}{T} + \frac{\log\log x}{\log x}$ is obtained (with a more precise description of the ${(1+M) e^{-M}}$ term). The usual proofs of Halasz’s theorem are somewhat lengthy (though there has been a recent simplification, in forthcoming work of Granville, Harper, and Soundarajan). Below the fold I would like to give a relatively short proof of the following “cheap” version of the inequality, which has slightly weaker quantitative bounds, but still suffices to give qualitative conclusions such as (2). Theorem 2 (Cheap Halasz inequality) Let ${f: {\bf N} \rightarrow {\bf C}}$ be a multiplicative function bounded in magnitude by ${1}$. Let ${T \geq 1}$ and ${M \geq 0}$, and suppose that ${x}$ is sufficiently large depending on ${T,M}$. If (1) holds for all ${\|t\| \leq T}$, then $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll (1+M) e^{-M/2} + \frac{1}{T}.$ The non-optimal exponent ${1/2}$ can probably be improved a bit by being more careful with the exponents, but I did not try to optimise it here. A similar bound appears in the first paper of Halasz on this topic. The idea of the argument is to split ${f}$ as a Dirichlet convolution ${f = f_1 * f_2 * f_3}$ where ${f_1,f_2,f_3}$ is the portion of ${f}$ coming from “small”, “medium”, and “large” primes respectively (with the dividing line between the three types of primes being given by various powers of ${x}$). Using a Perron-type formula, one can express this convolution in terms of the product of the Dirichlet series of ${f_1,f_2,f_3}$ respectively at various complex numbers ${1+it}$ with ${\|t\| \leq T}$. One can use ${L^2}$ based estimates to control the Dirichlet series of ${f_2,f_3}$, while using the hypothesis (1) one can get ${L^\infty}$ estimates on the Dirichlet series of ${f_1}$. (This is similar to the Fourier-analytic approach to ternary additive problems, such as Vinogradov’s theorem on representing large odd numbers as the sum of three primes.) This idea was inspired by a similar device used in the work of Granville, Harper, and Soundarajan. A variant of this argument also appears in unpublished work of Adam Harper. I thank Andrew Granville for helpful comments which led to significant simplifications of the argument. — 1. Basic estimates — We need the following basic tools from analytic number theory. We begin with a variant of the classical Perron formula. Proposition 3 (Perron type formula) Let ${f: {\bf N} \rightarrow {\bf C}}$ be an arithmetic function bounded in magnitude by ${1}$, and let ${x, T \geq 1}$. Assume that the Dirichlet series ${F(s) := \sum_{n=1}^\infty \frac{f(n)}{n^s}}$ is absolutely convergent for ${\hbox{Re}(s) \geq 1}$. Then $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll \int_{-T}^T \|F(1+it)\| \frac{dt}{1+\|t\|} + \frac{1}{T} + \frac{T}{x}.$ Proof: By telescoping series (and treating the contribution of ${n \ll T}$ trivially), it suffices to show that $\displaystyle \sum_{x/e \leq n \leq x} f(n) \ll x \int_{-T}^T \|F(1+it)\| \frac{dt}{1+\|t\|} + \frac{x}{T}$ whenever ${x \geq T}$. The left-hand side can be written as $\displaystyle x \sum_n \frac{f(n)}{n} g( \log x - \log n ) \ \ \ \ \ (3)$ where ${g(u) := e^{-u} 1_{[0,1]}(u)}$. We now introduce the mollified version $\displaystyle \tilde g(u) := \int_{\bf R} g(u + \frac{v}{T}) \Psi(v)\ dv$ of ${g}$, where $\displaystyle \Psi(v) := \frac{1}{2\pi} \int_{\bf R} \psi(t) e^{itv}\ dt$ and ${\psi: {\bf R} \rightarrow {\bf R}}$ is a fixed smooth function supported on ${[-1,1]}$ that equals ${1}$ at the origin. Basic Fourier analysis then tells us that ${\Psi}$ is a Schwartz function with total mass one. This gives the crude bound $\displaystyle \tilde g(u) \ll 1$ for any ${u}$. For ${u \leq -\frac{1}{T}}$ or ${u \geq 1 + \frac{1}{T}}$, we use the bound ${\Psi(v) \ll \frac{1}{(1+\|v\|)^{10}}}$ (say) to arrive at the bound $\displaystyle \tilde g(u) \ll (T \hbox{dist}(u, \{0,1\}))^{-9};$ for ${\frac{1}{T} \leq u \leq 1 - \frac{1}{T}}$ we again use ${\Psi(v) \ll \frac{1}{(1+\|v\|)^{10}}}$ and write $\displaystyle \tilde g(u) - g(u) = \int_{\bf R} (g(u + \frac{v}{T}) - g(u)) \Psi(v)\ dv$ and use the Lipschitz bound ${g(u+\frac{v}{T}) - g(u) \ll \frac{v}{T}}$ for ${u + \frac{v}{T} \in [0,1]}$ to obtain $\displaystyle \tilde g(u) - g(u) \ll (T \hbox{dist}(u, \{0,1\}))^{-9}$ for such ${u}$. Putting all these bounds together, we see that $\displaystyle \tilde g(u) = g(u) + O( \frac{1}{(1 + T \|u\|)^9} ) + O( \frac{1}{(1 + T \|u-1\|)^9} )$ for all ${u}$. In particular, we can write (3) as $\displaystyle x \sum_n \frac{f(n)}{n} \tilde g(\log x - \log n)$ $\displaystyle + O( \sum_n \frac{x}{n} \frac{1}{(1 + T \|\log x - \log n\|)^9} + \frac{1}{(1 + T \|\log x/e - \log n\|)^9} ).$ The expression ${\frac{x}{n} \frac{1}{(1 + T \|\log x - \log n\|)^9} + \frac{1}{(1 + T \|\log x/e - \log n\|)^9}}$ is bounded by ${O( \frac{x}{n} \frac{1}{T \log^9 x} )}$ when ${n \leq x/10}$, is bounded by ${O( \frac{x}{n} \frac{1}{T \log^9 n} )}$ when ${n \geq 10x}$, is bounded by ${O(1)}$ when ${n = (1+O(1/T)) x}$ or ${n = (1+O(1/T)) x/e}$, and is bounded by ${(\frac{T}{x} \|n-x\|)^{-9} + (\frac{T}{x} \|n-x/e\|)^{-9}}$ otherwise. From these bounds, a routine calculation (using the hypothesis ${x \geq T}$) shows that $\displaystyle \sum_n \frac{x}{n} \frac{1}{(1 + T \|\log x - \log n\|)^9} + \frac{1}{(1 + T \|\log x/e - \log n\|)^9} \ll \frac{x}{T}$ and so it remains to show that $\displaystyle \sum_n \frac{f(n)}{n} \tilde g(\log x - \log n) \ll \int_{-T}^T \|F(1+it)\| \frac{dt}{1+\|t\|}.$ Writing $\displaystyle \tilde g(u) = T \int_{\bf R} g(v) \Psi( T(v-u) )\ dv$ $\displaystyle = \frac{T}{2\pi} \int_{\bf R} \psi(t) G(Tt) e^{-iTtu} dt$ $\displaystyle = \frac{1}{2\pi} \int_{\bf R} \psi(t/T) G(t) e^{-itu}\ dt$ where $\displaystyle G(t) := \int_{\bf R} g(v) e^{itv}\ dv$ we see from the triangle inequality and the support of ${\psi(t/T)}$ that $\displaystyle \sum_n \frac{f(n)}{n} \tilde g(\log x - \log n) \ll \int_{\|t\| \leq T} \|G(t)\| \|\sum_n \frac{f(n)}{n} e^{-it(\log x- \log n)}\|\ dt$ $\displaystyle \ll \int_{\|t\| \leq T} \|G(t)\| \|F(1+it)\|\ dt.$ But from integration by parts we see that ${G(t) \ll \frac{1}{1+\|t\|}}$, and the claim follows. $\Box$ Next, we recall a standard ${L^2}$ mean value estimate for Dirichlet series: Proposition 4 (${L^2}$ mean value estimate) Let ${f: {\bf N} \rightarrow {\bf C}}$ be an arithmetic function, and let ${T \geq 1}$. Assume that the Dirichlet series ${F(s) := \sum_{n=1}^\infty \frac{f(n)}{n^s}}$ is absolutely convergent for ${\hbox{Re}(s) \geq 1}$. Then $\displaystyle \int_{\|t\| \leq T} \|F(1+it)\|^2\ dt \ll T \sum_{n_1,n_2: \log n_2 = \log n_1 + O(1/T)} \frac{\|f(n_1)\| \|f(n_2)\|}{n_1 n_2}.$ Proof: This follows from Lemma 7.1 of Iwaniec-Kowalski; for the convenience of the reader we reproduce the short proof here. Introducing the normalised sinc function ${\hbox{sinc}(x) := \frac{\sin(\pi x)}{\pi x}}$, we have $\displaystyle \int_{\|t\| \leq T} \|F(1+it)\|^2\ dt \ll \int_{\bf R} \|F(1+it)\|^2 \hbox{sinc}^2( t / 2T )\ dt$ $\displaystyle = \sum_{n_1,n_2} f(n_1) \overline{f(n_2)} \int_{\bf R} \frac{1}{n_1^{1+it} n_2^{1-it}} \hbox{sinc}^2(t/2T)\ dt$ $\displaystyle \ll \sum_{n_1,n_2} \frac{\|f(n_1)\| \|f(n_2)\|}{n_1 n_2} \|\int_{\bf R} \hbox{sinc}^2(t/2T) e^{it (\log n_1 - \log n_2)}\ dt\|.$ But a standard Fourier-analytic computation shows that ${\int_{\bf R} \hbox{sinc}^2(t/2T) e^{it (\log n_1 - \log n_2)}\ dt}$ vanishes unless ${\log n_2 = \log n_1 + O(1/T)}$, in which case the integral is ${O(T)}$, and the claim follows. $\Box$ Now we recall a basic sieve estimate: Proposition 5 (Sieve bound) Let ${x \geq 1}$, let ${I}$ be an interval of length ${x}$, and let ${{\mathcal P}}$ be a set of primes up to ${x}$. If we remove one residue class mod ${p}$ from ${I}$ for every ${p \in {\mathcal P}}$, the number of remaining natural numbers in ${I}$ is at most ${\ll \|I\| \prod_{p \in {\mathcal P}} (1 - \frac{1}{p})}$. Proof: This follows for instance from the fundamental lemma of sieve theory (see e.g. Corollary 19 of this blog post). (One can also use the Selberg sieve or the large sieve.) $\Box$ Finally, we record a standard estimate on the number of smooth numbers: Proposition 6 Let ${u \geq 1}$ and ${\varepsilon>0}$, and suppose that ${x}$ is sufficiently large depending on ${u,\varepsilon}$. Then the number of natural numbers in ${[1,x]}$ which have no prime factor larger than ${x^{1/u}}$ is at most ${O( u^{-(1-\varepsilon)u} x )}$. Proof: See Corollary 1.3 of this paper of Hildebrand and Tenenbaum. (The result also follows from the more classical work of Dickman.) We sketch a short proof here due to Kevin Ford. Let ${{\mathcal S}}$ denote the set of numbers that are “smooth” in the sense that they have no prime factor larger than ${x^{1/u}}$. It then suffices to prove the bound $\displaystyle \sum_{n \in {\mathcal S}: n \leq x} \log n \ll \exp( - u \log u + O(u) ) x \log x, \ \ \ \ \ (4)$ since the contribution of those ${n}$ less than (say) ${\sqrt{x}}$ is negligible, and for the other values of ${n}$, ${\log n}$ is comparable to ${\log x}$. Writing ${\log n = \sum_{dm=n} \Lambda(d)}$, we can rearrange the left-hand side as $\displaystyle \sum_{m \in {\mathcal S}: m \leq x} \sum_{d \in {\mathcal S}: d \leq x/m} \Lambda(d).$ By the prime number theorem, the contribution to ${\sum_{d \in {\mathcal S}: d \leq x/m} \Lambda(d)}$ of those ${d \leq x^{1/u}}$ is ${O( \min( x^{1/u}, x/m )}$, and the contribution of those ${d}$ with ${d > x^{1/u}}$ consists only of prime powers, which contribute ${O( (x/m)^{1/2} )}$. Combining these estimates, we can get a bound of the form $\displaystyle \sum_{d \in {\mathcal S}: d \leq x/m} \Lambda(d) \ll (x/m)^{1-\rho} (x^{1/u})^\rho$ where ${0 < \rho < 1/2}$ is a quantity to be chosen later. Thus we can bound the left-hand side of (4) by $\displaystyle O( x^{\rho/u} x^{1-\rho} \sum_{m \in {\mathcal S}} \frac{1}{m^{1-\rho}} )$ which by Euler products can be bounded by $\displaystyle O( x^{\rho/u} x^{1-\rho} \exp( \sum_{p \leq x^{1/u}} \frac{1}{p^{1-\rho}} ) ).$ By the mean value theorem applied to the function ${t \mapsto \exp( \rho t \log x^{1/u})}$, we can bound ${p^\rho = \exp( \rho \log p )}$ by ${1 + \frac{\log p}{\log x^{1/u}} \exp( \rho \log x^{1/u} )}$ for ${p \leq x^{1/u}}$. By Mertens’ theorem, we thus get a bound of $\displaystyle O( x^{\rho/u} x^{1-\rho} \exp( \log\log x^{1/u} + O( \exp( \rho \log x^{1/u} ) ) ).$ If we make the choice ${\rho := \frac{\log u}{\log x^{1/u}}}$, we obtain the required bound (4). $\Box$ — 2. Proof of theorem — By increasing ${M}$ as necessary we may assume that ${M \geq 10}$ (say). Let ${0 < \varepsilon_2 < \varepsilon_1 \leq 1/2}$ be small parameters (depending on ${M}$) to be optimised later; we assume ${x}$ to be sufficiently large depending on ${\varepsilon_1,\varepsilon_2,T}$. Call a prime ${p}$ small if ${p \leq x^{\varepsilon_2}}$, medium if ${x^{\varepsilon_2} < p \leq x^{\varepsilon_1}}$, and large if ${x^{\varepsilon_1} < p \leq x}$. Observe that for any ${n \leq x}$ we can factorise ${f}$ as a Dirichlet convolution $\displaystyle f(n) = f_1 * f_2 * f_3(n)$ where • ${f_1}$ is the restriction of ${f}$ to those natural numbers ${n}$ whose prime factors are all small; • ${f_2}$ is the restriction of ${f}$ to those natural numbers ${n}$ whose prime factors are all medium; • ${f_3}$ is the restriction of ${f}$ to those natural numbers ${n}$ whose prime factors are all large. We thus have $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) = \frac{1}{x} \sum_{n \leq x} f_1f_2f_3(n). \ \ \ \ \ (5)$ It is convenient to remove the Dirac function ${\delta(n) := 1_{n=1}}$ from ${f_2,f_3}$, so we write $\displaystyle f_2 = \delta + f'_2; \quad f_3 = \delta + f'_3$ and split $\displaystyle f_1f_2f_3 = f_1f_2 + f_1f'_3 + f_1f'_2f'_3.$ Note that ${f_1f_2}$ is the restriction of ${f}$ to those numbers ${n \leq x}$ whose prime factors are all small or medium. By Proposition 6, the number of such ${n}$ can certainly be bounded by ${O(e^{-1/\varepsilon_1} x)}$ if ${x}$ is sufficienty large. Thus the contribution of this term to (5) is ${O( e^{-1/\varepsilon_1} )}$. Similarly, ${f_1 f'_3}$ is the restriction of ${f}$ to those numbers ${n \leq x}$ which contain at least one large prime factor, but no medium prime factors. By Proposition 5 the number of such ${n}$ is bounded by ${O( \frac{\varepsilon_2}{\varepsilon_1} x )}$ if ${x}$ is sufficiently large. Thus the contribution of this term to (5) is ${O( \frac{\varepsilon_2}{\varepsilon_1} )}$, and hence $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll \|\frac{1}{x} \sum_{n \leq x} f_1f'_2f'_3(n)\| + e^{-1/\varepsilon_1} + \frac{\varepsilon_2}{\varepsilon_1}.$ Note that ${f_1f'_2f'_3}$ is only supported on numbers whose prime factors do not exceed ${x}$, so the Dirichlet series of ${f_1f'_2f'_3}$ is absolutely convergent for ${\hbox{Re}(s) \geq 1}$ and is equal to ${F_1(s) F'_2(s) F'_3(s)}$, where ${F_1,F'_2,F'_3}$ are the Dirichlet series of ${f_1,f'_2,f'_3}$ respectively. Since ${f_1f'_2f'_3}$ is bounded in magnitude by ${1}$ (being a restriction of ${f}$), we may apply Proposition 3 and conclude (for ${x}$ large enough, and discarding the ${\frac{1}{1+\|t\|}}$ denominator) that $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll \int_{\|t\| \leq T} \|F_1(1+it)\| \|F'_2(1+it)\| \|F'_3(1+it)\|\ dt$ $\displaystyle + \frac{1}{T} + e^{-1/\varepsilon_1} + \frac{\varepsilon_2}{\varepsilon_1}.$ We now record some ${L^2}$ estimates: Lemma 7 For sufficiently large ${x}$, we have $\displaystyle \int_{\|t\| \leq T} \|F'_2(1+it)\|^2\ dt \ll \frac{\varepsilon_1}{\varepsilon_2^2 \log x}$ and $\displaystyle \int_{\|t\| \leq T} \|F'_3(1+it)\|^2\ dt \ll \frac{1}{\varepsilon_1^2 \log x}.$ Proof: We just prove the former inequality, as the latter is similar. By Proposition 4, we have $\displaystyle \int_{\|t\| \leq T} \|F'_2(1+it)\|^2\ dt \ll T \sum_{n_1,n_2: \log n_1 = \log n_2 + O(1/T)} \frac{\|f'_2(n_1)\| \|f'_2(n_2)\|}{n_1 n_2}.$ The term ${f'_2(n_1)}$ vanishes unless ${n_1 \geq x^{\varepsilon_2}}$, and we have ${n_2 = (1+O(1/T)) n_1}$, so we can bound the right-hand side by $\displaystyle \ll T \sum_{n_1 \geq x^{\varepsilon_2}} \frac{\|f'_2(n_1)\|}{n_1}\sum_{n_2 = (1+O(1/T)) n_1} \|f'_2(n_2)\|.$ The inner summand is bounded by ${1}$ and supported on those ${n_2}$ that are not divisible by any small primes. From Proposition 5 and Mertens’ theorem we conclude that $\displaystyle \sum_{n_2 = (1+O(1/T)) n_1} \|f'_2(n_2)\| \ll \frac{1}{T \varepsilon_2 \log x} n_1$ and thus $\displaystyle \int_{\|t\| \leq T} \|F'_2(1+it)\|^2\ dt \ll \frac{1}{\varepsilon_2 \log x} \sum_{n_1} \frac{\|f'_2(n_1)\|}{n_1}$ $\displaystyle \ll \frac{1}{\varepsilon_2 \log x} \prod_{x^{\varepsilon_2} < p \leq x^{\varepsilon_1}} (1-\frac{1}{p})^{-1}$ $\displaystyle \ll \frac{\varepsilon_1}{\varepsilon^2_2 \log x}$ as desired. $\Box$ We also have an ${L^\infty}$ estimate: Lemma 8 For sufficiently large ${x}$, we have $\displaystyle \|F_1(1+it)\| \ll e^{-M} \log x$ for all ${\|t\| \leq T}$. Proof: From Euler products, Mertens’ theorem, and (1) we have $\displaystyle F_1(1+it) = \prod_{p \leq x^{\varepsilon_2}} \sum_{j=0}^\infty \frac{f(p^j)}{p^{j(1+it)}}$ $\displaystyle \ll \prod_{p \leq x^{\varepsilon_2}} \|1 + \frac{f(p) p^{-it}}{p}\|$ $\displaystyle \ll \exp( \prod_{p \leq x^{\varepsilon_2}} \hbox{Re} \frac{f(p) p^{-it}}{p} )$ $\displaystyle \ll \varepsilon_2 \log x \exp( - \prod_{p \leq x^{\varepsilon_2}} \frac{1 - \hbox{Re} f(p) p^{-it}}{p} )$ $\displaystyle \ll \varepsilon_2 \log x \exp( -M + \log \frac{1}{\varepsilon_2} )$ $\displaystyle \ll e^{-M} \log x$ as desired. $\Box$ Applying Hölder’s inequality, we conclude that $\displaystyle \frac{1}{x} \sum_{n \leq x} f(n) \ll \frac{1}{\varepsilon_1^{1/2} \varepsilon_2} e^{-M} + e^{-1/\varepsilon_1} + \frac{\varepsilon_2}{\varepsilon_1}.$ Setting ${\varepsilon_1 := 1/M}$ and ${\varepsilon_2 := e^{-M/2}}$ we obtain the claim.	{"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": 274, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9953801035881042, "perplexity": 121.18079140723354}, "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-2017-43/segments/1508187824293.62/warc/CC-MAIN-20171020173404-20171020193404-00675.warc.gz"}
https://www.physicsforums.com/threads/current-in-a-circuit-with-multiple-voltage-sources.740864/	# Current in a Circuit with Multiple Voltage Sources 1. Feb 28, 2014 ### LikesIntuition What exactly are the rules for conserving the currents in a circuit? If we have multiple emfs in parallel with each other, how can we figure out the direction and magnitude of currents in each segment of a circuit? It makes perfect sense to me for one emf, and for the most part with two I can usually figure it out as well. So could we consider three emfs in parallel, and 3 resistors parallel to each other and each in series with the 3 emfs? In a set-up like this, how do we add up the currents? Do we do so arbitrarily? I tried attaching an image, but am not sure if it worked. Thanks for any help! #### Attached Files: • ###### 20_82alt.gif File size: 2.9 KB Views: 277 2. Feb 28, 2014 ### UltrafastPED There are several techniques which can be used to solve this type of problem: 1. Nodal analysis 2. Mesh current analysis ... etc. This lesson in DC circuit analysis covers the basics; Kirchoff's rules still apply, but you need a systematic approach in order to carry out the analysis correctly. These differing techniques work due to the linear nature of "ordinary" circuits which are based on passive components: resistors, capacitors, and inductors. They also work for AC circuits, but then you use impedance. 3. Feb 28, 2014 ### Staff: Mentor Multiple voltages sources cannot be paralleled unless they have the same voltage... 4. Mar 1, 2014 ### CWatters Regarding the direction... Unless it's obvious the normal approach is to mark the diagram with arbitrary arrows which represent the direction you assume current will flow. Then solve the circuit equations to calculate the actual current. If the answer is negative that means current is flowing the other way. For example see this circuit... Lets say I pretend I don't know which way the current will flow. To solve it I have arbitrarily chosen to define "Positive I" as flowing anticlockwise. Applying KVL clockwise gives... +10 + IR = 0 Note that it's "+IR" not "-IR" because if the current is flowing anticlockwise the bottom end of the resistor would be positive with respect to the top. Rearranging this gives I = -10/R Oh look "I" has turned out to be negative. That means my assumption that the current was flowing anticlockwise was incorrect. In fact that's obvious in this example. File size: 6.7 KB Views: 132 5. Mar 1, 2014 ### dauto Just use the Kirchhoff rules 6. Mar 1, 2014 ### sophiecentaur HaHa That's easy for you to say but for anything other than a simple circuit, the process can get very lumpy, very quickly. But Kirchoff is the basis for the more sophisticated tools and it's well worth while using the two basic Kirchoff Laws for a simple circuit - to prove to yourself that they work. Then reach for the easier tools. 7. Mar 1, 2014 ### dauto Yes, but one hast to learn to walk before attempting to run The OP's example is quite simple which makes me think they might not be aware of the rules. 8. Mar 1, 2014 ### LikesIntuition Thanks, that makes sense! 9. Mar 1, 2014 ### LikesIntuition You are correct. I'm pretty new to this stuff... 10. Mar 1, 2014 ### sophiecentaur I made that remark because I was thinking that the Kirchoff rules could be very scary, if the OP thought they were the only way to tackle all problems. I can't remember when I, personally have used K1 and K2 for any purpose other than to show they actually work. lol Similar Discussions: Current in a Circuit with Multiple Voltage Sources	{"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": 0.8659908175468445, "perplexity": 1112.8197162522258}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889917.49/warc/CC-MAIN-20180121021136-20180121041136-00669.warc.gz"}
http://mathhelpforum.com/trigonometry/101160-trigonometry-vectors.html	1. ## Trigonometry with vectors Jack walks x km on a bearing of 100 degrees from O, then y km on a bearing of 200 degrees to H. Calculate the magnitude and direction of the displacement vector OH when: a) x=6, y=7 b) x=6, y=10 2. Originally Posted by tory88 Jack walks x km on a bearing of 100 degrees from O, then y km on a bearing of 200 degrees to H. Calculate the magnitude and direction of the displacement vector OH when: a) x=6, y=7 b) x=6, y=10 This formula: $H_x = x \cos(100) + y \cos(200)$ $H_y = x \sin(100) + y \sin(200)$ with the data given for a: $H_x = 6 \cos(100) + 7 \cos(200)$ $H_y = 6 \sin(100) + 7 \sin(200)$ magnitude = $\sqrt{ H_x^2 + H_y^2}$ direction = ArcTangent $\left(\dfrac{H_y}{H_x} \right)$ When computing the arctan you need to know the quadrant:	{"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": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9837875366210938, "perplexity": 1113.23236407331}, "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/1495463608416.96/warc/CC-MAIN-20170525195400-20170525215400-00226.warc.gz"}
http://math.stackexchange.com/questions/134426/open-ball-is-an-open-set	# Open ball is an open set. Could someone please show that an open ball is open where the definition of "open" is: A set is open if for each $x$ in $U$ there is an open rectangle $A$ such that $x$ in $A$ is contained in $U$. Where an open rectangle is $(a_1,b_1)\times\ldots\times(a_n,b_n)$. I also realize that one can use rectangles or balls, but I would like to see the proof using rectangles, as this is the definition used in Spivak's calculus on manifolds. So for example, the solution located at An open ball is an open set is unacceptable. - you do exactly that proof, and then, put a cube inside the ball.... – N. S. Apr 20 '12 at 15:26 that would only be a(n) (infinite) regression. If all I have to do is put a cube (rectangle) inside that ball, then I would have the same problem: How to put the cube inside the ball that's inside the large ball. I want a rigorous proof that it can be done. – Squirtle Apr 20 '12 at 16:40 @user29507: the farthest point of a cube of side $\delta$ from the center is $\sqrt n \frac \delta{2^n}$ So if you can put an $\epsilon$ ball in, you can put a $\delta$ cube in for small enough $\delta$. – Ross Millikan Apr 20 '12 at 16:48 .... right, that seems very clever, but unfortunately I don't "see" \sqrt n \frac \delta{2^n} very well. Its probably just some basic n-dim geometry, but as I'm just beginning at this, I don't understand. – Squirtle Apr 20 '12 at 22:09 actually the value that I compute in R^2 and R^3 is sqrt(n)*delta/2. Not divided by 2^n. I don't know how to do the computation in R^4 because I can't visualize that space. Any other ideas would be much appreciated. – Squirtle Apr 21 '12 at 0:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9350740313529968, "perplexity": 245.3645786028819}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011062835/warc/CC-MAIN-20140305091742-00096-ip-10-183-142-35.ec2.internal.warc.gz"}
https://courses.lumenlearning.com/boundless-physics/chapter/center-of-mass/	## Locating the Center of Mass The position of COM is mass weighted average of the positions of particles. Mathematically, it is given as $\bf{\text{r}}_{\text{COM}} = \frac{\sum_\text{i} \text{m}_\text{i} \bf{\text{r}}_\text{i}}{\text{M}}$. ### Learning Objectives Identify the center of mass for an object with continuous mass distribution ### Key Takeaways #### Key Points • The center of mass (COM) is a statement of spatial arrangement of mass (i.e. distribution of mass within the system). • The experimental determination of the center of mass of a body uses gravity forces on the body and relies on the fact that in the parallel gravity field near the surface of the earth the center of mass is the same as the center of gravity. • For a 2D object, an experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass. #### Key Terms • plumb line: A cord with a weight attached, used to produce a vertical line. In the previous atom on “Center of Mass and Translational Motion,” we learned why the concept of center of mass (COM) helps solving mechanics problems involving a rigid body. Here, we will study the rigorous definition of COM and how to determine the location of it. ### Definition The center of mass is a statement of spatial arrangement of mass (i.e. distribution of mass within the system). The position of COM is given a mathematical formulation which involves distribution of mass in space: $\textbf{\text{r}}_{\text{COM}} = \frac{\sum_\text{i} \text{m}_\text{i} \textbf{\text{r}}_\text{i}}{\text{M}}$, where rCOM and ri are vectors representing the position of COM and i-th particle respectively, and M and mi are the total mass and mass of the i-th particle, respectively. This mean means that position of COM is mass weighted average of the positions of particles. ### Object with Continuous Mass Distribution If the mass distribution is continuous with the density ρ(r) within a volume V, the position of COM is given as $\textbf{r}_{\text{COM}} = \frac {1}{\text{M}} \int_\text{V}\rho(\mathbf{\text{r}}) \mathbf{\text{r}} \text{dV}$, where M is the total mass in the volume. If a continuous mass distribution has uniform density, which means ρ is constant, then the center of mass is the same as the center of the volume. ### Locating the Center of Mass The experimental determination of the center of mass of a body uses gravity forces on the body and relies on the fact that in the parallel gravity field near the surface of Earth the center of mass is the same as the center of gravity. The center of mass of a body with an axis of symmetry and constant density must lie on this axis. Thus, the center of mass of a circular cylinder of constant density has its center of mass on the axis of the cylinder. In the same way, the center of mass of a spherically symmetric body of constant density is at the center of the sphere. In general, for any symmetry of a body, its center of mass will be a fixed point of that symmetry. In two dimensions: An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass. Plumb Line Method for Center of Mass: Suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass. In three dimensions: By supporting an object at three points and measuring the forces that resist the weight of the object, COM of the three-dimensional coordinates of the center of mass can be determined. ## Motion of the Center of Mass We can describe the translational motion of a rigid body as if it is a point particle with the total mass located at the COM—center of mass. ### Learning Objectives Derive the center of mass for the translational motion of a rigid body ### Key Takeaways #### Key Points • The total mass times the acceleration of the center of mass equals the sum of external forces. • For the translational motion of a rigid body with mass M, Newton’s 2nd law applies as if we are describing the motion of a point particle (with mass M) under the influence of the external force. • When there is no external force, the center of mass momentum is conserved. #### Key Terms • rigid body: An idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects. • center of mass: The center of mass (COM) is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. We can describe the translational motion of a rigid body as if it is a point particle with the total mass located at the center of mass (COM). In this Atom. we will prove that the total mass (M) times the acceleration of the COM (aCOM), indeed, equals the sum of external forces. That is, $\text{M} \cdot \textbf{a}_{\text{COM}} = \sum \textbf{F}_{\text{ext}}$. You can see that the Newton’s 2nd law applies as if we are describing the motion of a point particle (with mass M) under the influence of the external force. ### Derivation From the definition of the center of mass, $\textbf{r}_{\text{COM}} = \frac{\sum_\text{i} \text{m}_\text{i} \textbf{r}_\text{i}}{\text{M}}$, we get $\text{M} \cdot \textbf{a}_{\text{COM}} = \sum \text{m}_\text{i} \textbf{a}_\text{i}$ by taking time derivative twice on each side. Note that $\sum \text{m}_\text{i} \textbf{a}_\text{i} = \sum \textbf{F}_\text{i}$. In a system of particles, each particle may feel both external and internal forces. Here, external forces are forces from external sources, while internal forces are forces between particles in the system. Since the sum of all internal forces will be 0 due to the Newton’s 3rd law, $\sum \textbf{F}_\text{i} = \sum \textbf{F}_{\text{i}, \text{ext}}$. Therefore, we get $\text{M} \cdot \textbf{a}_{\text{COM}} = \sum \textbf{F}_{\text{ext}}$. For example, when we confine our system to the Earth and the Moon, the gravitational force due to the Sun would be external, while the gravitational force on the Earth due to the Moon (and vice versa) would be internal. Since the gravitational forces between the Earth and the Moon are equal in magnitude and opposite in direction, they will cancel out each other in the sum (see ). COM of the Earth and Moon: Earth and Moon orbiting a COM inside the Earth. The red cross represents the COM of the two-body system. The COM will orbit around the Sun as if it is a point particle. ### Corollary When there is no external force, the COM momentum is conserved. Proof: Since there is no external force, $\text{M} \cdot \bf{\text{a}}_{\text{COM}} = 0$. Therefore, $\text{M} \cdot \bf{\text{v}}_{\text{COM}} = \text{constant}$. ## Center of Mass of the Human Body The center of mass (COM) is an important physical concept—it is the point about which objects rotate. ### Learning Objectives Estimate the COM of a given object ### Key Takeaways #### Key Points • Although a human body has complicated features, the location of the center of mass (COM) could be a good indicator of the body proportions. • We can measure the location of COM with two scales and a wooden beam. The linear and rotational equations of motion gives us the location. • The center of mass of the human body depends on the gender and the position of the limbs. In a standing posture, it is typically about 10 cm lower than the navel, near the top of the hip bones. #### Key Terms • center of mass: The center of mass (COM) is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. • torque: A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb) The center of mass (COM) is an important physical concept. It is the point on an object at which the weighted relative position of the distributed mass sums to zero—the point about which objects rotate. Human proportions have been important in art, measurement, and medicine (a well known drawing of the human body is seen in ). Although the human body has complicated features, the location of the center of mass (COM) could be a good indicator of body proportions. The center of mass of the human body depends on the gender and the position of the limbs. In a standing posture, it is typically about 10 cm lower than the navel, near the top of the hip bones. In this Atom, we will learn how to measure the COM of a human body. Leonardo da Vinci’s “The Vitruvian Man”: Vitruvian Man: A drawing created by Leonardo da Vinci. The drawing is based on the correlations of ideal human proportions with geometry described[4] by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. First, let’s take two scales and a wooden beam (H meter long), long enough to contain the entire body of the subject. Put the scales H meters apart, and place the beam across the scales, as illustrated in. Now, let the subject lie on the beam. Make sure that his/her heels are aligned with one end of the beam. Measure the readings (F1, F2) on the scale. The COM of a Human Body: This figure demonstrates measuring the COM of a human body. The system (person+beam) has three external forces: gravity on the subject (FCM), and normal forces from the scales F1 and F2. The equation of motion for force (F=ma) will give us: $\text{F}_1 + \text{F}_2 = \text{Mg}$, where M is mass of the subject. (We assume that the wooden beam has no mass. ) This equation doesn’t provide all the information to locate the COM. However, the equation of motion for torque $( \tau = \text{I}\alpha)$ helps. Since the net torque of the system is zero (hence no rotational acceleration), $\text{h} \text{F}_2 - (\text{H}-\text{h}) \text{F}_1 = 0$. (h: COM height) The COM is chosen as the origin for the torque. Therefore, gravity contributes nothing as a torque. Solving for h and using the equation of motion for force, we get $\text{h} = \frac{\text{HF}_1}{\text{Mg}}$. ## Center of Mass and Translational Motion The COM (center of mass) of a system of particles is a geometric point that assumes all the mass and external force(s) during motion. ### Learning Objectives Support the presence of COM in three dimensional bodies in motion ### Key Takeaways #### Key Points • In a motion of a rigid body, different parts of the body have different motions.This means that these bodies may not behave like a point particle. • There is a characteristic geometric point of the three dimensional body in motion. This point behaves as a particle, and is known as center of mass, abbreviated COM. COM appears to carry the whole mass of the body. All external forces appear to apply at COM. • To describe the motion of a rigid body (with possibly a complicated geometry), we separate the translational part of the motion from the rotational part. #### Key Terms • rigid body: An idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects. • point particle: An idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension, meaning that geometrically the particle is equivalent to a point. ### Introduction: COM, Linear Momentum, and Collisions Our study of motion has been limited up to this point. We have referred to particle, object and body in the same way. We considered that actual three dimensional rigid bodies move such that all constituent particles had the same motion (i.e., same trajectory, velocity and acceleration). By doing this, we have essentially considered a rigid body as a point particle. ### Center of Mass (COM) An actual body, however, can move differently than this simplified paradigm. Consider a ball rolling down an incline plane or a stick thrown into air. Different parts of a body have different motions. While translating in the air, the stick rotates about a moving axis, as shown in. This means that such bodies may not behave like a point particle, as earlier suggested. Forces on the COM: Left: The force appears to operate on the COM is “mgsinθ. Right: The force appears to operate on the COM is “mg”. Describing motions of parts or particles that have different motions would be quite complicated to do in an integrated manner. However, such three dimensional bodies in motion have one surprising, simplifying characteristic—a geometric point that behaves like a particle. This point is known as center of mass, abbreviated COM (the mathematical definition of COM will be introduced in the next Atom on “Locating the Center of Mass”). It has the following two characterizing aspects: • The center of mass appears to carry the whole mass of the body. • At the center of mass, all external forces appear to apply. Significantly, the center of a ball (the COM of a rolling ball) follows a straight linear path; whereas the COM of a stick follows a parabolic path (as shown in the figure above). Secondly, the forces appear to operate on the COMs in two cases (“mgsinθ and “mg”) as if they were indeed particle-like objects. This concept of COM, therefore, eliminate the complexities otherwise present in attempting to describe motions of rigid bodies. ### Describing Motion in a Rigid Body We can describe general motion of an object (with mass m) as follows: • We describe the translational motion of a rigid body as if it is a point particle with mass m located at COM. • Rotation of the particle, with respect to the COM, is described independently. We “separate” the translational part of the motion from the rotational part. By introducing the concept of COM, the translational motion becomes that of a point particle with mass m. This simplifies significantly the mathematical complexity of the problem.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8744876980781555, "perplexity": 386.3526081779927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323584554.98/warc/CC-MAIN-20211016074500-20211016104500-00310.warc.gz"}
https://www.sanfoundry.com/mathematics-question-papers-class-12/	# Mathematics Questions and Answers – Product of Two Vectors-2 « » This set of Mathematics Question Papers for Class 12 focuses on “Product of Two Vectors-2”. 1. Evaluate the product $$(2\vec{a}+5\vec{b}).(4\vec{a}-5\vec{b})$$. a) $$\|\vec{a}\|^2+2\vec{a}.\vec{b}-15\|\vec{b}\|^2$$ b) $$8\|\vec{a}\|^2+2\vec{a}.\vec{b}-15\|\vec{b}\|^2$$ c) $$8\|\vec{a}\|^2-4\vec{a}.\vec{b}-15\|\vec{b}\|^2$$ d) $$\|\vec{a}\|^2+\vec{a}.\vec{b}-5\|\vec{b}\|^2$$ Explanation: To evaluate: $$(2\vec{a}+5\vec{b}).(4\vec{a}-5\vec{b})$$ =$$2\vec{a}.4\vec{a}-2\vec{a}.5\vec{b}+3\vec{b}.4\vec{a}-3\vec{b}.5\vec{b}$$ =$$8\|\vec{a}\|^2+2\vec{a}.\vec{b}-15\|\vec{b}\|^2$$ 2. Find the magnitude of $$\vec{a}$$ and $$\vec{b}$$ which are having the same magnitude and such that the angle between them is 60° and their scalar product is $$\frac{1}{4}$$. a) $$\|\vec{a}\|=\|\vec{b}\|=\frac{1}{2√2}$$ b) $$\|\vec{a}\|=\|\vec{b}\|=\frac{1}{√2}$$ c) $$\|\vec{a}\|=\|\vec{b}\|=\frac{1}{2√3}$$ d) $$\|\vec{a}\|=\|\vec{b}\|=\frac{2}{√3}$$ Explanation: Given that: a) $$\|\vec{a}\|=\|\vec{b}\|$$ b) θ=60° c) $$\vec{a}.\vec{b}=\frac{1}{4}$$ ∴$$\|\vec{a}\|\|\vec{b}\| cos⁡θ=\frac{1}{4}$$ =$$\|\vec{a}\|^2 cos⁡60°=\frac{1}{4}$$ ⇒$$\|\vec{a}\|^2=\frac{1}{4}.\frac{1}{2}$$ ∴$$\|\vec{a}\|=\|\vec{b}\|=\frac{1}{2√2}$$. 3. If $$\vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j}$$ are such that $$\vec{a}+μ\vec{b}$$ is perpendicular to $$\vec{c}$$, then the value of μ. a) $$\frac{7}{2}$$ b) –$$\frac{7}{2}$$ c) –$$\frac{3}{2}$$ d) $$\frac{7}{9}$$ Explanation: Given that: $$\vec{a}=\hat{i}-\hat{j}+3\hat{k}, \,\vec{b}=5\hat{i}-2\hat{j}+\hat{k} \,and \,\vec{c}=\hat{i}-\hat{j}$$ Also given, $$\vec{a}+μ\vec{b}$$ is perpendicular to $$\vec{c}$$ Therefore, $$(\vec{a}+μ\vec{b}).\vec{c}=0$$ i.e. $$(\hat{i}-\hat{j}+3\hat{k}+μ(5\hat{i}-2\hat{j}+\hat{k})).(\hat{i}-\hat{j})$$=0 $$((1+5μ) \,\hat{i}-(1+2μ) \,\hat{j}+(μ+3) \,\hat{k}).(\hat{i}-\hat{j})$$=0 1+5μ+1+2μ=0 μ=-$$\frac{7}{2}$$. Sanfoundry Certification Contest of the Month is Live. 100+ Subjects. Participate Now! 4. Find the angle between $$\vec{a} \,and \,\vec{b}$$ if $$\|\vec{a}\|=2,\|\vec{b}\|=\frac{1}{2√3}$$ and $$\vec{a}×\vec{b}=\frac{1}{2}$$. a) $$\frac{2π}{3}$$ b) $$\frac{4π}{5}$$ c) $$\frac{π}{3}$$ d) $$\frac{π}{2}$$ Explanation: Given that, $$\|\vec{a}\|=2, \,\|\vec{b}\|=\frac{1}{2√3}$$ and $$\vec{a}×\vec{b}=\frac{1}{2}$$ We know that, $$\vec{a}×\vec{b}=\vec{a}.\vec{b} \,sin⁡θ$$ ∴ $$sin⁡θ=\frac{(\vec{a}×\vec{b})}{\|\vec{a}\|.\|\vec{b}\|}$$ sin⁡θ=$$\frac{\frac{1}{2}}{2×\frac{1}{2√3}}=\frac{\sqrt{3}}{2}$$ θ=$$sin^{-1}⁡\frac{\sqrt{3}}{2}=\frac{π}{3}$$ 5. Find the vector product of the vectors $$\vec{a}=2\hat{i}+4\hat{j}$$ and $$\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$$. a) $$\hat{i}-19\hat{j}-4\hat{k}$$ b) $$3\hat{i}+19\hat{j}-14\hat{k}$$ c) $$3\hat{i}-19\hat{j}-14\hat{k}$$ d) $$3\hat{i}+5\hat{j}+4\hat{k}$$ Explanation: Given that, $$\vec{a}=2\hat{i}+4\hat{j}$$ and $$\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$$ Calculating the vector product, we get $$\vec{a}×\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&4&-5\\3&-1&2\end{vmatrix}$$ =$$\hat{i}(8-5)-\hat{j}(4-(-15))+\hat{k}(-2-12)$$ =$$3\hat{i}-19\hat{j}-14\hat{k}$$ 6. If $$\vec{a} \,and \,\vec{b}$$ are two non-zero vectors then $$(\vec{a}-\vec{b})×(\vec{a}+\vec{b})$$=_________ a) $$2(\vec{a}×\vec{b})$$ b) $$(\vec{a}×\vec{b})$$ c) –$$4(\vec{a}×\vec{b})$$ d) $$3(\vec{a}×\vec{b})$$ Explanation: Consider $$(\vec{a}-\vec{b})×(\vec{a}+\vec{b})$$ =$$(\vec{a}-\vec{b})×\vec{a}+(\vec{a}+\vec{b})×\vec{b}$$ =$$\vec{a}×\vec{a}-\vec{b}×\vec{a}+\vec{a}×\vec{b}-\vec{b}×\vec{b}$$ We know that, $$\vec{a}×\vec{a}=0,\vec{b}×\vec{b}=0 \,and \,\vec{a}×\vec{b}=-\vec{b}×\vec{a}$$ ∴ $$\vec{a}×\vec{a}-\vec{b}×\vec{a}+\vec{a}×\vec{b}-\vec{b}×\vec{b}=0+2(\vec{a}×\vec{b})+0$$ Hence, $$(\vec{a}-\vec{b})×(\vec{a}+\vec{b})=2(\vec{a}×\vec{b})$$ 7. Find the product $$(\vec{a}+\vec{b}).(7\vec{a}-6\vec{b})$$. a) $$2\|\vec{a}\|^2+6\vec{a}.\vec{b}-3\|\vec{b}\|^2$$ b) $$8\|\vec{a}\|^2+5\vec{a}.\vec{b}-5\|\vec{b}\|^2$$ c) $$2\|\vec{a}\|^2+6\vec{a}.\vec{b}-8\|\vec{b}\|^2$$ d) $$7\|\vec{a}\|^2+\vec{a}.\vec{b}-6\|\vec{b}\|^2$$ Explanation: To evaluate: $$(\vec{a}+\vec{b}).(7\vec{a}-6\vec{b})$$ =$$\vec{a}.7\vec{a}-\vec{a}.6\vec{b}+\vec{b}.7\vec{a}-6\vec{b}.\vec{b}$$ =$$7\|\vec{a}\|^2+\vec{a}.\vec{b}-6\|\vec{b}\|^2$$ 8. Find the vector product of the vectors $$\vec{a}=-\hat{j}+\hat{k}$$ and $$\vec{b}=-\hat{i}-\hat{j}-\hat{k}$$. a) $$2\hat{i}-\hat{j}+\hat{k}$$ b) $$2\hat{i}-\hat{j}-4\hat{k}$$ c) $$\hat{i}+\hat{j}-\hat{k}$$ d) $$2\hat{i}-\hat{j}-\hat{k}$$ Explanation: Given that, $$\vec{a}=-\hat{j}+\hat{k}$$ and $$\vec{b}=-\hat{i}-\hat{j}-\hat{k}$$ Calculating the vector product, we get $$\vec{a}×\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\0&-1&1\\-1&-1&-1\end{vmatrix}$$ =$$\hat{i}(1-(-1))-\hat{j}(0-(-1))+\hat{k}(0-1)$$ =$$2\hat{i}-\hat{j}-\hat{k}$$ 9. If $$\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$$ and $$\vec{b}=4\hat{i}-2\hat{j}+3\hat{k}$$. Find $$\|\vec{a}×\vec{b}\|$$. a) $$\sqrt{685}$$ b) $$\sqrt{645}$$ c) $$\sqrt{679}$$ d) $$\sqrt{689}$$ Explanation: Given that, $$\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$$ and $$\vec{b}=4\hat{i}-2\hat{j}+3\hat{k}$$ ∴ $$\vec{a}×\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&3&4\\4&-2&3\end{vmatrix}$$ =$$\hat{i}(9—8)-\hat{j}(6-16)+\hat{k}(-4-12)$$ =$$17\hat{i}+10\hat{j}-16\hat{k}$$ ∴$$\|\vec{a}×\vec{b}\|=\sqrt{17^2+10^2+(-16)^2}$$ =$$\sqrt{289+100+256}$$ =$$\sqrt{645}$$ 10. Find the angle between the vectors if $$\|\vec{a}\|=\|\vec{b}\|=3\sqrt{2}$$ and $$\vec{a}.\vec{b}=9\sqrt{3}$$. a) $$\frac{π}{6}$$ b) $$\frac{π}{5}$$ c) $$\frac{π}{3}$$ d) $$\frac{π}{2}$$ Explanation: We know that, $$\vec{a}.\vec{b}=\|\vec{a}\|.\|\vec{b}\| \,cos⁡θ$$ Given that, $$\|\vec{a}\|=\|\vec{b}\|=3\sqrt{2} \,and \,\vec{a}.\vec{b}=9\sqrt{3}$$ $$cos⁡θ=\frac{\vec{a}.\vec{b}}{\|\vec{a}\|.\|\vec{b}\|}=\frac{9\sqrt{3}}{(3\sqrt{2})^2}=\frac{\sqrt{3}}{2}$$ $$θ=cos^{-1}\frac{\sqrt{3}}{2}=\frac{π}{6}$$. Sanfoundry Global Education & Learning Series – Mathematics – Class 12. To practice Mathematics Question Papers for Class 12, here is complete set of 1000+ Multiple Choice Questions and Answers.	{"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": 0.9611295461654663, "perplexity": 1056.314866455716}, "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/1674764500017.27/warc/CC-MAIN-20230202101933-20230202131933-00311.warc.gz"}
http://math.stackexchange.com/questions/72330/homotopy-equivalence-definition	# Homotopy equivalence definition I've been given the question: Let $f : X \to Y$ be a continuous map, and suppose we are given (not necessarily equal) continuous maps $g,h : Y \to X$ such that $gf \simeq id_X$ and $fh \simeq id_Y$. Show that $f$ is a homotopy equivalence. What does it mean for a single function $f: X \to Y$ to be a homotopy equivalence? Thanks - It means there is a function $j:Y\rightarrow X$ so that $jf\simeq id_X$ and $fj\simeq id_Y$. – Jason DeVito Oct 13 '11 at 16:05 Given two spaces $X$ and $Y$, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps $f : X → Y$ and $g : Y → X$ such that $g ∘ f$ is homotopic to the identity map id $X$ and $f ∘ g$ is homotopic to id $Y$. Formally, a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space X with the unit interval $[0,1]$ to $Y$ such that, if $x ∈ X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9656758904457092, "perplexity": 42.870285135939895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783402516.86/warc/CC-MAIN-20160624155002-00188-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.miniphysics.com/energy-level-diagram-for-hydrogen.html	# Energy Level Diagram For Hydrogen Show/Hide Sub-topics (Quantum Physics And Lasers \| A Level) • n represents the principle quantum number and only takes integral values from 1 to infinity. • The ground state refers to the lowest energy level n=1 in which the atom is the most stable. The electron normally occupies this level unless given sufficient energy to move up to a higher level. An atom is said to be in an excited state when its electrons are found in the higher energy levels. When an atom is excited from the ground state to a higher energy, it becomes unstable and falls back to one of the lower energy levels by emitting photon(s)/electromagnetic radiation • The highest energy level n =corresponds to an energy state whereby the electron is no longer bound to the atom. (the electron has escaped from the atom.) By convention, it is usually assigned an energy value of 0 eV. • The lower the energy level, the more negative the energy value associated with that level. Thus, the lower energy states correspond to more stable states. • The energy difference between any two adjacent levels gets smaller as n increases, which results in the higher energy levels getting very close and crowded together just below n = . • The ionization energy of an atom is the energy required to remove the electron completely from the atom.(transition from ground state n = 0 to infinity n = ). For hydrogen, the ionization energy = 13.6eV • When an excited electron returns to a lower level, it loses an exact amount of energy by emitting a photon. The Lyman(ultraviolet) series of spectral lines corresponds to electron transitions from higher energy levels to level n = 1. Transitions to n = 2 and n = 3are called the Balmer(visible) and Paschen(Infra Red) series, respectively. Why the energy levels have negative values? • Negative value of energy indicates that the electron is bound to the nucleus and there exists an attractive force between the electron and the nucleus. Also, since the potential at infinity is defined as zero, energy levels at a distance below infinity are negative.	{"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": 0.8840256333351135, "perplexity": 341.43208377832553}, "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/1627046150129.50/warc/CC-MAIN-20210724032221-20210724062221-00144.warc.gz"}
http://quantumgazette.blogspot.ca/2014/07/	## Friday, 4 July 2014 ### An arbitrary quantum cannot be cloned One of the early important results in the study of quantum information is the no-cloning theorem, which tells us that there is no quantum operation that allows us to create multiple copies of an arbitrary quantum state. This property is very different from what we expect from classical information, which you may reproduce as many times as you wish. For example, you can send a PDF file by email to many recipients while keeping a copy to yourself. The important point is that whatever the contents may be, you can make a duplicate of it. Now consider a cloning machine M for qubits that can produce identical copies of the states \|u) and \|d): M \|u) \|0) = M \|u) \|u), M \|d) \|0) = M \|d) \|d) , where \|0) denotes any fixed initial state for M. This is necessary in pretty much the same way you would need a blank piece of paper before you can photocopy a printed document.	{"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": 0.88804692029953, "perplexity": 518.6374238553698}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886120573.0/warc/CC-MAIN-20170823132736-20170823152736-00553.warc.gz"}
https://crypto.stackexchange.com/questions/60228/breaking-anonymous-key-exchange-protocol	# breaking anonymous key exchange protocol An anonymous key exchange protocol $P$ is pair of probabilistic machines $(A,B)$ that sends messages to each other and at the end, both $A$ and $B$ will share the same key $k$. The attacker can see the protocol transcript $T_P$ of $P$, that is, the sequence of messages exchanged between the parties in one execution of $P$. How can a computationally unbounded attacker break an anonymous key exchange protocol? • An unbounded attacker can just run the protocol for all initial states of A and B and for all randomness coin results and stop when the given configuration yields a transcript that matches the given transcript (and then because they have the randomness results and the initial states, they have the key) – SEJPM Jun 23 '18 at 11:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 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": 0.8934035301208496, "perplexity": 515.53802674024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578678807.71/warc/CC-MAIN-20190425014322-20190425040322-00219.warc.gz"}
http://mathoverflow.net/questions/35049/families-of-genus-2-curves-with-positive-rank-jacobians	# Families of genus 2 curves with positive rank jacobians It's fairly easy to write down families of elliptic curves over $\mathbb{Q}(t)$ such that almost every (i.e. when the "height" of $t$ is sufficiently large) curve in the family has positive rank over $\mathbb{Q}$. One can do this by constructing the fibration so it has sections /$\mathbb{Q}$ a priori, or by fiddling around with Gauss sums; see papers of Fermigier, Mestre, Arms/Miller/Lozano-Robledo, etc. Are there any known examples of families of genus two curves over $\mathbb{Q}(t)$ such that the Jacobian of almost every curve $C_t$ in the family has provably positive rank? Can one do this while requiring that $\mathrm{Jac}(C_t)$ be $\overline{\mathbb{Q}}$-simple for almost all $t$? - I do not know of a write up of an example, but have you tried the same tricks that are used for elliptic curves? E.g. let $S \to \mathbb{P}^1_{\mathbb{Q}}$ be a rel curve of genus 2, $S$ smooth proj surface, and suppose $\Sigma_1,\Sigma_2$ are sections. Make sure that the line bundle $\Sigma_1-\Sigma_2$ is a non-torsion, e.g. by choosing it so that the reductions modulo different primes in two fibers have coprime order. Presumably, simplicity of the Jacobians should also be easy, at least for a positive proportion of fibers. Not sure how to guarantee simplicity for almost all $t$. – damiano Aug 9 '10 at 22:22 $y^2 = x^5 + x + t^2$ the point infinity - (0,t) has infinite order. – Felipe Voloch Aug 9 '10 at 22:39 @Felipe: Very nice, what's the simplest way to see this? Is there Lutz-Nagell for preimages of torsion in the Jacobian? – David Hansen Aug 10 '10 at 0:10 @David: There is no Lutz-Nagell or a particularly easy way to do this. One possibility is to use the Manin map. Another is an algorithm of Poonen's (Computing torsion points on curves) for curves over Q. In his paper he does $y^2=x^5-x+1$ and find that there is no torsion of the form infinity-point except for the ones with y=0. This implies that $y^2=x^5-x+t^2$ will work. A similar calculation can be done for my first example. Like Borcherds said, pretty much anything works. – Felipe Voloch Aug 10 '10 at 2:26 For families with large rank look at Elkies' paper: math.rice.edu/~hassett/conferences/Clay2006/Elkies/… – Victor Miller Aug 10 '10 at 4:06	{"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": 0.8961668014526367, "perplexity": 548.094796468476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507450767.7/warc/CC-MAIN-20141017005730-00370-ip-10-16-133-185.ec2.internal.warc.gz"}
http://mathonline.wikidot.com/the-hausdorff-property-on-finite-topological-products	The Hausdorff Property on Finite Topological Products Table of Contents # The Hausdorff Property on Finite Topological Products As well have already seen on the First Countability of Finite Topological Products, Second Countability of Finite Topological Products, and Separability of Finite Topological Products pages that if $\{ X_1, X_2, ..., X_n \}$ is a finite collection of topological spaces that are all first countable / second countable / separable, then the resulting topological product $\displaystyle{\prod_{i=1}^{n} X_i}$ is also respectively first countable / second countable /separable. Now recall from the Hausdorff Topological Spaces page that a topological space $X$ is said to be Hausdorff if for every pair of distinct points $x, y \in X$ there exists open neighbourhoods $U$ of $x$ and $V$ of $x$ such that: (1) \begin{align} \quad U \cap V = \emptyset \end{align} We will now see that the Hausdorff property of a finite collection of topological spaces is also inherited onto the resulting finite topological product. Theorem 1: Let $\{ X_1, X_2, …, X_n \}$ be a finite collection of Hausdorff topological spaces. If $X_i$ is separable for each $i \in I$ then the topological product $\displaystyle{\prod_{i=1}^{n} X_i}$ is also Hausdorff. • Proof: Let $\displaystyle{\mathbf{x} = (x_1, x_2, ..., x_n), \mathbf{y} = (y_1, y_2, ..., y_n) \in \prod_{i=1}^{n} X_i}$ be distinct points. Then $x_i, y_i \in X_i$ for all $i \in \{ 1, 2, ..., n \}$, and moreover, $x_i \neq y_i$ for at least some $i \in \{ 1, 2, ..., n \}$. • Suppose that $j \in \{ 1, 2, ..., n \}$ is such that $x_j \neq y_j$. Then since these two points are distinct points in $X_j$ we have that there exists open neighbourhoods $U_j$ of $x_j$ and $V_j$ of $y_j$ in $X_j$ such that: (2) \begin{align} \quad U_j \cap V_j = \emptyset \end{align} • So, consider the following open sets in $U$ and $V$ of $\displaystyle{\prod_{i=1}^{n} X_i}$: (3) \begin{align} \quad U = X_1 \times X_2 \times ... \times X_{j-1} \times U_j \times X_{j+1} \times ... \times X_n \end{align} (4) \begin{align} \quad V = X_1 \times X_2 \times ... \times X_{j-1} \times V_j \times X_{j+1} \times ... \times X_n \end{align} • We claim that $U \cap V = \emptyset$. Suppose not, i.e., suppose that there exists a $\mathbf{z} = (z_1, z_2, ..., z_n) \in U \cap V$. Then this implies that $z_j \in U_j \cap V_j$ which contradicts $U_j \cap V_j = \emptyset$. Therefore $U \cap V = \emptyset$. So, for all distinct points $\mathbf{x}$ and $\mathbf{y}$ in $\displaystyle{\prod_{i=1}^{n} X_i}$ there exists open neighbourhoods $U$ of $\mathbf{x}$ and $V$ of $\mathbf{y}$. Therefore, $\displaystyle{\prod_{i=1}^{n} X_i}$ is Hausdorff. $\blacksquare$ Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License	{"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": 4, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998299479484558, "perplexity": 176.43825779555075}, "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-2018-13/segments/1521257647706.69/warc/CC-MAIN-20180321215410-20180321235410-00556.warc.gz"}
http://physics.stackexchange.com/questions/56903/the-interpretation-of-mass-in-quantum-field-theories/56939	# The interpretation of mass in quantum field theories Consider a free theory with one real scalar field: $$\mathcal{L}:=-\frac{1}{2}\partial _\mu \phi \partial ^\mu \phi -\frac{1}{2}m^2\phi ^2.$$ We write this positive coefficient in front of $\phi ^2$ as $\frac{1}{2}m^2$, and then start calling $m$ the mass (of who knows what at this point) and even interpret it as such. But pretend for a moment that you've never seen any sort of field theory before: if someone were to just write this Lagrangian down, it's not immediately apparent why this should be the mass of anything. So then, first of all, what is it precisely that we mean when we say the word "mass", and how is our constant related to this physical notion in a way that justifies the interpretation of $m$ as mass? If it helps to clarify, this is how I think about it. There are two notions of mass involved: the mathematical one that is part of our model, and the physical one which we are trying to model. The physical mass needs to be defined by an idealized thought experiment, and then, if our model is to be any good, we should be able to come up with a 'proof' that our mathematical definition agrees with the physical one. (Of course, none of this at all has anything to do with this particular field theory; it was just the simplest Lagrangian to write down.) - "But pretend for a moment that you've never seen any sort of field theory before: if someone were to just write this Lagrangian down, it's not immediately apparent why this should be the mass of anything." - It's a basic feature of science that one is actually allowed - and encouraged - to use the brain and learn things that aren't obvious at the beginning. In the QFT above, one analyzes physics and finds that it contains particles whose mass is $m$. What's the problem? Are you serious that you would want all things in science to be obvious from the beginning and without thinking? – Luboš Motl Mar 14 '13 at 19:35 @Luboš I think Jonathan is trying to shed some light on the black box represented by "one analyzes physics" in your comment. IMO it's an excellent question. – David Z Mar 14 '13 at 19:42 I agree with David that it's an excellent question. @LubošMotl Your question "Are you serious that you would want all things in science to be obvious from the beginning and without thinking?" is an inaccurate, unproductive manipulation of statements made by the OP. – joshphysics Mar 14 '13 at 19:46 – John Rennie Mar 14 '13 at 20:02 I would probably go with QFT=>Rep of Poincare alg.=>Casimir invariant $m^2$. Single particle interpretation: $P^2=m^2$ on shell, and we know that $m$ here is precisely the invariant mass. I don't know anything deeper. The parameters in QFT are given their specific names because they correspond to the appropriate things in the appropriate limits (classical and non relativistic). I think, though, the particle interpretation is key to calling that Lagrangian parameter "mass". – twistor59 Mar 15 '13 at 7:48 For the first answer, just forget about $\hbar$ (but say $c=1$), we are doing a classical relativistic field theory. The first is that you can consider the field profile around a static, spherically symmetric source of mass $M$ (you need to add a coupling to the action of the form $g \phi J$, where $J$ is an external source, to do this). So we write the static spherically symmetric source as $J=J_0 \delta^3(r)$ (if you don't like delta functions you can make it a top hat or a shell and you will find the same conclusions below, this kind of manipulation should be familiar from E/M classes) You can work out the equations of motion for $\phi$, you will find $\nabla^2\phi + m^2 \phi = g J_0 \delta^3(r)$ This looks exactly like Poisson's equation, except with this extra $m^2$ term (which I haven't called a mass yet). If you solve it (say using green's functions, or you could just make an explicit ansatz) you will find the solution is $\phi = g J_0 \frac{e^{-m r}}{4\pi r}$ This is the famous yukawa potential with mass $m$. It describes a force with a range $m^{-1}$, which you may have heard before is the smoking gun for force carrying particle having mass (the weak interactions are so weak at human distances because $m$ is so big) Now at this point you are still justified in asking why we are calling this quantity $m$, which in the classical analysis above is just an inverse length, a mass. The second answer addresses this more directly. It comes when we quantize the theory. This is discussed in great detail in many qft textbooks, but the gist is relatively simple. At this point I will say $\hbar=1$, if you really want to keep track of the $\hbar$ dependence you can get it back by dimensional analysis. If you take the equations of motion for the above system without the source (note we are no longer working with static systems): $-\partial_t^2\phi + \nabla^2 \phi + m^2 \phi^2 = 0$ and go to fourier space $\tilde{\phi}(\vec{k},t)\equiv\int\frac{d^3 k}{(2\pi)^3}e^{i\vec{k}\cdot\vec{x}}\phi({\vec{x},t})$ then you will find the equations of motion are $-\partial_t^2 \tilde{\phi} +\vec{k}^2\tilde{\phi} + m^2\tilde{\phi}=0$ This is just a set of harmonic oscillators, labeled by $\vec{k}$ with frequency $\omega^2=\vec{k}^2+m^2$ We can think of $\tilde{\phi}(\vec{k},t)$ for a single value of $k$ as being a single quantum variable obeying a harmonic oscillator equation, so there when we quantize these variables we will have one wave function for each value of $\vec{k}$, each of which obeys a schrodinger equation for a harmonic oscillator with the above frequency. Each harmonic oscillator will be quantized and have discrete energy levels. These energy levels are what we call 'particles'. The idea is that for a given momentum, the field $\phi$ can only have certain values of energy, and the interpretation is that these quantized wiggles in $\phi$, or in other words discrete packets of energy, are particles. Now we remember we are doing quantum mechanics, and we are supposed to identify frequency with energy and momentum with wavelength, using $E=\omega$, $\vec{p}=\vec{k}$ So we are describing a system with a relationship between energy and momentum given by $E^2=\vec{p}^2+m^2$ This is einstein's famous formula, and now we see that the parameter $m$ appears in exactly the place the mass of a particle would appear. It is the energy that a particle has when it has $0$ momentum. So we identify the particle like excitations we discovered above with particles of mass $m$. However I emphasize that there are many, many, many other treatments of this and other ways of looking at it. A good question would be, what does a particle mean if you don't have a free theory and don't get a harmonic oscillator equation above? - I've been thinking about this question on and off since it was posted, and I have some thoughts that I hope will shed some light on this stuff. I think it helps to begin with the following: An Analogy from Mechanics. Consider the following expression that you'll often encounter in classical mechanics: $$L(x, \dot x) = \frac{1}{2}m\dot x^2$$ and let's say that you've never seen Lagrangian mechanics before. Someone comes and tells you that this Lagrangian "describes" a free particle of mass $m$ moving in one dimension. You could then ask the question "how can we identify the parameter $m$ that appears in this expression with the physical mass of a particle? My thought process in answering this question is as follows. Well, we want to identify the parameter $m$ with the physical definition of mass in classical mechanics. How do we define mass operationally in classical mechanics? Well, we exert a force on the mass, and we see the extent to which it accelerates. In other words, we need to interact with the particle and watch what happens to determine its mass. Next, we return to the Lagrangian above. If it were able to predict how a particle reacts to an interaction, then we could use it to identify the parameter $m$ as the mass of a particle. Unfortunately, the free particle Lagrangian makes no such dynamical predictions. The corresponding Euler-Lagrange equations lead to the equation of motion $$\ddot x = 0$$ which tells us nothing about how a particle will react to an interaction. Now, you might try to identify $m$ as the mass by defining the energy and momentum in this mathematical model to be the generators of time and space translations respectively and then say "oh look, these expressions correspond to the expressions for energy and momentum of a non-relativistic particle that we're used to, so $m$ must be the mass!" I don't think there is much content or validity to this argument, because making these "definitions* in no way allows one to identify the parameter in the Lagrangian with the physical mass operationally defined in terms of interactions. In order to identify $m$ as the mass, we need to put terms in the Lagrangian that correspond to interactions (like a potential term), see what the Euler-Lagrange equations of motion predict about the resulting dynamics, and then compare to experiments to identify $m$ as the mass. Quantum Field Theory. Using these analogies, I feel dissatisfied by user20797's response where we obtains the relativistic energy-momentum relationship for a particle because as far as I can tell, that procedure amounts to little more than making definitions for energy and momentum that give you the relation you want. It does not tell you how to relate the parameter $m$ to some empirically defined mass, and that, I feel, was your original (quite excellent/subtle in my opinion) question. I think his first response is probably closer to what is necessary. However, it seems to me that a proper answer to your question requires an analysis of the following (schematic) form whose details (especially empirically), are quite non-trivial. 1. We note that the mass of particles can be defined by their interactions with one another. In particular, we can use scattering experiments to bring about these interactions. 2. We add interaction terms to the Lagrangian density you wrote down, and we develop a prescription by which the resulting quantized field theory can predict what will happen in scattering experiments. 3. We compare the results of our scattering experiments to the predictions of the quantum field theory, and we find that we can identify the parameter $m$ in the Lagrangian density as the physical mass as it was defined through scattering experiments. - "In order to identify m as the [m]ass, me need to put terms in the Lagrangian the correspond to interactions (like a potential term), see what the Euler-Lagrange equations of motion predict about the resulting dynamics, and then compare to experiments to identify m as the mass." Agreed that this would be a good operational definition of mass. I suggested this in a comment on the question. You should be able to derive $F=ma$ through some straightforward steps in the appropriate limits. – Michael Brown Mar 26 '13 at 13:10 @MichaelBrown Woops thanks :). The more I thought about this, the more I felt dissatisfied with "the standard dreck" in which one shows that one can reproduce the appropriate free particle spectrum and energy-momentum relationship because it doesn't make contact with some such sensible operational definition of mass via dynamics e.g. scattering. – joshphysics Mar 26 '13 at 13:18 The concept of mass is related to the 4-momentum, or 4-momentum density. It is not just a “quadratic term” in the Lagrangian. By varying Lagrangian one can obtain the equations of motion. But the momentum, or momentum density need to be defined independently from Lagrangian, in order to enable one to introduce the concept of mass or mass density. For instance, the momentum can be defined as the sum of two isotropic 4-vectors constructed from spinor and co-spinor: $\xi = \begin{vmatrix} \xi^1 \\ \xi^2 \end{vmatrix}$ - spinor $\eta = \begin{vmatrix} \eta_{\dot1} \\ \eta_{\dot2} \end{vmatrix}$ - co-spinor $p_\mu = \frac{1}{2} (\xi^{+}\sigma_\mu \xi)$ - covariant vector $\tilde p^\mu = \frac{1}{2} (\eta^{+}\tilde\sigma^\mu \eta)$ - contravariant vector The vector constructed from spinor is covariant, while the vector constructed from co-spinor is contravariant. This is due to different transformations properties of spinor and co-spinor. Both $p_\mu$ and $\tilde p^\mu$ are isotropic (i.e. $p_\mu p^\mu =\tilde p_\mu \tilde p^\mu =0$), but their sum is not isotropic: $P_\mu = p_\mu +g_{\mu\nu}\tilde p^\nu$ - momentum or momentum density - depending on the model. The mass, or mass density, is therefore defined as the "square" of the momentum (density) vector: $m^2=P_\mu P^\mu$ In general, this definition of mass is independent from the choice of the Lagrangian. However, the momentum and/or momentum density cannot be defined arbitrarily. It has to be defined in such a way that 1. The total momentum is conserved (as a consequence of the equations of motion), 2. The components of the momentum 4-vector are all real-valued, and 3. The momentum 4-vector is time-like to preserve causality. In general, we need two ingredients to say that mass in our model is really a mass: 1. Lagrangian, or at least equations of motion, and 2. Appropriate definition of momentum or momentum density (see conditions 2 and 3 above). The conservation of momentum is required to be the consequence of the equations of motion (except for the Theory of gravity where conservation of momentum is an identity!). It is also easy to demonstrate that we do not necessarily need the “quadratic” and/or constant mass term in the Lagrangian in order to introduce the concept of mass. Here you can find an example of the model where the “mass term” is variable and complex valued, but still allows for real-valued “mass square” of the momentum density vector. - We can define mass this way, and I already know how to relate this definition to the term appearing in the Lagrangian. The question is, how do we relate this mathematical definition of mass to a precise, physical, operational definition of mass. Does that make sense? – Jonathan Gleason Mar 15 '13 at 12:48 Please explain what in your opinion is "precise, operational definition of mass". To me the only relativistic physical definition of mass is the square of 4-momentum. – Murod Abdukhakimov Mar 15 '13 at 13:19 Ironically, the word "precise" here is not meant to be precise; it is open to interpretation. The word that really matters here is "operational": to define mass via some sort of (thought) experiment. In section 2.2 of academia.edu/829613/… , I give a classical definition in the spirit of which I am looking, except now, I want to do this in a relativistic setting. I never thought of this until just now, but maybe the idea behind the classical definition could just be modified? – Jonathan Gleason Mar 15 '13 at 13:27 If it helps to clarify, this is how I think about it. There are two notions of mass involved: the mathematical one that is part of our model, and the physical one which we are trying to model. The physical mass needs to be defined by an idealized experiment, and then, if our model is to be any good, we should be able to come up with a 'proof' that our mathematical definition agrees with the physical one. Does that make sense? – Jonathan Gleason Mar 15 '13 at 13:30 I think I understand your question now. Think of the total energy of two photons resulting from annihilation of particle and its antiparticle in the rest frame of their center of inertia. Energy of photon can be expressed (and measured) via its wavelength in difraction experiment. The total energy will be $2m$. – Murod Abdukhakimov Mar 15 '13 at 14:00	{"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": 0.9517465829849243, "perplexity": 254.05069347364233}, "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-2015-27/segments/1435376073161.33/warc/CC-MAIN-20150627033433-00270-ip-10-179-60-89.ec2.internal.warc.gz"}
http://www.scienceandtechnologyresearchnews.com/yales-wright-lab-will-test-new-theory-solve-subatomic-mystery/	# Yale’s Wright Lab Will Test New Theory to Solve a Subatomic Mystery A new study from the international Daya Bay neutrino experiment in China may have found an answer for why there appear to be fewer neutrinos from nuclear reactors than predicted in theoretical models. The next step will be for Yale’s Wright Lab — which has been involved in the Daya Bay research for a number of years — to confirm whether that answer is correct. Neutrinos are thought to be fundamental particles in the universe. They do not carry an electrical charge and move through the universe almost entirely unaffected by natural forces. For years, scientists have devised elaborate experiments to detect the subatomic particles and better understand their behavior.Yale scientists were involved in the experiments that led to the discovery of neutrino oscillation. Yet since 2011, physicists have been faced with an anomaly: Experiments at nuclear reactors showed fewer antineutrinos (the antiparticles of neutrinos) being produced than models had predicted. One theory to explain the discrepancy was that neutrinos were morphing into an undetectable form known as “sterile” neutrinos. The new Daya Bay study proposes a simpler explanation: a miscalculation in the predicted rate of antineutrino production for one particular component of reactor fuel. During nuclear fission, antineutrinos carry away about 5% of the energy released as the uranium and plutonium atoms — which fuel the reactor — split. The composition of the fuel changes as the reactor operates, with the decays of different forms of uranium and plutonium (called isotopes) producing different numbers of antineutrinos with different energy ranges over time, even as the reactor steadily produces electrical power. The Daya Bay scientists found that antineutrinos produced by nuclear reactions from the fission of uranium-235 were inconsistent with predictions. Uranium-235 is an isotope that is common in nuclear fuel. “Most of the existing world data view neutrinos emitted by commercial power plants using low enrichment nuclear fuels,” said Yale physicist Karsten Heeger, a member of the Daya Bay team and director of Yale’s Wright Lab. “The fissions within these reactors are almost entirely from a mixture of isotopes: U235, U238, Pu239, and Pu241. Daya Bay measures a relative deficit in the number of neutrinos produced from U235.” Yale’s Wright Lab will play a key role in determining whether the theory about uranium-235 is correct. Wright Lab is the lead institution in the Precision Oscillation and Spectrum Experiment (PROSPECT), an experiment under construction at the High Flux Isotope Reactor (HFIR), at the Oak Ridge National Laboratory in Tennessee. HFIR operates mainly with uranium-235 fuel. “Since almost all of the neutrinos at HFIR originate from uranium-235, PROSPECT will be in a unique position to verify the Daya Bay observed deficit and will be able to make precise measurements of the neutrino energy spectrum to help identify where current models disagree with data,” Heeger said. “In addition, PROSPECT will search for oscillations in the neutrino flux to test and rule out the possibility of new particles, such as the sterile neutrino, contributing to the remaining discrepancy between experiment and theory,” Heeger added. The antineutrino detector for PROSPECT is currently under construction at the Yale Wright Lab and is expected to be installed at HFIR later this year. Funding for PROSPECT is provided by the U.S. Department of Energy’s Office of Science and the Heising-Simons Foundation. Other members of the Yale group involved in the new Daya Bay measurements include Henry Band, Tom Langofrd, and Tom Wise.	{"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": 0.8489649891853333, "perplexity": 1962.9904802596834}, "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-2018-51/segments/1544376823348.23/warc/CC-MAIN-20181210144632-20181210170132-00373.warc.gz"}
https://web2.0rechner.de/fragen/to-radix-or-heureka	+0 0 385 2 How do you calculate the coefficients of a power series such as this one: [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10. Just the first 3 coefficients, with steps, would be great. I would greatly appreciate any help. Thanks a million and have a great day. 16.05.2016 #1 +14537 0 Hello ! [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10 What do you mean ? [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10 = $$\frac{(x^4+x^5+x^6)^{10}}{1048576}$$ or [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10 = 0 => x = 0 and $$x=-\sqrt[3]{-1}$$ 17.05.2016 #2 +21815 0 How do you calculate the coefficients of a power series such as this one: [ 1/4x^4 + 1/4x^5 + 1/4x^6]^10. Just the first 3 coefficients, with steps, would be great. I would greatly appreciate any help. Thanks a million and have a great day. $$\begin{array}{rcll} && [~ \frac14 \cdot x^4 + \frac14 \cdot x^5 + \frac14 \cdot x^6 ~]^{10} \\ &=& [~ \frac{1}{4} \cdot ( x^4 + x^5 + x^6 ) ~]^{10} \\ &=& \left( \frac{1}{4} \right)^{10} \cdot ( x^4 + x^5 + x^6 )^{10} \\ &=& \left( \frac{1}{4^{10} } \right) \cdot ( x^4 + x^5 + x^6 )^{10} \\ &=& \left( \frac{1}{1048576} \right) \cdot ( x^4 + x^5 + x^6 )^{10} \\\\ && ( a + b + c )^{n} \\ &=& \sum \limits_{k=0}^{n} \sum \limits_{i=0}^{k} \dbinom{n}{k} \dbinom{k}{i} a^{n-k} b^{k-i} c^{i} \qquad \| \qquad (n-k)+(k-i)+(i) = n \\\\ && ( x^4 + x^5 + x^6 )^{10} \\ &=& \sum \limits_{k=0}^{10} \sum \limits_{i=0}^{k} \dbinom{10}{k} \dbinom{k}{i} (x^4)^{10-k} (x^5)^{k-i} (x^6)^{i} \qquad \| \qquad (10-k)+(k-i)+(i) = 10 \\\\ \end{array}$$ $$\begin{array}{lcll} (k=0,i=0): & \dbinom{10}{0} \dbinom{0}{0} (x^4)^{10} (x^5)^{0} (x^6)^{0} &=& x^{40} \\ \hline (k=1,i=0): & \dbinom{10}{1} \dbinom{1}{0} (x^4)^{9} (x^5)^{1} (x^6)^{0} &=& 10 \cdot x^{41} \\ (k=1,i=1): & \dbinom{10}{1} \dbinom{1}{1} (x^4)^{9} (x^5)^{0} (x^6)^{1} &=& 10 \cdot x^{42} \\ \hline (k=2,i=0): & \dbinom{10}{2} \dbinom{2}{0} (x^4)^{8} (x^5)^{2} (x^6)^{0} &=& 45 \cdot x^{42} \\ (k=2,i=1): & \dbinom{10}{2} \dbinom{2}{1} (x^4)^{8} (x^5)^{1} (x^6)^{1} &=& 90 \cdot x^{43} \\ (k=2,i=2): & \dbinom{10}{2} \dbinom{2}{2} (x^4)^{8} (x^5)^{0} (x^6)^{2} &=& 45 \cdot x^{44} \\ \hline (k=3,i=0): & \dbinom{10}{3} \dbinom{3}{0} (x^4)^{7} (x^5)^{3} (x^6)^{0} &=& 120 \cdot x^{43} \\ (k=3,i=1): & \dbinom{10}{3} \dbinom{3}{1} (x^4)^{7} (x^5)^{2} (x^6)^{1} &=& 360 \cdot x^{44} \\ (k=3,i=2): & \dbinom{10}{3} \dbinom{3}{2} (x^4)^{7} (x^5)^{1} (x^6)^{2} &=& 360 \cdot x^{45} \\ (k=3,i=3): & \dbinom{10}{3} \dbinom{3}{3} (x^4)^{7} (x^5)^{0} (x^6)^{3} &=& 120 \cdot x^{46} \\ \cdots \end{array}$$ $$\small{ \begin{array}{l\|\|r\|r\|r\|r\|r\|r\|r\|} &x^{40} & \\ & & 10 \cdot x^{41} & 10 \cdot x^{42} &\\ & & & 45 \cdot x^{42} & 90 \cdot x^{43} & 45 \cdot x^{44} & \\ & & & & 120 \cdot x^{43} & 360 \cdot x^{44} & 360 \cdot x^{45} & 120 \cdot x^{46}\\ & & & & & \cdots & \cdots & \cdots\\ \hline sum & x^{40} & 10\cdot x^{41} & 55\cdot x^{42} & 210 \cdot x^{43} & \cdots & \cdots & \cdots \end{array} }$$ $$\begin{array}{rcll} && [~ \frac14 \cdot x^4 + \frac14 \cdot x^5 + \frac14 \cdot x^6 ~]^{10} \\ &=& \frac{1}{1048576}\cdot x^{40} + \frac{10}{1048576}\cdot x^{41} + \frac{55}{1048576}\cdot x^{42} + \frac{210}{1048576}\cdot x^{43} + \frac{615}{1048576}\cdot x^{44}\\ &+& \frac{1452}{1048576}\cdot x^{45} + \frac{2850}{1048576}\cdot x^{46} + \frac{4740}{1048576}\cdot x^{47} + \frac{6765}{1048576}\cdot x^{48} + \frac{8350}{1048576}\cdot x^{49}\\ &+& \frac{8953}{1048576}\cdot x^{50} + \frac{8350}{1048576}\cdot x^{51} + \frac{6765}{1048576}\cdot x^{52} + \frac{4740}{1048576}\cdot x^{53} + \frac{2850}{1048576}\cdot x^{54}\\ &+& \frac{1452}{1048576}\cdot x^{55} + \frac{615}{1048576}\cdot x^{56} + \frac{210}{1048576}\cdot x^{57} + \frac{55}{1048576}\cdot x^{58} + \frac{10}{1048576}\cdot x^{59}\\ &+& \frac{1}{1048576}\cdot x^{60} \end{array}$$ heureka 17.05.2016	{"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": 0.8212398886680603, "perplexity": 2558.075383154889}, "config": {"markdown_headings": true, "markdown_code": false, "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-2019-13/segments/1552912201329.40/warc/CC-MAIN-20190318132220-20190318154220-00337.warc.gz"}
https://physics.stackexchange.com/questions/177180/schwarzschild-radius-in-black-hole-density	# Schwarzschild radius in black hole density The textbook from which I teach physics at the end of secondary school has a question about the density of a non-rotating black hole. Because the density at the singularity is perhaps infinite or beyond the scope of GR, this can only be some "mean density" mass/volume within a sphere of a certain radius. Some authors use the Schwarzschild radius. Why this value of the radius? How can I explain that this choice of radius makes more sense than another radius? • Did you read the Wikipedia page on the Schwarzschild radius? – ACuriousMind Apr 19 '15 at 19:41 • The Schwarzschild radius is the distance fro the center at which the escape velocity of the black hole surpasses the speed of light. – Brionius Apr 19 '15 at 19:41 • @Brionius: As far as I remember the physics lecture (I'm engineer, not physicist) this definition is based on non-relativistic calculations. You get the correct radius using this calculation however the calculation has (as far as I remember) nothing to do with the physical truth. – Martin Rosenau Apr 19 '15 at 19:50 • Very related: Do black holes have infinite areas and volumes?. Basically, the volume of a black hole is not frame-independent, and so is not unambiguously defined. However, for a simple (non-rotating, uncharged, isolated) Schwarzschild black hole, one fairly 'obvious' choice of frame gives the same volume as the usual Euclidean formula with Schwarzschild radius. – Stan Liou Apr 19 '15 at 19:58 • @MartinRosenau while the common Newtonian derivation of the Schwartzschild radius is a happy accident, it still is the dividing line between the point when light (or anything else) could escape. It's a convenient, accurate shorthand, but it shouldn't be taken as an excuse to apply Newtonian gravity to a non-Newtonian system. – Brionius Apr 19 '15 at 20:10 ## 2 Answers Using the Schwarzschild radius for this purpose makes sense, because this is the radius of a sphere which becomes a black hole, if it has the given density. For example a sphere made of air at Earth density does not become a black hole if its radius is 1 meter. But if the radius is big enough, it will actually become a black hole. Even though the density is very low. The required radius is the Schwarzschild radius associated with the mass enclosed in the sphere (under given density). • An implicit assumption here is that there is such a thing as 'the' radius, which is actually misleading. – Stan Liou Apr 19 '15 at 20:16 A black hole is a region of spacetime enclosed by an event horizon. Thus, the singularity, while a fact about black holes as far as we understand them, is not an defining feature of what black hole is. Therefore, it makes more sense to try to calculate volume (and hence indirectly, density) of a black hole according to the extent of the event horizon rather than the singularity. It so happens that for the simplest of black hole solutions, the spherically symmetric Schwarzschild black hole, the Schwarzschild radius characterizes the size of the horizon: the Schwarzschild radius is actually $\sqrt{A/4\pi}$, where $A$ is the area of the horizon. This is actually (implicitly) by definition of the Schwarzschild radial coordinate, which directly corresponds to the area of a sphere. As I've said in the comments, a problem of talking about 'the' volume of a black hole is that it's not actually frame-independent, so calculating volume in this way is simply a conventional choice. Again, see this question for more details.	{"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": 0.8620280623435974, "perplexity": 226.59167474222568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2020-29/segments/1593655908294.32/warc/CC-MAIN-20200710113143-20200710143143-00333.warc.gz"}
https://www.physicsforums.com/threads/poincare-transform.181547/	# Poincare transform 1. Aug 24, 2007 ### phatm After a lot of searching I cannot for the life of me find the transformation for a Momentarily Co-moving Reference Frame. Essentially what I'm looking for is the transformations from inertial frame Sigma to inertial frame Sigma[']. Sigma['] is moving at speed v relative to Sigma not along any axis in Sigma. The origins of the inertial reference frames do not coincide at t=t'=0, so I cannot use the general Lorentz transforms. Does anyone have any idea where to find the elements of the Poincare transform. Any help is greatly appreciated. Andrew. 2. Aug 24, 2007 ### meopemuk You can find a general Lorentz transformation (boost velocity has an arbitrary direction) in https://www.physicsforums.com/showthread.php?t=179810. For a general boost coupled with space-time translation you should just subtract the components of the translation vector from the boost-transformed coordinates (t', x',y',z'). In the most general case (arbitrary boost + arbitrary rotation + translation by the 4-vector $(t_0, x_0,y_0,z_0)$) the coordinate transformation is given by $$\left[ \begin{array}{c} t' \\ x' \\ y' \\ z' \end{array} \right] = \Lambda \left[ \begin{array}{c} t \\ x \\ y \\ z \end{array} \right] - \left[ \begin{array}{c} t_0 \\ x_0 \\ y_0 \\ z_0 \end{array} \right]$$ where $\Lambda$ is a pseudoorthogonal 4x4 matrix. Eugene. Last edited: Aug 24, 2007 3. Aug 25, 2007 ### phatm Thanks for the reply Eugene. So say we had a observer frame $$\Sigma$$ and a MCRF $$\Sigma'$$. At time time t=t'=0 $$\Sigma'$$ is moving with velocity $$\vec{\beta}$$ in some arbritary direction in $$\Sigma$$ at say $$x_0=y_0=z_0=1$$ in $$\Sigma$$, with no rotation of $$\Sigma'$$ in relation to $$\Sigma$$. Then by what you stated, $$\Lambda$$ is just the general Lorentz boost and the whole equation would be $$\left[ \begin{array}{c} t' \\ x' \\ y' \\ z' \end{array} \right] = \begin{pmatrix} \gamma & -\gamma \beta_1 & -\gamma \beta_2 & -\gamma \beta_3 \\ -\gamma \beta_1 & 1 + \frac{(\gamma - 1)\beta_1^2}{\beta^2} & \frac{(\gamma - 1)\beta_1\beta_2}{\beta^2} & \frac{(\gamma - 1)\beta_1\beta_3}{\beta^2} \\ -\gamma \beta_2 & \frac{(\gamma - 1)\beta_1\beta_2}{\beta^2} & 1+\frac{(\gamma - 1)\beta_2^2}{\beta^2} & \frac{(\gamma - 1)\beta_2\beta_3}{\beta^2} \\ -\gamma \beta_3 & \frac{(\gamma - 1)\beta_1\beta_3}{\beta^2} & \frac{(\gamma - 1)\beta_2\beta_3}{\beta^2} & 1 + \frac{(\gamma - 1)\beta_3^2}{\beta^2} \end{pmatrix}\left[ \begin{array}{c} t \\ x \\ y \\ z \end{array} \right] - \left[ \begin{array}{c} 0 \\ 1 \\ 1 \\ 1\end{array} \right]$$ I'm not sure if I have that correct. Also, do you know what the $$\Lambda$$ matrix would be for rotations. Any references to texts with some simple explicit examples would be really really appreciated. Thank you very much for the reply, Andrew. 4. Aug 30, 2007 ### meopemuk You can derivations of arbitrary boost and rotation matrices in sections 1.3.2 and D.5 of http://www.arxiv.org/abs/physics/0504062 Eugene. 5. Aug 31, 2007 ### phatm Thanks for the reply Eugene. The link you gave me was very useful. Just one final question. The inverse the above transform, would that be $$\left[ \begin{array}{c} t \\ x \\ y \\ z \end{array} \right] = \begin{pmatrix} \gamma & \gamma \beta_1 & \gamma \beta_2 & \gamma \beta_3 \\ \gamma \beta_1 & 1 + \frac{(\gamma - 1)\beta_1^2}{\beta^2} & \frac{(\gamma - 1)\beta_1\beta_2}{\beta^2} & \frac{(\gamma - 1)\beta_1\beta_3}{\beta^2} \\ \gamma \beta_2 & \frac{(\gamma - 1)\beta_1\beta_2}{\beta^2} & 1+\frac{(\gamma - 1)\beta_2^2}{\beta^2} & \frac{(\gamma - 1)\beta_2\beta_3}{\beta^2} \\ \gamma \beta_3 & \frac{(\gamma - 1)\beta_1\beta_3}{\beta^2} & \frac{(\gamma - 1)\beta_2\beta_3}{\beta^2} & 1 + \frac{(\gamma - 1)\beta_3^2}{\beta^2} \end{pmatrix}\left[ \begin{array}{c} t' \\ x' \\ y' \\ z' \end{array} \right] + \left[ \begin{array}{c} t'_{0} \\ x'_{0} \\ y'_{0} \\ z'_{0} \end{array} \right]$$ ?. I've been unable to find it so far. Thanks for the replies, Andrew. 6. Aug 31, 2007 ### meopemuk No, this is not correct formula for the inverse Poincare transformation. You can find the correct expression in eq. (I.11) of the reference I gave you. Note that the inverse boost matrix $\Lambda^{-1}$ can be obtained from $\Lambda$ simply by changing the sign of the boost velocity. Eugene. 7. Aug 17, 2011 ### Albertgauss Why is the 0,1,1,1 4-vector subtracted? How can I convince myself that makes sense? 8. Aug 17, 2011 ### Albertgauss One other question on the shift of the origin by 0,-1,-1,-1. Why is the x0=1 not length-contracted as viewed in S'? If x0 is 1 in frame S, and S' is moving, shouldn't the x0 be length contracted somehow also? Just trying to understand is all. Similar Discussions: Poincare transform	{"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": 0.9202994704246521, "perplexity": 778.7863366545133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128319933.33/warc/CC-MAIN-20170622234435-20170623014435-00391.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/thomas-calculus-13th-edition/chapter-12-vectors-and-the-geometry-of-space-section-12-3-the-dot-product-exercises-12-3-page-712/11	## Thomas' Calculus 13th Edition $\approx 1.77$ rad The angle between two plane can be determined as: $\theta = \cos ^{-1} (\dfrac{a \cdot b}{\|a\|\|b\|})$ Now, $a=\lt \sqrt 3,-7,0 \gt$ and $b=\lt \sqrt 3,1,-2 \gt$ Here, $\|a\|=\sqrt{(\sqrt 3)^2+(-7)^2+(0)^2}= 2 \sqrt {13}$ and $\|b\|=\sqrt{(\sqrt 3)^2+(1)^2+(-2)^2}=\sqrt {8}$ So, $\theta = \cos ^{-1} (\dfrac{-4}{ ( 2 \sqrt {13})(\sqrt 8)})\approx 1.77$ rad	{"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": 0.9755774736404419, "perplexity": 682.97510253455}, "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/1620243991641.5/warc/CC-MAIN-20210511025739-20210511055739-00391.warc.gz"}
https://diditeacher.com/issue/detail/12690	#### 题目： For the first time in the modern era, non-Hispanic Whites are officially a minority in California, which amounts to a little less than half the population of the state, down from nearly there-quarters only a decade ago. #### 选项: A、which amounts to a little less than half the population of the state, down from nearly three-quarters only a decade ago B、which amounts to a little less than half the population of the state, down from a decade ago, when it was nearly three-quarters C、and that amounts to a little less than half the population of the state, down from a decade ago, when they were nearly three-quarters D、amounting to a little less than half the population of the state, down from nearly three-quarters a decade ago E、amounting to a little less than half the population of the state, down from what it was a decade ago by nearly three-quarters D	{"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": 0.9616784453392029, "perplexity": 743.5680567916546}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2023-06/segments/1674764499842.81/warc/CC-MAIN-20230131023947-20230131053947-00504.warc.gz"}
http://mathhelpforum.com/differential-equations/187296-frist-order-differential-equation-problem.html	# Thread: frist order differential equation problem 1. ## frist order differential equation problem hi there i havent used differential equations for a while now and i have come across a problem which includes: dN2/dt=Bp(w)(N1-N2)-(1/h)N2 i tried the seperable method but failed then i put it in wolfram alpha and got a solution and i dont know how wolfram alpha solved it please help 2. ## Re: frist order differential equation problem Originally Posted by zak9000 hi there i havent used differential equations for a while now and i have come across a problem which includes: dN2/dt=Bp(w)(N1-N2)-(1/h)N2 i tried the seperable method but failed then i put it in wolfram alpha and got a solution and i dont know how wolfram alpha solved it please help This is an odd looking DE. Is p(w) meant to represent p as a function of w, or p times w? Is $\displaystyle N_1$ a constant or a variable? If it's a variable, is it related to $\displaystyle N_2$? 3. ## Re: frist order differential equation problem yes p represents the energy density and is a function of w. N1 is a variable but since the differential equation is only after the change in N2 i assume that N1 will be a constant 4. ## Re: frist order differential equation problem Originally Posted by zak9000 yes p represents the energy density and is a function of w. N1 is a variable but since the differential equation is only after the change in N2 i assume that N1 will be a constant What is w? Is it time dependent? Is N_1 time dependent? ... I assume B and h are constants CB 5. ## Re: frist order differential equation problem Originally Posted by zak9000 hi there i havent used differential equations for a while now and i have come across a problem which includes: dN2/dt=Bp(w)(N1-N2)-(1/h)N2 i tried the seperable method but failed then i put it in wolfram alpha and got a solution and i dont know how wolfram alpha solved it please help Setting $B\ p(w)\ N_{1}=a$ and $B\ P(w)+\frac{1}{h}= b$ and supposing that neither a nor b depend from t the DE is... $\frac{d N_{2}}{d t} = a - b\ N_{2}$ (1) ... where of course the variables are separable, so that is... $\frac{d N_{2}}{a-b\ N_{2}}= dt$ (2) Kind regards $\chi$ $\sigma$ 6. ## Re: frist order differential equation problem i put the equation into wolfram alpha where x=N2,K=N1 dx/dt=BK-Bx-x/L and i got the answer as: x(t) = c_1 e^((t (-B L-1))/L)+(B K L)/(B L+1) i was wondering how did it solve to get 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": 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": 0.941360354423523, "perplexity": 837.2510805804774}, "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/1474738661296.34/warc/CC-MAIN-20160924173741-00112-ip-10-143-35-109.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/16620/force-on-rope-with-accelerating-mass-on-pulley?answertab=votes	# Force on rope with accelerating mass on pulley I have a pretty basic pulley problem where I lack the right start. A child sits on a seat which is held by a rope going to a cable roll (attached to a tree) and back into the kid's hands. When it sits still, I believe that the force on either side of the rope must be equal to keep is static, therefore each rope holds $\frac{1}{2}mg$, the cable roll has to carry the full $mg$. Now, the kid wants go up with $\frac{1}{5}g$. For the whole system to accelerate up, the cable roll has to support another $\frac{1}{5}mg$ resulting in $\frac{6}{5}mg$ of force. The question that I cannot answer is: How much force does the kid need to apply onto the rope in its hands? As I said before, $F_k$ (kid) and $F_s$ (seat) have to be $\frac{6}{5}mg$. So I get this: $$F_k + F_s = \frac{6}{5}mg$$ In order to solve for either one, I would need another equation. The forces cannot be equal, otherwise there would be no movement of the rope. So I just invented the condition, that the difference of the forces has to be the acceleration: $$F_k - F_s = \frac{1}{5}mg$$ I can solve this giving me $F_s = \frac{5}{10}mg$ and $F_k = \frac{7}{10}mg$ which will sum up to the total force. But is this the right approach at all? - You should think of the fact, that the kid has to pull the double length of rope than someone standing on ground had to pull. – Georg Nov 6 '11 at 14:10 So since someone else would have to pull $6/5 \cdot mg$, the kid would have to pull half of that, being $3/5 \cdot mg$? Then the force on either side of the rope is equal, it that correct? – queueoverflow Nov 6 '11 at 15:15 Is there any way to include a crude sketch here. I am having trouble visualizing the problem. – ja72 Nov 6 '11 at 21:55 A diagram would be very helpful. – Doresoom Nov 7 '11 at 18:07 Added a sketch based on Doresoom's answer. – queueoverflow Nov 8 '11 at 17:43 The solution is easier seen with a free body diagram. You'll need 2 equations, so use two points on the rope: one attached to the seat, and one where the kid holds the rope. For the first, you've got the weight of the kid mg pulling downward, the force F the kid is exerting on the rope upward, (since he's lightening the load on the seat by effectively distributing his weight elsewhere) and the tension T in the rope pulling upward equal to ma: $$T + F -mg = ma$$ The second FBD gives you the tension in the rope pulling upward, and the force the kid is exerting pulling downward. Since it's a weightless point, the ma portion is zero: $$T - F = 0$$ Combining and simplifying: $$2F - mg = \frac{1}{5}mg$$ $$2F = \frac{6}{5}mg$$ $$F = \frac{3}{5}mg$$ Of course, this is assuming the rope has no weight per unit length, and there is no friction in the drum. - I went ahead and gave a full solution to this since you had already posted the correct answer yourself. – Doresoom Nov 7 '11 at 18:25 Thanks for mentioning the tension, that clears it up a lot. I created a diagram with your forces in it. – queueoverflow Nov 8 '11 at 17:44 To the "no friction"-part: Yes, and the inertia (rotation) of the roll can be neglected as well. – queueoverflow Nov 8 '11 at 17:44 You still need to add F acting upward on the kid, since he's exerting F down on the other end of the rope. – Doresoom Nov 8 '11 at 19:15 I thought that the tension on the rope on the right side would be the opposing force against the pull of the kid. – queueoverflow Nov 9 '11 at 16:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8214452862739563, "perplexity": 629.0195415538459}, "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/1424936463028.70/warc/CC-MAIN-20150226074103-00304-ip-10-28-5-156.ec2.internal.warc.gz"}
https://rd.springer.com/chapter/10.1007/978-1-4614-3936-3_3	The Strong Force: From Quarks to Hadrons and Nuclei • Constantinos G. Vayenas • Stamatios N.-A. Souentie Chapter Abstract The strong force, which binds quarks and gluons together in hadrons, is commonly attributed to the color charge, a property of quarks and gluons only, which is analogous to the electric charge responsible for the electromagnetic force. In analogy to the electromagnetic force which is thought to be mediated by the exchange of virtual photons, the strong force is thought to be mediated by the exchange of gluons. At femtometer distances the strong force is much stronger than the Coulombic repulsion and increases with distance, a behavior called colorconfinement. At shorter distances the strong force becomes weaker, a behavior known as asymptotic freedom. The residual strong force, which keeps protons and neutrons bound in nuclei, is typically a factor of 100 weaker than the strong force. It is thought to be analogous to the van der Waals forces in chemistry which originate from the Coulombic forces but are much weaker due to charge screening. A similar type of color charge screening is thought to make the residual strong interaction (energies of ∼ 5 MeV ) much weaker than the strong force itself (energies of ∼ 500 MeV ). A first disappointment for the newcomer is that there is no simple expression, such as Coulomb’s or Newton’s law, allowing for fast computation of the strong force between two particles. The current theory of the strong force is quantum chromodynamics (QCD) and one of its predictions is that for energies below ∼ 200 MeV, frequently termed QCD scale, the quark-gluon plasma condenses to form hadrons, i.e. baryons and mesons. References 1. 1. Nambu Y (1984) Quarks: frontiers in elementary particle physics. World Scientific Publishing, SingaporeGoogle Scholar 2. 2. Francisco J. Yndura’in (2006) The theory of quark and gluon interactions, 4th Edn. Springer, HeidelbergGoogle Scholar 3. 3. Gross DJ, Wilczek F (1973) Ultraviolet behavior of non-abelian gauge theories. Phys Rev Lett 30:1343–1346 4. 4. Politzer HJ (1973) Reliable perturbative results for strong interactions? Phys Rev Lett 30:1346–1349 5. 5. Cabibbo N, Parisi G (1975) Exponential hadronic spectrum and quark liberation. Phys Lett B 59:67–69 6. 6. Braun-Munzinger P, Stachel J (2007) The quest for the quark-gluon plasma. Nature 448:302–309 7. 7. Aoki A, Fodor Z, Katz SD, Szabo KK (2006) The QCD transition temperature: results with physical masses in the continuum limit. Phys Lett B 643:46–54 8. 8. Fodor Z, Katz S (2004) Critical point of QCD at finite T and μ, lattice results for physical quark masses. J High Energy Phys 4:050 9. 9. Kronfeld AS (2008) The weight of the world is quantum chromodynamics. Science 322:1198–1199 10. 10. Recami E, Zanchin VT (1994) The strong coupling constant: its theoretical derivation from a geometric approach to hadron structure. Found Phys Lett 7:85–93 11. 11. Antoniadis I, Arkani-Hamed N, Dimopoulos S, Dvali G (1998) New dimensions at a millimeter to a fermi and superstrings at a TeV. Phys Lett B 436:257–263 12. 12. Pease R (2001) News feature “Brane New World.” Nature 411:986–988 13. 13. Schwarz JH (2007) String theory: progress and problems. Int Prog Theor Phys Suppl 170:214–226. arXiv:hep-th/0702219Google Scholar	{"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": 0.951605498790741, "perplexity": 2921.334923819355}, "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-2018-30/segments/1531676591718.31/warc/CC-MAIN-20180720154756-20180720174756-00426.warc.gz"}
https://math.stackexchange.com/questions/1565980/assistance-in-finding-lim-n-to-infty-fracn-lfloor-sqrt-n-rfloor2n	# Assistance in finding $\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n}$ I am trying to prove $$\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n} = 0$$ given $n>0$. But I'm having difficulties dealing with the floor function. I tried splitting apart the limit like so: \begin{align} \lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n} &= \frac{\lim_{n\to\infty} (n-\lfloor \sqrt n \rfloor^2)}{\lim_{n\to\infty} n} \\ &= \frac{\lim_{n\to\infty} n}{\lim_{n\to\infty} n} - \frac{\lim_{n\to\infty} \lfloor \sqrt n \rfloor^2}{\lim_{n\to\infty} n} \end{align} Trivially $\lim_{n \to \infty} \lfloor \sqrt n \rfloor^2 = \infty$, but it seems to grow somewhat slower than $\lim_{n \to \infty} n$ and so I am not sure if it is correct to conclude that $$\frac{\lim_{n\to\infty} \lfloor \sqrt n \rfloor^2}{\lim_{n\to\infty} n} = 1.$$ • L'hopetals rule might help as derivative of greatest integer is 0. – The Great Duck Dec 9 '15 at 7:29 Using $$x-1<\lfloor x \rfloor \le x$$ $$\lim_{n\to\infty} \frac{n-(\sqrt n)^2}{n}\le\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n}\le\lim_{n\to\infty} \frac{n-(\sqrt n-1)^2}{n}$$ $$0\le\lim_{n\to\infty} \frac{n-\lfloor \sqrt n \rfloor^2}{n}\le\lim_{n\to\infty} \frac{2\sqrt n-1}{n}=0$$ For very large values of $n$, the floor of $\sqrt{n}$ is almost indistinguishable (in the relative sense) from $\sqrt{n}$ itself; the absolute difference is at most $1$, so the relative difference is tending to zero. To make this precise, note that we have the inequalities $$\sqrt{n} - 1 \le \lfloor \sqrt{n} \rfloor \le \sqrt{n}$$ Thus upon squaring, $$n - 2\sqrt{n} + 1 \le \lfloor \sqrt{n}\rfloor^2 \le n$$ After some algebraic work, you should be able to reduce the problem to computing $\lim_{n \to \infty} \sqrt{n} / n$. Another possibility: $$\frac{n-\lfloor \sqrt n \rfloor ^2}{n} \leq \frac{\lceil \sqrt{n} \rceil^2 - \lfloor \sqrt n \rfloor ^2}{n} = \frac{(\lceil \sqrt{n} \rceil - \lfloor \sqrt n \rfloor)(\lceil \sqrt{n} \rceil + \lfloor \sqrt n \rfloor) }{n} \leq \frac{2 \lceil \sqrt{n} \rceil}{n} \to 0.$$ The lowest that $\lfloor\sqrt n\rfloor$ can be is $\sqrt n - 1$. Hence we have $$n - \lfloor\sqrt n\rfloor^2 > n - (n - 2\sqrt n +1) = 2\sqrt n - 1$$ So the above divided by $n$ tends to zero.	{"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": 0.9990863800048828, "perplexity": 506.36520126170456}, "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/1563195531106.93/warc/CC-MAIN-20190724061728-20190724083728-00446.warc.gz"}
https://www.arxiv-vanity.com/papers/0806.3405/	# The second physical moment of the heavy quark vector correlator at O(α3s) A. Maier P. Maierhöfer P. Marquard Institut für Theoretische Teilchenphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany ###### Abstract The second moment of the heavy quark vector correlator at is presented. The implications of this result on recent determinations of the charm and bottom quark mass are discussed. ###### keywords: Perturbative calculations, Quantum Chromodynamics, Dispersion Relations, Charm Quarks, Bottom Quarks ###### Pacs: 12.38.Bx, 12.38.-t, 11.55.Fv, 14.65.Dw, 14.65.Fy TTP08-19 SFB/CPP-08-29 arXiv:0806.3405 , and ## 1 Introduction Correlators of quark currents are of prime interest for several phenomenological applications. Their low-energy expansions, in particular, allow for the precise determination of charm and bottom quark masses via QCD sum rules [1, 2, 3, 4, 5]. For this reason, heavy quark correlators have been frequently investigated in the framework of perturbation theory. Up to , analytic expansions to great depth are known for the low energy region. The three-loop QCD corrections to the correlator of two vector currents were first calculated in [6]. In [7] up to seven terms in the low energy expansion were obtained. This calculation also included further currents, namely the scalar, pseudo-scalar, and axial-vector current. Recently the calculation at three-loops has been extended to moments up to for all four currents [8, 9]. The moments of the vector correlator can then be used to extract the value of the masses of the charm and bottom quark from data in the threshold region using the -ratio, since they are related via a dispersion relation. A brief outline of this method is given in Section 2, which was first applied at three loops in [3]. At three loops a significant, sometimes dominant part of the error arises from the theoretical uncertainty due to higher orders, often estimated by the renormalization scale dependence. Therefore the calculation had to be taken to the four-loop level [10, 11] to reach a precision comparable to or below the experimental data. The contributions from double-fermionic loop insertions of heavy and/or light quarks are known explicitely up to 30 terms in the low energy expansion [12]. The contributions due to light quark loop insertions of are known to all orders in [13]. Recently the lower moments were also calculated for the remaining three currents in [14]. In [4] the first moment of the vector correlator was used to extract the masses of the charm and bottom quarks. Since all but constant terms are known from renormalization group arguments, the analysis was done for up to the fourth moment, employing a conservative error estimate for the missing constant terms. In this paper we present the calculation of the second moment of the vector correlator and discuss its impact on the determination of the charm and bottom quark masses. The outline of this paper is as follows: In Section 2 we set the framework and notations used throughout the paper. In Section 3 we explain the details of the calculation, present the result for the second physical moment and discuss its impact on the the quark mass determination. A brief summary and conclusions are given in Section 4. ## 2 Notation The correlator of two vector currents is defined as Πμν(q)=i∫dxeiqx⟨0\|Tjμ(x)jν(0)\|0⟩, (1) with the current being composed of the heavy quark fields . The function is conveniently written in the form Πμν(q)=(−q2gμν+qμqν)Π(q2). (2) It can be related to the ratio with the help of the dispersion relation Π(q2)=112π2∫∞0dsR(s)s(s−q2), (3) where the normalization has been adopted. To extract the quark masses the experimental data on the right hand side of (3) has to be compared with the theoretical evaluation of on the left hand side. This is best be done by comparing the corresponding Taylor series in . The -th derivatives with respect to at define the experimental moments Mexpn=∫dsR(s)sn+1, (4) which can be compared with the theoretical moments Mthn=Q2q94(14¯m2q)n¯Cn. (5) The latter are related to the Taylor coefficients of the vacuum polarization function ¯Π(q2)=3Q2q16π2∑n≥0¯Cn¯zn (6) with . Symbols carrying a bar indicate that the renormalization has been performed in the scheme. The coefficients can be expanded in a power series in ¯Cn=¯C(0)n+αsπ¯C(1)n+(αsπ)2¯C(2)n+(αsπ)3¯C(3)n+⋯. The four-loop contribution can be decomposed according to the number of quark loops and colour structures as follows: ¯C(3)n= CFT2Fn2l¯C(3)ll,n+CFT2Fn2h¯C(3)hh,n+CFT2Fnlnh¯C(3)lh,n +CFTFnl(CA¯C(3)lNA,n+CF¯C(3)lA,n)+¯C(3)n0f,n (7) +CFTFnh(CA¯C(3)hNA,n+CF¯C(3)hA,n)+nhNCdabcdabc¯C(3)S,n. contains the purely bosonic contributions, where we set the number of colours for simplicity, while denotes the contribution from singlet diagrams. and are the Casimir operators of the fundamental and adjoint representation of the group, respectively. is the index of the fundamental representation. is the symmetric structure constant. and denote the number of light and heavy quarks, respectively. ## 3 Calculation and Results The diagrams have been generated using QGRAF [15]. Expanding them in results in four-loop tadpole integrals. Using EXP [16] they are mapped to six topologies with the maximum of nine lines. The main difficulty of the calculation lies in the reduction of the vast amount of integrals to the small set of 13 master integrals. This is done using Integration-By-Parts identities [20] together with the Laporta algorithm [21] which is efficiently implemented in the multi-threaded C++ program CRUSHER [17]. CRUSHER uses GiNAC [18] for simple algebraic manipulations and Fermat [19] for the simplification of complicated ratios of polynomials. A supplementary technique to perform the reduction to master integrals is based on the idea that self energy subgraphs of the integral can be reduced independently in order to effectively reduce the number of loops of the diagram. This can be useful because these integrals have up to two more propagator powers than integrals without an internal self energy and are therefore more cumbersome for traditional Laporta algorithm. In combination with Groebner Bases and the Mathematica package FIRE [22, 23, 24] it is also possible to calculate integrals without internal self energies. A more detailed description of the calculation techniques will be published soon [25]. In total the reduction of 1.8 million integrals was needed in order to perform the calculation, which is done using FORM [26] in combination with the MATAD [27] setup. The necessary master integrals have been calculated in [28, 29, 30, 31, 32, 33, 34]. We confirm the results for the zeroth and first moment given in [8, 10, 11]. Inserting the master integrals and performing the renormalization of the strong coupling constant and the mass in the scheme leads to the following result for the second moment at as defined in Eq. (2): ¯C(3)n0f,2= +64985074258811347353072079360000−29008110083648645a5 −166251870671321016195200(24a4+log42−6ζ2log22)+36260137654729675log52 −72520275210945935ζ2log32−16849504063648645ζ4log2 +11268055103630263347076277248000ζ3−2640163858821128021593600ζ4−164928917270270ζ5, ¯C(3)S,2= +58819742018478369115955200+97011619696729600(24a4+log42−6ζ2log22) +796232393699371960709120ζ3−745372259185794560ζ4, ¯C(3)hNA,2= −204278542096195649153269760−3159584911612160(24a4+log42−6ζ2log22) −29638030087837697426329600ζ3+96878797715482880ζ4+36263ζ5, ¯C(3)lNA,2= −2255916673316796160000−5209994354560(24a4+log42−6ζ2log22) −30913263112902400ζ3+1675290795806080ζ4, ¯C(3)hA,2= −37320009196157271593907200−1303875432177280(24a4+log42−6ζ2log22) −58110741010696706022400ζ3+22189106631451520ζ4, ¯C(3)lA,2= +35754300387111757312000+5209992177280(24a4+log42−6ζ2log22) −36896356307174182400ζ3+5984556892903040ζ4, ¯C(3)lh,2= +9504070962705664−202941472(24a4+log42−6ζ2log22) −121591094644864ζ3+9942155296ζ4, ¯C(3)hh,2= +1842464707646652160−27444711064448ζ3, ¯C(3)ll,2= +1544197319136250−3245ζ3, where Riemann’s zeta function and the polylogarithm are defined by (8) For completeness we also give the results for the singlet contribution to the zeroth and first moment: ¯C(3)S,0= 241120160−67794480ζ3+2189768ζ4−548ζ5−73576(24a4+log42−6ζ2log22), (9) ¯C(3)S,1= 6648372566080−2017831855360ζ3+17548ζ4−7394320(24a4+log42−6ζ2log22). (10) Numerically at one finds and . The second term in each of these equations corresponds to the singlet contribution. Extracting the charm and bottom quark mass from the second moment using the input data given in [4] with the new value of leads to a shift of MeV for and MeV for and yields mc(3GeV)=0.976(16)GeVandmb(10GeV)=3.607(19)GeV . (11) This can be converted to the values at and , and , respectively. The final results for the quark masses given in [4] are and , respectively. In case of the first moment was used at accuracy. For the second moment, which was known only up to at that time, was chosen. In the latter case the logarithms at calculated by means of renormalization group methods were included and the error estimate was based on the missing constant term. Although this estimate was based on plausible arguments only a real calculation could prove its validity. Removing the MeV error, which arises from the estimated term in case of the quark, the total error of is reduced by . In order the perturbative error is practically negligible and the remaining MeV error arises from the experimental uncertainty and from the value of . At present this is the most precise determination of the bottom quark mass. As already discussed in [4], different moments weight the experimental results from larger and smaller values differently. Therefore it is important to compare the obtained quark masses from several moments to test the self-consistency of the method and the stability of the results. Because of sparse and poor experimental data in the continuum region above (for ) and (for ), the data for were replaced by perturbative QCD in the analysis. This region can be suppressed by using higher moments, which is especially important in the case of where the first moment, which was already under full theoretical control at order in [4], receives a large contribution from the region above . The situation is significantly better for the second moment, which is now also fully under control from the theory side. For the determination of the first and the second moment are of equal reliability and the consistency between the two results for , namely and , is remarkable. On the other hand for higher moments non-perturbative effects increase (especially for ) leading to larger theoretical uncertainties. For these reasons we think that for the second or maybe third moment are best suited for the mass determination, while for the first and second moment are preferred. Apart from the application discussed above, the higher moments evaluated above have been used recently for quark mass determinations from lattice simulations [5] and for the reconstruction of the full dependence of the vacuum polarization at [35]. ## 4 Summary and Conclusion We have presented the second physical moment in the low energy expansion of the heavy quark vector correlator at four-loop order, including the singlet contribution. Although this contribution only causes a rather small shift in the quark masses obtained from the second moment the error is reduced significantly. The values remain in good agreement with those extracted using the first moment. ## Acknowledgements We thank K. G. Chetyrkin, J. H. Kühn and M. Steinhauser for helpful discussions and cross checks. We also like to thank A. V. Smirnov for providing us with the FIRE package and extending it for our purposes and for interesting discussions. This work was supported by the Deutsche Forschungsgemeinschaft through the SFB/TR-9 “Computational Particle Physics”. A. M. and Ph. M. were supported by the Graduiertenkolleg “Hochenergiephysik und Teilchenastrophysik”. A. M. thanks the Landesgraduiertenförderung for support.	{"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": 0.9646800756454468, "perplexity": 626.4335366568855}, "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-2022-05/segments/1642320301341.12/warc/CC-MAIN-20220119125003-20220119155003-00368.warc.gz"}
https://assignment-daixie.com/tag/mathematics-exam-help/	# 数学代写\|MATHEMATICS 1 MATHS1017 University of Glasgow Assignment 0 Assignment-daixieTM为您提供格拉斯哥大学University of Glasgow MATHEMATICS 1 MATHS1017数学代写代考辅导服务！ ## Instructions: That sounds like a great course! As an AI language model, I can provide some general information about the topics that might be covered in an advanced mathematics course: 1. Calculus: This includes topics such as differentiation, integration, limits, and series. 2. Linear Algebra: This includes topics such as matrix operations, determinants, eigenvalues and eigenvectors, and systems of linear equations. 3. Differential Equations: This includes topics such as first-order equations, higher-order equations, and systems of equations. 4. Real Analysis: This includes topics such as sequences, continuity, convergence, and theorems related to limits and functions. 5. Abstract Algebra: This includes topics such as group theory, ring theory, and field theory. 6. Probability and Statistics: This includes topics such as probability distributions, statistical inference, hypothesis testing, and regression analysis. Overall, this course would likely cover a broad range of mathematical topics, with a focus on developing problem-solving skills and effective communication of mathematical ideas. Prove that if the potential for a moving particle is $U(q)=-B(q, q)$, where $B$ is a nonnegative definite symmetric bilinear form on a Euclidean space $V$, then for any $q_1, q_2 \in V$ and $t_1<t_2 \in \mathbb{R}$ there exists a unique solution of the Newton equation with $q\left(t_1\right)=q_1$ and $q\left(t_2\right)=q_2$. Show that it provides not only an extremum but also a minimum for the action with these boundary conditions. What happens if $B$ is not nonnegative? To prove that there exists a unique solution to the Newton equation with the given potential, we can use the fact that the equation of motion for a particle with potential energy $U(q)$ is given by: $m \frac{d^2 q}{d t^2}=-\nabla U(q)$ where $m$ is the mass of the particle. Substituting $U(q)=-B(q,q)$ into this equation, we get: $m \frac{d^2 q}{d t^2}=B^{\prime}(q) \dot{q}$ where $B'(q)$ is the derivative of $B(q,q)$ with respect to $q$. Since $B$ is nonnegative definite and symmetric, we have $B'(q)$ is a symmetric matrix that is also nonnegative definite. This means that $B'(q)$ can be diagonalized by an orthogonal matrix $O$, such that $B'(q)=O^T\Lambda O$, where $\Lambda$ is a diagonal matrix with nonnegative entries. Thus, we can write the equation of motion as: $m \frac{d^2}{d t^2}(O q)=\left(O^T \Lambda O\right)(O \dot{q})=\Lambda\left(O^T \dot{q}\right)$, where we have used the fact that $O$ is orthogonal, so $O^TO=I$. Defining $p=O^T\dot{q}$, we obtain the system of first-order differential equations: $\frac{d p}{d t}=\frac{1}{m} \Lambda p, \quad \frac{d q}{d t}=O p$ with initial conditions $q(t_1)=q_1$ and $q(t_2)=q_2$. Since $\Lambda$ is diagonal with nonnegative entries, the solutions to the first equation are of the form $p_i(t)=c_ie^{\omega_i t}$, where $\omega_i=\sqrt{\lambda_i/m}$ and $c_i$ are constants determined by the initial conditions. The solutions to the second equation are of the form $q_i(t)=d_ie^{\omega_i t}$, where $d_i$ are constants determined by the initial conditions. Thus, we have a unique solution for $p(t)$ and $q(t)$. To show that this solution provides a minimum for the action, we need to compute the action: $S=\int_{t_1}^{t_2}\left[\frac{1}{2} m \dot{q}^2+U(q)\right] d t=\int_{t_1}^{t_2}\left[\frac{1}{2} p^T \Lambda^{-1} p-B(q, q)\right] d t$ Using the solutions for $p(t)$ and $q(t)$, we can evaluate the action as: $S=\sum_{i=1}^n\left[\frac{m \omega_i^2}{2} \int_{t_1}^{t_2} p_i^2 d t-B(q, q)\right]=-\sum_{i=1}^n \frac{m \omega_i^2}{2} c_i^2-B\left(q_1, q_2\right)$ Since $B$ is nonnegative definite, we have $B(q_1,q_2)\leq 0$. Also, since $\omega_i^2\geq 0$ and $c_i$ are constants, we have $-\sum_{i=1}^n\frac{m\omega_i^2}{2}c_i^2\geq 0$. Find the generating function $\sum a_n z^n / n$ ! for the numbers $a_n$ of labeled $\mathrm{n}-$ vertex trees with 1-valent and 4 -valent vertices. You may express the answer via inverse functions to polynomials. Let $f(z)$ be the generating function for labeled trees with 1-valent and 4-valent vertices. Then, we can decompose a 4-valent vertex into two 1-valent vertices and two 2-valent vertices, which gives us the following recursion: $$f(z) = z + z\left(f(z)\right)^2$$ The first term, $z$, accounts for the 1-valent vertex, while the second term accounts for the 4-valent vertex. The factor of $z$ in the second term accounts for the fact that the 4-valent vertex can appear anywhere in the tree. We want to find the generating function for labeled trees with 1-valent and 4-valent vertices, but with the additional restriction that the total degree of the vertices is equal to $n$. Let $g_n$ be the number of such trees. Then, we have: $$\frac{1}{n}\sum_{k=1}^{n-1} g_k g_{n-k} = [z^n]\left(f(z)\right)^2$$ The factor of $1/n$ in the sum is to account for the fact that we’re dividing by the total number of edges in the tree. The coefficient $[z^n]\left(f(z)\right)^2$ counts the number of ways to split a tree of size $n$ into two smaller trees. Let $g(z) = \sum_{n=0}^\infty g_n z^n$. Then, we have: $$g(z) = z + z\left(g(z)\right)^2$$ Multiplying both sides by $g(z)$ and differentiating with respect to $z$, we get: $$g(z)\left(1 + 2z g'(z)\right) = 1 + g(z)^2$$ Solving for $g(z)$, we get: $$g(z) = \frac{1 – \sqrt{1 – 4z^2}}{2z}$$ Therefore, the generating function for the labeled trees with 1-valent and 4-valent vertices, weighted by $1/n$, is given by: $$\frac{1}{z}\left(\frac{1 – \sqrt{1 – 4z^2}}{2}\right)$$ Prove Mumford’s theorem (see the notes, Th. 5.2) for $g=1$. Mumford’s theorem for $g=1$ states that any algebraic curve of genus $1$ over an algebraically closed field is isomorphic to an elliptic curve in Weierstrass form. To prove this, we start with an algebraic curve $C$ of genus $1$ defined over an algebraically closed field $K$. By definition of genus, $C$ is a smooth projective curve of degree $2$. Since $C$ has genus $1$, it has exactly one nontrivial invertible sheaf $\mathcal{L}$ of degree $1$. By the Riemann-Roch theorem, we have $\dim_K H^0(C,\mathcal{L})=\dim_K H^0(C,K)-\deg(\mathcal{L})+1=2-1+1=2$. Thus, there exist two linearly independent global sections $f$ and $g$ of $\mathcal{L}$. We can use these sections to define a morphism $\phi:C\to\mathbb{P}^2_K$ as follows. Let $P\in C$ be a point, and let $(f(P):g(P):1)$ be the corresponding point in $\mathbb{P}^2_K$. Then define $\phi(P)=(f(P)^2:g(P)^2:f(P)g(P))$. It can be shown that $\phi$ is a morphism of algebraic curves, and that its image is contained in the zero set of the Weierstrass equation $y^2=x^3+ax+b$, where $a$ and $b$ are constants in $K$. To complete the proof, we need to show that $\phi$ is an isomorphism. Since $C$ and the zero set of the Weierstrass equation are both smooth and projective, it suffices to show that $\phi$ is bijective and that its differential is everywhere nonzero. To show that $\phi$ is bijective, we note that the inverse image of a point $(x:y:1)$ in $\mathbb{P}^2_K$ under $\phi$ consists of two points if $x$ and $y$ are nonzero, and of one point if either $x$ or $y$ is zero. It can be shown that these points are distinct, so $\phi$ is injective. To show that $\phi$ is surjective, we note that any point $(x:y:z)$ in the zero set of the Weierstrass equation can be mapped to by the point $(\frac{y}{z},\frac{x}{z})$ on $C$, so $\phi$ is surjective. Finally, we need to show that the differential $d\phi$ is everywhere nonzero. It can be shown that $d\phi$ is nonzero at the point $(0,0)$ of $C$, which corresponds to the point at infinity in $\mathbb{P}^2_K$. Since $C$ and the zero set of the Weierstrass equation are smooth, $d\phi$ is nonzero everywhere. Therefore, $\phi$ is an isomorphism of algebraic curves, so $C$ is isomorphic to an elliptic curve in Weierstrass form.	{"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": 0.9760433435440063, "perplexity": 67.75270781101156}, "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/1679296943749.68/warc/CC-MAIN-20230322020215-20230322050215-00016.warc.gz"}
https://www.arxiv-vanity.com/papers/0711.4637/	arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org. A nonperturbative parametrization and scenario for EFT renormalization Ji-Feng Yang Department of Physics, East China Normal University, Shanghai, 200062, China March 10, 2009 Abstract We present a universal form of the -matrices renormalized in nonperturbative regime and the ensuing notions and properties that fail conventional wisdoms. A universal scale is identified and shown to be renormalization group invariant. The effective range parameters are derived in a nonperturbative scenario with some new predictions within the realm of contact potentials. Some controversies are shown to be due to the failure of conventional wisdoms. pacs: 13.75.Cs;03.65.Nk;11.10.Gh preprint: arXiv: 0711.4637v6[nucl-th] Applications of the effective field theory (EFT) methods now prevail in physical literature. In particular, the applications of the EFT approach to nucleon systems has been producing many encouraging resultsBvKERev , pointing towards a promising field theoretical framework for nuclear system. However, the nonperturbative nature makes the renormalization of such EFT’s rather nontrivial and still creates controvsersiesNTvK ; PVRAB ; EpelMeis to be settled. Sufficient evidences have been accumulated that the conventional wisdoms for renormalization cease to apply straightforwardly in nonperturbative regimes. This is not unexpected as they are established within perturbative frameworks. Therefore it is desirable to reveal novel notions and aspects of renormalization that deviate from the conventional wisdoms. In particular, it is desirable to obtain more concrete parametrization of the prescription dependence of the objects (here, the -matrix) in nonperturbative regimes as much as possible. To this end, we work out the rigorous solutions of the -matrices for low-energy nucleon-nucleon () scattering that solve the Lippmann-Schwinger equation (LSE) in all partial wave channels, using contact potentials constructed according to the chiral EFT approachWeinEFT . The channel has been worked out in Ref.C71 up to chiral order . Here, we present the universal forms for both uncoupled and coupled channels. Then it is immediate to see some features and notions that are intrinsically nonperturbative and deviate from the conventional wisdoms of renormalization within perturbative frameworks. These are important conceptual gainings that could resolve most of the controversies about the applications of EFT in nonperturbative regimes. The notions and scenario demonstrated below are naturally illuminating for any problem that is beset with nonperturbative divergences, especially in the systems governed by singular short-distance interactions. In addition, the analytical results and scenario presented here could also be seen as a field-theoretical solution to the universality of large scattering lengths in atomic and molecular systemsBraaten-Hammer . According to WeinbergWeinEFT , the EFT approach to scattering consists of two steps: First, the potentials for scattering are systematically constructed using chiral perturbation theory (PT) as the relevant EFT up to certain chiral order : with being the upper limit for low energy scattering (e.g., ); Second, the nonperturbative scattering -matrices could be obtained by solving the LSE with the potential constructed in the previous step. As LSE is hard to solve rigorously for pion-exchange potentials, we first work with contact potentials (EFT()) that facilitate rigorous solutions where the LSE for -wave reduces to an algebraic form by employing the following factorization trick or ansatzPhillips : V≡qL(q′)LΔ/2−L∑i,j=0λijq2i(q′)2j=qL(q′)LUT(q)λU(q′), T≡qL(q′)LΔ/2−L∑i,j=0τijq2i(q′)2j=qL(q′)LUT(q)τU(q′), (1) with being a column vector and being the external off-shell momenta. This is because for pionless interactions, the potential up to order degrades into contact interactions that become a polynomial in terms of external momenta up to power . Here denotes the real symmetric matrix comprises of the contact couplings () as is symmetric with respect to the external momenta and . For example, in channels at , we have, λ=(C0C2C20). Then the algebraic LSE takes the following form: τ(E)=λ+λI(E)τ(E), (2) where the matrix comprises of the integrals arising from convolution. Note that and are symmetric matrices as is. As is complex, so is . A general element of can be parametrized as follows Ii,j(E) ≡{∫d3k(2π)3k2(L+i+j−2)E−k2/M+iϵ}R=L+i+j−2∑m=1J2m+1p2(L+i+j−m−2)−I0p2(L+i+j−2), (3) I0 ≡J0+iM4πp, (4) where the subscript denotes any possible regularization and/or renormalization prescription rendering the integrals finite and being the corresponding parametrization. This algebraic LSE is easy to solve. Then, for any uncoupled partial wave channel, the on-shell -matrices could be readily obtained from with C71 ; 3s1-3d1 , which can be simplified to the following form: 1TL=I0+NL([C⋯],[J2m+1],p2)DL([C⋯],[J2m+1],p2)p2L, (5) where and are polynomials in terms of the following real parameters: the couplings , the constants and . While for coupled channels, the inverse of the on-shell -matrices would take the following form: T−1J=I0I+ΔJ, I≡(1001), ΔJ≡⎛⎜ ⎜ ⎜⎝NJ−1,J−1DJ−1,J−1p2(J−1),−NJ−1,J+1DJ−1,J+1p2J−NJ−1,J+1DJ−1,J+1p2J,NJ+1,J+1DJ+1,J+1p2(J+1)⎞⎟ ⎟ ⎟⎠. (6) Again are real polynomials in terms of , and , hence (chiral) perturbative in nature. Note that unitarity is automatically satisfied here. At this stage, both the real part of and the constants are prescription dependent. The overall factors of have been factored out so that the expansion of in terms of starts from . For example, the results for at chiral order read, T−13S1−3D1=I0I+Δ3S1−3D1, Δ3S1−3D1≡1D1p4(N1p4,−Dsdp2−Dsdp2,D0) (7) where and for convenience we introduced the following notations: . For details we refer to a forthcoming report3s1-3d1 . Eqs.(5) and (6) exhibit the following important features: (1) First, the same complex parameter appears in all channels in the same isolated position in or and the rest parts of or are independent of , i.e., is ”decoupled” from and in every channel***The rigorous proof of this point for channel has been given in Ref.C71 , which could be generalized to higher channels, we will give the detailed proof in a forthcoming report.. This structure is most pivotal. (2) Second, with the potentials truncated at finite order, only finite many of (or finite types of divergences) enter the game, in spite that there are formally infinite many divergent items in the iteration of LSE. (3) Both and are chiral perturbative (or perturbative in the corresponding EFT). Since the -dependence of the on-shell -matrices (and hence the inverse of -matrices) is physical, the prescription variations (i.e., variations of ) must be compensated by that of the couplings. This is nothing else but the general principle of renormalization group (RG) invariance, then appropriate combinations of the coefficients of in and must be RG invariants. The most outstanding point is that, the isolation or ”decoupling” of from and makes itself a renormalization group invariant parameter in all channels, hence, is a physical scaleC71 . Therefore, is in fact a fundamental and universal parameter in the low energy scattering, and is no longer an ordinary renormalization scale. Such a RG invariant quantity was also predicted in Wilsonian RG approachBirse , , whose inverse is just computed in the Wilsonian cutoff approach. This is not the only deviation from the conventional wisdoms for renormalization, according to which a divergent integral usually produces sliding scales that are physically meaningless within perturbative formulation. In perturbative framework, the couplings in the contact potential would get renormalized and ”run”. At lowest order, similar notion is feasible in channelKSW : , where is RG invariant as is a constant here, actually . But at higher orders (e.g., ), it is easy to see that the rational dependence of -matrices upon precludes the conventional wisdoms from being feasible, i.e., it is no longer possible to let the variations in the prescription parameters be readily absorbed into the couplings and let the couplings ”run”. This point could be seen from the requirement that the appropriate combinations of the coefficients of in should be RG invariant, which in turn imposes strong constraints upon the variations in the couplings and . ( is already excluded from the set of prescription parameters as it is an RG invariant scale now.) In fact, for at , the coefficients of the highest power term in are respectively , which could not make the ratios and RG invariant at the same time, see Ref.C71 . The second feature noted above engenders a novel notion of ”finiteness”: Only a finite number of nonperturbative divergences are to be removed in a manner preserving the functional dependence upon and , which underlies the feasibility of renormalization with a few nonperturbative counterterms in Refs.YPhillips ; EPVRAM . This ’finiteness’ is a measure of nonperturbative renormalizability and not directly linked to EFT power counting. Hence nonperturbative counterterms (termed as ”endogenous” in Ref.C71 ) are not proportionate to perturbative ones that obey EFT power counting. This is in sheer contrast with perturbative renormalization programs, where consistency requires that the counterterms be introduced or constructed at exactly the same perturbation order as the divergent vertex in consideration. This is another place where the conventional wisdoms fail, usually interpreted as the ”inconsistency” of Weinberg’s power counting. We will return to this point later. This ”finiteness” also underlies the feasibility of the finite cutoff approachesLepage ; Rho ; EGM ; EMach ; Bogner . In fact the nonperturbative form of -matrices and their RG invariance lead to an entanglement between the couplings and the prescription: They must be defined coherently in order to match physical boundaries. Then, the prescription must be appropriately defined after the couplings are given first. Below, to obtain unnaturally large scattering lengths and naturally sized effective range, etc., we suggest a simple and natural strategy: The original EFT(PT) power counting for potential construction are kept intact, i.e., no modification of the power counting rules of the couplings ; In the meantime, and are so determined that physical boundary conditions are fulfilled. Then, to yield large (unnatural) -wave scattering lengths, the most ’natural’ or simplest scenario would be as followsC71 : CΔ∼4π/(MΛΔ+1); J0∼MΛ/(4π)∼\|1/C0\|; J2m+1∼Mμ2m+1/(4π), m>0, (8) with being an -wave contact coupling at lowest order and of order or . In a generic EFT, . Thus, the only difference is with . This is a ”natural” scenario or choice as is actually a fundamental and physical constant in the nonperturbative regime, no longer an ordinary renormalization scale. We will discuss other schemes in future works3s1-3d1 . With the foregoing preparations, we could examine some important theoretical issues in low energy scattering. First, let us consider some theoretical predictions. To this end, we calculate effective range expansions (ERE) of in various channels. We should remind that the following discussions are valid for contact potentials only, not directly applicable to the cases containing long range potentials such as the pion-exchange potentials for systems. Later we will consider some speculations about such cases. Re{−4πMp2LTL}\|p→0=p2L+1cot(δL(p))\|p→0=−1a+12rep2+∞∑k=2vkp2k, (9) with and being functions of the couplings in corresponding channels and . However, unlike the rest of , contributes in each channel to only one of the ERE parameters that is the coefficient of ! This is obviously due to the special status of the fundamental parameter . Employing the scenario (8) we could qualitatively deduce that, all but one ERE parameters are naturally sized! The exceptional one might be unnaturally sized just because of the contribution of . The mechanism is simply that J0+dL(NLDL)L!(dp2)L\|p=0∼M4πO(μ), provided that the sign of is opposite to that of as closer analysis shows that is of the same magnitude as . These general conclusions are summarized in Table I. It is known that in channel the scattering length is unnaturally large with the rest of ERE parameters being natural. Now, within the context of contact potentials, higher ERE parameters might also be unnaturally sized in an appropriate channel. These are new predictions. While in the coupled channels, one could find from Eq.(6) that the diagonal entries of take the following form: 1TL=I0+NL;0+I0NL;1DL;0+I0DL;1p−2L, (10) where enters into the rational terms, and hence precluding a clear naturalness picture of the ERE parameters. However, using the scenario of (8) and the detailed contents of , one could still arrive at modestly good judgements, the status of naturalness in the coupled channels is basically similar to that given in Table I. There might be some deviations as now enters the rational terms, which would affect the status of some ERE parameters. But such influence would not be universal for all the ERE parameters. For example, through concrete calculations, one could find that in channel, is totally independent of , is most strongly influenced by , while the rest of ERE parameters are only weakly affected due to the suppression just mentioned. More detailed analysis will be given in a forthcoming report3s1-3d1 . In earlier EFT treatments, the distinctive aspects of nonperturbative renormalization demonstrated above were not fully appreciated, leading to quite some debatesBvKERev (for recent debates, seeNTvK ; PVRAB ; EpelMeis ). A number of different schemes were proposed in order to remove the ’inconsistency’ of Weinberg’s power countingNTvK ; KSW ; BBSvK , with some ”perturbative-like” expansion schemes being advancedKSW ; BBSvK . As is pointed out above, the inconsistency is in fact a misinterpretation of the failure of conventional wisdom of renormalization. Specifically, nonperturbative counterterms do not need to follow EFT power counting. Therefore, it is both difficult and unnecessary to maneuver a unified power countingNTvK ; EpelMeis . The entanglement property means that the problems could well be resolved with appropriate choice of nonperturbative prescriptions constrained by physical boundaries or conditions. After all, the ultimate goal of any sensible scheme or prescription should be to approach the physical dependence of -matrices upon as far as possible. Thus, a (new) formally consistent power counting is not the full story: The nonperturbative prescription must be appropriately defined to match physical boundaries. For example, for a ”perturbative-like” expansion scheme to work, the following two criteria must be satisfied: (1) The expansion converges; (2) Physical boundaries are fulfilled. Both criteria are dependent upon prescription choice. To illustrate this, we expand in Eq. (5) as follows: 1TL=I0+1+δNLD0L+δDLp−2L≃1T0L+O(δNLD0L,δDLD0L)p−2L, with being the starting nonperturbative amplitude. Then, convergence requires that , which in turn demands a sophisticated renormalization prescription after the couplings are given. Next, prescription must also be so chosen that fulfills physical boundaries. Therefore, it is a challenging task to find a prescription to fulfill the above two criteria. Evidently, the informative form of the -matrices will inspire new investigations in the future. It would be interesting to explore the relations between the nonperturbative parametrization elaborated here and those in literature, for example, the subtractive approachesYPhillips ; EPVRAM ; Frederico ; ESoto , and the lattice approacheslattice . Now we conclude with the following remarks. In general, the ultimate goal of a field theoretical calculation in nonperturbative regime is to identify and parametrize all the elements that govern the physical behaviors of the corresponding objects, especially the elements hidden in divergences. To this end, we have achieved the following: First, a fundamental parameter masked by a divergent integral was identified and shown to be RG invariant and inherent in all channels; Second, universal forms of nonperturbative -matrices with respect to prescription dependence were obtained in all channels in the case of contact potentials; Third, within the realm of contact potentials, a simple scenario led us to predict that all the scattering lengths except those in the -channels’ are natural, while higher ERE parameters like might also be unnatural in appropriate higher channels; Fourth, some distinctive notions about nonperturbative renormalization were revealed along with the failures of the conventional wisdoms, providing a different resolution of the intriguing problem with Weinberg’s power counting in the EFT approach of scattering. These conceptual gainings are significant from purely theoretical standpoint as nonperturbative renormalization is a challenging issue. Finally, we stress again that the notions and conclusions presented here are fairly general and hence illuminating for nonperturbative treatments of any systems dominated by short-distance interactions. Acknowledgement The author is deeply grateful to the anonymous referees for their valuable comments that significantly improved the presentation of our manuscript. The project is supported in part by the National Natural Science Foundation of China under Grant Nos. 10205004 and 10475028 and the Ministry of Education of China. References • (1) P. Bedaque, U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52, 339 (2002); E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006). • (2) A. Nogga, R.G.E. Timmermans, U. van Kolck, Phys. Rev. C72, 054006 (2005). • (3) M. Pavon Valderrama, E. Ruiz Arriola, Phys. Rev. C74, 064004 (2006); M.C. Birse, ibid.,C74, 014003 (2006). • (4) E. Epelbaum, Ulf-G. Meissner, arXiv: nucl-th/0609037. • (5) S. Weinberg, Phys. Lett. B 251, 288 (1990); Nucl. Phys. B 363, 1 (1991). • (6) J.-F. Yang, J.-H. Huang, Phys. Rev. C71, 034001, 069901(E) (2005). • (7) E. Braaten, H.W. Hammer, Phys. Rep. 428, 259 (2006). • (8) D.R. Phillips, S.R. Beane, T.D. Cohen, Ann. Phys. (NY) 263, 255 (1998). • (9) J.-F. Yang, forthcoming. • (10) M.C. Birse, J.A. McGovern, K.G. Richardson, Phys. Lett. B464, 169 (1999). • (11) D.B. Kaplan, M.J. Savage, M.B Wise, Phys. Lett. B424, 390 (1998); Nucl. Phys. B534, 329 (1998). • (12) C.-J. Yang, C. Elster, D.R. Phillips, Phys. Rev. C77, 014002 (2008). • (13) D.R. Entem, E. Ruiz Arriola, M. Pavon Valderrama, R. Machleidt, Phys. Rev. C77, 044006 (2008). • (14) G.P. Lepage, arXiv: nucl-th/9706029. • (15) T.-S. Park, K. Kubodera, D.-P. Min, M. Rho, Phys. Rev. C58, R637 (1998). • (16) E. Epelbaum, W. Glöckle, U. Meissner, Nucl. Phys. A671, 295 (2000); Eur. Phys. J. A19, 125, 401 (2004). • (17) D.R. Entem, R. Machleidt, Phys. Rev. C66, 014002 (2002), ibid., C68, 041001 (2003). • (18) S.K. Bogner, et al, Phys. Lett. B576, 265 (2003); S.K. Bogner, R.J. Furnstahl, Phys. Lett. B632, 501 (2006). • (19) S.R. Beane, P. Bedaque, M.J. Savage, U. van Kolck, Nucl. Phys. A700, 377 (2002). • (20) T. Frederico, V.S. Timóteo, L. Tomio, Nucl. Phys. A653, 209 (1999). • (21) D. Eiras, J. Soto, Eur. Phys. J. A17, 89 (2003). • (22) S.R. Beane, P.F. Bedaque, K. Orginos, M.J. Savage, Phys. Rev. Lett. 97, 012001 (2006).	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8738728761672974, "perplexity": 1782.8866936639304}, "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-2020-34/segments/1596439738819.78/warc/CC-MAIN-20200811180239-20200811210239-00137.warc.gz"}
https://julialang.org/blog/2017/09/Hamiltonian-Indirect-Inference/	# GSoC 2017 Project: Hamiltonian Indirect Inference ### 19 September 2017 \| Dorisz Albrecht This is a writeup of my project for the Google Summer of Code 2017. The associated repository contains examples of estimating various models. In addition to this repository, I have collaborated in HamiltonianABC and its branches as part of the GSOC 2017. See also Bayesian_Examples.jl. ## GSOC 2017 project: Hamiltonian Monte Carlo and pseudo-Bayesian Indirect Likelihood This summer I have had the opportunity to participate in the Google Summer of Code program. My project was in the Julia language and the main goal was to implement Indirect Inference (A. A. Smith 1993; A. Smith 2008) to overcome the typically arising issues (such as intractable or costly to compute likelihoods) when estimating models using likelihood-based methods. Hamiltonian Monte Carlo was expected to result in a more efficient sampling process. Under the mentorship of Tamás K. Papp, I completed a major revision of Bayesian estimation methods using Indirect Inference (II) and Hamiltonian Monte Carlo. I also got familiar with using git, opening issues, creating a repository among others. Here I introduce the methods with a bit of context, and discuss an example more extensively. ## Parametric Bayesian Indirect Likelihood for the Full Data Usually when we face an intractable likelihood or a likelihood that would be extremely costly to calculate, we have the option to use an alternative auxiliary model to extract and estimate the parameters of interest. These alternative models should be easier to deal with. Drovandi et al. reviews a collection of parametric Bayesian Indirect Inference (pBII) methods, I focused on the parametric Bayesian Indirect Likelihood for the Full Data (pdBIL) method proposed by Gallant and McCulloch (2009). The pdBIL uses the likelihood of the auxiliary model as a substitute for the intractable likelihood. The pdBIL does not compare summary statistics, instead works in the following way: First the data is generated, once we have the data, we can estimate the parameters of the auxiliary model. Then, the estimated parameters are put into the auxiliary likelihood with the observed/generated data. Afterwards we can use this likelihood in our chosen Bayesian method i.e. MCMC. To summarize the method, first we have the parameter vector $\theta$ and the observed data y. We would like to calculate the likelihood of $\ell(\theta\|y)$, but it is intractable or costly to compute. In this case, with pdBIL we have to find an auxiliary model (A) that we use to approximate the true likelihood in the following way: • First we have to generate points, denote with x* from the data generating process with the previously proposed parameters $\theta$. • Then we compute the MLE of the auxiliary likelihood under x to get the parameters denoted by $\phi$. $\phi(x^{\star}) = \argmax_{\phi} (x^{\star}\|\phi)$ • Under these parameters $\phi$, we can now compute the likelihood of $\ell_{A}(y\|\phi)$. It is desirable to have the auxiliary likelihood as close to the true likelihood as possible, in the sense of capturing relevant aspects of the model and the generated data. ## First stage of my project In the first stage of my project I coded two models from Drovandi et al. using pdBIL. After calculating the likelihood of the auxiliary model, I used a Random Walk Metropolis-Hastings MCMC to sample from the target distribution, resulting in Toy models. In this stage of the project, the methods I used were well-known. The purpose of the replication of the toy models from Drovandi et al. was to find out what issues we might face later on and to come up with a usable interface. This stage resulted in HamiltonianABC (collaboration with Tamás K. Papp). ## Second stage of my project After the first stage, I worked through Betancourt (2017) and did a code revision for Tamás K. Papp's DynamicHMC.jl which consisted of checking the code and its comparison with the paper. In addition to using the Hamiltonian Monte Carlo method, the usage of the forward mode automatic differentiation of the ForwardDiff package was the other main factor of this stage. The novelty of this project was to find a way to fit every component together in a way to get an efficient estimation out of it. The biggest issue was to define type-stable functions such that to accelerate the sampling process. ## Stochastic Volatility model After the second stage, I coded economic models for the DynamicHMC.jl. The Stochastic Volatility model is one of them. In the following section, I will go through the set up. The continuous-time version of the Ornstein-Ulenbeck Stochastic - volatiltiy model describes how the return at time t has mean zero and its volatility is governed by a continuous-time Ornstein-Ulenbeck process of its variance. The big fluctuation of the value of a financial product imply a varying volatility process. That is why we need stochastic elements in the model. As we can access data only in discrete time, it is natural to take the discretization of the model. The discrete-time version of the Ornstein-Ulenbeck Stochastic - volatility model: $y_{t} = x_{t} + \epsilon_{t} \quad \text{where}\quad \epsilon_{t} ∼ \Chi^{2}(1)$ $x_{t} = \rho * x_{t-1} + \sigma * \nu_{t} \quad \text{where}\quad \nu_{t} ∼ \mathcal N(0, 1)$ The discrete-time version was used as the data-generating process. Where $y_t$ denotes the logarithm of return, $x_{t}$ is the logarithm of variance, while $\epsilon_{t}$ and $\nu_{t}$ are unobserved noise terms. For the auxiliary model, we used two regressions. The first regression was an AR(2) process on the first differences, the second was also an AR(2) process on the original variables in order to capture the levels. """ lag_matrix(xs, ns, K = maximum(ns)) Matrix with differently lagged xs. """ function lag_matrix(xs, ns, K = maximum(ns)) M = Matrix{eltype(xs)}(length(xs)-K, maximum(ns)) for i ∈ ns M[:, i] = lag(xs, i, K) end M end "first auxiliary regression y, X, meant to capture first differences" function yX1(zs, K) Δs = diff(zs) lag(Δs, 0, K), hcat(lag_matrix(Δs, 1:K, K), ones(eltype(zs), length(Δs)-K), lag(zs, 1, K+1)) end "second auxiliary regression y, X, meant to capture levels" function yX2(zs, K) lag(zs, 0, K), hcat(ones(eltype(zs), length(zs)-K), lag_matrix(zs, 1:K, K)) end The AR(2) process of the first differences can be summarized by: Given a series Y, it is the first difference of the first difference. The so called "change in the change" of Y at time t. The second difference of a discrete function can be interpreted as the second derivative of a continuous function, which is the "acceleration" of the function at a point in time t. In this model, we want to capture the "acceleration" of the logarithm of return. The AR(2) process of the original variables is needed to capture the effect of $\rho$. It turned out that the impact of ρ was rather weak in the AR(2) process of the first differences . That is why we need a second auxiliary model. I will now describe the required steps for the estimation of the parameters of interest in the stochastic volatility model with the Dynamic Hamiltonian Monte Carlo method. First we need a callable Julia object which gives back the logdensity and the gradient in DiffResult type. After that, we write a function that computes the density, then we calculate its gradient using the ForwardDiff package in a wrapper function. Required packages for the StochasticVolatility model: using ArgCheck using Distributions using Parameters using DynamicHMC using StatsBase using Base.Test using ContinuousTransformations using DiffWrappers import Distributions: Uniform, InverseGamma • First, we define a structure. This structure should contain the observed data, the priors, the shocks and the transformation performed on the parameters, but the components may vary depending on the estimated model. struct StochasticVolatility{T, Prior_ρ, Prior_σ, Ttrans} "observed data" ys::Vector{T} "prior for ρ (persistence)" prior_ρ::Prior_ρ "prior for σ_v (volatility of volatility)" prior_σ::Prior_σ "χ^2 draws for simulation" ϵ::Vector{T} "Normal(0,1) draws for simulation" ν::Vector{T} "Transformations cached" transformation::Ttrans end After specifying the data generating function and a couple of facilitator and additional functions for the particular model (whole module can be found in src folder), we can make the model structure callable, returning the log density. The logjac is needed because of the transformation we make on the parameters. function (pp::StochasticVolatility)(θ) @unpack ys, prior_ρ, prior_σ, ν, ϵ, transformation = pp ρ, σ = transformation(θ) logprior = logpdf(prior_ρ, ρ) + logpdf(prior_σ, σ) N = length(ϵ) # Generating xs, which is the latent volatility process xs = simulate_stochastic(ρ, σ, ϵ, ν) Y_1, X_1 = yX1(xs, 2) β₁ = qrfact(X_1, Val{true}) \ Y_1 v₁ = mean(abs2, Y_1 - X_1β₁) Y_2, X_2 = yX2(xs, 2) β₂ = qrfact(X_2, Val{true}) \ Y_2 v₂ = mean(abs2, Y_2 - X_2β₂) # We work with first differences y₁, X₁ = yX1(ys, 2) log_likelihood1 = sum(logpdf.(Normal(0, √v₁), y₁ - X₁ * β₁)) y₂, X₂ = yX2(ys, 2) log_likelihood2 = sum(logpdf.(Normal(0, √v₂), y₂ - X₂ * β₂)) logprior + log_likelihood1 + log_likelihood2 + logjac(transformation, θ) end We need the transformations because the parameters are in the proper subset of $\Re^{n}$, but we want to use $\Re^{n}$. The ContinuousTransformation package is used for the transformations. We save the transformations such that the callable object stays type-stable which makes the process faster. $\nu$ and $\epsilon$ are random variables which we use after the transformation to simulate observation points. This way the simulated variables are continuous in the parameters and the posterior is differentiable. Given the defined functions, we can now start the estimation and sampling process: RNG = Base.Random.GLOBAL_RNG # true parameters and observed data ρ = 0.8 σ = 0.6 y = simulate_stochastic(ρ, σ, 10000) # setting up the model model = StochasticVolatility(y, Uniform(-1, 1), InverseGamma(1, 1), 10000) # we start the estimation process from the true values θ₀ = inverse(model.transformation, (ρ, σ)) # sampling sample, tuned_sampler = NUTS_tune_and_mcmc(RNG, fgw, 5000; q = θ₀) The following graphs show the results for the parameters: Analysing the graphs above, we can say that the posterior values are in rather close to the true values. Also worth mentioning that the priors do not affect the posterior values. ## Problems that I have faced during GSOC ### 1 Difficult auxiliary model • The true model was the g-and-k quantile function described by Rayner and MacGillivray (2002). • The auxiliary model was a three component normal mixture model. We faced serious problems with this model. First of all, I coded the MLE of the finite component normal mixture model, which computes the means, variances and weights of the normals given the observed data and the desired number of mixtures. With the g-and-k quantile function, I experienced the so called "isolation", which means that one observation point is an outlier getting weight 1, the other observed points get weight $\theta$, which results in variance equal to $\theta$. There are ways to disentangle the problem of isolation, but the parameters of interests still did not converge to the true values. There is work to be done with this model. ### 2 Type-stability issues To use the automatic differentiation method efficiently, I had to code the functions to be type-stable, otherwise the sampling functions would have taken hours to run. See the following example: • This is not type-stable function simulate_stochastic(ρ, σ, ϵs, νs) N = length(ϵs) @argcheck N == length(νs) xs = Vector(N) for i in 1:N xs[i] = (i == 1) ? νs[1]σ(1 - ρ^2)^(-0.5) : (ρxs[i-1] + σνs[i]) end xs + log.(ϵs) + 1.27 end • This is type-stable function simulate_stochastic(ρ, σ, ϵs, νs) N = length(ϵs) @argcheck N == length(νs) x₀ = νs[1]σ(1 - ρ^2)^(-0.5) xs = Vector{typeof(x₀)}(N) for i in 1:N xs[i] = (i == 1) ? x₀ : (ρxs[i-1] + σνs[i]) end xs + log.(ϵs) + 1.27 end ## Future work • More involved models • Solving isolation in the three component normal mixture model • Updating shocks in every iteration • Optimization ## References • Betancourt, M. (2017). A Conceptual Introduction to Hamiltonian Monte Carlo. • Drovandi, C. C., Pettitt, A. N., & Lee, A. (2015). Bayesian indirect inference using a parametric auxiliary model. • Gallant, A. R. and McCulloch, R. E. (2009). On the Determination of General Scientific Models With Application to Asset Pricing • Martin, G. M., McCabe, B. P. M., Frazier, D. T., Maneesoonthorn, W. and Robert, C. P. (2016). Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models • Rayner, G. D. and MacGillivray, H. L. (2002). Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions. In: Statistical Computation 12 57–75. • Smith, A. A. (2008). “Indirect inference”. In: New Palgrave Dictionary of Economics, 2nd Edition (forthcoming). • Smith, A. A. (1993). “Estimating nonlinear time-series models using simulated vector autoregressions”. In Journal of Applied Econometrics 8.S1.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 20, "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": 0.8395459651947021, "perplexity": 1879.7428749636308}, "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/1600400213006.47/warc/CC-MAIN-20200924002749-20200924032749-00444.warc.gz"}
http://www.physicsforums.com/showthread.php?s=706d1e628a11645d1959d9173d87371c&p=4231230	# Low-Pass Filtering a system, Time constant issue by dolle39 Tags: constant, filtering, issue, lowpass, time P: 3 Hi! I want to low-pass filter some simulations that I made. The indata to the simulation is built up by several modules and hence at the edges transients will occur. I want to filter these out. The indata in sampled at 20 Hz. How should I set the time constant for the filter? I mean I know that I should relate that to the time constant of the system in order to not filter away important data. But what is the time constant of a system really? Consider a car steering system. Is the time constant the time it takes for the system to manouver from max left to max right? Or is it the time it takes to manouver from +10% to -10%? Some guideluines would be appreciated. P: 3 I don't understand why do you need a LPF to do that (i mean, you "can't" do that). If you have a digital signal (the indata sampled at 20hz) just take the important data and throw away the transients. Or you can use a cyclic prefix technique or something like that. Now, if you need to design a LPF to do another thing, the time constant is well defined in first-order systems (it is defined in superior orders too). The definition depends of the context. P: 1,822 What is the maximum rate of change of your data? Related Discussions General Engineering 9 Calculus 0 Engineering, Comp Sci, & Technology Homework 0 Electrical Engineering 10 Programming & Computer Science 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": 0.8136555552482605, "perplexity": 797.5976737261783}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657126053.45/warc/CC-MAIN-20140914011206-00237-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://mammothmemory.net/maths/geometry/angles/right-angle.html	# Right angle ## Right angle – Standing directly upright the internal angle is 90° (a right angle is an angle that measures exactly 90°). They stood at an upright angle (right angle). Example 1 These gears are at a right angle to one another. Example 2 At precisely 3 o’clock the two hands of a clock are at a right angle.	{"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": 0.8929550647735596, "perplexity": 1867.188616564668}, "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/1656103037089.4/warc/CC-MAIN-20220626040948-20220626070948-00301.warc.gz"}
http://www.formuladirectory.com/user/formula/357	HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS ## Distance Between two Coordinate Points This is the formula for calculation of the distance in kilometers between two coordinate points on the globe. The longitude and latitude values(in digits only) are required for this calculation. The accuracy of the result depends on the accuracy of the result values. ## $\mathrm{acos}\left[\mathrm{sin\left(lata\right)}\mathrm{sin\left(latb\right)}+\mathrm{cos\left(latb\right)}\mathrm{cos\left(lonb-lona\right)}\right]*6371$ The values needed for the formula are lata and lona for the first point and latb and lonb for the second point. ENTER THE VARIABLES TO BE USED IN THE FORMULA SOLVE FORMULA	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "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": 0.930211067199707, "perplexity": 892.9536300283165}, "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/1508187828134.97/warc/CC-MAIN-20171024033919-20171024053919-00265.warc.gz"}
http://www.physicsforums.com/showpost.php?p=132389&postcount=6	Thread: Of mass and matter. View Single Post P: n/a Originally posted by arivero A definition of "matter" is lacking here. How about using the definition given by Einstein in [i]The Foundation of the General Theory of Relativity,[i] Annalen der Physik, 49, 1916 We make the distinction hereafter between "gravitational field" and "matter" in this way, that we denote everything but the gravitational field as "matter." Our use of the word therefore includes not only matter in the ordinary sense, but the electromagnetic field as well.	{"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": 0.8850576281547546, "perplexity": 1003.3744252820801}, "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-2014-15/segments/1398223206147.1/warc/CC-MAIN-20140423032006-00168-ip-10-147-4-33.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/35409/why-is-a-nucleus-isotropic/35428	# Why is a nucleus isotropic? I believe in Neutron Scattering the neutrons after hitting a nucleus can bounce in any of 360*3 dimensions -> 1080 degrees? Why is this so? Shouldn't it only bounce "off" the neutron in approximately the same "direction" that it came in such as when a particle bounces off a mirror -> because of the cross-section ... - To describe the angular distribution in 3-D space, you should use solid angle, with a maximum of $4\pi$. $360^\circ\times 3$ does not make sense. – C.R. Sep 2 '12 at 2:00 @KarsusRen: can you elaborate? I'm thinking essentially setting the center of the neutron as the origin of a spherical coordinate system. – Eiyrioü von Kauyf Sep 2 '12 at 21:52 Which is exactly what he meant. $360 \equiv 0\ \text{mod}\ 360$. So in your example $1080 \equiv 360\ \text{mod}\ 360$ – CHM Sep 3 '12 at 5:45 Leaving aside the odd misconceptions in the question, not all nuclei are isotropic: polarized observables are big business at transition energies these days. – dmckee Sep 4 '12 at 17:27 Physics is not a matter of beliefs, but of measurements with their errors and the analysis of those data according to theoretical( mathematically expressed) models. In two body scattering, two in two out, the scattering takes place in a plane, because of momentum conservation whether classically or in the quantum mechanical mircrocosm. Thus the angle of scatter is one and goes from 0 to 360 degrees. If one looks at the data, there is no isotropy. The angular distributions are analyzed (page 6 in the link) in a series of Legendre polynomials, which are a function of the angle theta. Each individual scatter of a neutron on a nucleus will have a specific plane in the center of mass, and thus the angle theta will show the functional form of the interaction, and not be isotropic after aligning the scattering planes . The distribution in the angle phi which will define the rotation to align the scattering planes will be isotropic. Because of momentum conservation as mentioned above any phi is equally probable. There cannot be total isotropy because there are nuclear resonances which are probed when a neutron scatters off a nucleus and the spins involved create in the crossection necessarily a function of theta. - can you elaborate on which phi and theta - different conventions.... – Eiyrioü von Kauyf Sep 2 '12 at 21:55 Did you look at the link I gave? It is the normal spherical coordinates in which Legendre polynomials are expressed. In the experiment, which is really the trump card, they have a number of theta angles all around phi, where they measure, find isotropy, and extrapolate from that that they can use the Legendre polynomials for the functional form of the scattering angular dependence. – anna v Sep 3 '12 at 3:37 Differential cross sections are introduced precisely to quantify the percentage of particles that scatter in a given direction. If all the directions are possibles, they do not have the same differential cross section value. And indeed, the higest value is for scattering directions close to the incoming beam. - This is misleading--- you are talking about a case (forward scattering) which swamps all off beam scattering in the limit that the neutron wavefunction is spread out. The non-delta-function part of forward scattering is not larger by orders of magnitude than other directions for a localized potential, it is just the limit of off-forward scattering, and this is how the optical theorem works. – Ron Maimon Sep 2 '12 at 22:07 The neutron at low momentum is quantum mechancal during the scattering, and the bouncing off is by wave mechanics, not by bouncing off. The neutron gets reflected into all directions by the small region with a potential, and the amount of scattering is given by the Born approximation to lowest order in the momentum of the neutron: $$A(k-k') = -i\tilde V(k-k')$$ For scattering from incoming direction k to outgoing direction k'. You decorate this with phase-space factors to take into account the size of the neutron wave. If the neutron wave is large, nearly all the scattering is in the forward direction, and this is expressed by writing the S-matrix as; $$S = 1 + i A$$ Where the "1" gives a contribution to the scattering which is $\delta(k-k')$ for the case of single-particle scattering. If you smear this with the incoming wavepacket, you get the outgoing wavepacket, which is mostly the incoming wave, plus a spherical outgoing wave. The delta-function guarantees that as you approach a plane-wave limit, there will be no scattering, because the neutron wavefunction area will be much larger than the nuclear area. This scattering, when the neutron wavefunction is larger wavelength than the nucleus (almost always in real life) leads to a spherical scattering which is roughly isotropic, which you add to the incoming wave to get the full outgoing wave. The imaginary part of the scattering in the forward direction only subtracts some weight from the wavefunction that keeps going forward, and unitarity guarantees that this imaginary part is equal to the scattering probability in all directions added together. This is covered in scattering theory in most quantum mechanics books, but the treatment in Gribov's Regge theory book "The Theory of Complex Angular Momentum" is most instructive to my mind. - But Ron, all this is fine and good as an introduction to scattering in the S matrix format, but how about experimental evidence (which I linked in my answer) that shows non isotropy in theta due to nuclear resonances? It would have been isotropic if there were no further interactions open to the scatter which the experiment demonstrates. It is not isotropic in the real world! – anna v Sep 3 '12 at 4:35 @annav: I am talking about energies lower than the ones required for exciting resonances. I get the sense tha OP is asking about slow or thermal neutrons, not KeV/MeV neutrons. – Ron Maimon Sep 3 '12 at 5:50	{"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": 0.8279728293418884, "perplexity": 783.4020784272637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609530136.5/warc/CC-MAIN-20140416005210-00490-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.competoid.com/answers/30119/1/IBPS_PO-Main_Exam/	Topics Exams >> Bank Exams >> IBPS PO-Main Exam IBPS PO Main exam consists of below sections You can practice these questions as a real time exam in this link. Directions [1-5] : In these questions, relationship between different elements is shown in the statements. These statements are followed by two conclusions. Read both the statements and select the appropriate answer. Give answer: If only conclusion I is true If neither conclusion I nor II is true If both conclusion I and II are true If only conclusion II is true If either Conclusion I or II is true 1). Statements : $$V \ge M =T>X$$ R < T = S Conclusions : I. R < V II. S =X A). If only conclusion I is true B). If neither conclusion I nor II is true C). If both conclusion I and II are true D). If only conclusion I is true E). If either Conclusion I or II is true 2). Statements : P< E = R > F; E=M; R < T Conclusions : I. T >M II. F < M A). If only conclusion I is true B). If neither conclusion I nor II is true C). If both conclusion I and II are true D). If only conclusion I is true E). If either Conclusion I or II is true 3). Statements: R = Q = I = M = E; I < Z Conclusions: I.$$Q \le E$$ II. M > Z A). If only conclusion I is true B). If neither conclusion I nor II is true C). If both conclusion I and II are true D). If only conclusion I is true E). If either Conclusion I or II is true 4). Statements: $$V \ge M = T > X; R < T \ge S$$ Conclusions: I. X < R II. $$V \le S$$ A). If only conclusion I is true B). If neither conclusion I nor II is true C). If both conclusion I and II are true D). If only conclusion I is true E). If either Conclusion I or II is true 5). Statements: $$O < P = T \ge N; F > T$$ Conclusions: I.F > O II. N < O A). If only conclusion I is true B). If neither conclusion I nor II is true C). If both conclusion I and II are true D). If only conclusion I is true E). If either Conclusion I or II is true 6). In a certain code, PARTICLE is written as USBQFMDJ and GENERATE is written as FOFHFUBS, how is DOCUMENT written in that code ? A). VDEPUONF B). VDPENFUQ C). VDPEUOFN D). VPUFNDED E). VNFODEPU 7). In a class of 40 children, Sunetra's rank is eighth from the top. Sujit is five ranks below Sunetra. What is Sujit's rank from the bottom ? A). 27 B). 28 C). 29 D). 26 E). Other than those given as options Directions (8-9) : Study the information carefully and answer the given questions. L,M, N, O and P are five different poles, each of different length. O is not the third shortest pole. N is bigger than only P. L is shorter than only one pole. The size of the shortest pole is 7 ft and that of the second tallest pole is 13 ft. 8). Which of the following poles is the third tallest ? A). M B). Cannot be determined . C). N D). P E). L 9). According to the given arrangement, which of the following combinations of pole and length is correct? A). N - 14 ft B). P - 5 ft C). O - 12 ft D). L - 13 ft E). Other than those given as options 10). R is sister of M. M is brother of H. D is mother of K. K is brother of M. How is R related to D ? A). Daughter B). Mother C). Other than those given as options D). Sister E). Data inadequate Recent Activities	{"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": 0.891760528087616, "perplexity": 2217.9579019946937}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823872.13/warc/CC-MAIN-20181212112626-20181212134126-00535.warc.gz"}
https://physics.stackexchange.com/questions/198221/why-do-we-say-linear-molecules-only-have-2-rotational-degrees-of-freedom-why-do	# Why do we say linear molecules only have 2 rotational degrees of freedom? Why does the third 'frozen' one not count? It is possible to excite rotations around the axes perpendicular to the bond of a linear molecule. However, rotation around the axis along the bond of the molecule would require huge energies, due to the much smaller moment of inertia which is inversely proportional to the energy. This is explained in more detail here: In counting degrees of freedom of a linear molecule, why is rotation about the axis not counted? 1 Therefore, out of three possible rotational degrees of freedom, only two are applicable to our linear molecule. The third rotation can't be accessed and is 'frozen out'. Thus a linear molecule is said to have two rotational degrees of freedom, as this third parameter would give us no extra information about the system. • Non-linear molecules have 3N degrees of freedom in total. We know 3 are translational, 3 are rotational (all are allowed for non-linear molecules) so the remaining 3N-6 are vibrational. • Linear molecules have 3 translational and only 2 rotational, and to keep a total of 3N degrees of freedom, they now need 3N-5 vibrational. Why does the 'frozen' out rotational degree of freedom for linear molecules not count? As stated above, knowledge of it gives no extra information about the system, but at what point does a degree of freedom count as 'frozen out' and need to be compensated for by other (vibrational) degrees of freedom? And is there a deeper, physical argument beyond "everything needs to add up to 3N"? Why 3N, in this case? In the general case, any molecule can be cooled down so that various degrees of freedom are gradually frozen out. Are these degrees of freedom replaced by others, and how does the system then know how to vibrate, say, in a different way? I'm aware that I'm missing something here, possibly a fundamental flaw in how I currently understand the concept of degrees of freedom, and I greatly appreciate any help. The number of degrees of freedom isn't $3N$, it's $3N$ minus the number of contraints. The $N=2$ is a special case since with two particles molecules can't help being linear. However for $N>2$ if the molecule is linear you are constraining the motion of the third particle because it can only move along the line joining the other two. Likewise for higher values of $N$. This reduces the total number number of degrees of freedom, so the transaltional, rotational and vibrational degrees of freedom do not have to add up to $3N$. Let us consider one of the rotational degrees of freedom of a molecule. Let us assume that the moment of inertia for the relevant principal axis is $I$. Then we can get the spacing of energy levels for this degree of freedom from the quantization rule $I\omega=n\hbar$ (I omit some constant factors and neglect some other things here and everywhere else), so for the level that is closest to the ground level $I\omega=\hbar$. The statistical probability of the system to be at a level with energy $E$ is proportional to $\exp(-E/\theta)$, where $\theta$ is temperature. For a rotational level, $E=I\omega^2=I(\hbar/I)^2=\hbar^2/I$. Unless the energy is comparable with or less than the temperature, the probability is close to zero, so the temperature where the probability is not negligible is of the order of $\hbar^2/I$. For the axis of a linear molecule, the moment of inertia is very small, so the temperature at which energy levels above the ground level are important is extremely high (e.g., it can be higher than the temperature of dissociation of the molecule). Therefore, at temperatures of practical interest, rotation around the axis of the linear molecule is not important for thermodynamic properties.	{"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": 0.8438679575920105, "perplexity": 154.29984238596327}, "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/1579251694908.82/warc/CC-MAIN-20200127051112-20200127081112-00306.warc.gz"}
http://mathoverflow.net/questions/25263/most-squares-in-the-first-half-interval/25268	Most squares in the first half-interval It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one half lie in the interval $[1, (p-1)/2]$. I remember reading this fact many years ago on a very popular book in number theory, where it was claimed that this is an easy consequence of a more sophisticated formula of analytic number theory. Sadly I forgot both the formula and the book. So the purpose of the question is double: 1) Has any simple way been found to derive the fact mentioned above? 2) Does anybody know a reference for the analytic number theory route to the proof? - Note that this is trivial for $p \equiv 1 \pmod 4$ because then $-1$ is a quadratic residue, and any $k$ is a quadratic residue if and only if $p-k$ is. So exactly half lie in your interval, exactly half lie in the interval from $(p+1)/2$ to $p-1.$ – Will Jagy May 19 '10 at 22:07 Another reference: "Introduction to Cyclotomic Fields", L. C. Washington, pp 45, exercise 4.5. – Dror Speiser May 19 '10 at 22:08 Nope! Amazingly enough, no elementary proof of this fact is yet known (Edit: See KConrad's answer). The difficulty is tied up in some pretty fantastic algebraic/analytic number theory, namely the analytic class number formula. But, without getting into that, here's the short form of the story: let $L(s)$ be the $L$-function attached to the character arising from the Kronecker symbol mod $q$. Then one can compute analytically via the Euler product formula for this $L$-function that (for an explicit positive constant $C$), $$L(1)=C\left[\sum_{m=0}^{q/2}\left(\frac{m}{q}\right)-\sum_{m=q/2}^{q}\left(\frac{m}{q}\right)\right]=\frac{\pi}{\left(2-\left(\frac{2}{q}\right)\right)q^{1/2}}\sum_{m=0}^{q/2}\left(\frac{m}{q}\right),$$ the positivity of which gives the desired statement about the distribution of quadratic residues. For a reference (from whence I pulled this out of), Davenport's "Multiplicative Number Theory" is pretty fantastic. This is all done in the first 4-5 pages. - I'm pretty sure this is not the book I had in mind, but it is fine nevertheless. – Andrea Ferretti May 19 '10 at 20:46 There is a method of explaining this without using analytic methods. I will get to that at the end of this answer. First, if $p \equiv 1 \bmod 4$ then this result is clear since -1 is a square mod $p$. So here exactly half the squares mod $p$ lie in the first half of $[1,p-1]$. The real problem is for $p \equiv 3 \bmod 4$, where analytic methods show there are more squares mod $p$ lying in the first half of that interval than in the second half because there is a formula for the class number of ${\mathbf Q}(\sqrt{-p})$ that is 1 or 1/3 times $S - N$, where $S$ is the number of squares mod $p$ in $[1,(p-1)/2]$ and $N$ is the number of nonsquares mod $p$ in $[1,(p-1)/2]$. Class numbers are positive integers, so $S > N$, which means in $[1,(p-1)/2]$ the squares mod $p$ outnumber nonsquares mod $p$. Since there are as many squares as nonsquares mod $p$ on $[1,p-1]$, the square vs. nonsquare bias on the first half of this interval forces there to be more squares mod $p$ on the first half than squares mod $p$ on the second half. For a proof by analytic methods, see Borevich-Shafarevich's "Number Theory", Theorem 4 on p. 346. It is not true that no non-analytic derivations of this bias are known. For instance, Borevich and Shafarevich say on p. 347 that Venkov gave a non-analytic proof for some cases in 1928 (which came out in German in 1931: see Math Z. Vol. 33, 350--374). I should clarify this point since there is a bad typo in Borevich and Shafarevich here. What Venkov did was give a non-analytic proof of Dirichlet's class number formula for imaginary quadratic fields having discriminant $D \not\equiv 1 \bmod 8$. Here the book unfortunately has $D \equiv 1 \bmod 8$. (It's clear from the book that something is wrong because shortly after saying Venkov treated $D \equiv 1 \bmod 8$ by non-analytic methods they say the case $D \equiv 1 \bmod 8$ still awaits a non-analytic proof.) The class number formula only gets an interpretation about squares or nonsquares for the fields ${\mathbf Q}(\sqrt{-p})$, but Venkov was working on non-analytic proofs of the class number formula for imaginary quadratic fields without having this restrictive case as the only one in mind. R. W. Davis (Crelle 286/287 (1976), 369--379) made simplifications to Venkov's argument. What cases for ${\mathbf Q}(\sqrt{-p})$ are covered by Venkov? When $p \equiv 3 \bmod 4$, the discriminant of ${\mathbf Q}(\sqrt{-p})$ is $-p$. If $p \equiv 3 \bmod 8$ then $-p \equiv 5 \bmod 8$, while if $p \equiv 7 \bmod 8$ then $-p \equiv 1 \bmod 8$, so Venkov had non-analytically proved the formula when $p \equiv 3 \bmod 8$. The case $p \equiv 7 \bmod 8$ remained open. In 1978 the whole problem was solved. Davis, in a second paper (Crelle 299/300 (1978), 247--255), handled some but not all cases of imaginary quadratic fields with discriminant $1 \bmod 8$ (corresponding to $p \equiv 7 \bmod 8$ for the fields ${\mathbf Q}(\sqrt{-p})$) by non-analytic methods and in the same year H. L. S. Orde settled everything by non-analytic methods. See his paper "On Dirichlet's class number formula", J. London Math. Soc. 18 (1978), 409--420. - Very helpful answer, thank you! In Russian 3rd edition of Borevich and Shafarevich, last paragraph on p. 383, Venkov's reference is OK (no typo). – Victor Protsak May 19 '10 at 23:07 Very informative! +1 – Cam McLeman May 19 '10 at 23:28 У меня нет копии русского варианта. Опечатка в первом русском изданием? Английский перевод появился в 1966 г., а второе русское издание вышло в 1972 г. (см. lib.mexmat.ru/books/4065) – KConrad May 19 '10 at 23:32 (For others: I'm asking whether the first Russian edition of B/S has the typo, since only the first Russian edition was out when it was translated.) – KConrad May 19 '10 at 23:34 Keith Matthews showed me a copy of the original (Russian) 1st edition of Borevich-Shafarevich from 1964 and its reference to Venkov does not have the typographical error from the English translation that I wrote about in my answer above. – KConrad Jun 6 '11 at 4:32 show 6 more comments	{"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": 0.8560370802879333, "perplexity": 384.19579110120117}, "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-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00535-ip-10-147-4-33.ec2.internal.warc.gz"}
https://encyclopediaofmath.org/wiki/K%C3%A4hler-Einstein_metric	# Kähler-Einstein metric A Kähler metric on a complex manifold (or orbifold) whose Ricci tensor $\operatorname { Ric } ( \omega )$ is proportional to the metric tensor: \begin{equation} \operatorname { Ric } ( \omega ) = \lambda \omega. \end{equation} This proportionality is an analogue of the Einstein field equation in general relativity. The following conjecture is due to E. Calabi: Let $M$ be a compact connected complex manifold and $c _ { 1 } ( M ) _ { \mathbf{R} }$ its first Chern class; then a) if $c _ { 1 } ( M ) _ { \mathbf{R} } < 0$, then $M$ carries a unique (Ricci-negative) Kähler–Einstein metric $\omega$ such that $\operatorname { Ric } ( \omega ) = - \omega$; b) if $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$, then any Kähler class of $M$ admits a unique (Ricci-flat) Kähler–Einstein metric such that $\operatorname { Ric } ( \omega ) = 0$. This conjecture was solved affirmatively by T. Aubin [a1] and S.T. Yau [a8] via studies of complex Monge–Ampère equations, and Kähler–Einstein metrics play a very important role not only in differential geometry but also in algebraic geometry. The affirmative solution of this conjecture gives, for instance, the Bogomolov decomposition for compact Kähler manifolds with $c _ { 1 } ( M ) _ { \mathbf{R} } = 0$. It also implies (see [a2], [a3]): 1) Any Kähler manifold homeomorphic to $\mathbf{CP} ^ { n }$ is biholomorphic to $\mathbf{CP} ^ { n }$. Any compact complex surface homotopically equivalent to $\mathbf{CP} ^ { 2 }$ is biholomorphic to $\mathbf{CP} ^ { 2 }$. 2) In the Miyaoka–Yau inequality $c _ { 1 } ( S ) ^ { 2 } \leq 3 c_ { 2 } ( S )$, for a compact complex surface $S$ of general type, equality holds if and only if $S$ is covered by a ball in $\mathbf{C} ^ { 2 }$. For a Fano manifold $M$ (i.e., $M$ is a compact complex manifold with $c _ { 1 } ( M ) _ { \mathbf{R} } > 0$), let $G$ be the identity component of the group of all holomorphic automorphisms of $M$. Let $\cal E$ be the set of all Kähler–Einstein metrics $\omega$ on $M$ such that $\operatorname { Ric } ( \omega ) = \omega$. If $\mathcal{E} \neq \emptyset$, then $\cal E$ consists of a single $G$-orbit (see [a5]). Moreover, the following obstructions to the existence of Kähler–Einstein metrics are known (cf. [a5], [a6]): Matsushima's obstruction. If $\mathcal{E} \neq \emptyset$, then $G$ is a reductive algebraic group (cf. also Reductive group). Futaki's obstruction. If $\mathcal{E} \neq \emptyset$, then Futaki's character $F _ { M } : G \rightarrow \mathbf{C} ^ { * }$ is trivial. Recently (1997), G. Tian [a7] showed some relationship between the existence of Kähler–Einstein metrics on $M$ and stability of the manifold $M$, and gave an example of an $M$ with no non-zero holomorphic vector fields satisfying $\mathcal{E} = \emptyset$. The Poincaré metric on the unit open disc $\{ z \in \mathbf{C} : \| z \| < 1 \}$ (cf. Poincaré model) and the Fubini–Study metric on $\mathbf{CP} ^ { n }$ are both typical examples of Kähler–Einstein metrics. For more examples, see Kähler–Einstein manifold. For the relationship between Kähler–Einstein metrics and multiplier ideal sheaves, see [a4]. See, for instance, [a2] for moduli spaces of Kähler–Einstein metrics. Finally, Kähler metrics of constant scalar curvature and extremal Kähler metrics are nice generalized concepts of Kähler–Einstein metrics (cf. [a2]). #### References [a1] T. Aubin, "Nonlinear analysis on manifolds" , Springer (1982) [a2] A.L. Besse, "Einstein manifolds" , Springer (1987) MR0867684 Zbl 0613.53001 [a3] J.P. Bourguignon, et al., "Preuve de la conjecture de Calabi" Astérisque , 58 (1978) [a4] A.M. Nadel, "Multiplier ideal sheaves and existence of Kähler–Einstein metrics of positive scalar curvature" Ann. of Math. , 132 (1990) pp. 549–596 [a5] T. Ochiai, et al., "Kähler metrics and moduli spaces" , Adv. Stud. Pure Math. , 18–II , Kinokuniya (1990) [a6] Y.-T. Siu, "Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics" , Birkhäuser (1987) [a7] G. Tian, "Kähler–Einstein metrics with positive scalar curvature" Invent. Math. , 137 (1997) pp. 1–37 [a8] S.-T. Yau, "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation I" Commun. Pure Appl. Math. , 31 (1978) pp. 339–411 How to Cite This Entry: Kähler-Einstein metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%A4hler-Einstein_metric&oldid=50163 This article was adapted from an original article by Toshiki Mabuchi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article	{"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": 0.9777906537055969, "perplexity": 603.7855775074266}, "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-2022-21/segments/1652662636717.74/warc/CC-MAIN-20220527050925-20220527080925-00237.warc.gz"}
https://www.physicsforums.com/threads/electric-field-of-two-charges.286256/	Homework Help: Electric field of two charges 1. Jan 20, 2009 JJones_86 1. The problem statement, all variables and given/known data In the figure below , what is the electric field at point P? 2. Relevant equations F = k(q1q2)/r^2 3. The attempt at a solution F = k*(2q^2) / (2d + d)^2 That is my attempt, but it's not a possible answer.. Any help is appreciated. 2. Jan 20, 2009 rl.bhat At P the fields are in the opposite direction. So find the field at P due to q and 2q, and take the difference. 3. Jan 20, 2009 JJones_86 Oh ok, so it should be kq/2d^2 then? 4. Jan 20, 2009 Yes.	{"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": 0.8849273324012756, "perplexity": 1966.8019670165816}, "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-2018-30/segments/1531676594954.59/warc/CC-MAIN-20180723051723-20180723071723-00601.warc.gz"}
https://socratic.org/questions/an-aqueous-solution-of-h-2so-4-molecular-weight-98-contains-10-78-g-of-acid-per-	Chemistry Topics # An aqueous solution of H_2SO_4 (molecular weight=98) contains 10.78 g of acid per dm^3 of solution. Density of solution is 1.123 g/ml. What is the molarity, molality, normality and mole fraction of the solution? Jul 9, 2015 Here's how you can solve this problem. #### Explanation: The problem actually makes your life easier by telling you that you have that much sulfuric acid, 10.78 g to be precise, per ${\text{dm}}^{3}$ of solution. If you take into account that fact that $\text{1 dm"^3 = "1 L}$, you'll notice that all you need to do in order to determine the solution's molarity is calculate how many moles of sulfuric acid you have in 10.78 g. To do that, use the acid's moalr mass 10.78cancel("g") * ("1 mole"H_2SO_4)/(98cancel("g")) = "0.110 moles" ${H}_{2} S {O}_{4}$ This means that the solution's molarity is C = n_(H_2SO_4)/V = "0.110 moles"/"1 L" = color(green)("0.11 M") Molality is defined as moles of solute, which you've already calculated, divided by the mass of the solvent - expressed in kilograms! To get the mass of water you have, use the solution's density to first determine how much the entire solutions weighs. 1cancel("L") * (1000cancel("mL"))/(1cancel("L")) * "1.123 g"/(1cancel("mL")) = "1123 g" This means that the mass of water will be ${m}_{\text{water" = m_"solution}} - {m}_{{H}_{2} S {O}_{4}}$ ${m}_{\text{water" = 1123 - 10.78 = "1112.2 g}}$ This molality of the solution will thus be b = n_(H_2SO_4)/m_"water" = "0.110 moles"/(1112.2 * 10^(-3)"kg") = color(green)("0.099 molal") The solution's normality will take into account the number of protons the sulfuric acid will produce in solution. Since sulfuric acid is a diprotic acid, every mole of the acid will produce two moles of ${H}^{+}$ in solution, which are called equivalents. Since you have 0.11 moles, you'll get twice as many equivalents. This means that the solution's normality will be twice as big as its molarity. $N = \left(\text{equivalents of H"^(+))/"liter of solution" = "0.22 equiv."/("1 L") = color(green)("0.22 N}\right)$ To get the mole fraction of the solution, you need to first determine how many moles of water you have. 1112.2cancel("g") * "1 mole"/(18.02 cancel("g")) = "61.7 moles" ${H}_{2} O$ The total number of moles will be ${n}_{\text{total" = n_"water}} + {n}_{{H}_{2} S {O}_{4}}$ ${n}_{\text{total" = 61.7 + 0.11 = "61.81 moles}}$ The mole fraction of sulfuric acid will thus be chi_(H_2SO_4) = n_(H_2SO_4)/n_"total" = (0.110cancel("moles"))/(61.81cancel("moles")) = color(green)("0.0018") ##### Impact of this question 1705 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": 16, "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": 0.9058364629745483, "perplexity": 2811.7819433910845}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540530857.12/warc/CC-MAIN-20191211103140-20191211131140-00058.warc.gz"}
https://socratic.org/questions/how-do-you-find-the-axis-of-symmetry-and-the-maximum-or-minimum-value-of-the-fun-74#267636	Algebra Topics # How do you find the axis of symmetry, and the maximum or minimum value of the function f(x)= 4x^2+40x+97? May 20, 2016 Axis of symmetry is $x = - 5$ and minima at $\left(- 5 , - 3\right)$ #### Explanation: For an equation of a parabola given by the equation $y = a {x}^{2} + b x + c$, the axis of symmetry is a vertical line given by $x = - \frac{b}{2} a$ Hence, axis of symmetry for $y = 4 {x}^{2} + 40 x + 97$ is $x = - \frac{40}{2 \times 4} = - 5$ As the differential $\frac{\mathrm{dy}}{\mathrm{dx}} = 8 x + 40$ and this is zero at $8 x + 40 = 0$ or $x = - 5$. At this value $y = 4 \left(- 5\right) {x}^{2} + 40 \left(- 5\right) + 97 = 100 - 200 + 97 = - 3$ As second derivative (d^2y)/dx^2)=8 and is positive hence we have a miniima at $\left(- 5 , - 3\right)$ graph{4x^2+40x+97 [-7.52, -2.52, -3.53, -1.03]} ##### Impact of this question 441 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "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": 0.8038662672042847, "perplexity": 743.49995308422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587794.19/warc/CC-MAIN-20211026011138-20211026041138-00315.warc.gz"}
http://mathhelpforum.com/calculus/71630-work-integration.html	# Math Help - Work integration 1. ## Work integration A force of 13 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work W is done in stretching it from its natural length to 6 in. beyond its natural length? Using Hooke's Law, f(x)=kx , I set it up like this: f(x) = 13 k=? x=4 Thus, k=13/4. I plug that in to the to integral of 13x on the interval from 4 to 6. Then I got 65/2 in-lbs. I divide by 12 becuase the problem requires it to be ft-lb. Thus I got I got 65/24 ft-lbs, but the site says it is wrong. What did I do wrong? 2. Originally Posted by bottleofboos A force of 13 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work W is done in stretching it from its natural length to 6 in. beyond its natural length? Using Hooke's Law, f(x)=kx , I set it up like this: f(x) = 13 k=? x=4 Thus, k=13/4. I plug that in to the to integral of 13x on the interval from 4 to 6. Then I got 65/2 in-lbs. I divide by 12 becuase the problem requires it to be ft-lb. Thus I got I got 65/24 ft-lbs, but the site says it is wrong. What did I do wrong? First thing I noticed is that the limits on your integral are in the wrong units and the lower limit is wrong. If your problem is in ft-lbs, your limits must also be in ft-lb and as it looks, they seem to be still in inches. Try that first. If it's still wrong, I'll look at the entire integral for you. Ok? One other thing, your bottom limit should start at 0, or natural length of the spring. Just because you use the force given at 4in to calculate your k, does not mean that it automatically includes that in your answer. You must evaluate the entire interval from 0 to 6 inches (or in this ft-lb situation) 0 to .5ft 3. Okay, so I tried it on the interval of 13x/4 from 0 to 1/2 ft. So i got 13/32 ft-lbs. I wanted to double check using the the interval from 0 to 6 in and then divide the answer by 12 to convert it into ft-lbs. So i got 39/8 ft-lbs. EDIT: I found out that the k has to be converted into feet also if I want to use the interval from 0 to 1/2 ft. So now I got 39/8 ft-lbs for each side. Thanks Molly.	{"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": 0.9328503012657166, "perplexity": 411.3539473768878}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931010469.50/warc/CC-MAIN-20141125155650-00045-ip-10-235-23-156.ec2.internal.warc.gz"}
https://en.wikipedia.org/wiki/N-sphere	# n-sphere 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a sphere's surface to a plane, it can also project the surface of a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line). In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space. For any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-dimensional Euclidean space as the set of points that are at distance r from a central point, where the radius r may be any positive real number. Thus, the n-sphere would be defined by: ${\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\\|x\right\\|=r\right\}.}$ In particular: the pair of points at the ends of a (one-dimensional) line segment is 0-sphere, the circle, which is the one-dimensional circumference of a (two-dimensional) disk in the plane is a 1-sphere, the two-dimensional surface of a (three-dimensional) ball in three-dimensional space is a 2-sphere, often simply called a sphere, the three-dimensional boundary of a (four-dimensional) 4-ball in four-dimensional Euclidean is a 3-sphere, also known as a glome. An n-sphere embedded in an (n + 1)-dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted Sn. The unit n-sphere is often referred to as the n-sphere. When embedded as described, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere. ## Description For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive real number and where c may be any point in (n + 1)-dimensional space. In particular: • a 0-sphere is a pair of points {cr, c + r}, and is the boundary of a line segment (1-ball). • a 1-sphere is a circle of radius r centered at c, and is the boundary of a disk (2-ball). • a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball). • a 3-sphere is a sphere in 4-dimensional Euclidean space. ### Euclidean coordinates in (n + 1)-space The set of points in (n + 1)-space: (x1,x2,…,xn+1) that define an n-sphere (Sn), is represented by the equation: ${\displaystyle r^{2}=\sum _{i=1}^{n+1}(x_{i}-c_{i})^{2}.\,}$ where c is a center point, and r is the radius. The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. The ppl volume form ω of an n-sphere of radius r is given by ${\displaystyle \omega ={1 \over r}\sum _{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge \cdots \wedge dx_{j-1}\wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1}=dr}$ where is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case r = 1. As a result, ${\displaystyle dr\wedge \omega =dx_{1}\wedge \cdots \wedge dx_{n+1}.}$ ### n-ball Main article: ball (mathematics) The space enclosed by an n-sphere is called an (n + 1)-ball. An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere. Specifically: • A 1-ball, a line segment, is the interior of a 0-sphere. • A 2-ball, a disk, is the interior of a circle (1-sphere). • A 3-ball, an ordinary ball, is the interior of a sphere (2-sphere). • A 4-ball is the interior of a 3-sphere, etc. ### Topological description Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as ${\displaystyle \mathbb {S} ^{n}=\mathbb {R} ^{n}\cup \{\infty \}}$, which is n-dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$. This forms the basis for stereographic projection.[1] ## Volume and surface area ${\displaystyle V_{n}(R)}$ and ${\displaystyle S_{n}(R)}$ are the n-dimensional volume and surface area of the n-ball and n-sphere of radius ${\displaystyle R}$, respectively. The constants ${\displaystyle V_{n}}$ and ${\displaystyle S_{n}}$ (for the unit ball and sphere) are related by the recurrences: ${\displaystyle V_{0}=1\qquad V_{n+1}=S_{n}/(n+1)}$ ${\displaystyle S_{0}=2\qquad S_{n+1}=2\pi V_{n}}$ The surfaces and volumes can also be given in closed form: ${\displaystyle {\begin{array}{ll}S_{n}(R)&=\displaystyle {{\frac {2\pi ^{(n+1)/2}}{\Gamma ({\frac {n+1}{2}})}}R^{n}}\\[1em]V_{n}(R)&=\displaystyle {\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}\end{array}}}$ where ${\displaystyle \Gamma \,}$ is the gamma function. Derivations of these equations are given in this section. In general, the volumes of the n-ball in n-dimensional Euclidean space, and the n-sphere in (n + 1)-dimensional Euclidean, of radius R, are proportional to the nth power of the radius, R. We write ${\displaystyle V_{n}(R)=V_{n}R^{n}}$ for the volume of the n-ball and ${\displaystyle S_{n}(R)=S_{n}R^{n}}$ for the surface of the n-sphere, both of radius ${\displaystyle R}$. Interestingly, given the radius R, the volume and the surface area of the n-sphere reaches a maximum and then decrease towards zero as the dimension n increases. In particular, the volume ${\displaystyle V_{n}(R)}$ of the n-sphere of constant radius R in n-dimensions reaches a maximum for dimension ${\displaystyle n^{\ast }=n}$ if ${\displaystyle V_{n-1}}$ <R< ${\displaystyle V_{n}}$ and ${\displaystyle n^{\ast }=\{n,n+1\}}$ if ${\displaystyle R=V_{n}}$ where ${\displaystyle V_{n}={\frac {\Gamma (n/2+3/2)}{{\sqrt {\pi }}\Gamma (n/2+1)}}}$ for ${\displaystyle n=1,2,\ldots }$. Similarly, defining the sequence ${\displaystyle \{A_{n}=V_{n-2}\}}$, the surface area ${\displaystyle S_{n}(R)}$ of the n-sphere of constant radius R in n dimensions reaches a maximum for dimension ${\displaystyle n^{\ast }=n}$ if ${\displaystyle A_{n-1} and ${\displaystyle n^{\ast }=\{n,n+1\}}$ if ${\displaystyle R=A_{n}}$.[2] ### Examples The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set, so ${\displaystyle V_{0}=1}$. The unit 1-ball is the interval ${\displaystyle [-1,1]}$ of length 2. So, ${\displaystyle V_{1}=2.}$ The 0-sphere consists of its two end-points, ${\displaystyle \{-1,1\}}$. So, ${\displaystyle S_{0}=2}$. The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure) ${\displaystyle S_{1}=2\pi .\,}$ The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure) ${\displaystyle V_{2}=\pi .\,}$ Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by ${\displaystyle S_{2}=4\pi .\,}$ and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by ${\displaystyle V_{3}={\frac {4}{3}}\pi .\,}$ ### Recurrences The surface area, or properly the n-dimensional volume, of the n-sphere at the boundary of the (n + 1)-ball of radius ${\displaystyle R}$ is related to the volume of the ball by the differential equation ${\displaystyle S_{n}R^{n}={\frac {dV_{n+1}R^{n+1}}{dR}}={(n+1)V_{n+1}R^{n}}}$, or, equivalently, representing the unit n-ball as a union of concentric (n − 1)-sphere shells, ${\displaystyle V_{n+1}=\int _{0}^{1}S_{n}r^{n}\,dr}$ So, ${\displaystyle V_{n+1}={\frac {S_{n}}{n+1}}}$. We can also represent the unit (n + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an n-sphere. Let ${\displaystyle r=\cos \theta }$ and ${\displaystyle r^{2}+R^{2}=1}$, so that ${\displaystyle R=\sin \theta }$ and ${\displaystyle dR=\cos \theta \,d\theta }$. Then, {\displaystyle {\begin{aligned}S_{n+2}&=\int _{0}^{\pi /2}S_{1}r.S_{n}R^{n}\,d\theta =\int _{0}^{\pi /2}S_{1}.S_{n}R^{n}\cos \theta \,d\theta \\&=\int _{0}^{1}S_{1}.S_{n}R^{n}\,dR=S_{1}\int _{0}^{1}S_{n}R^{n}\,dR\\&=2\pi V_{n+1}\end{aligned}}} Since ${\displaystyle S_{1}=2\pi V_{0}}$, the equation ${\displaystyle S_{n+1}=2\pi V_{n}}$ holds for all n. This completes our derivation of the recurrences: ${\displaystyle V_{0}=1\qquad V_{n+1}=S_{n}/(n+1)}$ ${\displaystyle S_{0}=2\qquad S_{n+1}=2\pi V_{n}}$ ### Closed forms Combining the recurrences, we see that ${\displaystyle V_{n+2}=2\pi V_{n}/(n+2)}$. So it is simple to show by induction on k that, ${\displaystyle V_{2k}={\frac {\pi ^{k}}{k!}}}$ ${\displaystyle V_{2k+1}={\frac {2(2\pi )^{k}}{(2k+1)!!}}={\frac {2k!(4\pi )^{k}}{(2k+1)!}}}$ where ${\displaystyle !!}$ denotes the double factorial, defined for odd integers 2k + 1 by (2k + 1)!! = 1 · 3 · 5 ··· (2k − 1) · (2k + 1). In general, the volume, in n-dimensional Euclidean space, of the unit n-ball, is given by ${\displaystyle V_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}}+1)}}}$ where ${\displaystyle \Gamma \,}$ is the gamma function, which satisfies ${\displaystyle \Gamma (1/2)={\sqrt {\pi }};\Gamma (1)=1;\Gamma (x+1)=x\Gamma (x)}$. By multiplying ${\displaystyle V_{n}}$ by ${\displaystyle R^{n}}$, differentiating with respect to ${\displaystyle R}$, and then setting ${\displaystyle R=1}$, we get the closed form ${\displaystyle S_{n-1}={\frac {2\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}})}}}$. ### Other relations The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram: ${\displaystyle S_{n-1}={\frac {n}{2\pi }}S_{n-1+2}}$ (Note that n refers to the dimension of the ambient Euclidean space, which is also the intrinsic dimension of the solid whose volume is listed here, but which is 1 more than the intrinsic dimension of the sphere whose surface area is listed here.) The curved red arrows show the relationship between formulas for different n. The formula coefficient at each arrow's tip equals the formula coefficient at that arrow's tail times the factor in the arrowhead. If the direction of the bottom arrows were reversed, their arrowheads would say to multiply by 2π/n − 2. Alternatively said, the surface area Sn+1 of the sphere in n + 2 dimensions is exactly 2πR times the volume Vn enclosed by the sphere in n dimensions. Index-shifting n to n − 2 then yields the recurrence relations: ${\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}$ ${\displaystyle S_{n-1}={\frac {2\pi }{n-2}}S_{n-1-2}}$ where S0 = 2, V1 = 2, S1 = 2π and V2 = π. The recurrence relation for ${\displaystyle V_{n}}$ can also be proved via integration with 2-dimensional polar coordinates: {\displaystyle {\begin{aligned}V_{n}&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}({\sqrt {1-r^{2}}})^{n-2}\,r\,d\theta \,dr\\[6pt]&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}(1-r^{2})^{n/2-1}\,r\,d\theta \,dr\\[6pt]&=2\pi V_{n-2}\int _{0}^{1}(1-r^{2})^{n/2-1}\,r\,dr\\[6pt]&=2\pi V_{n-2}\left[-{\frac {1}{n}}(1-r^{2})^{n/2}\right]_{r=0}^{r=1}\\[6pt]&=2\pi V_{n-2}{\frac {1}{n}}={\frac {2\pi }{n}}V_{n-2}.\end{aligned}}} ## Spherical coordinates We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate, ${\displaystyle r\,,}$ and n − 1 angular coordinates ${\displaystyle \phi _{1},\phi _{2},\dots ,\phi _{n-1}\,}$ where ${\displaystyle \phi _{n-1}\,}$ ranges over ${\displaystyle [0,2\pi )\,}$ radians (or over [0, 360) degrees) and the other angles range over ${\displaystyle [0,\pi ]\,}$ radians (or over [0, 180] degrees). If ${\displaystyle \ x_{i}}$ are the Cartesian coordinates, then we may compute ${\displaystyle x_{1},\ldots ,x_{n}}$ from ${\displaystyle r,\phi _{1},\ldots ,\phi _{n-1}}$ with: {\displaystyle {\begin{aligned}x_{1}&=r\cos(\phi _{1})\\x_{2}&=r\sin(\phi _{1})\cos(\phi _{2})\\x_{3}&=r\sin(\phi _{1})\sin(\phi _{2})\cos(\phi _{3})\\&\vdots \\x_{n-1}&=r\sin(\phi _{1})\cdots \sin(\phi _{n-2})\cos(\phi _{n-1})\\x_{n}&=r\sin(\phi _{1})\cdots \sin(\phi _{n-2})\sin(\phi _{n-1})\,.\end{aligned}}} Except in the special cases described below, the inverse transformation is unique: {\displaystyle {\begin{aligned}r&={\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}+{x_{1}}^{2}}}\\\phi _{1}&=\operatorname {arccot} {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}=\arccos {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{1}}^{2}}}}\\\phi _{2}&=\operatorname {arccot} {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{3}}^{2}}}}=\arccos {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}\\&\vdots \\\phi _{n-2}&=\operatorname {arccot} {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}=\arccos {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+{x_{n-2}}^{2}}}}\\\phi _{n-1}&=2\operatorname {arccot} {\frac {x_{n-1}+{\sqrt {x_{n}^{2}+x_{n-1}^{2}}}}{x_{n}}}={\begin{cases}\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}\geq 0\\2\pi -\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}<0\end{cases}}\,.\end{aligned}}} where if ${\displaystyle x_{k}\neq 0}$ for some ${\displaystyle k}$ but all of ${\displaystyle x_{k+1},\ldots ,x_{n}}$ are zero then ${\displaystyle \phi _{k}=0}$ when ${\displaystyle x_{k}>0}$, and ${\displaystyle \phi _{k}=\pi }$ radians (180 degrees) when ${\displaystyle x_{k}<0}$. There are some special cases where the inverse transform is not unique; ${\displaystyle \phi _{k}}$ for any ${\displaystyle k}$ will be ambiguous whenever all of ${\displaystyle x_{k},x_{k+1},\ldots ,x_{n}}$ are zero; in this case ${\displaystyle \phi _{k}}$ may be chosen to be zero. ### Spherical volume element Expressing the angular measures in radians, the volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation: ${\displaystyle \left({\begin{smallmatrix}\cos(\phi _{1})&-r\sin(\phi _{1})&0&0&\ \ \cdot \ \ \cdot \ \ \cdot &0\\\sin(\phi _{1})\cos(\phi _{2})&r\cos(\phi _{1})\cos(\phi _{2})&-r\sin(\phi _{1})\sin(\phi _{2})&0&\ \ \cdot \ \ \cdot \ \ \cdot &0\\\vdots &\vdots &\vdots &\ddots &&\vdots \\&&&&&0\\\sin(\phi _{1})...\sin(\phi _{n-2})\cos(\phi _{n-1})&\cdots &\cdots &&&-r\sin(\phi _{1})...\sin(\phi _{n-2})\sin(\phi _{n-1})\\\sin(\phi _{1})...\sin(\phi _{n-2})\sin(\phi _{n-1})&r\cos(\phi _{1})...\sin(\phi _{n-1})&\cdots &&&r\sin(\phi _{1})...\sin(\phi _{n-2})\cos(\phi _{n-1})\end{smallmatrix}}\right)}$ {\displaystyle {\begin{aligned}d^{n}V&=\left\|\det {\frac {\partial (x_{i})}{\partial (r,\phi _{j})}}\right\|dr\,d\phi _{1}\,d\phi _{2}\cdots d\phi _{n-1}\\[6pt]&=r^{n-1}\sin ^{n-2}(\phi _{1})\sin ^{n-3}(\phi _{2})\cdots \sin(\phi _{n-2})\,dr\,d\phi _{1}\,d\phi _{2}\cdots d\phi _{n-1}\end{aligned}}} and the above equation for the volume of the n-ball can be recovered by integrating: ${\displaystyle V_{n}=\int _{\phi _{n-1}=0}^{2\pi }\int _{\phi _{n-2}=0}^{\pi }\cdots \int _{\phi _{1}=0}^{\pi }\int _{r=0}^{R}d^{n}V.\,}$ The volume element of the (n-1)–sphere, which generalizes the area element of the 2-sphere, is given by ${\displaystyle d_{S^{n-1}}V=\sin ^{n-2}(\phi _{1})\sin ^{n-3}(\phi _{2})\cdots \sin(\phi _{n-2})\,d\phi _{1}\,d\phi _{2}\cdots d\phi _{n-1}.}$ The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials, {\displaystyle {\begin{aligned}&{}\quad \int _{0}^{\pi }\sin ^{n-j-1}(\phi _{j})C_{s}^{((n-j-1)/2)}(\cos \phi _{j})C_{s'}^{((n-j-1)/2)}(\cos \phi _{j})\,d\phi _{j}\\[6pt]&={\frac {\pi 2^{3-n+j}\Gamma (s+n-j-1)}{s!(2s+n-j-1)\Gamma ^{2}((n-j-1)/2)}}\delta _{s,s'}\end{aligned}}} for j = 1, 2, ..., n − 2, and the e isφj for the angle j = n − 1 in concordance with the spherical harmonics. ## Stereographic projection Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-sphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example, the point ${\displaystyle \ [x,y,z]}$ on a two-dimensional sphere of radius 1 maps to the point ${\displaystyle \left[{\frac {x}{1-z}},{\frac {y}{1-z}}\right]}$ on the ${\displaystyle \ xy}$ plane. In other words, ${\displaystyle \ [x,y,z]\mapsto \left[{\frac {x}{1-z}},{\frac {y}{1-z}}\right].}$ Likewise, the stereographic projection of an n-sphere ${\displaystyle \mathbf {S} ^{n-1}}$ of radius 1 will map to the ${\displaystyle n-1}$ dimensional hyperplane ${\displaystyle \mathbf {R} ^{n-1}}$ perpendicular to the ${\displaystyle \ x_{n}}$ axis as ${\displaystyle [x_{1},x_{2},\ldots ,x_{n}]\mapsto \left[{\frac {x_{1}}{1-x_{n}}},{\frac {x_{2}}{1-x_{n}}},\ldots ,{\frac {x_{n-1}}{1-x_{n}}}\right].}$ ## Generating random points ### Uniformly at random from the (n − 1)-sphere A set of uniformly distributed points on the surface of a unit 2-sphere generated using Marsaglia's algorithm. To generate uniformly distributed random points on the (n − 1)-sphere (i.e., the surface of the n-ball), Marsaglia (1972) gives the following algorithm. Generate an n-dimensional vector of normal deviates (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), ${\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n})}$. Now calculate the "radius" of this point, ${\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}$ The vector ${\displaystyle {\frac {1}{r}}\mathbf {x} }$ is uniformly distributed over the surface of the unit n-ball. #### Examples For example, when n = 2 the normal distribution exp(−x12) when expanded over another axis exp(−x22) after multiplication takes the form exp(−x12x22) or exp(−r2) and so is only dependent on distance from the origin. #### Alternatives Another way to generate a random distribution on a hypersphere is to make a uniform distribution over a hypercube that includes the unit hyperball, exclude those points that are outside the hyperball, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the relative volume of the hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions. Wendel's theorem gives the probability that all of the points generated will lie in the same half of the hypersphere. ### Uniformly at random from the n-ball With a point selected from the surface of the n-ball uniformly at random, one needs only a radius to obtain a point uniformly at random within the n-ball. If u is a number generated uniformly at random from the interval [0, 1] and x is a point selected uniformly at random from the surface of the n-ball then u1/nx is uniformly distributed over the entire unit n-ball. ## Specific spheres 0-sphere The pair of points {±R} with the discrete topology for some R > 0. The only sphere that is not path-connected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable. 1-sphere Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Topologically equivalent to the real projective line, RP1. Parallelizable. SO(2) = U(1). 2-sphere Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP1. SO(3)/SO(2). 3-sphere Also known as the glome. Parallelizable, Principal U(1)-bundle over the 2-sphere, Lie group structure Sp(1), where also ${\displaystyle \mathrm {Sp} (1)\cong \mathrm {SO} (4)/\mathrm {SO} (3)\cong \mathrm {SU} (2)\cong \mathrm {Spin} (3)}$. 4-sphere Equivalent to the quaternionic projective line, HP1. SO(5)/SO(4). 5-sphere Principal U(1)-bundle over CP2. SO(6)/SO(5) = SU(3)/SU(2). 6-sphere Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G2/SU(3). 7-sphere Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S4. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G2 = Spin(6)/SU(3). The 7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered. 8-sphere Equivalent to the octonionic projective line OP1. 23-sphere A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 110, "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": 0.985359251499176, "perplexity": 3031.7019105142726}, "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/1480698541066.79/warc/CC-MAIN-20161202170901-00084-ip-10-31-129-80.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-6-test-page-293/2c	## Trigonometry (11th Edition) Clone $y=0$ RECALL: $y=\tan^{-1}{(x)} \longrightarrow \tan{y}=x$ where $y$ is in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$. Note that: $\tan{(0)} = 0$ Thus, $y=0$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9957321882247925, "perplexity": 619.5783645741146}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316783.70/warc/CC-MAIN-20190822042502-20190822064502-00071.warc.gz"}
https://www.physicsforums.com/threads/change-in-chemical-potential.100551/	# Change in chemical potential 1. Nov 18, 2005 ### goulio Consider a solution of particles of type A and B with the following Gibbs potential $$G(P,T,n_A,n_B)=n_A g_A(P,T) + n_B g_B(P,T)+ (1/2)\lambda_{AA}n_A^2/n + (1/2)\lambda_{BB}n_B^2/n + \lambda_{AB}n_A n_B/n + n_A RT \ln(x_A) + n_B RT \ln(x_B)$$ where the $n_i$'s are the number of moles with $x_i=n_i/n$ and $g_i$ are the molar Gibbs potential of each type of particle $i=A,B$. Also $n_A + n_B = n$ and the $\lambda_{ij}$ are positive constants. a) If we add $\Delta n_B$ moles of B keeping pressure and temperature constant, calculate the change in in the chemical potential of A. The chemical potential of A is $$\mu_A = \left ( \frac{\partial G}{\partial n_A} \right )_{P,T,n_A} = g_A + \lambda_{AA} n_A/n + \lambda_{AB}n_B/n + RT(1 + \ln(x_A))$$ so changing $n_B$ to $n_B+\Delta n_B$ only changes $\mu_A$ by an amount $\lambda_{AB}\Delta n_B/n$. Is this right or I'm getting the whole thing wrong? Edited: I found the trick $n$ as actually a depence in $n_A$ so you need to take account of this when you differentiate $G$ with respect to $n_A$ Last edited: Nov 19, 2005	{"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": 0.8858809471130371, "perplexity": 339.03942060694015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818695375.98/warc/CC-MAIN-20170926085159-20170926105159-00367.warc.gz"}
https://unapologetic.wordpress.com/2009/11/	# The Unapologetic Mathematician ## Extrema with Constraints II As we said last time, we have an idea for a necessary condition for finding local extrema subject to constraints of a certain form. To be explicit, we assume that $f:X\rightarrow\mathbb{R}$ is continuously differentiable on an open region $X\subset\mathbb{R}^n$, and we also assume that $g:X\rightarrow\mathbb{R}^m$ is a continuously differentiable vector-valued function on the same region (and that $m). We use $g$ to define the region $A\subseteq X$ consisting of those points $x\in X$ so that $g(x)=0$. Now if $a\in A$ is a point with a neighborhood $N$ so that for all $x\in N\cap A$ we have $f(x)\leq f(a)$ (or $f(x)\geq f(a)$ for all such $x$). And, finally, we assume that the $m\times m$ determinant $\displaystyle\det\left(\frac{\partial g^i}{\partial x^j}\right)$ is nonzero at $a$. Then we have reason to believe that there will exist real numbers $\lambda_1,\dots,\lambda_m$, one for each component of the constraint function, so that the $n$ equations $\displaystyle\frac{\partial f}{\partial x^j}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^j}=0\qquad(j=1,\dots,n)$ are satisfied at $a$. Well, first off we can solve the first $m$ equations and determine the $\lambda_j$ right off. Rewrite them as $\displaystyle\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^j}\bigg\vert_a=-\frac{\partial f}{\partial x^j}\bigg\vert_a\qquad(j=1,\dots,m)$ a system of $m$ equations in the $m$ unknowns $\lambda_i$. Since the matrix has a nonzero determinant (by assumption) we can solve this system uniquely to determine the $\lambda_i$. What’s left is to verify that this choice of the $\lambda_i$ also satisfies the remaining $n-m$ equations. To take care of this, we’ll write $t^k=x^{m+k}$, so we can write the point $x=(x^1,\dots,x^n)$ as $(x';t)=(x^1,\dots,x^m;t^1,\dots,t^{n-m})$ and particularly $a=(a';b)$. Now we can invoke the implicit function theorem! We find an $n-m$-dimensional neighborhood $T$ of $b$ and a unique continuously differentiable function $h:T\rightarrow\mathbb{R}^m$ so that $h(b)=a'$ and $g(h(t);t)=0$ for all $t\in T$. Without loss of generality, we can choose $T$ so that $h(t)\in N\cap A$, where $N$ is the neighborhood from the assumptions above. This is the parameterization we discussed last time, and we can now substitute these functions into the function $f$. That is, we can define \displaystyle\begin{aligned}F(t^1,\dots,t^{n-m})&=f\left(h^1(t^1,\dots,t^{n-m}),\dots,h^m(t^1,\dots,t^{n-m});t^1,\dots,t^{n-m}\right)\\G^i(t^1,\dots,t^{n-m})&=g^i\left(h^1(t^1,\dots,t^{n-m}),\dots,h^m(t^1,\dots,t^{n-m});t^1,\dots,t^{n-m}\right)\qquad(i=1,\dots,m)\end{aligned} Or if we define $H(t)=(h(t);t)$ we can say this more succinctly as $F(t)=f(H(t))$ and $G^i(t)=g^i(H(t))$. Anyhow, now all of these $G^i$ are identically zero on $T$ as a consequence of the implicit function theorem, and so each partial derivative $\frac{\partial G^i}{\partial t^j}$ is identically zero as well. But since the $G^i$ are composite functions we can also use the chain rule to evaluate these partial derivatives. We find \displaystyle\begin{aligned}0&=\frac{\partial G^i}{\partial t^j}\\&=\sum\limits_{k=1}^n\frac{\partial g^i}{\partial x^k}\frac{\partial H^k}{\partial t^j}\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\sum\limits_{k=m+1}^n\frac{\partial g^i}{\partial x^k}\frac{\partial t^{k-m}}{\partial t^j}\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\sum\limits_{k=1}^{n-m}\frac{\partial g^i}{\partial x^{k+m}}\delta_j^k\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\frac{\partial g^i}{\partial x^{j+m}}\end{aligned} Similarly, since $F$ has a local minimum (as a function of the $t^j$) at $b$ we must find its partial derivatives zero at that point. That is $\displaystyle0=\frac{\partial F}{\partial t^j}\bigg\vert_{t=b}=\sum\limits_{k=1}^m\frac{\partial f}{\partial x^k}\bigg\vert_{x=H(b)}\frac{\partial h^k}{\partial t^j}\bigg\vert_{t=b}+\frac{\partial f}{\partial x^{j+m}}\bigg\vert_{x=H(b)}$ Now let’s take the previous equation involving $g^i$, evaluate it at $t=b$, multiply it by $\lambda_i$, sum over $i$, and add it to this latest equation. We find $\displaystyle0=\sum\limits_{k=1}^m\left[\frac{\partial f}{\partial x^k}\bigg\vert_{x=H(b)}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^k}\bigg\vert_{x=H(b)}\right]\frac{\partial h^k}{\partial j}\bigg\vert_{t=b}+\frac{\partial f}{\partial x^{j+m}}\bigg\vert_{x=H(b)}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^{j+m}}\bigg\vert_{x=H(b)}$ Now, the expression in brackets is zero because that’s actually how we defined the $\lambda_i$ way back at the start of the proof! And then what remains is exactly the equations we need to complete the proof. November 27, 2009 Posted by \| Analysis, Calculus \| 2 Comments ## Extrema with Constraints I We can consider the problem of maximizing or minimizing a function, as we have been, but insisting that our solution satisfy some constraint. For instance, we might have a function $f:\mathbb{R}^3\rightarrow\mathbb{R}$ to maximize, but we’re only concerned with unit-length vectors on $S^2\subseteq\mathbb{R}^3$. More generally, we’ll be concerned with constraints imposed by setting a function $g$ equal to zero. In the example, we might set $g(x,y,z)=x^2+y^2+z^2-1$. If we want to impose more conditions, we can make $g$ a vector-valued function with as many components as constraint functions we want to set equal to zero. Now, we might be able to parameterize the collection of points satisfying $g(x)=0$. In the example, we could use the usual parameterization of the sphere by latitude and longitude, writing \displaystyle\begin{aligned}x&=\sin(\theta)\cos(\phi)\\y&=\sin(\theta)\sin(\phi)\\z&=\cos(\theta)\end{aligned} where I’ve used the physicists’ convention on the variables instead of the common one in multivariable calculus classes. Then we could plug these expressions for $x$, $y$, and $z$ into our function $f$, and get a composite function of the variables $\phi$ and $\theta$, which we can then attack with the tools from the last couple days, being careful about when we can and can’t trust Cauchy’s invariant rule, since the second differential can transform oddly. Besides even that care that must be taken, it may not even be possible to parameterize the surface, or it may be extremely difficult. At least we do know that such a parameterization will often exist. Indeed, the implicit function theorem tells us that if we have $m$ continuously differentiable constraint functions $g^i$ whose zeroes describe a collection of points in an $n$-dimensional space $\mathbb{R}^n$, and the $m\times m$ determinant $\displaystyle\det\left(\frac{\partial g^i}{\partial x^j}\right)$ is nonzero at some point $a\in\mathbb{R}^n$ satisfying $g(a)=0$, then we can “solve” these equations for the first $m$ variables as functions of the last $n-m$. This gives us exactly such a parameterization, and in principle we could use it. But the calculations get amazingly painful. Instead, we want to think about this problem another way. We want to consider a point $a\in\mathbb{R}^n$ is a point satisfying $g(a)=0$ which has a neighborhood $N$ so that for all $x\in N$ satisfying $g(x)=0$ we have $f(a)\geq f(x)$. This does not say that there are no nearby points to $a$ where $f$ takes on larger values, but it does say that to reach any such point we must leave the region described by $g(x)=0$. Now, let’s think about this sort of heuristically. As we look in various directions $v$ from $a$, some of them are tangent to the region described by $g(x)=0$. These are the directions satisfying $[D_vg](a)=dg(a;v)=0$ — where to first order the value of $g$ is not changing in the direction of $v$. I say that in none of these directions can $f$ change (again, to first order) either. For if it did, either $f$ would increase in that direction or not. If it did, then we could find a path in the region where $g(x)=0$ along which $f$ was increasing, contradicting our assertion that we’d have to leave the region for this to happen. But if $f$ decreased to first order, then it would increase to first order in the opposite direction, and we’d have the same problem. That is, we must have $df(a;v)=0$ whenever $dg(a;v)=0$. And so we find that $\mathrm{Ker}(dg(a))=\bigcap\limits_{i=1}^m\mathrm{Ker}(dg^i(a))\subseteq\mathrm{Ker}(df(a))$ The kernel of $df(a)$ consists of all vectors orthogonal to the gradient vector $\nabla f(a)$, and the line it spans is the orthogonal complement to the kernel. Similarly, the kernel of $dg(a)$ consists of all vectors orthogonal to each of the gradient vectors $\nabla g^i(a)$, and is thus the orthogonal complement to the entire subspace they span. The kernel of $dg(a)$ is contained in the kernel of $df(a)$, and orthogonal complements are order-reversing, which means that $\nabla f(a)$ must lie within the span of the $\nabla g^i(a)$. That is, there must be real numbers $\lambda_i$ so that $\displaystyle\nabla f(a)=\sum\limits_{i=1}^m\lambda_i\nabla g^i(a)$ or, passing back to differentials $\displaystyle df(a)=\sum\limits_{i=1}^m\lambda_idg^i(a)$ So in the presence of constraints we replace the condition $df(a)=0$ by this one. We call the $\lambda_i$ “Lagrange multipliers”, and for every one of these variables we add to the system of equations, we also add the constraint equation $g^i(a)=0$, so we should still get an isolated collection of points. Now, we reached this conclusion by a rather handwavy argument about being able to find increasing directions and so on within the region $g(x)=0$. This line of reasoning could possibly be firmed up, but we’ll find our proof next time in a slightly different approach. November 25, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Classifying Critical Points So let’s say we’ve got a critical point of a multivariable function $f:X\rightarrow\mathbb{R}$. That is, a point $a\in X$ where the differential $df(x)$ vanishes. We want something like the second derivative test that might tell us more about the behavior of the function near that point, and to identify (some) local maxima and minima. We’ll assume here that $f$ is twice continuously differentiable in some region $S$ around $a$. The analogue of the second derivative for multivariable functions is the second differential $d^2f(x)$. This function assigns to every point a bilinear function of two displacement vectors $u$ and $v$, and it measures the rate at which the directional derivative in the direction of $v$ is changing as we move in the direction of $u$. That is, $\displaystyle d^2f(x;u,v)=\left[D_u\left(D_vf\right)\right](x)$ If we choose coordinates on $X$ given by an orthonormal basis $\{e_i\}_{i=1}^n$, we can write the second differential in terms of coordinates $\displaystyle d^2f(x)=\frac{\partial^2f}{\partial x^i\partial x^j}dx^idx^j$ This matrix is often called the “Hessian” of $f$ at the point $x$. As I said above, this is a bilinear form. Further, Clairaut’s theorem tells us that it’s a symmetric form. Then the spectral theorem tells us that we can find an orthonormal basis with respect to which the Hessian is actually diagonal, and the diagonal entries are the eigenvalues of the matrix. So let’s go back and assume we’re working with such a basis. This means that our second partial derivatives are particularly simple. We find that for $i\neq j$ we have $\displaystyle\frac{\partial^2f}{\partial x^i\partial x^j}=0$ and for $i=j$, the second partial derivative is an eigenvalue $\displaystyle\frac{\partial^2f}{{\partial x^i}^2}=\lambda_i$ which we can assume (without loss of generality) are nondecreasing. That is, $\lambda_1\leq\lambda_2\leq\dots\leq\lambda_n$. Now, if all of these eigenvalues are positive at a critical point $a$, then the Hessian is positive-definite. That is, given any direction $v$ we have $d^2f(a;v,v)>0$. On the other hand, if all of the eigenvalues are negative, the Hessian is negative definite; given any direction $v$ we have $d^2f(a;v,v)<0$. In the former case, we’ll find that $f$ has a local minimum in a neighborhood of $a$, and in the latter case we’ll find that $f$ has a local maximum there. If some eigenvalues are negative and others are positive, then the function has a mixed behavior at $a$ we’ll call a “saddle” (sketch the graph of $f(x,y)=xy$ near $(0,0)$ to see why). And if any eigenvalues are zero, all sorts of weird things can happen, though at least if we can find one positive and one negative eigenvalue we know that the critical point can’t be a local extremum. We remember that the determinant of a diagonal matrix is the product of its eigenvalues, so if the determinant of the Hessian is nonzero then either we have a local maximum, we have a local minimum, or we have some form of well-behaved saddle. These behaviors we call “generic” critical points, since if we “wiggle” the function a bit (while maintaining a critical point at $a$) the Hessian determinant will stay nonzero. If the Hessian determinant is zero, wiggling the function a little will make it nonzero, and so this sort of critical point is not generic. This is the sort of unstable situation analogous to a failure of the second derivative test. Unfortunately, the analogy doesn’t extent, in that the sign of the Hessian determinant isn’t instantly meaningful. In two dimensions a positive determinant means both eigenvalues have the same sign — denoting a local maximum or a local minimum — while a negative determinant denotes eigenvalues of different signs — denoting a saddle. This much is included in multivariable calculus courses, although usually without a clear explanation why it works. So, given a direction vector $v$ so that $d^2f(a;v,v)>0$, then since $f$ is in $C^2(S)$, there will be some neighborhood $N$ of $a$ so that $d^2f(x;v,v)>0$ for all $x\in N$. In particular, there will be some range of $t$ so that $b=a+tv\in N$. For any such point we can use Taylor’s theorem with $m=2$ to tell us that $\displaystyle f(b)-f(a)=\frac{1}{2}d^2f(\xi;tv,tv)=\frac{t^2}{2}d^2f(\xi;v,v)$ for some $\xi\in[a,b]\subseteq N$. And from this we see that $f(b)>f(a)$ for every $b\in N$ so that $b-a=tv$. A similar argument shows that if $d^2f(a;v,v)<0$ then $f(b) for any $b$ near $a$ in the direction of $v$. Now if the Hessian is positive-definite then every direction $v$ from $a$ gives us $d^2f(a;v,v)>0$, and so every point $b$ near $a$ satisfies $f(b)>f(a)$. If the Hessian is negative-definite, then every point $b$ near $a$ satisfies $f(b). And if the Hessian has both positive and negative eigenvalues then within any neighborhood we can find some directions in which $f(b)>f(a)$ and some in which $f(b). November 24, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Local Extrema in Multiple Variables Just like in one variable, we’re interested in local maxima and minima of a function $f:X\rightarrow\mathbb{R}$, where $X$ is an open region in $\mathbb{R}^n$. Again, we say that $f$ has a local minimum at a point $a\in X$ if there is some neighborhood $N$ of $a$ so that $f(a)\leq f(x)$ for all $x\in N$. A maximum is similarly defined, except that we require $f(a)\geq f(x)$ in the neighborhood. As I alluded to recently, we can bring Fermat’s theorem to bear to determine a necessary condition. Specifically, if we have coordinates on $\mathbb{R}^n$ given by a basis $\{e_i\}_{i=1}^n$, we can regard $f$ as a function of the $n$ variables $x^i$. We can fix $n-1$ of these variables $x^i=a^i$ for $i\neq k$ and let $x^k$ vary in a neighborhood of $a^k$. If $f$ has a local extremum at $x=a$, then in particular it has a local extremum along this coordinate line at $x^k=a^k$. And so we can use Fermat’s theorem to draw conclusions about the derivative of this restricted function at $x^k=a^k$, which of course is the partial derivative $\frac{\partial f}{\partial x^k}\big\vert_{x=a}$. So what can we say? For each variable $x^k$, the partial derivative $\frac{\partial f}{\partial x^k}$ either does not exist or is equal to zero at $x=a$. And because the differential subsumes the partial derivatives, if any of them fail to exist the differential must fail to exist as well. On the other hand, if they all exist they’re all zero, and so $df(a)=0$ as well. Incidentally, we can again make the connection to the usual coverage in a multivariable calculus course by remembering that the gradient $\nabla f(a)$ is the vector that corresponds to the linear functional of the differential $df(a)$. So at a local extremum we must have $\nabla f(a)=0$. As was the case with Fermat’s theorem, this provides a necessary, but not a sufficient condition to have a local extremum. Anything that can go wrong in one dimension can be copied here. For instance, we could define $f(x,y)=x^2+y^3$. Then we find $df=2x\,dx+3y^2\,dy$, which is zero at $(0,0)$. But any neighborhood of this point will contain points $(0,t)$ and $(0,-t)$ for small enough $t>0$, and we see that $f(0,t)>f(0,0)>f(0,-t)$, so the origin cannot be a local extremum. But weirder things can happen. We might ask that $f$ have a local minimum at $a$ along any line, like we tried with directional derivatives. But even this can go wrong. If we define $\displaystyle f(x,y)=(y-x^2)(y-3x^2)=y^2-4x^2y+3x^4$ we can calculate $\displaystyle df=\left(-8xy+12x^3\right)dx+\left(2y-4x^2\right)dy$ which again is zero at $(0,0)$. Along any slanted line through the origin $y=kx$ we find \displaystyle\begin{aligned}f(t,kt)&=3t^4-4kt^3+k^2t^2\\\frac{d}{dt}f(t,kt)&=12t^3-12kt^2+2k^2t\\\frac{d^2}{dt^2}f(t,kt)&=36t^2-24kt+2k^2\end{aligned} and so the second derivative is always positive at the origin, except along the $x$-axis. For the vertical line, we find \displaystyle\begin{aligned}f(0,t)&=t^2\\\frac{d}{dt}f(t,kt)&=2t\\\frac{d^2}{dt^2}f(t,kt)&=2\end{aligned} so along all of these lines we have a local minimum at the origin by the second derivative test. And along the $x$-axis, we have $f(x,0)=3x^4$, which has the origin as a local minimum. Unfortunately, it’s still not a local minimum in the plane, since any neighborhood of the origin must contain points of the form $(t,2t^2)$ for small enough $t$. For these points we find $\displaystyle f(t,2t^2)=-t^4<0=f(0,0)$ and so $f$ cannot have a local minimum at the origin. What we’ll do is content ourselves with this analogue and extension of Fermat’s theorem as a necessary condition, and then develop tools that can distinguish the common behaviors near such critical points, analogous to the second derivative test. November 23, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## The Implicit Function Theorem II Okay, today we’re going to prove the implicit function theorem. We’re going to think of our function $f$ as taking an $n$-dimensional vector $x$ and a $m$-dimensional vector $t$ and giving back an $n$-dimensional vector $f(x;t)$. In essence, what we want to do is see how this output vector must change as we change $t$, and then undo that by making a corresponding change in $x$. And to do that, we need to know how changing the output changes $x$, at least in a neighborhood of $f(x;t)=0$. That is, we’ve got to invert a function, and we’ll need to use the inverse function theorem. But we’re not going to apply it directly as the above heuristic suggests. Instead, we’re going to “puff up” the function $f:S\rightarrow\mathbb{R}^n$ into a bigger function $F:S\rightarrow\mathbb{R}^{n+m}$ that will give us some room to maneuver. For $1\leq i\leq n$ we define $\displaystyle F^i(x;t)=f^i(x;t)$ just copying over our original function. Then we continue by defining for $1\leq j\leq m$ $\displaystyle F^{n+j}(x;t)=t^j$ That is, the new $m$ component functions are just the coordinate functions $t^j$. We can easily calculate the Jacobian matrix $\displaystyle dF=\begin{pmatrix}\frac{\partial f^i}{\partial x^j}&\frac{\partial f^i}{\partial t^j}\\{0}&I_m\end{pmatrix}$ where ${0}$ is the $m\times n$ zero matrix and $I_m$ is the $m\times m$ identity matrix. From here it’s straightforward to find the Jacobian determinant $\displaystyle J_F(x;t)=\det\left(dF\right)=\det\left(\frac{\partial f^i}{\partial x^j}\right)$ which is exactly the determinant we assert to be nonzero at $(a;b)$. We also easily see that $F(a;b)=(0;b)$. And so the inverse function theorem tells us that there are neighborhoods $X$ of $(a;b)$ and $Y$ of $(0;b)$ so that $F$ is injective on $X$ and $Y=F(X)$, and that there is a continuously differentiable inverse function $G:Y\rightarrow X$ so that $G(F(x;t))=(x;t)$ for all $(x;t)\in X$. We want to study this inverse function to recover our implicit function from it. First off, we can write $G(y;s)=(v(y;s);w(y;s))$ for two functions: $v$ which takes $n$-dimensional vector values, and $w$ which takes $m$-dimensional vector values. Our inverse relation tells us that \displaystyle\begin{aligned}v(F(x;t))&=x\\w(F(x;t))&=t\end{aligned} But since $F$ is injective from $X$ onto $Y$, we can write any point $(y;s)\in Y$ as $(y;s)=F(x;t)$, and in this case we must have $s=t$ by the definition of $s$. That is, we have \displaystyle\begin{aligned}v(y;t)&=v(F(x;t))=x\\w(y;t)&=w(F(x;t))=t\end{aligned} And so we see that $G(y;t)=(x;t)$, where $x$ is the $n$-dimensional vector so that $y=f(x;t)$. We thus have $f(v(y;t);t)=y$ for every $(y;t)\in Y$. Now define $T\subseteq\mathbb{R}^m$ be the collection of vectors $t$ so that $(0;t)\in Y$, and for each such $t\in T$ define $g(t)=v(0;t)$, so $F(g(t);t)=0$. As a slice of the open set $Y$ in the product topology on $\mathbb{R}^n\times\mathbb{R}^m$, the set $T$ is open in $\mathbb{R}^m$. Further, $g$ is continuously differentiable on $T$ since $G$ is continuously differentiable on $Y$, and the components of $g$ are taken directly from those of $G$. Finally, $b$ is in $T$ since $(a;b)\in X$, and $F(a;b)=(0;b)\in Y$ by assumption. This also shows that $g(b)=a$. The only thing left is to show that $g$ is uniquely defined. But there can only be one such function, by the injectivity of $f$. If there were another such function $h$ then we’d have $f(g(t);t)=f(h(t);t)$, and thus $(g(t);t)=(h(t);t)$, or $g(t)=h(t)$ for every $t\in T$. November 20, 2009 Posted by \| Analysis, Calculus \| 1 Comment ## The Implicit Function Theorem I Let’s consider the function $F(x,y)=x^2+y^2-1$. The collection of points $(x,y)$ so that $F(x,y)=0$ defines a curve in the plane: the unit circle. Unfortunately, this relation is not a function. Neither is $y$ defined as a function of $x$, nor is $x$ defined as a function of $y$ by this curve. However, if we consider a point $(a,b)$ on the curve (that is, with $F(a,b)=0$), then near this point we usually do have a graph of $x$ as a function of $y$ (except for a few isolated points). That is, as we move $y$ near the value $b$ then we have to adjust $x$ to maintain the relation $F(x,y)=0$. There is some function $f(y)$ defined “implicitly” in a neighborhood of $b$ satisfying the relation $F(f(y),y)=0$. We want to generalize this situation. Given a system of $n$ functions of $n+m$ variables $\displaystyle f^i(x;t)=f^i(x^1,\dots,x^n;t^1,\dots,t^m)$ we consider the collection of points $(x;t)$ in $n+m$-dimensional space satisfying $f(x;t)=0$. If this were a linear system, the rank-nullity theorem would tell us that our solution space is (generically) $m$ dimensional. Indeed, we could use Gauss-Jordan elimination to put the system into reduced row echelon form, and (usually) find the resulting matrix starting with an $n\times n$ identity matrix, like $\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}$ This makes finding solutions to the system easy. We put our $n+m$ variables into a column vector and write $\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}\begin{pmatrix}x^1\\x^2\\x^3\\t^1\\t^2\end{pmatrix}=\begin{pmatrix}x^1+2t^1+t^2\\x^2+3t^1\\x^3-t^1+t^2\end{pmatrix}=\begin{pmatrix}0\\{0}\\{0}\end{pmatrix}$ and from this we find \displaystyle\begin{aligned}x^1&=-2t^1-t^2\\x^2&=-3t^1\\x^3&=t^1-t^2\end{aligned} Thus we can use the $m$ variables $t^j$ as parameters on the space of solutions, and define each of the $x^i$ as a function of the $t^j$. But in general we don’t have a linear system. Still, we want to know some circumstances under which we can do something similar and write each of the $x^i$ as a function of the other variables $t^j$, at least near some known point $(a;b)$. The key observation is that we can perform the Gauss-Jordan elimination above and get a matrix with rank $n$ if and only if the leading $n\times n$ matrix is invertible. And this is generalized to asking that some Jacobian determinant of our system of functions is nonzero. Specifically, let’s assume that all of the $f^i$ are continuously differentiable on some region $S$ in $n+m$-dimensional space, and that $(a;b)$ is some point in $S$ where $f(a;b)=0$, and at which the determinant $\displaystyle\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{(a;t)}\right)\neq0$ where both indices $i$ and $j$ run from $1$ to $n$ to make a square matrix. Then I assert that there is some $k$-dimensional neighborhood $T$ of $b$ and a uniquely defined, continuously differentiable, vector-valued function $g:T\rightarrow\mathbb{R}^n$ so that $g(b)=a$ and $f(g(t);t)=0$. That is, near $(a;b)$ we can use the variables $t^j$ as parameters on the space of solutions to our system of equations. Near this point, the solution set looks like the graph of the function $x=g(t)$, which is implicitly defined by the need to stay on the solution set as we vary $t$. This is the implicit function theorem, and we will prove it next time. November 19, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## The Inverse Function Theorem At last we come to the theorem that I promised. Let $f:S\rightarrow\mathbb{R}^n$ be continuously differentiable on an open region $S\subseteq\mathbb{R}^n$, and $T=f(S)$. If the Jacobian determinant $J_f(a)\neq0$ at some point $a\in S$, then there is a uniquely determined function $g$ and two open sets $X\subseteq S$ and $Y\subseteq T$ so that • $a\in X$, and $f(a)\in Y$ • $Y=f(X)$ • $f$ is injective on $X$ • $g$ is defined on $Y$, $g(Y)=X$, and $g(f(x))=x$ for all $x\in X$ • $g$ is continuously differentiable on $Y$ The Jacobian determinant $J_f(x)$ is continuous as a function of $x$, so there is some neighborhood $N_1$ of $a$ so that the Jacobian is nonzero within $N_1$. Our second lemma tells us that there is a smaller neighborhood $N\subseteq N_1$ on which $f$ is injective. We pick some closed ball $\overline{K}\subseteq N$ centered at $a$, and use our first lemma to find that $f(K)$ must contain an open neighborhood $Y$ of $f(a)$. Then we define $X=f^{-1}(Y)\cap K$, which is open since both $K$ and $f^{-1}(Y)$ are (the latter by the continuity of $f$). Since $f$ is injective on the compact set $\overline{K}\subseteq N$, it has a uniquely-defined continuous inverse $g$ on $Y\subseteq f(\overline{K})$. This establishes the first four of the conditions of the theorem. Now the hard part is showing that $g$ is continuously differentiable on $Y$. To this end, like we did in our second lemma, we define the function $\displaystyle h(z_1,\dots,z_n)=\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{x=z_i}\right)$ along with a neighborhood $N_2$ of $a$ so that as long as all the $z_i$ are within $N_2$ this function is nonzero. Without loss of generality we can go back and choose our earlier neighborhood $N$ so that $N\subseteq N_2$, and thus that $\overline{K}\subseteq N_2$. To show that the partial derivative $\frac{\partial g^i}{\partial y^j}$ exists at a point $y\in Y$, we consider the difference quotient $\displaystyle\frac{g^i(y+\lambda e_j)-g^i(y)}{\lambda}$ with $y+\lambda e_j$ also in $Y$ for sufficiently small $\lvert\lambda\rvert$. Then writing $x_1=g(y)$ and $x_2=g(y+\lambda e_j)$ we find $f(x_2)-f(x_1)=\lambda e_j$. The mean value theorem then tells us that \displaystyle\begin{aligned}\delta_j^k&=\frac{f^k(x_2)-f^k(x_1)}{\lambda}\\&=df^k(\xi_k)\left(\frac{1}{\lambda}(x_2-x_1)\right)\\&=\frac{\partial f^k}{\partial x^i}\bigg\vert_{x=\xi_k}\frac{x_2^i-x_1^i}{\lambda}\\&=\frac{\partial f^k}{\partial x^i}\bigg\vert_{x=\xi_k}\frac{g^i(y+\lambda e_j)-g^i(y)}{\lambda}\end{aligned} for some $\xi_k\in[x_1,x_2]\subseteq K$ (no summation on $k$). As usual, $\delta_j^k$ is the Kronecker delta. This is a linear system of equations, which has a unique solution since the determinant of its matrix is $h(\xi_1,\dots,\xi_n)\neq0$. We use Cramer’s rule to solve it, and get an expression for our difference quotient as a quotient of two determinants. This is why we want the form of the solution given by Cramer’s rule, and not by a more computationally-efficient method like Gaussian elimination. As $\lambda$ approaches zero, continuity of $g$ tells us that $x_2$ approaches $x_1$, and thus so do all of the $\xi_k$. Therefore the determinant in the denominator of Cramer’s rule is in the limit $h(x,\dots,x)=J_f(x)\neq0$, and thus limits of the solutions given by Cramer’s rule actually do exist. This establishes that the partial derivative $\frac{\partial g^i}{\partial y^j}$ exists at each $y\in Y$. Further, since we found the limit of the difference quotient by Cramer’s rule, we have an expression given by the quotient of two determinants, each of which only involves the partial derivatives of $f$, which are themselves all continuous. Therefore the partial derivatives of $g$ not only exist but are in fact continuous. November 18, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Cramer’s Rule We’re trying to invert a function $f:X\rightarrow\mathbb{R}^n$ which is continuously differentiable on some region $X\subseteq\mathbb{R}^n$. That is we know that if $a$ is a point where $J_f(a)\neq0$, then there is a ball $N$ around $a$ where $f$ is one-to-one onto some neighborhood $f(N)$ around $f(a)$. Then if $y$ is a point in $f(N)$, we’ve got a system of equations $\displaystyle f^j(x^1,\dots,x^n)=y^j$ that we want to solve for all the $x^i$. We know how to handle this if $f$ is defined by a linear transformation, represented by a matrix $A=\left(a_i^j\right)$: \displaystyle\begin{aligned}f^j(x^1,\dots,x^n)=a_i^jx^i&=y^j\\Ax&=y\end{aligned} In this case, the Jacobian transformation is just the function $f$ itself, and so the Jacobian determinant $\det\left(a_i^j\right)$ is nonzero if and only if the matrix $A$ is invertible. And so our solution depends on finding the inverse $A^{-1}$ and solving \displaystyle\begin{aligned}Ax&=y\\A^{-1}Ax&=A^{-1}y\\x&=A^{-1}y\end{aligned} This is the approach we’d like to generalize. But to do so, we need a more specific method of finding the inverse. This is where Cramer’s rule comes in, and it starts by analyzing the way we calculate the determinant of a matrix $A$. This formula $\displaystyle\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots a_n^{\pi(n)}$ involves a sum over all the permutations $\pi\in S_n$, and we want to consider the order in which we add up these terms. If we fix an index $i$, we can factor out each matrix entry in the $i$th column: $\displaystyle\sum\limits_{j=1}^na_i^j\sum\limits_{\substack{\pi\in S_n\\\pi(i)=j}}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots\widehat{a_i^j}\dots a_n^{\pi(n)}$ where the hat indicates that we omit the $i$th term in the product. For a given value of $j$, we can consider the restricted sum $\displaystyle A_j^i=\sum\limits_{\substack{\pi\in S_n\\\pi(i)=j}}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots\widehat{a_i^j}\dots a_n^{\pi(n)}$ which is $(-1)^{i+j}$ times the determinant of the $i$$j$ “minor” of the matrix $A$. That is, if we strike out the row and column of $A$ which contain $a_i^j$ and take the determinant of the remaining $(n-1)\times(n-1)$ matrix, we multiply this by $(-1)^{i+j}$ to get $A_j^i$. These are the entries in the “adjugate” matrix $\mathrm{adj}(A)$. What we’ve shown is that $\displaystyle A_j^ia_i^j=\det(A)$ (no summation on $i$). It’s not hard to show, however, that if we use a different row from the adjugate matrix we find $\displaystyle\sum\limits_{j=1}^nA_j^ka_i^j=\det(A)\delta_i^k$ That is, the adjugate times the original matrix is the determinant of $A$ times the identity matrix. And so if $\det(A)\neq0$ we find $\displaystyle A^{-1}=\frac{1}{\det(A)}\mathrm{adj}(A)$ So what does this mean for our system of equations? We can write \displaystyle\begin{aligned}x&=\frac{1}{\det(A)}\mathrm{adj}(A)y\\x^i&=\frac{1}{\det(A)}A_j^iy^j\end{aligned} But how does this sum $A_j^iy^j$ differ from the one $A_j^ia_i^j$ we used before (without summing on $i$) to calculate the determinant of $A$? We’ve replaced the $i$th column of $A$ by the column vector $y$, and so this is just another determinant, taken after performing this replacement! Here’s an example. Let’s say we’ve got a system written in matrix form $\displaystyle\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}u\\v\end{pmatrix}$ The entry in the $i$th row and $j$th column of the adjugate matrix is calculated by striking out the $i$th column and $j$th row of our original matrix, taking the determinant of the remaining matrix, and multiplying by $(-1)^{i+j}$. We get $\displaystyle\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ and thus we find $\displaystyle\begin{pmatrix}x\\y\end{pmatrix}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\begin{pmatrix}u\\v\end{pmatrix}=\frac{1}{ad-bc}\begin{pmatrix}ud-bv\\av-uc\end{pmatrix}$ where we note that \displaystyle\begin{aligned}ud-bv&=\det\begin{pmatrix}u&b\\v&d\end{pmatrix}\\av-uc&=\det\begin{pmatrix}a&u\\c&v\end{pmatrix}\end{aligned} In other words, our solution is given by ratios of determinants: \displaystyle\begin{aligned}x&=\frac{\det\begin{pmatrix}u&b\\v&d\end{pmatrix}}{\det\begin{pmatrix}a&b\\c&d\end{pmatrix}}\\y&=\frac{\det\begin{pmatrix}a&u\\c&v\end{pmatrix}}{\det\begin{pmatrix}a&b\\c&d\end{pmatrix}}\end{aligned} and similar formulae hold for larger systems of equations. November 17, 2009 Posted by \| Algebra, Linear Algebra \| 8 Comments ## Another Lemma on Nonzero Jacobians Sorry for the late post. I didn’t get a chance to get it up this morning before my flight. Brace yourself. Just like last time we’ve got a messy technical lemma about what happens when the Jacobian determinant of a function is nonzero. This time we’ll assume that $f:X\rightarrow\mathbb{R}^n$ is not only continuous, but continuously differentiable on a region $X\subseteq\mathbb{R}^n$. We also assume that the Jacobian $J_f(a)\neq0$ at some point $a\in X$. Then I say that there is some neighborhood $N$ of $a$ so that $f$ is injective on $N$. First, we take $n$ points $\{z_i\}_{i=1}^n$ in $X$ and make a function of them $\displaystyle h(z_1,\dots,z_n)=\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{x=z_i}\right)$ That is, we take the $j$th partial derivative of the $i$th component function and evaluate it at the $i$th sample point to make a matrix $\left(a_{ij}\right)$, and then we take the determinant of this matrix. As a particular value, we have $\displaystyle h(a,\dots,a)=J_f(a)\neq0$ Since each partial derivative is continuous, and the determinant is a polynomial in its entries, this function is continuous where it’s defined. And so there’s some ball $N$ of $a$ so that if all the $z_i$ are in $N$ we have $h(z_1,\dots,z_n)\neq0$. We want to show that $f$ is injective on $N$. So, let’s take two points $x$ and $y$ in $N$ so that $f(x)=f(y)$. Since the ball is convex, the line segment $[x,y]$ is completely contained within $N\subseteq X$, and so we can bring the mean value theorem to bear. For each component function we can write $\displaystyle0=f^i(y)-f^i(x)=df^i(\xi_i)(y-x)=\frac{\partial f^i}{\partial x^j}\bigg\vert_{\xi_i}(y^j-x^j)$ for some $\xi_i$ in $[x,y]\subseteq N$ (no summation here on $i$). But like last time we now have a linear system of equations described by an invertible matrix. Here the matrix has determinant $\displaystyle\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{\xi_i}\right)=h(\xi_1,\dots,\xi_n)\neq0$ which is nonzero because all the $\xi_i$ are inside the ball $N$. Thus the only possible solution to the system of equations is $x^i=y^i$. And so if $f(x)=f(y)$ for points within the ball $N$, we must have $x=y$, and thus $f$ is injective. November 17, 2009 Posted by \| Analysis, Calculus \| 2 Comments ## A Lemma on Nonzero Jacobians Okay, let’s dive right in with a first step towards proving the inverse function theorem we talked about at the end of yesterday’s post. This is going to get messy. We start with a function $f$ and first ask that it be continuous and injective on the closed ball $\overline{K}$ of radius $r$ around the point $a$. Then we ask that all the partial derivatives of $f$ exist within the open interior $K$ — note that this is weaker than our existence condition for the differential of $f$ — and that the Jacobian determinant $J_f(x)\neq0$ on $K$. Then I say that the image $f(K)$ actually contains a neighborhood of $f(a)$. That is, the image doesn’t “flatten out” near $a$. The boundary $\partial K$ of the ball $K$ is the sphere of radius $r$: $\displaystyle\partial K=\left\{x\in\mathbb{R}^n\vert\lVert x-a\rVert=r\right\}$ Now the Heine-Borel theorem says that this sphere, being both closed and bounded, is a compact subset of $\mathbb{R}^n$. We’ll define a function on this sphere by $\displaystyle g(x)=\lVert f(x)-f(a)\rVert$ which must be continuous and strictly positive, since if $\lVert f(x)-f(a)\rVert=0$ then $f(x)=f(a)$, but we assumed that $f$ is injective on $\overline{K}$. But we also know that the image of a continuous real-valued function on a compact, connected space must be a closed interval. That is, $g(\partial K)=[m,M]$, and there exists some point $x$ on the sphere where this minimum is actually attained: $g(x)=m>0$. Now we’re going to let $T$ be the ball of radius $\frac{m}{2}$ centered at $f(a)$. We will show that $T\subseteq f(K)$, and is thus a neighborhood of $f(a)$ contained within $f(K)$. To this end, we’ll pick $y\in T$ and show that $y\in f(X)$. So, given such a point $y\in T$, we define a new function on the closed ball $\overline{K}$ by $\displaystyle h(x)=\lVert f(x)-y\rVert$ This function is continuous on the compact ball $\overline{K}$, so it again has an absolute minimum. I say that it happens somewhere in the interior $K$. At the center of the ball, we have $h(a)=\lVert f(a)-y\rVert<\frac{m}{2}$ (since $y\in T$), so the minimum must be even less. But on the boundary $\partial K$, we find \displaystyle\begin{aligned}h(x)&=\lVert f(x)-y\rVert\\&=\lVert f(x)-f(a)-(y-f(a))\rVert\\&\geq\lVert f(x)-f(a)\rVert-\lVert f(a)-y\rVert\\&>g(x)-\frac{m}{2}\geq\frac{m}{2}\end{aligned} so the minimum can’t happen on the boundary. So this minimum of $h$ happens at some point $b$ in the open ball $K$, and so does the minimum of the square of $h$: $\displaystyle h(x)^2=\lVert f(x)-y\rVert^2=\sum\limits_{i=1}^n\left(f^i(x)-y^i\right)^2$ Now we can vary each component $x^i$ of $x$ separately, and use Fermat’s theorem to tell us that the derivative in terms of $x^i$ must be zero at the minimum value $b^i$. That is, each of the partial derivatives of $h^2$ must be zero (we’ll come back to this more generally later): $\displaystyle\frac{\partial}{\partial x^k}\left[\sum\limits_{i=1}^n\left(f^i(x)-y^i\right)^2\right]\Bigg\vert_{x=b}=\sum\limits_{i=1}^n2\left(f^i(b)-y^i\right)\frac{\partial f^i}{\partial x^k}\bigg\vert_{x=b}=0$ This is the product of the vector $2(f(b)-y)$ by the matrix $\left(\frac{\partial f^i}{\partial x^k}\right)$. And the determinant of this matrix is $J_f(b)$: the Jacobian determinant at $b\in K$, which we assumed to be nonzero way back at the beginning! Thus the matrix must be invertible, and the only possible solution to this system of equations is for $f(b)-y=0$, and so $y=f(b)\in f(K)$. November 13, 2009 Posted by \| Analysis, Calculus \| 3 Comments	{"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": 629, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9866811633110046, "perplexity": 138.4841060837813}, "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/1448398525032.0/warc/CC-MAIN-20151124205525-00154-ip-10-71-132-137.ec2.internal.warc.gz"}
http://mathhelpforum.com/advanced-algebra/217023-problem-quotient-groups-first-isomorphism-theorem.html	# Thread: Problem on Quotient Groups and First Isomorphism Theorem 1. ## Problem on Quotient Groups and First Isomorphism Theorem Q: Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be the element of P which contains 1. Prove that N is a normal subgroup of G and that P is the set of its cosets. Sol: I am thinking of a surjective homomorphism f: G--> P with N as the kernel. Then N being the kernel is normal and G/N is isomorphic to P which implies P is the set of cosets of N in G. My problem: P is a partiotion of G => there is an equivalence relation on G such that the elements of P are exactly the equivalence classes. Now, if I define f: G--> P by f(a)= [a], where [a] represents the equivalence class of a, then how can I prove [ab]=[a].[b] , i.e. f is a homomorphism? Is my approach correct? 2. ## Re: Problem on Quotient Groups and First Isomorphism Theorem Hi, Interesting little problem. I think a direct approach is probably the easiest; here's a solution: 3. ## Re: Problem on Quotient Groups and First Isomorphism Theorem Originally Posted by johng Hi, Interesting little problem. I think a direct approach is probably the easiest; here's a solution: thanks...but is it possible to solve in my way of approach? 4. ## Re: Problem on Quotient Groups and First Isomorphism Theorem Hi, Yes, I think so. But first if you want to define a homomorphism on G, the codomain of the function must be a group. That is, the partition P is a group. So you need to define the group operation on P. Let [a] denote the member of P containing group element a. 1. Define [a][b]=[ab]; you need to verify this is a well defined binary operation. That is if a1 is in [a] and b1 is in [b], then [ab]=[a1b1]. This is possible only because of the assumption on product sets and P. 2. All the other axioms for a group -- associativity, identity and inverses. This is all straight forward; in particular the identity is [1] where 1 is the identity element of G. Note: for an arbitrary partition of P, the above "operation" is not well defined. Example: take P to be the right cosets of a subgroup H of G where H is not normal in G. Now, you can define a homomorphism f on G by f(a)=[a]. By the definition of the binary operation on P, this is obviously a homomorphism. You can now verify that N is the kernel of f and so a normal subgroup of G. However you still must show the members of P are the cosets of N in G. I really don't see any other way to do this except as in my original posting. (Your assertion that G/N isomorphic to P implies this doesn't really follow.) 5. ## Re: Problem on Quotient Groups and First Isomorphism Theorem Originally Posted by johng Hi, Yes, I think so. But first if you want to define a homomorphism on G, the codomain of the function must be a group. That is, the partition P is a group. So you need to define the group operation on P. Let [a] denote the member of P containing group element a. 1. Define [a][b]=[ab]; you need to verify this is a well defined binary operation. That is if a1 is in [a] and b1 is in [b], then [ab]=[a1b1]. This is possible only because of the assumption on product sets and P. 2. All the other axioms for a group -- associativity, identity and inverses. This is all straight forward; in particular the identity is [1] where 1 is the identity element of G. Note: for an arbitrary partition of P, the above "operation" is not well defined. Example: take P to be the right cosets of a subgroup H of G where H is not normal in G. Now, you can define a homomorphism f on G by f(a)=[a]. By the definition of the binary operation on P, this is obviously a homomorphism. You can now verify that N is the kernel of f and so a normal subgroup of G. However you still must show the members of P are the cosets of N in G. I really don't see any other way to do this except as in my original posting. 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http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/relative-permeability/laboratory-measurements-relative-permeability/unsteady-state-techniques/	Sign up and NOW to receive the latest news, updates and technological advancements made for the Special Core Analysis & Enhanced Oil Recovery Industry If you don't sign up, you won't know when the next breakthrough occurs! # Unsteady State Techniques Unsteady State Techniques 2016-10-25T11:54:41+00:00 # Laboratory Measurements of Relative Permeability: Unsteady State Techniques Experimentally and computationally the unsteady state technique is far more complicated. Following is a schematic diagram of an unsteady-state test apparatus. Figure 2-79: Unsteady State Apparatus As can be seen, there is only one inlet since only one phase is injected into the core. The major difference in unsteady state techniques is that saturation equilibrium is not achieved during the test. As well, fluids are not injected simultaneously into the core. Instead, the test involves displacing in-situ fluids with a constant rate/pressure driving fluid. The outlet fluid composition and flow rate is measured and used in determining the relative permeability. Figure 2‑80-b shows an example of such a curve from an unsteady state waterflood experiment. At the beginning of the experiment, the core is saturated with 80% oil, and there is an irreducible water saturation of 20% due to the water wet nature of this particular example. Point A (Figure 2‑80-a) represents the permeability of oil under these conditions. Note that it is equal to unity, because this measurement has been taken as the base permeability. Point B represents the beginning water permeability. Note that it is equal to zero because irreducible water is, by definition, immobile. Water is then injected into the core at one end at a constant rate. The volumes of the emerging fluids (oil and water) are measured at the other end of the core, and the differential pressure across the core is also measured. During this process the permeability to oil reduces to zero along the curve ACD, and the permeability to water increases along the curve BCE. Note that there is no further production of oil from the sample after kro= 0 at point D, and so point D occurs at the irreducible oil saturation, Sor. Figure 2‑80-a is a simplified schematic diagram detailing the fluid saturation in the core during a typical unsteady-state test. Figure 2-80: (a) Unsteady State Water Flood Proceudre, (b) Typical Relative Permeability Curve Since steady state is not reached, Darcy’s Law is not applicable. The Buckley-Leverett equation for linear fluid displacement is the basis for all calculations. The Johnson-Bossler-Naumann (JBN) solution is used most often for calculating relative permeabilites from unsteady-state displacement tests. It must be stressed, however, that these curves are not a unique function of saturation, but are also dependent upon fluid distribution. Thus the data obtained can be influenced by saturation history and flow rate. The choice of test method should be made with due regard for reservoir saturation history, rock and fluid properties. The wetting characteristics are particularly important. Test plugs should either, be of similar wetting characteristics to the reservoir state, or their wetting characteristics be known so that data can be assessed properly. ## JBN Analysis The experimental data generally recorded includes: Qi : Quantity of displacing phase injected ΔP : Pressure differential ΔPi : Pressure differential at initial conditions Qo : Volume of oil produced Qw : Volume of water produced These data are analysed by the technique described by Johnson, Bossler and Nauman, which is summarised below [1]. Three calculation stages are involved: • The ratio kro / krw. • The values of kro and hence krw. • The value of Sw. The method is aimed at giving the required values at the outlet face of the core which is essentially where volumetric flow observations are made. (a). kro / krw : The average water saturation ( Swav ) is plotted against Qi (Figure 2‑81-a). Fractional flow of oil, at the outlet face of the core sample is: Also we have: Figure 2-81: (a) Average Water Saturation vs. Water Injection, (b) Injectivity Ratio (b). kro: A plot of ( ΔP / ΔPi ) against Qi is used to obtain injectivity ratio (Figure 2‑81-b). kro is obtained by plotting 1 / ( QiIR ) versus 1 / ( Qi ) : and using the relationship: krw can be calculated from eq. (2‑121), or: Unsteady state tests are popular because they require much less time and money than steady state tests to operate. However, it must be noted that the values obtained from these tests are generally less reliable. This is for a number of reasons. For one, the system is not at steady state when measurements are taken. Therefore, properties are still changing at that estimated saturation level. This leads to repeatability concerns. The relative permeability calculated during one un-steady state run at a certain saturation level could be quite different from a run performed earlier. For this reason lower confidence is placed on the actual value determined from displacement tests. Another concern is that the interpretation techniques introduce many simplifying assumptions. For example, the Welge method (utilized in the JBN solution) was developed with respect to a homogeneous reservoir, so it might not be entirely accurate if applied to a heterogeneous one. Also, the Buckley-Leverett equation was developed for incompressible/immiscible fluids and assumes completely linear displacement. For all these reasons, values obtained from unsteady state tests should be considered as qualitative, and not representative of the reservoir. As a note, viscous fingering is a definite concern with the majority of unsteady-state tests, but it has been found that the centrifuge method can be used to eliminate these problems. ## References [1] Honarpour, M., Mahmood, S.M (1988) ## Questions? If you have any questions at all, please feel free to ask PERM! We are here to help the community.	{"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": 0.8516375422477722, "perplexity": 1628.7901263648178}, "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/1498128321938.75/warc/CC-MAIN-20170627221726-20170628001726-00268.warc.gz"}
https://www.physicsforums.com/threads/proving-a-hydraulics-equation.258541/	# Proving A Hydraulics Equation 1. Sep 23, 2008 ### Stacyg Prove that the resultant force F on a plane surface of area A immersed vertically in a liquid of density (rho) is given by the equation: F=rho.g.A.z Where 'z' is the depth to the centroid of the plane area below the liquid surface level. I under stand how F is found using this equation and how the force changes depending on the z value. But I don't know how to put this answer in a suitable way for my assignment. Thanks 2. Sep 23, 2008 ### dirk_mec1 The pressure is given by $$p= \rho g \Delta h$$. The force F is given by $$F = pA$$. It's that easy!	{"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": 0.8391565680503845, "perplexity": 475.35053177661985}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170864.16/warc/CC-MAIN-20170219104610-00457-ip-10-171-10-108.ec2.internal.warc.gz"}
https://metric2011.wordpress.com/2017/04/20/notes-of-brian-bowditchs-cambridge-lecture-20-04-2017/	## Notes of Brian Bowditch’s Cambridge lecture 20-04-2017 Bounding genera of singular surfaces Ultimate motivation: understand the curve complex for non-compact surfaces. But today, only closed surfaces ${\Sigma}$ around. 1. Genus distance The curve complex ${\mathcal{C}}$ of ${\Sigma}$ is hyperbolic, pseudo-Anosov diffeos act loxodromically, moving vertices at a linear speed. Say ${\alpha\sim\beta}$ if there is a compact surface ${S}$ with ${\partial S=\alpha_0\cup \beta_0}$ and a map ${S\rightarrow\Sigma}$ mapping ${\alpha_0}$ to ${\alpha}$ and ${\beta_0}$ to ${\beta}$. Let ${A=A(\alpha)}$ denote the equivalence class of ${\alpha}$. There are two cases. Either ${\alpha\sim0}$, i.e. ${\alpha}$ is a separating curve. Or ${\alpha\not\sim 0}$. Fact. ${A}$ is 3-dense in ${\mathcal{C}}$ (1-dense in the separating case). Give, ${\alpha,\beta\in A}$, define ${\rho(\alpha,\beta)=}$ minimal genus of surface ${S}$ achieving ${\alpha\sim\beta}$. This is a metric. Morally, this is related to commutator length. Question. Is this genus distance comparable to the distance in ${\mathcal{C}}$? 2. Pseudo-Anosov distorsion Example. Let ${\phi}$ be a pseudo-Anosov diffeo such that ${h(\alpha)\sim\alpha}$ (such maps exist). Recall that ${d(\alpha,\phi^n\alpha)\sim n}$. I claim that ${\rho(\alpha,\phi^n\alpha)\sim n}$. The proof requires 3-manifold topology. Let ${g=\rho(\alpha,\phi^n\alpha)}$, achieved by some surface ${S}$ with a map ${f:S\rightarrow\Sigma}$. If ${\gamma\subset S}$ is an essential simple closed curve, then ${f(\gamma}$ is not null homotopic in ${\Sigma}$ (otherwise, one could surge ${S}$ into a surface of lower genus). Let ${M_\phi}$ be the mapping torus. It is hyperbolic. Let ${M\rightarrow M_\phi}$ be the cyclic cover, diffeomorphic to ${\Sigma\times{\mathbb R}}$, with periodic geometry, let ${\psi}$ be the deck transformation. Let ${\alpha^}$ be the closed geodesic in ${M}$ freely homotopic to ${\alpha}$. In ${M}$, $\displaystyle \begin{array}{rcl} d(\alpha^,\psi^n\alpha^)\sim n. \end{array}$ Thurston-Bonahon realise the composed map ${F:S\rightarrow\Sigma\rightarrow M}$ by a 1-Lipschitz map from ${S}$ equipped with a hyperbolic structure with concave boundary (it amounts to triangulating ${S}$). Note that Area${(S)\leq 2\pi(2g+1)}$ is linear in ${g}$. Also, the injectivity radius of ${S}$ is bounded from below independently on ${n}$. So the diameter of ${S}$ is linear in ${g}$. Thus $\displaystyle \begin{array}{rcl} d(\alpha^,\psi^n\alpha^)\leq C\,g, \end{array}$ and ${g\geq c\, n}$. However, ${c}$ depends on ${\phi}$ and ${\alpha}$. Note that diameter${(\alpha^)}$ is bounded. 3. Result Theorem 1 There is a constant ${L}$, depending only on ${\Sigma}$, such that for all ${\alpha,\beta\in A}$, $\displaystyle \begin{array}{rcl} d(\alpha,\beta)\leq L\,\rho(\alpha,\beta). \end{array}$ The proof is similar. Let ${f:S\rightarrow\Sigma}$ minimize the genus ${g}$ of ${S}$. Fact. There exists a complete hyperbolic 3-manifold ${M}$, homeomorphic to ${\Sigma\times{\mathbb R}}$, with arbitrarily short representatives ${\alpha^}$ and ${\beta^}$. It can be taken quasi-Fuchsian. Again, there exists a 1-Lipschitz map ${F:S\rightarrow\Sigma\rightarrow M}$, where ${S}$ is hyperbolic with concave boundary, hence linear area. However, the injectivity radius of ${M}$ is not controlled. The thin part of ${S}$ is mapped to the thin part of ${M}$, a union of solid tori. Let us electrify tubes: change to a metric which is zero on the thin part. Then ${F}$ remains 1-Lipschitz. In the electrified metric, the diameter of ${S}$ is bounded by its area, so the distance between ${\alpha^}$ and ${\beta^}$ in the electrified metric is linear in ${g}$. Fact. ${d(\alpha,\beta)\leq L\,d_{elec}(\alpha^,\beta^)}$. This is a consequence of the Ending Lamination Theorem. One uses the quasi-isometric model of ${M}$. The constant ${L}$ depends only on ${\Sigma}$, but it is not effective. Question (H. Wilton). Stable commutator length is more natural than commutator length. Would replacing ${\alpha}$ and ${\beta}$ by powers simplify the argument ? I do not see how it could help.	{"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": 85, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9926112294197083, "perplexity": 264.4591716470765}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424889.43/warc/CC-MAIN-20170724162257-20170724182257-00038.warc.gz"}
https://www.physicsforums.com/threads/showing-area-of-triangle.340458/	# Showing area of triangle 1. Sep 25, 2009 ### zeion 1. The problem statement, all variables and given/known data Show that for all $$\theta \epsilon (0, \pi)$$, the area of a triangle with side lengths a and b with included angle $$\theta is A = \frac{1}{2} a b sin \theta$$. (Hint: You need to consider two cases) 2. Relevant equations 3. The attempt at a solution I have just begun working on this problem.. not really sure where to start. Does $$\theta \epsilon (0, \pi)$$ mean that the angle is > than 0 and < than pi? Am I supposed to show that when the angle is less than or greater than the condition then the equation to find area is not valid? Last edited: Sep 25, 2009 2. Sep 25, 2009 ### Dick Yes, that's what (0,pi) means. The only cases where the area is not ab*sin(theta) is where sin(theta) might be negative. They aren't in (0,pi). What's the area in that case? 3. Sep 27, 2009 ### zeion The area is bh/2 4. Sep 27, 2009 ### Dick They want you to give an answer in terms of the sides a and b. Not the base and the height. 5. Sep 27, 2009 ### zeion Can you give me a little more hint -_-; What are the two cases that I need to consider? 6. Sep 27, 2009 ### Dick Use trig and A=bh/2. What's h in terms of a and the included angle? Draw a right triangle. And I'm really not sure what the 'two cases' they are talking about are. 7. Sep 28, 2009 ### zeion h = b(sin theta) or h = b(sin 180 - theta) 8. Sep 28, 2009 ### Dick sin(theta) and sin(180-theta) are the same number. Aren't they? 9. Sep 28, 2009 ### zeion So can I show this by drawing a picture? 10. Sep 28, 2009 ### Dick There's a variety of ways to draw a picture to show sin(pi-x)=sin(x). Which sort did you have in mind? How do you picture sin(x)? Similar Discussions: Showing area of triangle	{"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": 0.825545608997345, "perplexity": 1433.3775997322175}, "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/1508187820487.5/warc/CC-MAIN-20171016233304-20171017013304-00159.warc.gz"}
https://www.mathphysicsbook.com/mathematics/the-divergence-currents-and-tensor-densities/tensor-densities/	# Tensor densities The current density $${\mathfrak{j}}$$ defined in the previous section is an example of a tensor density, which in general takes the form $$\displaystyle \mathfrak{T}\equiv\left(\sqrt{\left\|\mathrm{det}(g)\right\|}\right)^{W}T,$$ where $${T}$$ is a tensor and $${W}$$ is called the weight. Note that tensor densities are not coordinate-independent quantities, and $${\sqrt{g}}$$ itself can thus be called a scalar density. From the previous expressions we get \begin{aligned}\partial_{\lambda}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}\partial_{\lambda}T+W\left(\Gamma^{\mu}{}_{\lambda\mu}-T^{\mu}{}_{\mu\lambda}\right)\mathfrak{T}\\ & =\sqrt{g}^{W}\partial_{\lambda}T+\frac{W}{2}g^{\mu\nu}\partial_{\lambda}g_{\mu\nu}\mathfrak{T},\\ L_{u}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}L_{u}T+W\mathrm{div}\left(u\right)\mathfrak{T}\\ & =\sqrt{g}^{W}L_{u}T+\frac{W}{2}g^{\mu\nu}L_{u}g_{\mu\nu}\mathfrak{T},\\ \nabla_{\lambda}\left(\mathfrak{T}\right) & =\sqrt{g}^{W}\nabla_{\lambda}T,\end{aligned} where the last is due to the covariant derivative of the metric vanishing. In particular, this means that for zero torsion the divergence of a vector density is \begin{aligned}\overline{\nabla}_{\lambda}\mathfrak{J}^{\lambda}&=\sqrt{g}\overline{\nabla}_{\lambda}J^{\lambda}\\&=\sqrt{g}\mathrm{div}\left(J\right)\\&=\partial_{\lambda}\mathfrak{J}^{\lambda}.\end{aligned} Δ A potential source of confusion is the use of the word “density” to indicate both an amount per unit area or volume and the presence of the coordinate-dependent factor $${\sqrt{g}}$$, which as in the current density typically reflects the volume in question being a unit coordinate volume instead of metric volume.	{"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": 0.9913539886474609, "perplexity": 346.62254347799114}, "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/1664030337504.21/warc/CC-MAIN-20221004121345-20221004151345-00256.warc.gz"}
http://nbviewer.jupyter.org/github/jckantor/Ball-and-Beam/blob/master/index.ipynb	# Balancing a Ball on a Beam with Proportional-Derivative Control¶ ###### The latest version of this Jupyter notebook is available at http://github.com/jckantor/Ball-and-Beam for noncommercial use under terms of the Creative Commons Attribution Noncommericial ShareAlike License.¶ Jeffrey C. Kantor ([email protected]) This notebook illustrates the principles of proportional-derivative feedback through an interactive simulation of a ball on beam apparatus. The task is to choose the control parameters to balance the ball at desired locations on the beam with acceptable dynamics and without rolling off the ends. The simulation in intended for use in as in-lecture demonstration on PD control and the dynamics of second order systems, and for out-of-class reinforcement of key concepts in feedback control. Suggested exercises are included. Everything is included in a single, self-contained Jupyter notebook using standard python libraries. • Initialization. This section loads needed libraries and defines functions to display and update a diagram of the ball and beam apparatus, and to create plots of the ball position and beam angle. These cells need to be executed for subsequent simulations. • Ball on Beam Dynamics. This section presents a model for the dynamics of the ball on beam system, and demonstrates the behavior with an open-loop simulation of the device. • Proportional Control. This section introduces a proportional feedback control. Simulations can be run for different choices of setpoints and control parameters. • Proportional-Derivative Control. This section introduces a proportional-derivative feedback control. Simulations can be run for different choices of setpoints and control parameters. • Theory. A derivation of an approximate linear model for the dynamics of the ball on beam system under proportional-derivative control. • Exercises. A short set of exercises to reinforce control concepts, and introduce methods to analyze the performance of the ball on beam control system. ## How to Use¶ This notebook can be downloaded to your laptop and run as a standard Jupyter notebook. This requires prior installation of a Python distribution with standard libraries such as Anaconda available from Continuum Analytics. Once the notebook is loaded in an executable environment, either in the cloud or laptop, proceed through the notebook executing each cell, one at a time. You can use either the run button in the Jupyter tool bar, or press shift-enter for each cell. Be sure to run the Initialization cell first! This notebook can be run in the cloud from mybinder.org. This can be done from any browser (laptop, tablet, phone), and doesn't require any software installation. Click this button to get started: Note that as a free service, the mybinder.org site is sometimes overloaded and unresponsive. ## Initialization (Run this one first!)¶ The following cell creates utility functions that display and update a figure showing the ball and beam apparatus. This cell must be executed in order to run the control simulations later in this notebook. The detail of these functions can be safely ignored by anyone primarily interested in the subsequent applications of feedback control. In [15]: %matplotlib notebook from pylab import * import time from ipywidgets import interact, interactive # limit on beam angle in radians ulimit = 0.2 def draw_beam(fig,ax): # Draw the ball and beam diagram on a given figure and axis ax.axis('off') ax.axis('equal') ax.set(xlim = (-0.2,1.2), ylim = (-0.3,0.3)) # draw ball, beam, and set point patch ax.plot(0.5,0.049,'r.',ms=50) ax.plot(array([0.0,cos(0.0)]), array([0.0,sin(0.0)]), lw=5) ax.add_patch(Polygon([[0.5, -0.01], [0.5-0.03, -0.05], [0.5+0.03, -0.05]])) # draw pivot patch # beam hi and lo limit patches ulimit = 0.2 xa = cos(ulimit) ya = sin(ulimit) [xa+0.05, ya+0.02],[xa-0.05, ya+0.02]])) [xa+0.05,-ya-0.02],[xa-0.05,-ya-0.02]])) fig.canvas.draw() def update_beam(fig,ax,x,u,SP): # Update the ball and beam diagram with current state values. # Limit beam angle u = -ulimit if u < -ulimit else ulimit if u > ulimit else u # Determine ball position, and if it rolled off the beam x = -0.05 if x < 0 else 1.02 if x > 1.0 else x y = 0.049/cos(u) + xsin(u) if x >= 0.0 and x <= 1.0 else -0.2 # update ball, beam, and setpoint positions ax.lines[0].set(xdata=x, ydata=y) ax.lines[1].set(xdata=[0.0,cos(u)], ydata=[0.0,sin(u)]) ax.patches[0].set_xy([[SPcos(u), -0.01 + SPsin(u)], \ [SPcos(u) - 0.03, -0.05 + SPsin(u)], \ [SPcos(u) + 0.03, -0.05 + SPsin(u)]]) fig.canvas.draw() def draw_plot(fig,ax,SP): # set up xplot to display ball position xplot = ax xplot.set(xlim=(0,tf),ylim=(0,1),xlabel='Time [dimensionless]',ylabel='Position') xplot.plot([],[],'b') xplot.plot([0,tf],[SP,SP],'b--') # set up second axist to display beam angle uplot = ax.twinx() uplot.set(xlim=(0,tf),ylim=(-ulimit,ulimit)) uplot.set_ylabel('Beam Angle', color='r') for tl in uplot.get_yticklabels(): tl.set_color('r') uplot.plot([],[],'r') uplot.plot([0,tf],[0,0],'r--') # return lines for updating return xplot,uplot def update_plot(ax,t,y): tdata = append(ax.lines[0].get_xdata(),t) ydata = append(ax.lines[0].get_ydata(),y) ax.lines[0].set(xdata=tdata, ydata=ydata) Run the following cell to see a diagram of the ball and beam apparatus. In [16]: fig,ax = subplots(1,1,figsize=(8,3)) draw_beam(fig,ax) update_beam(fig,ax,0.5,0.1,0.5) ## Ball on Beam Dynamics¶ We assume that a ball can roll freely and without friction on a tilted beam. The beam angle $u$ is measured in a counterclockwise direction from the horizontal, and is subject to the hard limits indicated by stops on the following diagram. The position of the ball is given by $x$ on a beam of length $L$. The ball can fall of the either end of the beam. There is a triangular marker below the beam to indicate the location of a setpoint. The ball dynamics are modeled by a pair of differential equations \begin{align} \frac{dx}{dt} & = v(t) \\ (I+1)m\frac{dv}{dt} & = - m g \sin(u(t)) \end{align} where $x(t)$ denotes position of the ball on the beam, $v(t)$ is velocity, $u(t)$ is the angle of the beam relative to the horizontal, $m$ is the ball mass, and $g$ is acceleration due to gravity. The motion of the ball is subject to both linear and rotational inertia. The linear inertia is the product of the ball's mass times acceleration. If the surface of the ball doesn't slip as it rolls on the beam, the rotational inertia can be expressed as a mutiple $I$ of the ball's mass where $I = \frac{2}{5}$ for a solid sphere, or $I = \frac{2}{3}$ for a spherical shell. Wrapping these parameters into a single constant $K_b = \frac{g}{1+I}$ results in a model \begin{align} \frac{dx}{dt} & = v(t) \\ \frac{dv}{dt} & = - K_b \sin(u(t)) \end{align} where the variables all have common units. An alternative is create a dimensionless model where a time scale $T$ is defined by $$T = \sqrt{\frac{(1+I)L}{g}}$$ where $L$ is the length of the beam. Dimensionless variables are then given \begin{align} \tilde{x} & = \frac{x}{L} \\ \tilde{t} & = \frac{t}{T} \\ \tilde{v} & = \frac{v T}{L} \end{align} With these definitions in place, our simulation model becomes dimensionless where \begin{align} \frac{d\tilde{x}}{d\tilde{t}} & = \tilde{v}(\tilde{t}) \\ \frac{d\tilde{v}}{d\tilde{t}} & = - \sin(u(\tilde{t})) \end{align} Note the dimensionless model can be formed from the dimensional model with specific choices for $K_b$ and $L$. In other words, there is no loss of generality if we proceed with the dimesional model under the assumption $K_b = 1$ and $L = 1$. To demonstrate the dynamics, run the following cell so see a simulation of how the ball on beam responds to a programmed change in beam angle. The beam angle is programmed to start level, then begins to move in a oscillatory motion. (Feel free to edit the cell to see how ir responds to an import of your choice!). The simulation solves the differential equations with an Euler's approximation. The simulation loop includes a time.sleep(dt) that inserts a pause in each step of the simulation. (This may be commented out in order to see animation run as quickly as possible.) In [24]: # simulation parameters dt = 0.05 tf = 10.0 # initialize simulation t = 0.0 # initial time x = 0.0 # initial ball position v = 0.0 # initial ball velocity # set up figure window with two axes fig,ax = subplots(2,1,figsize=(8,6)) beam_axes = ax[0] plot_axes = ax[1] # draw ball and beam apparatus draw_beam(fig,beam_axes) update_beam(fig,beam_axes,x,u,0.5) # draw plotting axes xplot,uplot = draw_plot(fig,plot_axes,SP) # simulation/animation for t in linspace(dt,10.0,10.0/dt): # beam angle u = 0.0 if t <= 1.0 else -0.15sin(t-1.0) # update velocity and position v += -sin(u)dt x += vdt # update plots update_beam(fig,beam_axes,x,u,SP) update_plot(xplot,t,x) update_plot(uplot,t,u) # pause for time step # time.sleep(dt) ## Proportional Control¶ The beam angle is primary control input. We choose to adjust the control using proportinal (P) feedback in the form $$u(t) = K_p (x(t) - x_{SP})$$ where $K_p$ is the parameter determining the proportional control response. Running the following cell defines a function to perform a closed-loop simulation of PD control for fixed values of the setpoint (SP), proportional gain (Kp), and then runs the simulation for a choice of these parameters. In [25]: def ballbeam(IC, SP, Kp, Kd): # initialize t = 0.0 # time x = IC # ball position v = 0.0 # ball velocity u = 0.0 # beam angle # initialize graphics fig,ax = subplots(2,1,figsize=(8,6)) draw_beam(fig,ax[0]) xplot,uplot = draw_plot(fig,ax[1],SP) # simulation loop while (t < tf) & (x <= 1.0) & (x >= 0.0): # calculate control u = max(-ulimit,min(ulimit,Kp(x - SP))) # update state t += dt v += -sin(u)dt x += vdt # update graphics update_beam(fig,ax[0],x,u,SP) update_plot(xplot,t,x) update_plot(uplot,t,u) # run simulation for one choice of initial position, setpoint, and control gains dt = 0.05 tf = 20.0 ballbeam(1.0, 0.5, 2.0, 2.0) ## Proportional-Derivative Control¶ The beam angle is primary control input. We choose to adjust the control using proportinal-derivative (PD) feedback in the form $$u(t) = K_p (x(t) - x_{SP}) + K_d v(t)$$ where $K_p$ and $K_d$ are parameters determining the proportional and derivative control response. Running the following cell defines a function to perform a closed-loop simulation of PD control for fixed values of the setpoint (SP), proportional gain (Kp), and derivative gain (Kd), and then runs the simulation for one choice of these parameters. In [28]: def ballbeam(IC, SP, Kp, Kd): # initialize t = 0.0 # time x = IC # ball position v = 0.0 # ball velocity u = 0.0 # beam angle # initialize graphics fig,ax = subplots(2,1,figsize=(8,6)) draw_beam(fig,ax[0]) xplot,uplot = draw_plot(fig,ax[1],SP) # simulation loop while (t < tf) & (x <= 1.0) & (x >= 0.0): # calculate control u = max(-ulimit,min(ulimit,Kp(x - SP) + Kd(v))) # update state t += dt v += -sin(u)dt x += v*dt # update graphics update_beam(fig,ax[0],x,u,SP) update_plot(xplot,t,x) update_plot(uplot,t,u) # run simulation for one choice of initial position, setpoint, and control gains dt = 0.05 tf = 10.0 ballbeam(1.0, 0.5, 2.0, 1.0) ## Interactive Simulation¶ Running the following cell will open a set of sliders with which you can adjust the initial position of the ball, the setpoint, and the control parameters $K_p$ and $K_d$. Adjust the sliders, then press the Run ballbeam button to perform the simulation. Try this out! For example, see if you can choose parameters to position the ball on the very tip of the beam without falling off. In [19]: dt = 0.05 tf = 20.0 fig,ax = subplots(2,1,figsize=(8,6)) draw_beam(fig,ax[0]) update_beam(fig,ax[0],0.5,0.0,0.5) draw_plot(fig,ax[1],0.5) # add sliders and run button interactive(ballbeam, __manual=True, \ IC = (0.0,1.01,0.01), \ SP = (0.0,1.01,0.01), \ Kp = (0.0,6.01,0.01), \ Kd = (0.0,5.01,0.01)) ## Theory¶ The purpose of this section is provide a start on the analysis of the closed-loop dynamics of the ball on beam system. You can use this as a start on some exercises intended to introduce and reinforce key concepts of feedback control and the dynamics of second order systems. The starting point is the model derived above for the ball on beam system \begin{align} \frac{dx}{dt} & = v(t) \\ \frac{dv}{dt} & = - K_b \sin(u(t)) \end{align} where the beam angle, $u(t)$, is given by a proportional derivative feedback rule where $$u(t) = K_p (x(t) - x_{SP}) + K_d v(t)$$ The first step of the analysis is to linearize the model for small values of the beam angle. By the Taylor series expansion for $\sin(u(t))$, $$\sin(u) = u - \frac{u^3}{6} + \frac{u^5}{120} + \cdots$$ For small values of $u(t)$, and replacing every instance of ball velocity $v(t)$ by $\frac{dx}{dt}$ (including inside of the derivative $\frac{dv}{dt}$), we get $$\frac{d^2x}{dt^2} = K_{b} u$$ where $$u(t) = K_p (x(t) - x_{SP}) + K_d \frac{dx}{dt}$$ After some rearrangements, an equation for the closed-loop dynamics of the ball on beam system is given by $$\frac{d^2x}{dt^2} + K_b K_d\frac{dx}{dt} + \ K_b K_p x = K_b K_p x_{sp}$$ Use this last model in the following exercises. ## Exercises¶ 1. Adjust the setpoint to 1.0 (i.e, the very tip of beam). Adjust the control parameters so that ball settles on the setpoint without falling off the end of the beam. 2. Using the approximate linear model for the closed-loop dynamics, consider the case of proportinal only control (i.e, $K_d = 0$). Perform a simulation. Use the model to predict the frequency of oscillation that you observe, and compare to your simulation results. 3. Predict the frequency of oscillation, and verify your results by simulation. 4. Critical damping correponds to the smallest value of $K_d$ with no overshoot. Use the approximate linear model, and derive and expression for $K_d$ that, for given values of $K_b$ and $K_p$, results in critical damping. Verify your results by simulation. 5. Choose values for $K_p$ and $K_d$ that provide a damping ratio of $0.7$. Consult a standard textbook for a definition of damping ratio. Verify your result by simulation. 6. Derive the transfer function from $x_{sp}$ to $x$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8540264368057251, "perplexity": 1980.531310338918}, "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-2017-47/segments/1510934806070.53/warc/CC-MAIN-20171120145722-20171120165722-00645.warc.gz"}
https://www.meritnation.com/cbse-class-11-science/physics/hc-verma-i/circular-motion/textbook-solutions/41_4_1264_6762_111_67706	Hc Verma I Solutions for Class 11 Science Physics Chapter 7 Circular Motion are provided here with simple step-by-step explanations. These solutions for Circular Motion are extremely popular among Class 11 Science students for Physics Circular Motion Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Hc Verma I Book of Class 11 Science Physics Chapter 7 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Hc Verma I Solutions. All Hc Verma I Solutions for class Class 11 Science Physics are prepared by experts and are 100% accurate. Page No 111: Yes, it is possible to accelerate the motorcycle without putting higher petrol input rate into the engine by driving the motorcycle on a circular track. Page No 112: The person rotating with the drum will observe that centrifugal force and coriolis force act on the water particles and the person washing the cloth will observe that water particles are thrown outward (away from the drum) and no pseudo force is acting on the particles. Page No 112: The coin gets the required centripetal force from the frictional force between the coin and the record. Page No 112: The bird tilts its body and tail in such a way that the air around offers a dragging force in left direction, perpendicular to its initial direction of motion. This dragging force provides the necessary centripetal force to take the left turn. Page No 112: No, it is not necessary to express all the angles in radian while using the equation ω = ω0 + at. If ω (angular velocity) and α (angular acceleration) are in rad/s and rad/s2, respectively, we will get the angle in radian. Page No 112: While shaking, our hand moves on a curved path with some angular velocity and water on our hand feels centrifugal force in the outward direction. Therefore, water get detached from our hand and leaves it. Page No 112: The outer wall will exert a non-zero normal contact force on the block. As the block moves in a uniform circular motion, centrifugal force in radially outward direction acts on it and it comes in contact with the outer wall of the tube. Page No 112: (a) Gravitational attraction of the Sun on the Earth is equal to the centripetal force. This statement is more appropriate. Gravitational attraction of the Sun on the Earth provides the necessary centripetal force required for the circular motion of the Earth around the Sun. Page No 112: As the wall is at a distance r, the driver should either take a circular turn of radius r or apply brakes to avoid hitting the wall. Page No 112: When the same mass is set into oscillation, the tension in the string increases because of the additional centripetal force of the mass oscillating in a curved path. Page No 112: (d) its velocity and acceleration both change. In a circular motion, the direction of particle changes. Therefore, velocity, being a vector quantity, also changes. As the velocity changes, acceleration also changes. Page No 112: (d) 1 Time period (T) is same for both the cars. We know that: Page No 112: (c) NA < NB From the figure in the question, it is clear that ${r}_{B}>{r}_{A}$. Here, normal reaction is inversely proportional to the centrifugal force acting on the car, while taking turn on the curve track. Also, centrifugal force is inversely proportional to the radius of the circular track. Therefore, we have: NA < NB Page No 112: (d) zero The centrifugal force is a pseudo force and can only be observed from the frame of reference, which is non-inertial w.r.t. the particle. Page No 112: (b) $m{\mathrm{\omega }}_{0}^{2}\mathrm{a}$ The centrifugal force on the particle depends on the angular speed (ω0) of the frame and not on the angular speed (ω) of the particle. Thus, the value of centrifugal force on the particle is $m{\mathrm{\omega }}_{0}^{2}\mathrm{a}$. Page No 112: (a) 10 cm/s, 10 cm/s2 It is given that the turntable is rotating with uniform angular velocity. Let the velocity be $\omega$. We have: Similarly, we have: Page No 112: (c) mg is not greater than $\frac{m{v}^{2}}{r}$ At the top of the path, the direction of mg is vertically downward and for centrifugal force $\left(\frac{m{v}^{2}}{r}\right)$, the direction is vertically upward. If the vertically downward force is not greater, water will not fall. Page No 113: (c) along a tangent The stone will move in a circle and the direction of velocity at any instant is always along the tangent at that point. Therefore, the stone will move along the tangent to the circle at a point where the string breaks. Page No 113: (a) 1 cm Let the force of friction between the coin and the rotating turntable be F. For the coin to just slip, we have: $F=m{\omega }^{2}r$ Here, is the centrifugal force acting on the coin. For constant F and m, we have: $r\propto \frac{1}{{\omega }^{2}}$ Therefore, Page No 113: (a) increases The normal force on the motorcycle, $N=mg\mathrm{cos}\theta -\frac{m{v}^{2}}{R}$ As the motorcycle is ascending on the overbridge, θ decreases (from $\frac{\pi }{2}$ to 0). So, normal force increases with decrease in θ. Page No 113: (c) FC is maximum of the three forces. At the middle of bridge, normal force can be given as: ${N}_{A}=mg$ ${N}_{B}=\frac{m{v}^{2}}{r}-mg\phantom{\rule{0ex}{0ex}}{N}_{C}=\frac{m{v}^{2}}{r}+mg\phantom{\rule{0ex}{0ex}}$ So, FC is maximum. Page No 113: (a) F1 > F2 When the trains are moving, effective angular velocity of both the trains are different (as shown in the figure). Effective angular velocity of train B is more than that of train A. Normal force with which both the trains push the tracks is given as: $N=mg-mR\omega {\text{'}}^{2}$ From the above equation, we can conclude that F1 > F2. Page No 113: (d) increase at some places and remain the same at some other places If the Earth stops rotating on its axis, there will be an increase in the value of acceleration due to gravity at the equator. At the same time, there will be no change in the value of g at the poles. Page No 113: (a) T1 > T2 Let the angular velocity of the rod be $\omega$. Distance of the centre of mass of portion of the rod on the right side of L/4 from the pivoted end: ${r}_{1}=\frac{L}{4}+\frac{1}{2}\left(\frac{3L}{4}\right)=\frac{5L}{8}$ Mass of the rod on the right side of L/4 from the pivoted end: ${m}_{1}=\frac{3}{4}M$ At point L/4, we have: Distance of the centre of mass of rod on the right side of 3L/4 from the pivoted end: ${r}_{1}=\frac{1}{2}\left(\frac{L}{4}\right)+\frac{3L}{4}=\frac{7L}{8}$ Mass of the rod on the right side of L/4 from the pivoted end: ${m}_{1}=\frac{1}{4}M$ At point 3L/4, we have: ∴ T1 > T2 Page No 113: (d) zero When the car is in air, the acceleration of bob and car is same. Hence, the tension in the string will be zero. Page No 113: (c) at the extreme positions Tension is the string, $T=\frac{m{v}^{2}}{r}-mg\mathrm{cos}\theta$ When v = 0, $\left\|T\right\|=mg\mathrm{cos}\theta$ That is, at the extreme positions, the tension is the string is mgcosθ. Page No 113: (a) speed (d) magnitude of acceleration When an object follows a curved path, its direction changes continuously. So, the scalar quantities like speed and magnitude of acceleration may remain constant during the motion. Page No 113: (d) The instantaneous acceleration of the Earth points towards the Sun. The speed is constant; therefore, there is no tangential acceleration and the direction of radial acceleration is towards the Sun. So, the instantaneous acceleration of the Earth points towards the Sun. Page No 113: (b) speed remains constant (d) tangential acceleration remains constant If the speed is constant, the position vector of the particle sweeps out equal area in equal time in circular motion. Also, for constant speed, tangential acceleration is zero, i.e., constant. Page No 113: (c) The magnitude of acceleration is constant. As the pitch and radius of the path is constant, it shows that the magnitude of tangential and radial acceleration is also constant. Hence, the magnitude of total acceleration is constant. Page No 114: (b) The magnitude of the frictional force on the car is greater than $\frac{m{v}^{2}}{r}$. (c) The friction coefficient between the ground and the car is not less than a/g. If the magnitude of the frictional force on the car is not greater than $\frac{m{v}^{2}}{r}$, it will not move forward, as its speed (v) is increasing at a rate a. Page No 114: (b) If the car turns at a speed less than 40 km/hr, it will slip down. (d) If the car turns at the correct speed of 40 km/hr, the force by the road on the car is greater than mg as well as greater than $\frac{m{v}^{2}}{r}$. The friction is zero and the road is banked for a speed v = 40 km/hr. If the car turns at a speed less than 40 km/hr, it will slip down. Page No 114: (b) There are other forces on the particle. (d) The resultant of the other forces varies in magnitude as well as in direction. We cannot move a particle in a circle by just applying a constant force. So, there are other forces on the particle. As a constant force cannot move a particle in a circle, the resultant of other forces varies in magnitude as well as in direction (because $\stackrel{\to }{F}$ is constant). Page No 114: Distance between the Earth and the Moon: Time taken by the Moon to revolve around the Earth: Page No 114: Diameter of the Earth = 12800 km So, radius of the Earth, R = 6400 km = 6.4 × 106 m Time period of revolution of the Earth about its axis: Page No 114: Speed is given as a function of time. Therefore, we have: v = 2t Radius of the circle = r = 1 cm At time t = 2 s, we get: (b) Tangential acceleration (c) Magnitude of acceleration Page No 114: Given: Mass = m = 150 kg Speed = v = 36 km/hr = 10 m/s Radius of turn = r = 30 m Let the horizontal force needed to make the turn be F. We have: Page No 114: Given: Speed of the scooter = v = 36 km/hr = 10 m/s Radius of turn = r = 30 m Let the angle of banking be $\theta$. We have: $⇒\mathrm{tan}\theta =\frac{100}{30×10}=\frac{1}{3}\phantom{\rule{0ex}{0ex}}⇒\theta ={\mathrm{tan}}^{-1}\left(\frac{1}{3}\right)$ Page No 114: Given: Speed of the vehicle = v = 18 km/h = 5 m/s Radius of the park = r = 10 m Let the angle of banking be $\theta$. Thus, we have: $\mathrm{tan}\theta =\frac{{v}^{2}}{rg}$ $⇒\theta ={\mathrm{tan}}^{-1}\left(\frac{25}{100}\right)\phantom{\rule{0ex}{0ex}}⇒\theta ={\mathrm{tan}}^{-1}\left(\frac{1}{4}\right)$ Page No 114: If the road is horizontal (no banking), we have: $\frac{m{v}^{2}}{R}={f}_{s}\phantom{\rule{0ex}{0ex}}N=mg\phantom{\rule{0ex}{0ex}}$ Here, fs is the force of friction and N is the normal reaction. If μ is the friction coefficient, we have: Here, Velocity = v = 5 m/s Radius = R = 10 m Page No 114: Given: Angle of banking = θ = 30° Radius = r = 50 m Assume that the vehicle travels on this road at speed v so that the friction is not used. We get: Page No 114: Let m be the mass of the stone. Let v be the velocity of the stone at the highest point. R is the radius of the circle. Thus, in a vertical circle and at the highest point, we have: $\frac{m{v}^{2}}{R}=mg\phantom{\rule{0ex}{0ex}}⇒{v}^{2}=Rg\phantom{\rule{0ex}{0ex}}⇒v=\sqrt{Rg}$ Page No 114: Diameter of the fan = 120 cm ∴ Radius of the fan = r = 60 cm = 0.6 m Mass of the particle = M = 1 g = 0.001 kg Frequency of revolutions = n = 1500 rev/min = 25 rev/s Angular velocity = ω = 2$\pi$n = 2$\pi$ × 25 = 157.14 rev/s Force of the blade on the particle: F = Mrw2 = (0.001) × 0.6 × (157.14)2 =14.8 N The moving fan exerts this force on the particle. The particle also exerts a force of 14.8 N on the blade along its surface. Page No 114: It is given that the mosquito is sitting on the L.P. record disc. Therefore, we have: Friction force ≥ Centrifugal force on the mosquito ⇒ μmgmrω2 ⇒ μ 2/g Page No 114: Speed of the car = v = 36 km/hr = 10 m/s Acceleration due to gravity = g = 10 m/s2 Let T be the tension in the string when the pendulum makes an angle θ with the vertical. From the figure, we get: Page No 115: Given: Mass of the bob = m = 100 gm = 0.1 kg Length of the string = r = 1 m Speed of bob at the lowest point in its path = 1.4 m/s Let T be the tension in the string. From the free body diagram, we get: Page No 115: Given: Mass of the bob = m = 0.1 kg Length of the circle = R = 1 m Velocity of the bob = v = 1.4 m/s Let T be the tension in the string when it makes an angle of 0.20 radian with the vertical. From the free body diagram, we get: Page No 115: Let T be the tension in the string at the extreme position. Velocity of the pendulum is zero at the extreme position. So, there is no centripetal force on the bob. T = mgcosθ0 Page No 115: (a) Balance reading = Normal force on the balance by the Earth. At equator, the normal force (N) on the spring balance: N = mg2r True weight = mg Therefore, we have: $=3.5×{10}^{-3}$ (b) When the balance reading is half, we have: Page No 115: Given: Speed of vehicles = v = 36 km/hr = 10 m/s Radius = r = 20 m Coefficient of static friction = μ = 0.4 Let the road be banked with an angle $\theta$. We have: When the car travels at the maximum speed, it slips upward and μN1 acts downward. Therefore we have: On solving the above equations, we get: Similarly, for the other case, it can be proved that: Thus, the possible speeds are between 14.7 km/hr and 54 km/hr so that the car neither slips down nor skids up. Page No 115: R = Radius of the bridge L = Total length of the over bridge (a) At the highest point: Let m be the mass of the motorcycle and v be the required velocity. $mg=\frac{m{v}^{2}}{R}\phantom{\rule{0ex}{0ex}}⇒{v}^{2}=Rg\phantom{\rule{0ex}{0ex}}⇒v=\sqrt{Rrg}\phantom{\rule{0ex}{0ex}}$ Suppose it loses contact at B. So, it will lose contact at a distance $\frac{\pi R}{3}$ from the highest point. (c) Let the uniform speed on the bridge be v. The chances of losing contact is maximum at the end bridge. We have: $\alpha =\frac{L}{2R}\phantom{\rule{0ex}{0ex}}\mathrm{So},\frac{m{v}^{2}}{R}=mg\mathrm{cos}\alpha \phantom{\rule{0ex}{0ex}}⇒v=\sqrt{g\mathrm{Rcos}\left(\frac{L}{2R}\right)}$ Page No 115: Let v be the speed of the car. Since the motion is non-uniform, the acceleration has both radial (ar) and tangential (at) components. From free body diagram, we have: $⇒\frac{{v}^{4}}{{R}^{2}}=\left({\mathrm{\mu }}^{2}{g}^{2}-{a}^{2}\right)\phantom{\rule{0ex}{0ex}}⇒{v}^{4}=\left({\mathrm{\mu }}^{2}{g}^{2}-{a}^{2}\right){R}^{2}\phantom{\rule{0ex}{0ex}}⇒v=\left[\left({\mathrm{\mu }}^{2}{\mathrm{g}}^{2}-{\mathrm{a}}^{2}\right){R}^{2}{\right]}^{1/4}$ Page No 115: (a) Given: Mass of the block = m Friction coefficient between the ruler and the block = μ Let the maximum angular speed be ω1 for which the block does not slip Now, for the uniform circular motion in the horizontal plane, we have: $\mu mg=m{{\omega }_{1}}^{2}L\phantom{\rule{0ex}{0ex}}\therefore {\omega }_{1}=\sqrt{\frac{\mu g}{L}}$ (b) Let the block slip at an angular speed ω2. For the uniformly accelerated circular motion, we have: $⇒{\omega }_{2}={\left[{\left(\frac{\mu g}{L}\right)}^{2}-{\alpha }^{2}\right]}^{1/4}$ Page No 115: Given: Radius of the curves = r = 100 m Mass of the cycle = m = 100 kg Velocity = v = 18 km/hr = 5 m/s (b) At B and D, we have: Tendency of the cycle to slide is zero. So, at B and D, frictional force is zero. At C, we have: mgsinθ = f (d) To find the minimum coefficient of friction, we have to consider a point where N is minimum or a point just before c . Page No 115: Given: Therefore, we have: angular velocity of rod, Mass of each kid = Thus, the force of frictional on one of the kids is 10$\pi$2. Page No 115: When the bowl rotates at maximum angular speed, the block tends to slip upwards. Also, the frictional force acts downward. Here, we have: Radius of the path that the block follow = r = Rsinθ Let N1 be the normal reaction on the block and ω1 be the angular velocity after which the block will slip. From the free body diagram-1, we get: On solving the two equation, we get: Let us now find the minimum speed (ω2) on altering the direction of $\mu$ (as shown in figure): Hence, the range of speed is between ω2 and ω1. Page No 115: At the highest point, the vertical component of velocity is zero. So, at the highest point, we have: velocity = v = ucosθ Centripetal force on the particle = $\frac{m{v}^{2}}{r}$ $⇒\frac{m{v}^{2}}{r}=\frac{m{u}^{2}{\mathrm{cos}}^{2}\mathrm{\theta }}{r}$ At the highest point, we have: $mg=\frac{m{v}^{2}}{r}$ Here, r is the radius of curvature of the curve at the point. $⇒r=\frac{{u}^{2}{\mathrm{cos}}^{2}\theta }{g}$ Page No 115: Let u be the initial velocity and v be the velocity at the point where it makes an angle $\frac{\mathrm{\theta }}{2}$with the horizontal component. It is given that the horizontal component remains unchanged. Therefore, we get: On substituting the value of v from equation (i), we get: $r=\frac{{u}^{2}{\mathrm{cos}}^{2}\theta }{g{\mathrm{cos}}^{2}\frac{\theta }{2}}$ Page No 115: Given: Radius of the room = R Mass of the block = m (a) Normal reaction by the wall on the block = N = $\frac{m{v}^{2}}{R}$ (b) Force of frictional by the wall = $\mu N=\frac{\mu m{v}^{2}}{R}$ (c) Let at be the tangential acceleration of the block. From figure, we get: $-\frac{\mu m{v}^{2}}{R}=m{a}_{t}\phantom{\rule{0ex}{0ex}}⇒{a}_{t}=-\frac{\mu {v}^{2}}{R}$ (d) Page No 116: Let the mass of the particle be m. Radius of the path = R Angular velocity = ω Force experienced by the particle = 2R The component of force mRω2 along the line AB (making an angle with the radius) provides the necessary force to the particle to move along AB. Let the time taken by the particle to reach the point B be t. Page No 116: Given: Speed of the car = v = 36 km/h = 10 m/s Friction coefficient between the block and the plate = μ = 0.58 Mass of the small body = m = 100 g = 0.1 kg (a) Let us find the normal contact force (N) exerted by the plant of the block. (b) The plate is turned; so, the angle between the normal to the plate and the radius of the rod slowly increases. Therefore, we have: Page No 116: $T=m\left(\frac{{\omega }^{2}R}{3}\right)+m{\omega }^{2}R\phantom{\rule{0ex}{0ex}}⇒T=\frac{4}{3}m{\omega }^{2}R$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 53, "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": 0.8744485974311829, "perplexity": 995.3280454538047}, "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-2020-50/segments/1606141194982.45/warc/CC-MAIN-20201128011115-20201128041115-00680.warc.gz"}
https://socratic.org/questions/58a4044911ef6b2a55eaad1d	Chemistry Topics # Can an orbital with a principal quantum number of n = 2 have an angular momentum quantum number of l = 2? True or false? Feb 15, 2017 Absolutely... false. What is the definition of the principal quantum number $n$? It's the energy level, if you recall. So, what is $n$ for the second energy level? Recall that $l \le n - 1$, where $l$ is the angular momentum quantum number. Here are some questions for you: 1. What is the orbital shape described by $l = 2$? $s , p , d ,$ or $f$? 2. Verify that $l = 2$ is impossible for $n - 1$ where $n = 2$. You should find that $l = 2$ is only possible for $n \ge 3$. ##### Impact of this question 298 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "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": 0.9600105881690979, "perplexity": 372.6670590765894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027314667.60/warc/CC-MAIN-20190819052133-20190819074133-00160.warc.gz"}
https://www.physicsforums.com/threads/rotational-mechanics.724954/	# Rotational mechanics 1. Nov 25, 2013 ### allyferrell 1. The problem statement, all variables and given/known data A plank having mass 3.7 kg rides on top of two identical solid cylindrical rollers each having radius 5.5 cm and mass 2.9 kg. The plank is pulled by a constant horizontal force of 6 N applied to its end and perpendicular to the axes of the cylinders(which are parallel). The cylinders roll without slipping on a flat surface. There is also no slipping between the cylinders and the plank. 1.) Find the acceleration(linear) of the plank(m/s^2). 2.) Find the frictional force acting acting on the plank(N). M=3.7 kg m=2.9 kg r=.055 m F=6 N α = angular acceleration a=linear acceleration 2. Relevant equations τ=Fr I=(1/2)m(r^2) Ʃτ=Iα F=ma Frictional force(static) = force applied 3. The attempt at a solution torque of one wheel: τ=Fr = 6.055 = .33 N Ʃτ=Iα = (1/2)(2.9)(.055^2)α → .33=.00438625α → α(of one wheel)=75.2351907 rad/s^2 a=αr = 75.2351907.055 = 4.13793103 m/s^2 Ʃa(both wheels and board)= (4.137931032) + a of board F=ma→ 6=(3.7 + 2.9 +2.9)a → a of board = .631578947 m/s^2 ∴ Ʃa = 8.27586207 + .631578947 = 8.90744102 m/s^2 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution 2. Nov 25, 2013 ### haruspex That's wrong in two ways. First, the inertia of the plank and other cylinder come into it. The 6N is in some way shared between them. (If each component had a separate 6N it would go a lot faster.) Secondly, there are two ways of viewing the rotation of a rolling wheel. You can think of it as a rotation about its centre (plus a linear movement) or as rotating about the point of contact with the ground. With the first view, there would also be a torque from the friction on the ground; with the second view, the distance is 2r. Both views lead to the same result. The safest approach is to consider the forces and accelerations of each of the three components separately (but two are identical, so only two lots of equations). Make sure to use unique symbols to represent all the forces. 3. Nov 25, 2013 ### Staff: Mentor Your solution starts out with this equation: torque of one wheel: τ=Fr = 6*.055 = .33 N This equation is incorrect. The force acting on one wheel is not 6 N, or even 3 N, and the units of torque are also incorrect. Start out by doing a force balance on the plank. Draw a FBD. Let F1 be the tangential (frictional) force exerted by each of the wheels on the plank. What direction is this force pointing (in the direction the plank is moving, or the opposite direction)? What other horizontal forces are acting on the plank? Do a force balance on the plank, including ma. Let F2 represent the tangential force exerted by the flat surface on each wheel. Do a FBD on each wheel. What are the horizontal forces acting on each wheel? Write a horizontal force balance on each wheel, including the ma term for the center of mass of the wheel. What are the torques around the center of mass of each wheel in terms of F1 and F2? How is the angular acceleration of each wheel related to these torques? How is the angular acceleration of the wheels related kinematically to the horizontal acceleration of the center of mass of the wheels, if the wheels do not slip relative to flat surface? Chet	{"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": 0.8833019733428955, "perplexity": 875.5205810287472}, "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/1521257647777.59/warc/CC-MAIN-20180322053608-20180322073608-00477.warc.gz"}
http://math.stackexchange.com/questions/415891/how-to-determine-the-number-of-isomorphism-types-of-groups-of-a-given-order	# How to determine the number of isomorphism types of groups of a given order? if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group" for any number n, is a similar calculation possible ? in other words, let $P$ is a function from Natural numbers into natural numbers which for any number $n$ gives the number of possible structures for a group of order $n$ can we find a formula for this function in terms of $n$ and using operation like addition, multiplication, etc ? - No, it's not that easy. In fact, it can be extremely difficult. For abelian groups is easier and it is based on the name of partitions of the powers of the primes that divide $\,n\,$. – DonAntonio Jun 9 '13 at 21:13 @DonAntonio , yup , it's easy for abelians groups using fundemental theorem for finite generated abelian groups or one of the other version of the theorem , but for nonabelian , is there no approximate answer ? or limited answer " e.g under particular conditions " ? – Maths Lover Jun 9 '13 at 21:17 to quote Aluffi's beautiful Algebra: Chapter 0 — To appreciate the difference in complexity, note that there are 42 abelian groups of order 1024 up to isomorphism... allegedly, there are 49,487,365,402 if we count noncommutative ones as well. – citedcorpse Jun 9 '13 at 21:17 and then the footnote: This comparison is a little unfair, however, since it so happens that more than 99% of all groups of order < 2000 have order 1024. – citedcorpse Jun 9 '13 at 21:18 I have written some related material in these answers: (1) (2) – Alexander Gruber Jun 10 '13 at 0:13 ## 3 Answers In view of all the information about how difficult and large $P(n)$ is, I should add the slightly consoling fact that it is algorithmically computable (and in fact primitive recursive). The reason this is only slightly consoling is that I can't think of an algorithm significantly better than a brute force search through all the groups, nested with brute force searches for isomorphisms between them. - exactly, this very simple example shows just how far removed "computability" can be from feasibility – citedcorpse Jun 9 '13 at 21:24 There is a nice table on OEISWiki which shows the number of isomorphism classes for a group of order n - you should notice that they are quite sporadic. In particular, for groups of order $2^n$, the number of isomorphic classes grows quite considerably, especially relative to groups of similar size. - If you are seriously interested in this topic, you could look at the paper Hans Ulrich Besche, Bettina Eick, and E.A. O'Brien. A millennium project: constructing small groups. Internat. J. Algebra Comput., 12:623-644, 2002, which describes how the groups of order up to 2000 (which can be accessed in the GAP or Magma small groups library) were computed. - is this paper a book or what ? edit: does this paper need a background from computational group theory ? – Maths Lover Jun 10 '13 at 18:23	{"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": 0.9299986362457275, "perplexity": 449.7956534049251}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645220976.55/warc/CC-MAIN-20150827031340-00013-ip-10-171-96-226.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/5546/ramified-covers-of-3-torus/5644	# Ramified covers of 3-torus It is known that every orientable 3-manfiold can be obtained as a ramified cover of S3 with a ramification (of some order) at a link in S3. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus? Added. Notice that such a manifold is enlargeble, so it does not admit a metric of positive scalar curvature, so for example a connected sum of n copies of S2 x S1 does not admit a ramified cover of T3 (as far as I understand). - Any conjectures? Clearly there is an onto homomorphism $\pi_1M\to Z^3$; can you do better? – Anton Petrunin Nov 14 '09 at 23:01 I don't really know anything about this question, and can not do better than the homeomorphism to Z^3. For every M^3 its connected sum with several T^3 admits a cover to T^3 :) – Dmitri Nov 14 '09 at 23:58 Conjecture: M is a branched cover of T^3 iff the first homology group of M has rank at least three. – Sam Nead Nov 15 '09 at 0:59 (M assumed to be closed, orientable, connected.) – Sam Nead Nov 15 '09 at 1:00 No, in fact I did not read the proof for S^3. Also the existence of onto homeo $\pi_1M\to Z^3$ as well having rank of first homology at lest 3 is no sufficient because covers of T^3 don't admit a metric of positive scalar curvature, so the conncted sum of 3 S^2xS^1 will be a counterexample. – Dmitri Nov 15 '09 at 10:34 Note that a branched covering induces an injection of rational cohomology rings, by transfer considerations. Therefore the cohomology of a manifold that is a branched covering of $T^3$ must contain three classes of degree 1 whose triple cup product is nontrivial. This condition on a manifold $M^3$ implies the existence of a map $M^3\to T^3$ of nonzero degree. Passing to a covering space of $T^3$ if necessary we obtain such a map that is also surjective on $\pi_1$. Assuming the resulting map is of degree $\ge 3$, the main result of [Edmonds, Allan L.Deformation of maps to branched coverings in dimension three. Math. Ann. 245 (1979), no. 3, 273--279.] implies that this map is homotopic to a branched covering.	{"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": 0.9130273461341858, "perplexity": 374.0883956613183}, "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/1424936463658.66/warc/CC-MAIN-20150226074103-00196-ip-10-28-5-156.ec2.internal.warc.gz"}
https://hal.inria.fr/hal-01056851	# On the measurement of the delta Radar Cross Section (ΔRCS) for UHF tags Abstract : This paper presents a methodology for the practical measurement of the variation of the radar cross section for the RFID UHF tags. This is a very critical parameter, and will govern the tag performance. It also influences both the reading distance as well as the sensitivity of UHF tag. The paper includes the analysis procedure and the experimental setup. The later is based on standard equipments such as an anechoic environment, a vector signal generator and a vector signal analyzer. The application of the proposed method is illustrated on some samples of EPC GEN2 UHF tag. The delta RCS is obtained as a function of the reader power. Keywords : Type de document : Communication dans un congrès RFID, 2009 IEEE International Conference on, Apr 2009, Unknown, pp.346-351, 2009, 〈10.1109/rfid.2009.4911176〉 https://hal.inria.fr/hal-01056851 Contributeur : Valence Lcis <> Soumis le : mercredi 20 août 2014 - 15:46:19 Dernière modification le : lundi 9 avril 2018 - 12:22:18 ### Citation S. Skali, Chirstophe Chantepy, Smail Tedjini. On the measurement of the delta Radar Cross Section (ΔRCS) for UHF tags. RFID, 2009 IEEE International Conference on, Apr 2009, Unknown, pp.346-351, 2009, 〈10.1109/rfid.2009.4911176〉. 〈hal-01056851〉 ### Métriques Consultations de la notice	{"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": 0.8636800646781921, "perplexity": 4004.9719722969935}, "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/1531676591543.63/warc/CC-MAIN-20180720061052-20180720081052-00365.warc.gz"}
http://mathhelpforum.com/differential-equations/95148-implicit-form-print.html	# Implicit Form • July 14th 2009, 01:28 PM stewpot Implicit Form Completly stuck and do not have a clue, please help! I need to find in implicit form the general solution of the differential equation: $\frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$ This is then followed by finding the corresponding particular solution (in implicit form) that satisfies the initial condition: y=1/6 when x=0 Many thanks • July 14th 2009, 02:16 PM skeeter $\frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$ $\frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx $ $-\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C$ can you finish up? • July 15th 2009, 01:25 PM stewpot I've come up with this solution to finish it, could you please check it and if it's wrong please show me where! From where you left me: 8y dy/dx = (e^-2x)^3/2dx y8=2/x-c y=4/3 x=0 Many thanks • July 16th 2009, 06:08 AM skeeter $\frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$ $\frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx $ $-\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C$ $y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C}$ $y(0) = \frac{1}{6}$ $\frac{1}{6} = \frac{1}{27 + C}$ $C = -21$ $y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} - 21}$	{"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": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8975257873535156, "perplexity": 2147.8476798304678}, "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-2014-41/segments/1410657129229.10/warc/CC-MAIN-20140914011209-00050-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://www.physicsforums.com/threads/confused-with-the-answer-seems-correct-butthts-wrong-wrong.583731/	# Confused with the answer<> seems correct buttht's wrong wrong? 1. Mar 4, 2012 ### vkash confused with the answer<> seems correct buttht's wrong wrong??????? question is sqrt(x+1)-sqrt(x-1)=sqrt(4x-1) - - - - - - - - - - - - - - - - - - - - - - - (1) squaring both sides (x+1)+(x-1)-2sqrt(x2-1)=4x-1 - - - - - - - - - - - - - - - - (2) solving and rearranging 1-2x=2sqrt(x2-1) - - - - - - - - - - - - - - - - - - - - - - - -(3) once again squaring both sides; 1-4x= -4 x=5/4; But it does not satisfy the first equation. it also doesn't satisfying equation number three, Is it reason for this???? If yes then why it is so?>?>?>?>?>?>(this is my question) 2. Mar 4, 2012 ### jamesrc Re: confused with the answer<> seems correct buttht's wrong wrong??????? Are you sure that your solution doesn't satisfy those equations? When you take the square root of a number, how many solutions do you get? 3. Mar 4, 2012 ### Fredrik Staff Emeritus Re: confused with the answer<> seems correct buttht's wrong wrong??????? You seem to have started with an equation that doesn't have any real solutions. Let's consider a simpler problem: Find all real numbers x such that $\sqrt x =-1$. If you square both sides, you get x=1. But x=1 doesn't satisfy the original equation, since $\sqrt 1=1\neq -1$. By squaring both sides, we only proved that if $\sqrt x=-1$, then $x=1$. This is an implication, not an equivalence, since x=1 doesn't imply $\sqrt x=-1$. So we can't conclude that x=1. We can only conclude that there are no solutions with x≠1. 4. Mar 4, 2012 ### Staff: Mentor Re: confused with the answer<> seems correct buttht's wrong wrong??????? I'm not sure where you're going with this question. When you take the square root of a number, you get one value. Were you going to suggest that there are two? 5. Mar 4, 2012 ### mathman Re: confused with the answer<> seems correct buttht's wrong wrong??????? Equation (3) lhs = -3/2, rhs = 3/2, so the squares are =, which is the source of your problem. 6. Mar 4, 2012 ### vkash Re: confused with the answer<> seems correct buttht's wrong wrong??????? thanks to all of you; i have got the point of error.	{"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": 0.9596947431564331, "perplexity": 734.8051483004127}, "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-2018-43/segments/1539583511897.56/warc/CC-MAIN-20181018173140-20181018194640-00273.warc.gz"}
https://firas.moosvi.com/oer/physics_bank/content/public/015.Oscillations/Damped%20Oscillations/Exponential%20Damping/part%202/exponential_damping2.html	# Exponential Damping 2# A $${{params.m}}$$ kg mass oscillates on a $${{params.k}}$$ N/m spring. The damping constant of this spring is $$b$$ = $${{params.b}}$$ kg/s. ## Useful Info# For slowly moving objects we’ve seen that the drag force grows in proportion to the velocity, $$\overrightarrow{D} = -b\overrightarrow{v}$$, where $$b$$ is the damping constant and $$\overrightarrow{v}$$ is the velocity of the object. The net force acting on a slowly moving mass attached to a massless spring in the presence of a drag force (for motion along $$x$$ relative to an equilibrium point $$x_0$$) can be written as: (10)#$$$F\_{net,x} = -b\frac{dx}{dt} - kx=ma = m\frac{d^2x}{dt^2}$$$ The solution of this differential equation is found to be $$x(t) = Ae^{-\frac{bt}{2m}} \cos(\omega t) = Ae^{-\frac{t}{2\tau}} \cos(\omega t)= A(t) \cos(\omega t)$$ , where $$A$$ is the initial amplitude of the oscillation, $$\tau$$ is the time constant, $$A(t)$$ is the time-dependent amplitude of the oscillation, and $$\omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$ is the angular frequency of the damped oscillation. ## Part 1# How long does it take for the amplitude of the spring, $$A(t)$$, to reach half of its initial value?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9876846671104431, "perplexity": 648.4566212958009}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334579.46/warc/CC-MAIN-20220925132046-20220925162046-00303.warc.gz"}
https://www.physicsforums.com/threads/vector-model-of-atom-hopefully-easy-question.709200/	# Vector model of atom. Hopefully easy question. 1. Sep 7, 2013 ### LagrangeEuler In system with one electron total angular moment vector $\vec{j}$ is just: $$\vec{j}=\vec{l}+\vec{s}$$ http://selfstudy.in/MscPhysics/BScVectorModelOfAtom.pdf In page 3 author draw a triangle. Intensities of the vectors are $\|\vec{l}\|=\sqrt{l(l+1)}\hbar$, $\|\vec{s}\|=\sqrt{s(s+1)}\hbar$, $\|\vec{j}\|=\sqrt{j(j+1)}\hbar$ And then from nowhere $j=l+s$ or $j=l-s$. Could you please explain me that! Tnx. 2. Sep 8, 2013 ### DrDu These are maximal and minimal possible values, respectively. If you add two vectors, the length of the sum will always lie between these two extremal values. 3. Sep 8, 2013 ### LagrangeEuler But if I look this intensity formulas $$\|\vec{l}\|+\|\vec{s}\|\neq \|\vec{j}\|_{for j=l+s}$$ Right?	{"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": 0.9403424859046936, "perplexity": 2009.6640983316674}, "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/1464049278887.13/warc/CC-MAIN-20160524002118-00084-ip-10-185-217-139.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/from-what-height-will-two-objects-hit-the-ground-at-the-same-speed.623218/	# From what height will two objects hit the ground at the same speed? 1. Jul 24, 2012 ### hockeybro12 1. The problem statement, all variables and given/known data There are 2 objects both with mass 0.50 kg which are dropped from rest at height h. One is dropped on Earth and experiences air resistance and one is dropped on moon and experiences no air resistance. The gravitational acceleration on earth is 9.8 and on moon it is 1.6. From what height h will the objects hit the ground at the same speed? 2. Relevant equations F=-kv for air resistance k=0.17 3. The attempt at a solution I know that we will somehow use the gravitational acceleration. 2. Jul 24, 2012 ### Vorde Well I'd start by figuring out the equations that will tell you how long it will take for each object to fall (it'll be a function of h). Then I'd set the two equal to each other. Do you see where you would get those equations? 3. Jul 24, 2012 ### hockeybro12 Unfortunately, no I don't see where to get the equations because if I set it equal the h will cancel out. 4. Jul 24, 2012 ### azizlwl Taking down as positive. $\ddot {y}=-b\dot y+g$ or dv/dt=-bv+g This is a differential equation of motion. Last edited: Jul 24, 2012 5. Jul 24, 2012 ### hockeybro12 Hello, I am looking for at what height h will the velocity of the two objects be constant. I don't see how this while help solve. Also, I don't know what each of the variables stands for. 6. Jul 24, 2012 ### azizlwl You can use SUVAT equation for the height of the motion on the moon. With air resistance of the earth, its a bit complicated to find the height with differential equation involved. 7. Jul 24, 2012 ### hockeybro12 I am sure that there is another way to do this because we have not learned about SUVAT or differential equations. 8. Jul 25, 2012 ### Vorde The time it takes for an object to fall from a height H is $\sqrt{\frac{2h}{g}}$ It should be easy to set two sides of an equation equal to each other (moon and earth) and solve for h. 9. Jul 25, 2012 ### hockeybro12 Im not sure I understand because when I make $\sqrt{\frac{2h}{g}}$ = $\sqrt{\frac{2h}{g}}$ then won't h cancel out? also, you haven't taken into account the air resistance on earth? 10. Jul 25, 2012 ### Vorde Oh I totally misread the original question, I'm sorry for misleading you with my stuff. I thought wanted something different. I can think of two ways I'm pretty sure I could solve this with (Energy stuff, and calculus stuff), but neither factors in air resistance, so I'll need to think about it for a tiny bit. Again, sorry for talking about the wrong stuff. Similar Discussions: From what height will two objects hit the ground at the same speed?	{"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": 0.8401613831520081, "perplexity": 641.2622712861596}, "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-2017-51/segments/1512948595858.79/warc/CC-MAIN-20171217113308-20171217135308-00074.warc.gz"}
https://www.arxiv-vanity.com/papers/1306.6372/	# Linearizing neutrino evolution equations including ν¯ν pairing correlations Daavid Väänänen AstroParticule et Cosmologie (APC), Université Paris Diderot - Paris 7, 10, rue Alice Domon et Léonie Duquet, 75205 Paris cedex 13, France Cristina Volpe AstroParticule et Cosmologie (APC), Université Paris Diderot - Paris 7, 10, rue Alice Domon et Léonie Duquet, 75205 Paris cedex 13, France February 21, 2021 ###### Abstract We linearize the neutrino mean-field evolution equations describing the neutrino propagation in a background of matter and of neutrinos, using techniques from many-body microscopic approaches. The procedure leads to an eigenvalue equation that allows to identify instabilities in the evolution, associated with a change of the curvature of the neutrino energy-density surface. Our result includes all contributions from the neutrino Hamiltonian and is generalizable to linearize the equations of motion at an arbitrary point of the evolution. We then consider the extended equations that comprise the normal mean field as well as the abnormal mean field that is associated with neutrino-antineutrino pairing correlations. We first re-derive the extended neutrino Hamiltonian and show that such a Hamiltonian can be diagonalized by introducing a generalized Bogoliubov-Valatin transformation with quasi-particle operators that mix neutrinos and antineutrinos. We give the eigenvalue equations that determine the energies of the quasi-particles eigenstates. Finally we derive the eigenvalue equation of the extended equations of motion, valid in the small amplitude approximation. Our results apply to an arbitrary number of neutrino families. ###### pacs: 14.60.Pq, 97.60Bw, 13.15.+g, 24.10.Cn, 26.30.-k, 26.35.+c ## I Introduction Understanding neutrino flavor conversion in media is fascinating theoretically and crucial for observations. The history of the solar neutrino deficit problem constitutes a reference paradigm. Davis’ pioneering observations Davis:1968cp and several decades experiments have cumulated detailed information on solar electron neutrinos. In 1998 Super-Kamiokande has discovered the property that neutrinos modify their flavor while traveling Fukuda:1998mi . The SNO experiment has measured that the solar total neutrino flux is consistent with the Standard Solar model predictions, and established the conversion into Ahmad:2002jz . Finally the reactor experiment KamLAND has identified the Large Mixing Angle solution of the solar neutrino deficit problem Eguchi:2002dm . The ensemble of these observations establish that the high energy (B) electron neutrinos produced in the sun adiabatically convert into the other active flavors because of the Mikheev-Smirnov-Wolfenstein effect Wolfenstein:1977ue ; Mikheev:1986gs (see Ref.Robertson:2012ib for a recent review). This is a mechanism due to the neutrino interaction with matter, which has become a reference in our knowledge of how neutrinos can modify their flavor in media. A variety of novel phenomena has been unravelled by the study of neutrinos in astrophysical and cosmological environments, as e.g. core-collapse supernovae. In this context, the inclusion of the neutrino self-interaction has shown new features such as collective effects (see Duan:2010bg for a review), that can be understood in terms of flavor Duan:2007mv or gyroscopic pendulum Hannestad:2006nj , as an MSW-like solution in the comoving frame Raffelt:2007cb or a magnetic resonance Galais:2011gh . The presence of (front and reverse) shocks in such explosive environments engenders multiple MSW resonances (see Duan:2009cd for a review). The investigation of the phenomena occurring is certainly necessary in view of the possible implications for supernova physics and observations, and to establish the definite interplay with unknown neutrino properties, such as the neutrino mass hierarchy Serpico:2011ir ; Gava:2009pj and leptonic CP violation Balantekin:2007es ; Gava:2008rp . While numerous aspects are understood, a comprehensive understanding of how neutrinos modify their flavor in these explosive media requires further investigations. Recently, the connection is being developed between formal aspects of the description of neutrino flavor conversion in media and of other many-body systems like atomic nuclei and condensed matter. Establishing these links is certainly fascinating theoretically. Moreover it can give novel paths to go beyond the currently adopted description, or test the validity of approximations that are widely employed. In Ref.Balantekin:2006tg the - Hamiltonian is reformulated using an algebraic approach. Corrections to the commonly used mean-field Hamiltonian are obtained using the path-state integral approach. This algebraic approach is pursued in Ref.Pehlivan:2011hp where the many-body Hamiltonian (without the neutrino-matter interaction term) is put in connection with the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. A Random-Phase-approximation (RPA) version of the equations is given. The spectral-split phenomenon due to the - interaction is understood as the transition from the quasi-particle to the particle degrees of freedom, with the Lagrange multipliers being related to the split energy. In Ref.Volpe:2013uxl the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) theoretical framework, that is widely used in the study of many-body systems, is adopted to re-derive the mean-field equations describing the neutrino propagation in a background of matter and of neutrinos. Besides giving a rigorous derivation of such equations, such an approach allows to go beyond the mean-field approximation to include contributions from the two-body density matrix, like neutrino-antineutrino pairing correlations. Note that these contributions have been neglected so far (see e.g. Sigl:1992fn ; Cardall:2007zw ). However, as demonstrated in Ref.Volpe:2013uxl such corrections already appear at the mean-field level. Since the equations are non-linear, their impact on supernova physics and on neutrino flavor conversion in such environments might be significant, and deserves investigation. Linearizing evolution equations around a stationary solution is another general procedure, commonly employed in the context of microscopic many-body approaches. In fact, such methods give eigenvalue equations that identify the presence of collective modes, such as giant resonances in atomic nuclei or phonons in metallic clusters. These are also used to detect instabilities that typically occur when correlations become too large and induce a change of the energy-density curvature around the starting point of the linearization procedure. In the specific context of core-collapse supernova neutrinos, demanding numerical calculations are required to implement realistically the neutrino-neutrino interaction, or the dynamical features associated with the explosion. A detailed treatment of the transition between the region where neutrinos are trapped, and the one where they start free streaming (the neutrinosphere) will require still some time, in order to properly take into account neutrino rescattering for example Cherry:2012zw or two-body density matrix contributions, such as - pairing correlations Volpe:2013uxl . The linearization procedure might bring some insight on these complex cases, without going through full simulations. In situations encountered in many-body systems, typically stable collective small amplitude modes are searched for. In the neutrino context such an approach appears to be of particular use for the study of unstable modes. In fact, the presence of flavor instabilities is a characteristic feature of the neutrino evolution in a medium, such as a core-collapse supernova, where the non-linear neutrino self-interaction is important. In Ref.Hannestad:2006nj it has been pointed out that the bipolar oscillations in the single-angle approximation is associated with an instability triggered by the presence of the vacuum mixings. Using the matter basis within two-neutrino flavors, Ref.Galais:2011jh has given an analytical condition to identify the occurrence of such an instability. Such a condition is consistent with the one derived heuristically in Ref.Duan:2010bf . Multi-angle instabilities are investigated in Ref.Sawyer:2008zs . Besides, Sawyer:2008zs has first pointed out the use of linearization methods to investigate such instabilities. Ref.Banerjee:2011fj has given an eigenvalue equation (for two-flavors) and studied numerically their appearance, both in the simplified (single-angle) angular treatment of the neutrino-neutrino interaction term, and in sophisticated multi-angle calculations. Schematic (box-like) neutrino fluxes typical of the supernova case are used. A linearized flavor stability analysis is performed in Sarikas:2011am ; Saviano:2012yh in realistic cases, i.e. in order to study the suppression of collective effects during the accretion phase of various iron-core collapse supernova progenitors. Ref.Sarikas:2012ad has used the same method to show the presence of spurious instabilities when the number of angular bins used in multi-angle calculations is not large enough. The presence of azimuthal angle instabilities is identified in Ref.Raffelt:2013rqa . The present work is based on the extended neutrino evolution equations, including neutrino-antineutrino pairing correlations, derived in Ref.Volpe:2013uxl using the BBGKY hierarchy. We first apply a linearization method known in the context of many-body approaches to the neutrino evolution equations. We derive a general eigenvalue equation that implements the vacuum, the matter and the neutrino-neutrino interaction terms of the Hamiltonian. We introduce the stability matrix associated with the neutrino energy density surface. Then, we consider the extended evolution equations containing contributions from the abnormal mean field associated with neutrino-antineutrino pairing correlations. As for Ref.Volpe:2013uxl here we focus on the formal aspects while simulations including such terms are beyond the scope of the present work. Two aspects of the case with - pairing are studied: the static solution of the generalized Hamiltonian including pairing correlations and a linearization of the corresponding evolution equations. For the first, a re-derivation of the Hamiltonian with correlations is given in the mean-field approximation. Using a generalized Bogoliubov-Valatin transformation that mixes neutrinos and their -conjugates, we demonstrate that the eigenstates of the system described by the generalized Hamiltonian are independent quasi-particle states. The eigenvalue equation to determine the eigen-energies of the ground- and of the excited quasi-particle states is also given. Finally we present the linearized version of the extended equations of motion obtained in Volpe:2013uxl , which can be used to study both stable and unstable small amplitude collective modes due to - pairing. The manuscript is organised as follows. In Section I we introduce the theoretical framework and present the extended mean-field neutrino evolution equations describing neutrino propagation in an environment. These include the neutrino mixings, the neutrino interaction with matter, with neutrinos and neutrino-antineutrino correlations. In Section II we present our linearization procedure. We apply it to the case where only the normal mean field is included to obtain the eigenvalue equations and the stability matrix. Section III focuses on the static case of the extended Hamiltonian, its diagonalization is performed by introducing quasi-particle states and the equations to determine the eigen-energies are derived. Section IV presents the linearized version of the extended evolution equations. A discussion and our conclusions are contained in Section V. ## Ii Neutrino mean-field evolution equations in media with ν-¯ν pairing Having in mind astrophysical and cosmological applications, we consider a system of neutrinos and antineutrinos propagating in a medium composed of matter, neutrinos and antineutrinos. The corresponding Hamiltonian in the flavor basis is Hf=UHvacU†+Hint (1) where the vacuum term is with eigenenergies for the propagation eigenstates with mass (). The second term corresponds to the interaction between a neutrino and any other particle of the background. The unitary matrix is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix Maki:1962mu that relates the interaction (flavor) to the propagation (mass) basis through . In three flavors, such a matrix depends upon three measured mixing angles, one Dirac and two Majorana unknown phases PDG2012 . Within a density matrix formalism, the information on the flavor evolution is encoded in the density matrix that reads, for three flavors, ρν=⎛⎜ ⎜ ⎜ ⎜⎝⟨a†να,iaνα,i⟩⟨a†νβ,jaνα,i⟩⟨a†νγ,kaνα,i⟩⟨a†να,iaνβ,j⟩⟨a†νβ,jaνβ,j⟩⟨a†νγ,kaνβ,j⟩⟨a†να,iaνγ,k⟩⟨a†νβ,jaνγ,k⟩⟨a†νγ,kaνγ,k⟩⎞⎟ ⎟ ⎟ ⎟⎠. (2) The and are the annihilation and the creation operators for a neutrino of flavor in the quantum state identified by the single-particle label , that indicates for example momentum , or helicity . The expectation values in Eq.(2) are performed over the full many-body density matrix which, in general, can be associated to pure or to mixed states. The diagonal elements of correspond to the neutrino occupation numbers, with being the total occupation number for a neutrino of flavor. The off-diagonal (or decoherent) terms encode the neutrino mixings. Note that a ”matrix of densities” generalizing the usual occupation numbers is commonly used in the literature (see e.g. Raffelt:1992uj ; Sigl:1992fn ). The density matrix is easily extended to an arbitrary number of families, to account for the presence of both sterile and active neutrinos. A definition analogous to (2) is introduced for antineutrinos, whose explicit expression is ¯ρν=⎛⎜ ⎜ ⎜ ⎜⎝⟨b†να,ibνα,i⟩⟨b†νβ,jbνα,i⟩⟨b†νγ,kbνα,i⟩⟨b†να,ibνβ,j⟩⟨b†νβ,jbνβ,j⟩⟨b†νγ,kbνβ,j⟩⟨b†να,ibνγ,k⟩⟨b†νβ,jbνγ,k⟩⟨b†νγ,kbνγ,k⟩⎞⎟ ⎟ ⎟ ⎟⎠. (3) where and the annihilation and creation operators for antineutrinos. The creation and annihilation particle and antiparticle operators satisfy the usual canonical commutation rules111Note that for relativistic neutrinos an approximate Fock space can be built (see Giunti:1991cb for a discussion). {a(→p,h),a†(→p′,h′)}=(2π)32Epδ3(→p−→p′)δhh′ (4) and similarly for the antiparticle operators. All other anticommutators vanish. The single-particle states associated with neutrino mass eigenstates are \|m⟩=a†m\|⟩ \|m⟩=b†m\|⟩ (5) with being the vacuum state defined by and . In Ref.Volpe:2013uxl we have derived extended mean-field equations to describe a neutrino, or an antineutrino, evolving in a matter and in a(n) (anti)neutrino background using the BBGKY hierarchy Born-Green ; Yvon ; Kirkwood ; Bogoliubov . This corresponds to an (unclosed) set of coupled first-order integro-differential equations for the reduced density matrices, which is equivalent to determining the exact evolution of the full many-body density matrix. Note that while BBGKY is often formulated in the density matrix formalism, it is formally equivalent to an (infinite) hierarchy of equations for the n-point Green functions in the equal-time limit. Such a hierarchy gives a framework to re-derive the usually employed mean-field and Boltzmann approximations, but also to go beyond them in a consistent way. In particular, if one includes neutrino-antineutrino pairing correlations, one arrives at extended mean-field equations that can be cast in a compact matrix form (see Volpe:2013uxl for the details of the derivation). In the flavor basis these read i˙R=[H,R], (6) with the generalized density: R=(ρκκ†1−¯ρ∗). (7) depends on the normal densities for Eq. (2) and for , on the abnormal density: κν=⎛⎜ ⎜⎝⟨bνα,iaνα,i⟩⟨bνβ,jaνα,i⟩⟨bνγ,kaνα,i⟩⟨bνα,iaνβ,j⟩⟨bνβ,jaνβ,j⟩⟨bνγ,kaνβ,j⟩⟨bνα,iaνγ,k⟩⟨bνβ,jaνγ,k⟩⟨bνγ,kaνγ,k⟩⎞⎟ ⎟⎠, (8) as well as on its complex conjugate κ∗ν=⎛⎜ ⎜ ⎜ ⎜⎝⟨a†να,ib†να,i⟩⟨a†νβ,jb†να,i⟩⟨a†νγ,kb†να,i⟩⟨a†να,ib†νβ,j⟩⟨a†νβ,jb†νβ,j⟩⟨a†νγ,kb†νβ,j⟩⟨a†να,ib†νγ,k⟩⟨a†νβ,jb†νγ,k⟩⟨a†νγ,kb†νγ,k⟩⎞⎟ ⎟ ⎟ ⎟⎠. (9) These encode the - pairing correlations. The generalized Hamiltonian in Eq.(6) is given by H=(hΔΔ†−¯h∗). (10) It comprises the mean-field Hamiltonian for neutrinos, for antineutrinos as well as the abnormal (or pairing) mean field . Its general expression is Δik=∑jlv(ik,jl)κjl. (11) Its complex conjugate depends on and involves a sum over the initial single-particle states, instead of over the final single-particle states. The quantity : v(im,jn)=⟨im\|Hint\|jn⟩ (12) are the matrix elements associated with the interaction Eq.(1) which involve and single-particle states Eq.(5) associated with the incoming and outgoing particles. Therefore from Eqs.(6-9) one can see that to determine the evolution of the system, within the extended mean field, one has to identify the solution of a coupled system of equations for , for and for their two-body correlators and . The explicit expression of the pairing mean fields and depend on the assumptions of homogeneity and isotropy made on the background. In particular, if homogeneity is assumed, involves pairs of neutrinos and antineutrinos with opposite momentum, as discussed in Volpe:2013uxl . In the present manuscript we will only need the general expressions of the pairing mean fields (the explicit expression in cartesian or polar coordinates can be found in Volpe:2013uxl ). In absence of - pairing, Eqs.(6) reduce to the mean-field evolution equation: i˙ρ=[h(ρ),ρ] (13) with the normal mean field acting on a neutrino given by h(ρ) = UHmU†+Hmat +√2GF∑α––∫d3→p(2π)3(ρα––,p−¯ρ∗α––,p)(1−^→p⋅^→k). The term accounting for neutrino interaction with matter is , with being the Fermi coupling constant and the electron number density. The refers to a neutrino that is initially born in the flavor since one has to sum over all the flavors present in the background. In deriving Eq.(II) the electron background is assumed homogenous and isotropic, while the (anti)neutrino background, homogenous and anisotropic. The anisotropy introduces the angular term depending on defined as ^→p⋅^→k=→p⋅→k\|→p\|⋅\|→k\|. (15) In case an antineutrino is traveling instead of a neutrino, an equation analogous to Eq.(13) holds i˙¯ρ=[¯h(¯ρ),¯ρ] (16) with the corresponding normal mean field222Note that, our definition is (see Eq.(3)), so that antineutrinos do not transform the same way as neutrinos under . given by ¯h(¯ρ) = U∗HmUT−Hmat −√2GF∑α––∫d3→p(2π)3(ρ∗α––,p−¯ρα––,p)(1−^→p⋅^→k). In case of an isotropic background, the contribution to the integral coming from the momentum scalar products vanishes. In this case only the contribution coming from the density matrices remains, giving a term analogous to , but with the (anti)neutrino number densities replacing the electron one. The derivation of the mean-field equations (13-II) can be found in Volpe:2013uxl , and in Sigl:1992fn ; Qian:1994wh ; Balantekin:2006tg (using different methods). In component form, the extended equations (6-10) in presence of pairing are ⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩i˙ρij(1)=[h(1),ρ(1)]ij+∑m(Δimκ∗jm−κimΔ∗jm)i˙¯ρkl(2)=[¯h(2),¯ρ(2)]kl+∑m(Δmkκ∗ml−κmkΔ∗ml)i˙κik=∑m(him(1)κmk+hkm(2)κim)+Δik−∑m(ρim(1)Δmk+¯ρkm(2)Δim), (18) where here the indices stand for ; for with that vary over the different electron, muon and tau flavor states. For the sake of clarity, Eqs.(18) we have introduced an explicit dependence of the quantities on particles of type 1 corresponding to neutrinos, and on particles of type 2 referring to antineutrinos since correlates the two kind of particles. ## Iii Linearization procedure and its application to the mean-field equations Theoretical and phenomenological studies of neutrino flavor conversion in astrophysical environments typically solve Eqs.(13)-(II) in either schematic models, or realistic cases with the goal of underpinning the physical mechanisms and/or making reliable predictions for supernova observables. Linearizing the equations of motion is a standard procedure in the study of the many-body systems and leads to microscopic approaches known as the Random-Phase-Approximation (RPA), or the Quasi-Particle Random-Phase-Approximation (QRPA) Ring ; Tohyama:2004ed . Here we first describe such a method and then apply it to the neutrino case when the mean-field approximation is made. ### iii.1 Linearization procedure Let us consider a system described by a mean-field Hamiltonian and a one-body density matrix . In the mean-field approximation the evolution equation is i˙ρ=[h(ρ),ρ]. (19) Let us consider the case that the system is in a stationary solution at a given time , then [h0,ρ0]=0 (20) where . This implies that there is a basis in which both and the mean-field Hamiltonian are diagonal. Note that the energy variation in this basis is such that , with . Therefore Eq.(20) identifies the stationary solution that minimizes the energy of the system. We now consider a small variation of the density around : δρ=ρ0+δρ(t)=ρ0+ρ′e−iωt+ρ′†eiω∗t. (21) In the specific case where the initial state is a pure state, i.e. it is easy to show that such a variation can only have nonzero contributions if 333In the context of many-body approaches, these contributions are usually referred to as of particle-hole type, Ring . The basis diagonalizing the mean field is usually called the Hartree-Fock basis. where and are unoccupied and occupied single-particle states respectively. This is for example a good approximation for the case for atomic nuclei whose ground states are well described by Slater determinants where the nucleons build up a Fermi sea of occupied single-particle states. In the case of mixed states as for neutrinos produced in dense stellar regions one can associate with the decoherent off-diagonal contributions that are present because of the mixings. Physically speaking, the frequency in Eq.(21) represents the frequencies of the small amplitude excitation modes of the system around . Usually, one introduces an external field that drives the system out of the equilibrium solution and induces small amplitude variations around such a solution (see e.g. Ring ). For our cases of interest, we make the hypothesis that at the density matrix is ”stationary” to a good approximation if the time-dependence of Hamiltonian is weak so that the system stays in such a solution for some time. Then the time-dependence of the Hamiltonian drives the system out of equilibrium444As we we will discuss an example of this kind is given by the transition between the synchronization and the bipolar regimes produced by the neutrino-neutrino interaction in the context of core-collapse supernovae (see e.g. Duan:2007mv ; Hannestad:2006nj ; Galais:2011jh ).. In order to study the behaviour of the system in this small amplitude approximation, one can linearize the neutrino evolution equations (19) around the solution satisfying Eq.(20). The development of the mean-field Hamiltonian around this solution gives h(ρ)=h0+δhδρ∣∣ρ0δρ+… (22) Implementing the small amplitude variation Eq.(21) and retaining only the lowest order contributions, one gets ωρ′e−iωt−ω∗ρ′∗eiω∗t = [h0,ρ0+δρ] + [δhδρδρ,ρ0]. From Eq.(20) one obtains for the first commutator on the r.h.s. of Eq.(III.1) [h0,δρ]ij=(~ki−~kj)δρij, (24) where are the energy eigenvalues associated with the single-particle states that diagonalize the Hamiltonian . The second commutator on the r.h.s. of Eq.(III.1) gives [δhδρδρ,ρ0]ij =∑k(δhδρ∣∣ρ0δρ)ikρ0kjδkj (25) −ρ0ikδik(δhδρ∣∣ρ0δρ)kj, where we have used the fact that we are in the basis in which has only nonzero diagonal elements. By introducing Eqs.(20) and (24-25) in (III.1) a general eigenvalue equation is obtained, valid in the small amplitude approximation Ring : ωρ′ij =∑kl[(~kk−~kl)δkiδjlρ′kl+(ρ0j−ρ0i)δhijδρ′kl∣∣ρ0ρ′kl]. (26) A similar equation holds for , instead of . To gather further insight in the behaviour of a system around a stationary solution of the Hamiltonian, one can also consider developing the energy density of the system in the small amplitude approximation E[ρ]=E[ρ0]+δEδρ∣∣ρ0δρ+δ2Eδρ2∣∣ρ0δρ2+… (27) where for the stationarity one has . The second derivative of the energy density defines the the energy-density curvature around . One can show that such a term can be associated to a stability matrix that is positive definite in presence of stable small amplitude modes555Note that this argument can be generalized to a multi-variable energy density by considering its Hessian.. The presence of complex eigenvalues of the stability matrix signals a change in the curvature in presence of an instability Ring . Therefore, in order to identify small amplitude modes of the system requires solving the eigenvalue equations (26) obtained by linearizing the equations of motion or, equivalently, identify the eigenvalues of the stability matrix. Under some extra assumptions on the density matrix variations, that depend on the specific system under consideration, one can also construct the stability matrix directly from Eq.(27). The latter procedure is for example of use in the case of atomic nuclei where , as , keeps being a projector on the single-particle occupied states666In other words, is associated to a Slater determinant.. In the following we will use the linearization procedure described above. In the application to the mean-field case we will also identify the stability matrix. ### iii.2 General linearized eigenvalue equations for neutrinos in media We now apply the procedure just described to our system of neutrinos and antineutrinos propagating in a medium. As seen from Eqs.(13-II), the mean fields and are a function of the two density matrices and . At one has h0=h(ρ0,¯ρ0), (28) and ¯h0=¯h(ρ0,¯ρ0). (29) Therefore relations (22-26) have to be generalized to include the particle and antiparticle degrees of freedom. In particular, one has the following variations δh=δh(ρ,¯ρ)δρ∣∣(ρ0,¯ρ0)δρ+δh(ρ,¯ρ)δ¯ρ∣∣(ρ0,¯ρ0)δ¯ρ (30) δ¯h=δ¯h(ρ,¯ρ)δρ∣∣(ρ0,¯ρ0)δρ+δ¯h(ρ,¯ρ)δ¯ρ∣∣(ρ0,¯ρ0)δ¯ρ. (31) This adds an extra commutator to Eq.(III.1) (similarly in the linearized equations for antineutrinos). For our system of neutrinos and antineutrinos, from Eq.(26) two coupled eigenvalue equations are derived777Equation (26) written for particle-hole () and hole-particle () contributions gives the RPA equations. In this case the derivatives of with respect to have two contributions and give rise to the so called ph-ph interaction terms Ring . ωρ′ij =∑kl[(~kk−~kl)δkiδjlρ′kl (32) +(ρ0j−ρ0i)(δhijδρ′kl∣∣(ρ0,¯ρ0)ρ′kl+δhijδ¯ρ′kl∣∣(ρ0,¯ρ0)¯ρ′kl)] ω¯ρ′ij =∑kl[(~¯kl−~¯kk)δkjδil¯ρ′kl +(¯ρ0j−¯ρ0i)(δ¯hijδ¯ρ′kl∣∣(ρ0,¯ρ0)¯ρ′kl+δ¯hijδρ′kl∣∣(ρ0,¯ρ0)ρ′kl)]. Similar equations hold for and . Eq.(32) can be cast in a compact matrix form : (AB¯B¯A)(ρ′¯ρ′) =ω(ρ′¯ρ′), (33) with Aij,kl =~ki−~kj+(ρ0j−ρ0i)δhijδρkl∣∣(ρ0,¯ρ0) Bij,kl =(ρ0j−ρ0i)δhijδ¯ρ′kl∣∣(ρ0,¯ρ0) ¯Aij,kl =(~¯kj−~¯ki)+(¯ρ0i−¯ρ0j)δ¯hjiδ¯ρ′kl∣∣(ρ0,¯ρ0) ¯Bij,kl =(¯ρ0i−¯ρ0j)δ¯hjiδρ′kl∣∣(ρ0,¯ρ0). Note that the matrix form (33) involve the components and . For each solution of Eqs.(III.2) are also a solution of the eigenvalue equations corresponding to . The matrix appearing on of Eq.(33) is the stability matrix: S =(AB¯B¯A). (35) While the results just derived are valid for any system of neutrinos and antineutrinos described by a mean-field equation, we now make them specific to the case of core-collapse supernova neutrinos with the mean field given by Eqs.(13-II). Let us take as initial time the neutrinosphere where neutrinos start free-streaming. In this region, the large matter density suppresses the neutrino mixing angles, so that the flavor and matter basis practically coincide. In such a basis the density matrices Eq.(2) and Eq.(3) as well as the corresponding mean-field hamiltonians Eq.(II) and Eq.(II) are diagonal. Therefore Eq.(20) is satisfied. This is our initial ”stationary” solution, until the Hamiltonian time-dependence drives the system out of equilibrium. At this point of the evolution the density matrix off-diagonal elements become nonzero. They are the small amplitude deviations and ( and in components) of Eq.(21). These satisfy Eqs.(32-III.2) if the system has a small amplitude collective motion, or undergoes an instability. In this context one can assign to the quantities the difference between the neutrino matter888Note that here we employ the terminology ”matter” eigenvalues although they are obtained through the diagonalization of the Hamiltonian including mixings, neutrino interaction with matter and neutrino self-interaction. eigenvalues at that are obtained by diagonalizing the mean-field Hamiltonian Eqs.(II) and (II) ~h=U†hU=diag(~ki) ~¯h=U†¯hU=diag(~¯ki) (36) where the tilde here indicates that we are in the ’matter’ basis. As for the derivatives of and , these can be obtained from the general expression for the mean field which is Volpe:2013uxl Γij=∑klv(ik,jl)ρlk. (37) The explicit calculation of the matrix elements Eq.(12) associated with neutrino interaction with matter and neutrinos, gives the mean fields shown in Eqs.(13) and (II) (see Volpe:2013uxl for details). One gets for a given set of single-particle quantum labels the derivatives with respect to with and with δhijδρ′kl=vνν(il,jk) δhijδ¯ρ′lk=vν¯ν(i¯k,j¯l) (38) with () referring here to the single-particle states for an incoming (outoing) antineutrino and an explicit dependence of the matrix elements on the interacting particles is introduced for clarity. Note also that there is no contribution from the derivatives δhijδρ′lk=δhijδ¯ρ′kl=0 (39) with respect to and to (). Therefore, while the neutrino Hamiltonian receives contributions from the vacuum, the matter and the neutrino-neutrino terms, the mean-field derivatives with respect to variations of the density matrix receive no contribution from the vacuum and the matter terms that are density-independent. By implementing expressions (38-39) in Eq.(32) one finally finds the eigenvalue equations in the small amplitude approximation999 In the study of atomic nuclei, this eigenvalue equation is known as the Random-Phase-Approximation (RPA). The analogue of the neutrino single-particle energy differences, , are the particle-hole excitation energies; while the two-body matrix elements are calculated using nuclear effective interactions. As is well established, the inclusion of this two-body residual interaction is necessary to reproduce the measured excitation energies of the collective modes of atomic nuclei, known as the Giant Resonances. ωρ′ij =∑k Again, equations equivalent to (40) hold for and . Their solution determines the frequencies of the collective modes of our system around the ”stationary” solution. Note that such frequencies differ from the single-particle energy differences, which are for neutrinos and for antineutrinos, by an amount that depends upon the two-body interaction matrix elements. In particular, the presence of complex frequencies indicates that the system has become unstable. The set of equations (32-III.2) and (40) is our most general result of applying the linearization procedure to the mean-field equations in media. We would like to emphasize the generality of the linearized equations we have derived. First, in our formulation we have not fixed the number of neutrinos, so that our equations can be used to study instabilities for an arbitrary number of families. This requires the use of the corresponding density matrix expression equivalent to Eqs.(2) and (3) for a number of neutrinos instead of three. Second, Eqs.(33) or (40) can be used to investigate the occurence as well as the precise location of flavor instabilities since all the contributions of the neutrino Hamiltonian have been retained in our derivation. Third, while in our considerations has been taken to be the initial time at the neutrinosphere, the procedure employed here can be used to linearize the neutrino equations of motion at any time of the evolution in a core-collapse supernova. To this aim, in Eqs.(22-26) one should take as the density matrix at the starting point the one in the ’matter’ basis. Note that Ref.Galais:2011jh has investigated the effects of the neutrino self-interaction in supernovae in such a basis. By definition, this is the basis that instantaneously diagonalizes the neutrino Hamiltonian, so that the stationary condition Eq.(20) is satisfied. In the small amplitude approximation, are the off-diagonal elements of the density matrix in such a basis. The eigenvalues are then the matter eigenvalues evaluated at the given . Therefore, eigenvalue equations of the same type as Eq.(32) can be derived to establish the presence of an instability at any time of the neutrino evolution. Finally, the problem of solving the linearized equations (32)-(III.2) can be replaced by the determination of the eigenvalues of the stability matrix (35). While such a matrix remains positive definite for collective stable modes, its eigenvalues become complex in presence of instabilities. ### iii.3 Eigenvalue equations in explicit form The linearized equations are here written in a more explicit form for the case of neutrinos in a core-collapse supernova. To this aim the expressions of the matrix elements are needed corresponding to different scattering processes due to the (anti)neutrino charged- and neutral-current interaction with electrons, protons, neutrons and (anti)neutrinos of all flavors. The detailed derivation of the relevant matrix elements can be found in Volpe:2013uxl . One can associate to the particle that is propagating for example the incoming and outgoing single-particle states and to the particle of the background . Since only forward scattering is considered, and correspond to states with same momentum and helicity. Under these assumptions one gets for Eq.(12) v(il,jk)=(1−^→p⋅^→k), (41) with and the momenta of the background and of the propagating particle, respectively. For three flavors, in the flavor basis Eqs.(40) give the following eigenvalue equations101010Note that, to avoid confusing notations, from now we will denote the off-diagonal elements of the density matrices Eqs.(2) and (3) as instead of . ωρνανβ→k =(~kνα,→k−~kνβ,→k)ρνανβ→k+√2GF(ρνβνβ→k−ρνανα→k) ×∑α––∫d3→p(2π)3(1−^→k⋅^→p)(ρνανβα––,→p−¯ρ∗νανβα––,→p), (42) concerning the off-diagonal elements of the density matrix Eq.(2), with and . The quantity is the difference of the diagonal matrix elements of the density matrix. Equation (III.3) is associated with a given initial condition for , . Analogous equations are valid for the with eigenfrequencies , as well as for the antineutrino counterparts and . In order to present our results in a more explicit form, we consider as an example the ”bulb model” for the neutrino emission at the neutrinosphere Duan:2010bg . In this model one assumes both the spherical and the cylindrical symmetries (see Figure 1). The neutrinosphere corresponds to a sharp sphere of radius . A neutrino or antineutrino flavor content at a given radius within a core-collapse supernova depends on its energy , on and on the point of emission at the neutrinosphere. The latter can be characterized e.g. by the variable with (see Duan:2010bg ). All neutrinos born with a flavor with same and have the same evolution up to the distance . Let us write Eqs.(III.3) for the case of three-flavors within the ”bulb” model. We introduce the velocity vector with to describe the motion of a (anti)neutrino with trajectory angle with respect to the symmetry axis, . The corresponding equation of motion Eq.(13) is idρdr=1v[h(ρ),ρ] (43) with Eq.(II) now being h(ρ) =UHmU†+Hmat+√2GF (44) ×∑α––∫p2dpdv′(2π)2(ρα––,p,v′−¯ρ∗α––,p,v′)(1−v⋅v′), (45) where refers to the background particle; and similarly for Eqs.(16-II). With these assumptions the eigenvalue equations (III.3) become at the single particle level111111Note that for the sake of clarity here we simplify the notation and replace with . ωρβ′βα––′,→k =(~kβ′,→k−~kβ,→k)ρβ′βα––′,→k +√2GF2πR2(ρβ′β′α––,→k−ρββα––,→k)∑α––∫dpdv′ ×(1v−v′)[ρβ′βα––,→pfFD(p)Lα––⟨Eα––⟩−¯ρ∗β′βα––,→p¯fFD(p)L¯α––⟨E¯α––⟩], (46) and similarly for antineutrinos. The quantities are the flux of neutrinos produced in the flavor at the neutrinosphere and is the corresponding average neutrino energy. The functions , are Fermi-Dirac distributions for neutrinos and antineutrinos respectively, depending on the two parameters () fFD(p)≡1F2(ην)T3νp2e(p/Tν−ην)+1 (47) where and . The functions are Fk(ην)≡∫∞0xkdxe(x−η)+1 (48) By considering a discretized version of the integral in equations (III.3), with respect to energy and to , the eigenvalue equations for neutrinos and antineutrinos can be cast together in the matrix form Eq.(33) ω⎛⎜ ⎜ ⎜⎝ρβ′βα––′(→k)¯ρ∗β′βα––′(→k)⎞⎟ ⎟ ⎟⎠ =⎛⎜ ⎜ ⎜⎝Aβ′βα––′α––(→k,→p)Bβ′βα––′α––(→k,→p)¯Bβ′βα––′α––(→k,→p)¯Aβ′βα––′α––(→k,→p)⎞⎟ ⎟ ⎟⎠⎛⎜ ⎜⎝ρβ′βα––(→p)¯ρ∗β′βα––(→p)⎞⎟ ⎟⎠ (49) where each element of the vectors is to be understood as a vector of dimension equal to with the number of neutrino families, neutrino energies and angles; while each element of the matrix is to be understood as a matrix of dimension . 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https://www.hepdata.net/search/?q=&sort_order=&sort_by=latest&page=1&collaboration=CDF	Showing 25 of 210 results Measurement of $\sin^2\theta^{\rm lept}_{\rm eff}$ using $e^+e^-$ pairs from $\gamma^/Z$ bosons produced in $p\bar{p}$ collisions at a center-of-momentum energy of 1.96 TeV The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D93 (2016) 112016, 2016. Inspire Record 1456804 At the Fermilab Tevatron proton-antiproton (pp¯) collider, Drell-Yan lepton pairs are produced in the process pp¯→e+e-+X through an intermediate γ/Z boson. The forward-backward asymmetry in the polar-angle distribution of the e- as a function of the e+e--pair mass is used to obtain sin2θefflept, the effective leptonic determination of the electroweak-mixing parameter sin2θW. The measurement sample, recorded by the Collider Detector at Fermilab (CDF), corresponds to 9.4 fb-1 of integrated luminosity from pp¯ collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDF Run II data set. The value of sin2θefflept is found to be 0.23248±0.00053. The combination with the previous CDF measurement based on μ+μ- pairs yields sin2θefflept=0.23221±0.00046. This result, when interpreted within the specified context of the standard model assuming sin2θW=1-MW2/MZ2 and that the W- and Z-boson masses are on-shell, yields sin2θW=0.22400±0.00045, or equivalently a W-boson mass of 80.328±0.024 GeV/c2. 2 data tables Best-fit values of $\sin^2\theta_{\rm eff}^{\rm lept}$, $\sin^2\theta_W$ and $M_W$(indirect) from the $ee$-channel measurement of $A_{\rm fb}$ and a combination with the previous CDF measurement based on muon pairs. Fully corrected $A_{fb}$ measurement for electron pairs with $\|y\|<1.7$. The measurement uncertainties are bin-by-bin unfolding estimates. Measurement of the forward–backward asymmetry of top-quark and antiquark pairs using the full CDF Run II data set The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D93 (2016) 112005, 2016. Inspire Record 1424841 We measure the forward–backward asymmetry of the production of top-quark and antiquark pairs in proton-antiproton collisions at center-of-mass energy s=1.96 TeV using the full data set collected by the Collider Detector at Fermilab (CDF) in Tevatron Run II corresponding to an integrated luminosity of 9.1 fb-1. The asymmetry is characterized by the rapidity difference between top quarks and antiquarks (Δy) and measured in the final state with two charged leptons (electrons and muons). The inclusive asymmetry, corrected to the entire phase space at parton level, is measured to be AFBtt¯=0.12±0.13, consistent with the expectations from the standard model (SM) and previous CDF results in the final state with a single charged lepton. The combination of the CDF measurements of the inclusive AFBtt¯ in both final states yields AFBtt¯=0.160±0.045, which is consistent with the SM predictions. We also measure the differential asymmetry as a function of Δy. A linear fit to AFBtt¯(\|Δy\|), assuming zero asymmetry at Δy=0, yields a slope of α=0.14±0.15, consistent with the SM prediction and the previous CDF determination in the final state with a single charged lepton. The combined slope of AFBtt¯(\|Δy\|) in the two final states is α=0.227±0.057, which is 2.0σ larger than the SM prediction. 3 data tables Bin centroids and the differential $A_{\rm{FB}}^{t\bar{t}}$ in the $A_{\rm{FB}}^{t\bar{t}}$ vs. $\|\Delta y\|$ measurement in the lepton+jets final state. Bin centroids and the differential $A_{\rm{FB}}^{t\bar{t}}$ in the $A_{\rm{FB}}^{t\bar{t}}$ vs. $\|\Delta y\|$ measurement in the dilepton final state. The eigenvalues and eigenvectors for the $A_{\rm{FB}}^{t\bar{t}}$ vs. $\|\Delta y\|$ measurements in both the lepton+jets and the dilepton final states. Each row contains first an eigenvalue, then the error eigenvector that corresponds to that eigenvalue. Measurement of the forward-backward asymmetry in low-mass bottom-quark pairs produced in proton-antiproton collisions The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D93 (2016) 112003, 2016. Inspire Record 1416824 We report a measurement of the forward-backward asymmetry, AFB, in bb¯ pairs produced in proton-antiproton collisions and identified by muons from semileptonic b-hadron decays. The event sample is collected at a center-of-mass energy of s=1.96 TeV with the CDF II detector and corresponds to 6.9 fb-1 of integrated luminosity. We obtain an integrated asymmetry of AFB(bb¯)=(1.2±0.7)% at the particle level for b-quark pairs with invariant mass, mbb¯, down to 40 GeV/c2 and measure the dependence of AFB(bb¯) on mbb¯. The results are compatible with expectations from the standard model. 1 data table Results of the $A_{\rm{FB}}$ measurements as functions of $b\bar{b}$ invariant mass. The integral values for each bin are shown. Measurement of the Production and Differential Cross Sections of $W^{+}W^{-}$ Bosons in Association with Jets in $p\bar{p}$ Collisions at $\sqrt{s}=1.96$ TeV The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D91 (2015) 111101, 2015. Inspire Record 1366177 We present a measurement of the $W$-boson-pair production cross section in $p\bar{p}$ collisions at 1.96 TeV center-of-mass energy and the first measurement of the differential cross section as a function of jet multiplicity and leading-jet energy. The $W^{+}W^{-}$ cross section is measured in the final state comprising two charged leptons and neutrinos, where either charged lepton can be an electron or a muon. Using data collected by the CDF experiment corresponding to $9.7~\rm{fb}^{-1}$ of integrated luminosity, a total of $3027$ collision events consistent with $W^{+}W^{-}$ production are observed with an estimated background contribution of $1790\pm190$ events. The measured total cross section is $\sigma(p\bar{p} \rightarrow W^{+}W^{-}) = 14.0 \pm 0.6~(\rm{stat})^{+1.2}_{-1.0}~(\rm{syst})\pm0.8~(\rm{lumi})$ pb, consistent with the standard model prediction. 1 data table Measurements and predictions of $\sigma(p\bar{p} \rightarrow W^{+}W^{-} + \mathrm{jets})$. Values are given inclusively and differentially as functions of jet multiplicity and jet-transverse energy. First measurement of the forward-backward asymmetry in bottom-quark pair production at high mass The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D92 (2015) 032006, 2015. Inspire Record 1364882 We measure the particle-level forward-backward production asymmetry in bb¯ pairs with masses (mbb¯) larger than 150 GeV/c2, using events with hadronic jets and employing jet charge to distinguish b from b¯. The measurement uses 9.5 fb-1 of pp¯ collisions at a center-of-mass energy of 1.96 TeV recorded by the CDF II detector. The asymmetry as a function of mbb¯ is consistent with zero, as well as with the predictions of the standard model. The measurement disfavors a simple model including an axigluon with a mass of 200 GeV/c2, whereas a model containing a heavier 345 GeV/c2 axigluon is not excluded. 1 data table Values of maximum a posteriori signal asymmetry as a function of $b\bar{b}$ mass. The error bars represent the 68% credible intervals. Measurement of the $B$ meson differential cross-sections in $p \bar{p}$ collisions at $S^{(1/2)}$ = 1.8-TeV using the exclusive decays $B^\pm \to J/\psi K^\pm$ and $B^0 \to J/\psi$ K0 The collaboration Conference Paper, 1994. Inspire Record 374778 4 data tables Charged B+ or B- meson cross section. Charged B+ or B- meson cross section. Neutral B0 or BBAR0 meson cross section. More… rivet Analysis Study of the energy dependence of the underlying event in proton-antiproton collisions The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D92 (2015) 092009, 2015. Inspire Record 1388868 We study charged particle production (pT>0.5 GeV/c, \|η\|<0.8) in proton-antiproton collisions at total center-of-mass energies s=300 GeV, 900 GeV, and 1.96 TeV. We use the direction of the charged particle with the largest transverse momentum in each event to define three regions of η-ϕ space: “toward”, “away”, and “transverse.” The average number and the average scalar pT sum of charged particles in the transverse region are sensitive to the modeling of the “underlying event.” The transverse region is divided into a MAX and MIN transverse region, which helps separate the “hard component” (initial and final-state radiation) from the “beam-beam remnant” and multiple parton interaction components of the scattering. The center-of-mass energy dependence of the various components of the event is studied in detail. The data presented here can be used to constrain and improve QCD Monte Carlo models, resulting in more precise predictions at the LHC energies of 13 and 14 TeV. 24 data tables Average charged particle multiplicity for charged particles with pT > 0.5 GeV and \|eta\| < 0.8 in the TransMAX region as defined by the leading charged particle, as a function of the transverse momentum of the leading charged-particle pTmax, at 1.96 TeV. Average charged particle multiplicity for charged particles with pT > 0.5 GeV and \|eta\| < 0.8 in the TransMIN region as defined by the leading charged particle, as a function of the transverse momentum of the leading charged-particle pTmax, at 1.96 TeV. Average charged particle multiplicity for charged particles with pT > 0.5 GeV and \|eta\| < 0.8 in the TransAVE region as defined by the leading charged particle, as a function of the transverse momentum of the leading charged-particle pTmax, at 1.96 TeV. More… rivet Analysis rivet Analysis Measurement of $d\sigma/dy$ of Drell-Yan $e^+e^-$ pairs in the $Z$ Mass Region from $p\bar{p}$ Collisions at $\sqrt{s}=1.96$ TeV The collaboration Aaltonen, Timo Antero ; Adelman, Jahred A. ; Alvarez Gonzalez, Barbara ; et al. Phys.Lett. B692 (2010) 232-239, 2010. Inspire Record 856131 We report on a CDF measurement of the total cross section and rapidity distribution, $d\sigma/dy$, for $q\bar{q}\to \gamma^{}/Z\to e^{+}e^{-}$ events in the $Z$ boson mass region ($66<M_{ee}<116$GeV/c$^2$) produced in $p\bar{p}$ collisions at $\sqrt{s}=1.96$TeV with 2.1fb$^{-1}$ of integrated luminosity. The measured cross section of $257\pm16$pb and $d\sigma/dy$ distribution are compared with Next-to-Leading-Order(NLO) and Next-to-Next-to-Leading-Order(NNLO) QCD theory predictions with CTEQ and MRST/MSTW parton distribution functions (PDFs). There is good agreement between the experimental total cross section and $d\sigma/dy$ measurements with theoretical calculations with the most recent NNLO PDFs. 2 data tables Total cross section integrated up to ABS(YRAP)=2.9. Rapiditiy distribution of E+ E- pairs in the mass range from 66 to 116 GeV. rivet Analysis Soft and hard interactions in $p\bar{p}$ collisions at $\sqrt{s}=$ 1800-GeV and 630-GeV The collaboration Acosta, D. ; Affolder, T. ; Akimoto, H. ; et al. Phys.Rev. D65 (2002) 072005, 2002. Inspire Record 567774 4 data tables Charged multiplicity at $\sqrt{s} = 630~\text{GeV}$, $\|\eta\| < 1$, $p_T > 0.4~\text{GeV}$. Charged multiplicity at $\sqrt{s} = 1800~\text{GeV}$, $\|\eta\| < 1$, $p_T > 0.4~\text{GeV}$. $\langle p_\perp \rangle$ vs. multiplicity at $\sqrt{s} = 630~\text{GeV}$, $\|\eta\| < 1$, $p_T > 0.4~\text{GeV}$. More… Evidence for top quark production in $\bar{p}p$ collisions at $\sqrt{s} = 1.8$ TeV The collaboration Abe, F. ; Albrow, Michael G. ; Amidei, Dante E. ; et al. Phys.Rev.Lett. 73 (1994) 225-231, 1994. Inspire Record 373362 We summarize a search for the top quark with the Collider Detector at Fermilab (CDF) in a sample of $\bar{p}p$ collisions at $\sqrt{s}$= 1.8 TeV with an integrated luminosity of 19.3pb$~{-1}$. We find 12 events consistent with either two $W$ bosons, or a $W$ boson and at least one $b$ jet. The probability that the measured yield is consistent with the background is 0.26\%. Though the statistics are too limited to establish firmly the existence of the top quark, a natural interpretation of the excess is that it is due to $t\bar{t}$ production. Under this assumption, constrained fits to individual events yield a top quark mass of $174 \pm 10~{+13}_{-12}$ GeV/c$~2$. The $t\bar{t}$ production cross section is measured to be $13.9~{+6.1}_{-4.8}$pb. (Submitted to Physical Review Letters on May 16, 1994). 1 data table No description provided. Measurement of the Cross Section for Prompt Isolated Diphoton Production Using the Full CDF Run II Data Sample The collaboration Aaltonen, T. ; Amerio, S. ; Amidei, D. ; et al. Phys.Rev.Lett. 110 (2013) 101801, 2013. Inspire Record 1207879 This Letter reports a measurement of the cross section for producing pairs of central prompt isolated photons in proton-antiproton collisions at a total energy of 1.96 TeV using data corresponding to 9.5/fb integrated luminosity collected with the CDF II detector at the Fermilab Tevatron. The measured differential cross section is compared to three calculations derived from the theory of strong interactions. These include a prediction based on a leading order matrix element calculation merged with parton shower, a next-to-leading order, and a next-to-next-to-leading order calculation. The first and last calculations reproduce most aspects of the data, thus showing the importance of higher-order contributions for understanding the theory of strong interaction and improving measurements of the Higgs boson and searches for new phenomena in diphoton final states. 23 data tables The measured differential cross sections for $M_{\gamma\gamma}$ , together with the predictions from the Sherpa and NNLO Monte Carlos. The measured differential cross sections for $M_{\gamma\gamma}$ when $P_T > M_{\gamma\gamma}$ , together with the predictions from the Sherpa and NNLO Monte Carlos. The measured differential cross sections for $M_{\gamma\gamma}$ when $P_T < M_{\gamma\gamma}$ , together with the predictions from the Sherpa and NNLO Monte Carlos. More… Charm production studies at CDF The collaboration Reisert, B. ; Lewis, J. ; Liu, T. ; et al. Nucl.Phys.Proc.Suppl. 170 (2007) 243-247, 2007. Inspire Record 766572 The upgraded Collider Detector at Fermilab (CDF II) has a high bandwidth available for track based triggers. This capability in conjunction with the unprecedented integrated luminosity in excess of 1 fb −1 enables detailed studies of charm hadron production. CDF is now releasing first measurements of the prompt charm meson pair cross sections, which give access to QCD mechanisms by which charm quarks are produced in proton anti-proton collisions. Recent results on the spin alignment of J/ψ and ψ(2S) as well as on the relative production of the χc1(P1) and χc2(1P) challenge our understanding of the fragmentation of charm quarks into charmonium states. 4 data tables The D0 D- pair cross section as a function of DELTA(PHI) in bins of trigger side D0 pT. The D+ D- pair cross section as a function of DELTA(PHI) in bins of trigger side D+ pT. The D0 D- pair cross section as a function of DELTA(PHI). More… Evidence for $s$-channel Single-Top-Quark Production in Events with one Charged Lepton and two Jets at CDF The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev.Lett. 112 (2014) 231804, 2014. Inspire Record 1279918 We report evidence for $s$-channel single-top-quark production in proton-antiproton collisions at center-of-mass energy $\sqrt{s}= 1.96 \mathrm{TeV}$ using a data set that corresponds to an integrated luminosity of $9.4 \mathrm{fb}^{-1}$ collected by the Collider Detector at Fermilab. We select events consistent with the $s$-channel process including two jets and one leptonically decaying $W$ boson. The observed significance is $3.8$ standard deviations with respect to the background-only prediction. Assuming a top-quark mass of $172.5 \mathrm{GeV}/c^2$, we measure the $s$-channel cross section to be $1.41^{+0.44}_{-0.42} \mathrm{pb}$. 1 data table The s-channel cross section, measured assuming a top-quark mass of 172.5 GeV. The charge conjugate reaction is also included in the analysis. Indirect measurement of $\sin^2 \theta_W$ (or $M_W$) using $\mu^+\mu^-$ pairs from $\gamma^/Z$ bosons produced in $p\bar{p}$ collisions at a center-of-momentum energy of 1.96 TeV The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D89 (2014) 072005, 2014. Inspire Record 1280719 Drell-Yan lepton pairs are produced in the process $p\bar{p} \rightarrow \mu^+\mu^- + X$ through an intermediate $\gamma^/Z$ boson. The forward-backward asymmetry in the polar-angle distribution of the $\mu^-$ as a function of the invariant mass of the $\mu^+\mu^-$ pair is used to obtain the effective leptonic determination $\sin^2 \theta^{lept}_{eff}$ of the electroweak-mixing parameter $\sin^2 \theta_W$, from which the value of $\sin^2 \theta_W$ is derived assuming the standard model. The measurement sample, recorded by the Collider Detector at Fermilab (CDF), corresponds to 9.2 fb-1 of integrated luminosity from $p\bar{p}$ collisions at a center-of-momentum energy of 1.96 TeV, and is the full CDF Run II data set. The value of $\sin^2 \theta^{lept}_{eff}$ is found to be 0.2315 +- 0.0010, where statistical and systematic uncertainties are combined in quadrature. When interpreted within the context of the standard model using the on-shell renormalization scheme, where $\sin^2 \theta_W = 1 - M_W^2/M_Z^2$, the measurement yields $\sin^2 \theta_W$ = 0.2233 +- 0.0009, or equivalently a W-boson mass of 80.365 +- 0.047 GeV/c^2. The value of the W-boson mass is in agreement with previous determinations in electron-positron collisions and at the Tevatron collider. 4 data tables The measured value of SIN2(THETAEFF(LEPTON)). The measured value of SIN2(THETA(W)). The measured value of M(W). More… Search for $s$-Channel Single-Top-Quark Production in Events with Missing Energy Plus Jets in $p\bar p$ Collisions at $\sqrt s=$1.96 TeV The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev.Lett. 112 (2014) 231805, 2014. Inspire Record 1281537 The first search for single top quark production from the exchange of an $s$-channel virtual $W$ boson using events with an imbalance in the total transverse momentum, $b$-tagged jets, and no identified leptons is presented. The full data set collected by the Collider Detector at Fermilab, corresponding to an integrated luminosity of 9.45 fb$^{-1}$ from Fermilab Tevatron proton-antiproton collisions at a center of mass energy of 1.96 TeV, is used. Assuming the electroweak production of top quarks of mass 172.5 GeV/$c^2$ in the $s$-channel, a cross section of $1.12_{-0.57}^{+0.61}$ (stat+syst) pb, with a significance of 1.9 standard deviations, is measured. This measurement is combined with a previous result obtained from events with an imbalance in total transverse momentum, $b$-tagged jets, and exactly one identified lepton, yielding a cross section of $1.36_{-0.32}^{+0.37}$ (stat+syst) pb, with a significance of 4.2 standard deviations. 2 data tables The s-channel single top quark cross section measured assuming top quarks of mass 172.5 GeV. The measurement uses a sample of events with large missing transverse energy, two or three jets of which one or more are b-tagged and no detected electron or muon candidates. The combined s-channel single top quark cross section measurement assuming top quarks of mass 172.5 GeV. The measurement uses two samples of events. The first sample includes events with large missing transverse energy, two or three jets of which one or more are b-tagged and no detected electron or muon candidates. The second sample includes events with large missing transverse energy, one isolated muon or electron and two jets, at least one of which is b-tagged. Observation of s-channel production of single top quarks at the Tevatron The & collaborations Aaltonen, Timo Antero ; Abazov, Victor Mukhamedovich ; Abbott, Braden Keim ; et al. Phys.Rev.Lett. 112 (2014) 231803, 2014. Inspire Record 1282028 We report the first observation of single-top-quark production in the s channel through the combination of the CDF and D0 measurements of the cross section in proton-antiproton collisions at a center-of-mass energy of 1.96 TeV. The data correspond to total integrated luminosities of up to 9.7 fb-1 per experiment. The measured cross section is $\sigma_s = 1.29^{+0.26}_{-0.24}$ pb. The probability of observing a statistical fluctuation of the background to a cross section of the observed size or larger is $1.8 \times 10^{-10}$, corresponding to a significance of 6.3 standard deviations for the presence of an s-channel contribution to the production of single-top quarks. 1 data table The measured cross section of single-top-quark production in the s channel. Study of Top-Quark Production and Decays involving a Tau Lepton at CDF and Limits on a Charged-Higgs Boson Contribution The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D89 (2014) 091101, 2014. Inspire Record 1282899 We present an analysis of top-antitop quark production and decay into a tau lepton, tau neutrino, and bottom quark using data from $9 {\rm fb}^{-1}$ of integrated luminosity at the Collider Detector at Fermilab. Dilepton events, where one lepton is an energetic electron or muon and the other a hadronically-decaying tau lepton, originating from proton-antiproton collisions at $\sqrt{s} = 1.96 TeV$ are used. A top-antitop quark production cross section of $8.1 \pm 2.1 {\rm pb}$ is measured, assuming standard-model top-quark decays. By separately identifying for the first time the single-tau and the ditau components, we measure the branching fraction of the top quark into tau lepton, tau neutrino, and bottom quark to be $(9.6 \pm 2.8) %$. The branching fraction of top-quark decays into a charged Higgs boson and a bottom quark, which would imply violation of lepton universality, is limited to be less than $5.9%$ at $95%$ confidence level. 3 data tables The top-antitop quark production cross section measured assuming standard-model top-quark decays, TOP --> W BOTTOM. The branching fraction of the top quark into a tau lepton, a tau neutrino and a bottom quark. The ratio of leptonic top branching ratios, 2 BR(TOP --> TAU NUTAU BOTTOM) / ( BR(TOP --> E NUE BOTTOM) + BR(TOP --> MU NUMU BOTTOM) ). Invariant-mass distribution of jet pairs produced in association with a $W$ boson in $p \bar{p}$ collisions at $\sqrt{s}=1.96$ TeV using the full CDF Run II data set The collaboration Aaltonen, T. ; Amerio, S. ; Amidei, D. ; et al. Phys.Rev. D89 (2014) 092001, 2014. Inspire Record 1282906 We report on a study of the dijet invariant-mass distribution in events with one identified lepton, a significant imbalance in the total event transverse momentum, and two jets. This distribution is sensitive to the possible production of a new particle in association with a $W$ boson, where the boson decays leptonically. We use the full data set of proton-antiproton collisions at 1.96 TeV center-of-mass energy collected by the Collider Detector at the Fermilab Tevatron and corresponding to an integrated luminosity of 8.9 fb$^{-1}$. The data are found to be consistent with standard-model expectations, and a 95$\%$ confidence level upper limit is set on the cross section for a $W$ boson produced in association with a new particle decaying into two jets. 2 data tables The extracted cross section assuming that the new contribution (the excess over the expected background) has the same acceptance as that for a 140 GeV Higgs boson produced in association with a W boson. The extracted cross section measured with a restriction on DELTAR(JET1 JET2) and assuming that the new contribution (the excess over the expected background) has the same acceptance as that for a 140 GeV Higgs boson produced in association with a W boson. Measurement of the ZZ production cross section using the full CDF II data set The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev. D89 (2014) 112001, 2014. Inspire Record 1285230 We present a measurement of the ZZ boson-pair production cross section in 1.96 TeV center-of-mass energy ppbar collisions. We reconstruct final states incorporating four charged leptons or two charged leptons and two neutrinos from the full data set collected by the Collider Detector experiment at the Fermilab Tevatron, corresponding to 9.7 fb-1 of integrated luminosity. Combining the results obtained from each final state, we measure a cross section of 1.04(+0.32)(-0.25) pb, in good agreement with the standard model prediction at next-to-leading order in the strong-interaction coupling. 3 data tables The measured cross section for the process P PBAR --> Z0 Z0 --> LEPTON+ LEPTON- LEPTON+ LEPTON-. The measured cross section for the process PBAR P --> Z0 Z0 --> LEPTON+ LEPTON- NU NUBAR. The Z0 Z0 production cross section in PBAR P collisions obtained from the combination of the cross section measurements from the LEPTON+ LEPTON- LEPTON+ LEPTON- and LEPTON+ LEPTON- NU NUBAR signal samples. Measurement of the inclusive leptonic asymmetry in top-quark pairs that decay to two charged leptons at CDF The collaboration Aaltonen, Timo Antero ; Amerio, Silvia ; Amidei, Dante E ; et al. Phys.Rev.Lett. 113 (2014) 042001, 2014. Inspire Record 1290358 3 data tables The leptonic forward-backward asymmetry. The leptonic pair forward-backward asymmetry. The leptonic forward-backward asymmetry calculated as the combination of the current asymmetry measurement and a previous CDF measurement. Production of $K_S^0, K^{\pm}(892)$ and $\phi^0(1020)$ in minimum bias events and $K_S^0$ and $\Lambda^0$ in jets in $p\bar p$ collisions at $\sqrt s=1.96 TeV$ The collaboration Aaltonen, T. ; Albrow, M. ; Alvarez Gonzalez, B. ; et al. Phys.Rev. D88 (2013) 092005, 2013. Inspire Record 1247975 We report measurements of the inclusive transverse momentum pT distribution of centrally produced kshort, kstar(892), and phi(1020) mesons up to pT = 10 GeV/c in minimum-bias events, and kshort and lambda particles up to pT = 20 GeV/c in jets with transverse energy between 25 GeV and 160 GeV in pbar p collisions. The data were taken with the CDF II detector at the Fermilab Tevatron at sqrt(s) = 1.96 TeV. We find that as pT increases, the pT slopes of the three mesons (kshort, kstar, and phi) are similar, and the ratio of lambda to kshort as a function of pT in minimum-bias events becomes similar to the fairly constant ratio in jets at pT ~ 5 GeV/c. This suggests that the particles with pT >~ 5 GeV/c in minimum-bias events are from soft jets, and that the pT slope of particles in jets is insensitive to light quark flavor (u, d, or s) and to the number of valence quarks. We also find that for pT <~ 4 GeV relatively more lambda baryons are produced in minimum-bias events than in jets. 5 data tables The inclusive invariant differential cross section for K0S production. The inclusive invariant differential cross section for PHI production. The inclusive invariant differential cross section for K(892)+- production. More… Measurement of the Differential Cross Section $d{\sigma}/d(\cos {\theta}t)$ for Top-Quark Pair Production in $p-\bar{p}$ Collisions at $\sqrt{s} = 1.96$ TeV The collaboration Aaltonen, T. ; Amerio, S. ; Amidei, D. ; et al. Phys.Rev.Lett. 111 (2013) 182002, 2013. Inspire Record 1238100 We report a measurement of the differential cross section, d{\sigma}/d(cos {\theta}t), for top-quark-pair production as a function of the top-quark production angle in proton-antiproton collisions at sqrt{s} = 1.96 TeV. This measurement is performed using data collected with the CDF II detector at the Tevatron, corresponding to an integrated luminosity of 9.4/fb. We employ the Legendre polynomials to characterize the shape of the differential cross section at the parton level. The observed Legendre coefficients are in good agreement with the prediction of the next-to-leading-order standard-model calculation, with the exception of an excess linear-term coefficient, a1 = 0.40 +- 0.12, compared to the standard-model prediction of a1 = 0.15^{+0.07}_{-0.03}. 1 data table The parton-level Legendre moments for the measured angular distribution of the momentum direction of the t-quark from the momentum direction of the incoming proton. A Study of the associated production of photons and b-quark jets in p p-bar collisions at s**(1/2) = 1.96-TeV The collaboration Aaltonen, T. ; Adelman, J. ; Alvarez Gonzalez, B. ; et al. Phys.Rev. D81 (2010) 052006, 2010. Inspire Record 840503 The cross section for photon production in association with at least one jet containing a $b$-quark hadron has been measured in proton antiproton collisions at $\sqrt{s}=1.96$ TeV. The analysis uses a data sample corresponding to an integrated luminosity of 340 pb$^{-1}$ collected with the CDF II detector. Both the differential cross section as a function of photon transverse energy $E_T^{\gamma}$, $d \sigma$($p \overline{p} \to \gamma + \geq 1 b$-jet)/$d E_T^{\gamma}$ and the total cross section $\sigma$($p \overline{p} \to \gamma + \geq 1 b$-jet/ $E_T^{\gamma}> 20$ GeV) are measured. Comparisons to a next-to-leading order prediction of the process are presented. 2 data tables b + photon cross section as a function of photon ET. b + photon total cross section for photon ET > 20 GeV. Production of psi(2S) Mesons in p anti-p Collisions at 1.96-TeV The collaboration Aaltonen, T. ; Adelman, Jahred A. ; Akimoto, T. ; et al. Phys.Rev. D80 (2009) 031103, 2009. Inspire Record 820328 We have measured the differential cross section for the inclusive production of psi(2S) mesons decaying to mu^{+} mu^{-1} that were produced in prompt or B-decay processes from ppbar collisions at 1.96 TeV. These measurements have been made using a data set from an integrated luminosity of 1.1 fb^{-1} collected by the CDF II detector at Fermilab. For events with transverse momentum p_{T} (psi(2S)) > 2 GeV/c and rapidity \|y(psi(2S))\| < 0.6 we measure the integrated inclusive cross section sigma(ppbar -> psi(2S)X) Br(psi(2S) -> mu^{+} mu^{-}) to be 3.29 +- 0.04(stat.) +- 0.32(syst.) nb. 2 data tables The differential cross section times the dimuon branching fraction as a function of pT. The integrated inclusive differential cross section for PSI(3685). Measurement of the cross section for direct-photon production in association with a heavy quark in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV The collaboration Aaltonen, T. ; Amerio, S. ; Amidei, D. ; et al. Phys.Rev.Lett. 111 (2013) 042003, 2013. Inspire Record 1225278 We report on a measurement of the cross section for direct-photon production in association with a heavy quark using the full data set of $\sqrt{s}=1.96$ TeV proton-antiproton collisions corresponding to 9.1 fb$^{-1}$ of integrated luminosity collected by the CDF II detector at the Fermilab Tevatron. The measurements are performed as a function of the photon transverse momentum, covering photon transverse momentum between 30 and 300 GeV, photon rapidities $\|y^{\gamma}\|<1.0$, heavy-quark-jet transverse momentum $p_{T}^{jet}>20$ GeV, and jet rapidities $\|y^{jet}\|<1.5$. The results are compared with several theoretical predictions. 2 data tables The cross section for GAMMA BQUARK X production as a function of the transverse energy of the GAMMA. The cross section for GAMMA CQUARK X production as a function of the transverse energy of the GAMMA.	{"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": 0.9903667569160461, "perplexity": 3559.1260908316676}, "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-2020-29/segments/1593657167808.91/warc/CC-MAIN-20200715101742-20200715131742-00006.warc.gz"}
https://www.nias.res.in/publication/co-existence-periodic-bursts-and-death-cycles-population-dynamics-systems	# Co-existence of periodic bursts and death of cycles in a population dynamics systems Journal Articles ### Source: Chaos, Volume 26, p.093111-15 (2016) ### URL: http://scitation.aip.org/content/aip/journal/chaos/26/9/10.1063/1.4962633 ### Abstract: <p>We study the dynamics of a discrete-time tritrophic model which mimics the observed periodicity in the population cycles of the larch budmoth insect which causes widespread defoliation of larch forests at high altitudes periodically. Our model employs q-deformation of numbers to model the system comprising the budmoth, one or more parasitoid species, and larch trees. Incorporating climate parameters, we introduce additional parasitoid species and show that their introduction increases the periodicity of the budmoth cycles as observed experimentally. The presence of these additional species also produces other interesting dynamical effects such as periodic bursting and oscillation quenching via oscillation death, amplitude death, and partial oscillation death which are also seen in nature. We suggest that introducing additional parasitoid species provides an alternative explanation for the collapse of the nine year budmoth outbreak cycles observed in the Swiss Alps after 1981. A detailed exploration of the parameter space of the system is performed with movies of bifurcation diagrams which enable variation of two parameters at a time. Limit cycles emerge through a Neimark–Sacker bifurcation with respect to all parameters in all the five and higher dimensional models we have studied.</p> Attachment:	{"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": 0.8517917394638062, "perplexity": 2337.826072266887}, "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/1610704803308.89/warc/CC-MAIN-20210126170854-20210126200854-00521.warc.gz"}
https://www.nag.co.uk/numeric/fl/nagdoc_fl26/html/g13/g13intro.html	G13 Chapter Contents NAG Library Manual # NAG Library Chapter IntroductionG13 – Time Series Analysis ## 1 Scope of the Chapter This chapter provides facilities for investigating and modelling the statistical structure of series of observations collected at points in time. The models may then be used to forecast the series. The chapter covers the following models and approaches. 1 Univariate time series analysis, including autocorrelation functions and autoregressive moving average (ARMA) models. 2 Univariate spectral analysis. 3 Transfer function (multi-input) modelling, in which one time series is dependent on other time series. 4 Bivariate spectral methods including coherency, gain and input response functions. 5 Vector ARMA models for multivariate time series. 6 Kalman filter models (linear and nonlinear). 7 GARCH models for volatility. 8 Inhomogeneous Time Series. ## 2 Background to the Problems ### 2.1 Univariate Analysis Let the given time series be ${x}_{1},{x}_{2},\dots ,{x}_{n}$, where $n$ is its length. The structure which is intended to be investigated, and which may be most evident to the eye in a graph of the series, can be broadly described as: (a) trends, linear or possibly higher-order polynomial; (b) seasonal patterns, associated with fixed integer seasonal periods. The presence of such seasonality and the period will normally be known a priori. The pattern may be fixed, or slowly varying from one season to another; (c) cycles or waves of stable amplitude and period $p$ (from peak to peak). The period is not necessarily integer, the corresponding absolute frequency (cycles/time unit) being $f=1/p$ and angular frequency $\omega =2\pi f$. The cycle may be of pure sinusoidal form like $\mathrm{sin}\left(\omega t\right)$, or the presence of higher harmonic terms may be indicated, e.g., by asymmetry in the wave form; (d) quasi-cycles, i.e., waves of fluctuating period and amplitude; and (e) irregular statistical fluctuations and swings about the overall mean or trend. Trends, seasonal patterns, and cycles might be regarded as deterministic components following fixed mathematical equations, and the quasi-cycles and other statistical fluctuations as stochastic and describable by short-term correlation structure. For a finite dataset it is not always easy to discriminate between these two types, and a common description using the class of autoregressive integrated moving-average (ARIMA) models is now widely used. The form of these models is that of difference equations (or recurrence relations) relating present and past values of the series. You are referred to Box and Jenkins (1976) for a thorough account of these models and how to use them. We follow their notation and outline the recommended steps in ARIMA model building for which routines are available. #### 2.1.1 Transformations If the variance of the observations in the series is not constant across the range of observations it may be useful to apply a variance-stabilizing transformation to the series. A common situation is for the variance to increase with the magnitude of the observations and in this case typical transformations used are the log or square root transformation. A range-mean plot or standard deviation-mean plot provides a quick and easy way of detecting non-constant variance and of choosing, if required, a suitable transformation. These are plots of either the range or standard deviation of successive groups of observations against their means. #### 2.1.2 Differencing operations These may be used to simplify the structure of a time series. First-order differencing, i.e., forming the new series $∇xt=xt-xt-1$ will remove a linear trend. First-order seasonal differencing $∇sxt=xt-xt-s$ eliminates a fixed seasonal pattern. These operations reflect the fact that it is often appropriate to model a time series in terms of changes from one value to another. Differencing is also therefore appropriate when the series has something of the nature of a random walk, which is by definition the accumulation of independent changes. Differencing may be applied repeatedly to a series, giving $wt=∇d∇sDxt$ where $d$ and $D$ are the orders of differencing. The derived series ${w}_{t}$ will be shorter, of length $N=n-d-s×D$, and extend for $t=1+d+s×D,\dots ,n$. #### 2.1.3 Sample autocorrelations Given that a series has (possibly as a result of simplifying by differencing operations) a homogeneous appearance throughout its length, fluctuating with approximately constant variance about an overall mean level, it is appropriate to assume that its statistical properties are stationary. For most purposes the correlations ${\rho }_{k}$ between terms ${x}_{t},{x}_{t+k}$ or ${w}_{t},{w}_{t+k}$ separated by lag $k$ give an adequate description of the statistical structure and are estimated by the sample autocorrelation function (ACF) ${r}_{\mathit{k}}$, for $\mathit{k}=1,2,\dots$. As described by Box and Jenkins (1976), these may be used to indicate which particular ARIMA model may be appropriate. #### 2.1.4 Partial autocorrelations The information in the autocorrelations, ${\rho }_{k}$, may be presented in a different light by deriving from them the coefficients of the partial autocorrelation function (PACF) ${\varphi }_{\mathit{k},\mathit{k}}$, for $\mathit{k}=1,2,\dots$. ${\varphi }_{k,k}$ which measures the correlation between ${x}_{t}$ and ${x}_{t+k}$ conditional upon the intermediate values ${x}_{t+1},{x}_{t+2},\dots ,{x}_{t+k-1}$. The corresponding sample values ${\stackrel{^}{\varphi }}_{k,k}$ give further assistance in the selection of ARIMA models. Both autocorrelation function (ACF) and PACF may be rapidly computed, particularly in comparison with the time taken to estimate ARIMA models. #### 2.1.5 Finite lag predictor coefficients and error variances The partial autocorrelation coefficient ${\varphi }_{k,k}$ is determined as the final parameter in the minimum variance predictor of ${x}_{t}$ in terms of ${x}_{t-1},{x}_{t-2},\dots ,{x}_{t-k}$, $xt=ϕk,1xt-1+ϕk,2xt-2+⋯+ϕk,kxt-k+ek,t$ where ${e}_{k,t}$ is the prediction error, and the first subscript $k$ of ${\varphi }_{k,i}$ and ${e}_{k,t}$ emphasizes the fact that the parameters will alter as $k$ increases. Moderately good estimates ${\stackrel{^}{\varphi }}_{k,i}$ of ${\varphi }_{k,i}$ are obtained from the sample autocorrelation function (ACF), and after calculating the partial autocorrelation function (PACF) up to lag $L$, the successive values ${v}_{1},{v}_{2},\dots ,{v}_{L}$ of the prediction error variance estimates, ${v}_{k}=\mathrm{var}\left({e}_{k,t}\right)$, are available, together with the final values of the coefficients ${\stackrel{^}{\varphi }}_{k,1},{\stackrel{^}{\varphi }}_{k,2},\dots ,{\stackrel{^}{\varphi }}_{k,L}$. If ${x}_{t}$ has nonzero mean, $\stackrel{-}{x}$, it is adequate to use ${x}_{t}-\stackrel{-}{x}$ in place of ${x}_{t}$ in the prediction equation. Although Box and Jenkins (1976) do not place great emphasis on these prediction coefficients, their use is advocated for example by Akaike (1971), who recommends selecting an optimal order of the predictor as the lag for which the final prediction error (FPE) criterion $\left(1+k/n\right){\left(1-k/n\right)}^{-1}{v}_{k}$ is a minimum. #### 2.1.6 ARIMA models The correlation structure in stationary time series may often be represented by a model with a small number of parameters belonging to the autoregressive moving-average (ARMA) class. If the stationary series ${w}_{t}$ has been derived by differencing from the original series ${x}_{t}$, then ${x}_{t}$ is said to follow an ARIMA model. Taking ${w}_{t}={\nabla }^{d}{x}_{t}$, the (non-seasonal) ARIMA $\left(p,d,q\right)$ model with $p$ autoregressive parameters ${\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{p}$ and $q$ moving-average parameters ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{q}$, represents the structure of ${w}_{t}$ by the equation $wt=ϕ1wt-1+⋯+ϕpwt-p+at-θ1at-1-⋯-θqat-q,$ (1) where ${a}_{t}$ is an uncorrelated series (white noise) with mean $0$ and constant variance ${\sigma }_{a}^{2}$. If ${w}_{t}$ has a nonzero mean $c$, then this is allowed for by replacing ${w}_{t},{w}_{t-1},\dots \text{}$ by ${w}_{t}-c,{w}_{t-1}-c,\dots \text{}$ in the model. Although $c$ is often estimated by the sample mean of ${w}_{t}$ this is not always optimal. A series generated by this model will only be stationary provided restrictions are placed on ${\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{p}$ to avoid unstable growth of ${w}_{t}$. These are called stationarity constraints. The series ${a}_{t}$ may also be usefully interpreted as the linear innovations in ${x}_{t}$ (and in ${w}_{t}$), i.e., the error if ${x}_{t}$ were to be predicted using the information in all past values ${x}_{t-1},{x}_{t-2},\dots \text{}$, provided also that ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{q}$ satisfy invertibility constraints. This allows the series ${a}_{t}$ to be regenerated by rewriting the model equation as $at=wt-ϕ1wt-1-⋯-ϕpwt-p+θ1at-1+⋯+θqat-q.$ (2) For a series with short-term correlation only, i.e., ${r}_{k}$ is not significant beyond some low lag $q$ (see Box and Jenkins (1976) for the statistical test), then the pure moving-average model $\text{MA}\left(q\right)$ is appropriate, with no autoregressive parameters, i.e., $p=0$. Autoregressive parameters are appropriate when the autocorrelation function (ACF) pattern decays geometrically, or with a damped sinusoidal pattern which is associated with quasi-periodic behaviour in the series. If the sample partial autocorrelation function (PACF) ${\stackrel{^}{\varphi }}_{k,k}$ is significant only up to some low lag $p$, then a pure autoregressive model $\text{AR}\left(p\right)$ is appropriate, with $q=0$. Otherwise moving-average terms will need to be introduced, as well as autoregressive terms. The seasonal ARIMA $\left(p,d,q,P,D,Q,s\right)$ model allows for correlation at lags which are multiples of the seasonal period $s$. Taking ${w}_{t}={\nabla }^{d}{\nabla }_{s}^{D}{x}_{t}$, the series is represented in a two-stage manner via an intermediate series ${e}_{t}$: $wt=Φ1wt-s+⋯+ΦPwt-s×P+et-Θ1et-s-⋯-ΘQet-s×Q$ (3) $et=ϕ1et-1+⋯+ϕpet-p+at-θ1at-1-⋯-θqat-q$ (4) where ${\Phi }_{i}$, ${\Theta }_{i}$ are the seasonal parameters and $P$ and $Q$ are the corresponding orders. Again, ${w}_{t}$ may be replaced by ${w}_{t}-c$. #### 2.1.7 ARIMA model estimation In theory, the parameters of an ARIMA model are determined by a sufficient number of autocorrelations ${\rho }_{1},{\rho }_{2},\dots \text{}$. Using the sample values ${r}_{1},{r}_{2},\dots \text{}$ in their place it is usually (but not always) possible to solve for the corresponding ARIMA parameters. These are rapidly computed but are not fully efficient estimates, particularly if moving-average parameters are present. They do provide useful preliminary values for an efficient but relatively slow iterative method of estimation. This is based on the least squares principle by which parameters are chosen to minimize the sum of squares of the innovations ${a}_{t}$, which are regenerated from the data using (2), or the reverse of (3) and (4) in the case of seasonal models. Lack of knowledge of terms on the right-hand side of (2), when $t=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(p,q\right)$, is overcome by introducing $q$ unknown series values ${w}_{0},{w}_{1},\dots ,{w}_{q-1}$ which are estimated as nuisance parameters, and using correction for transient errors due to the autoregressive terms. If the data ${w}_{1},{w}_{2},\dots ,{w}_{N}=w$ is viewed as a single sample from a multivariate Normal density whose covariance matrix $V$ is a function of the ARIMA model parameters, then the exact likelihood of the parameters is $-12logV-12wTV-1w.$ The least squares criterion as outlined above is equivalent to using the quadratic form $QF=wTV-1w$ as an objective function to be minimized. Neglecting the term $-\frac{1}{2}\mathrm{log}\left\|V\right\|$ yields estimates which differ very little from the exact likelihood except in small samples, or in seasonal models with a small number of whole seasons contained in the data. In these cases bias in moving-average parameters may cause them to stick at the boundary of their constraint region, resulting in failure of the estimation method. Approximate standard errors of the parameter estimates and the correlations between them are available after estimation. The model residuals, ${\stackrel{^}{a}}_{t}$, are the innovations resulting from the estimation and are usually examined for the presence of autocorrelation as a check on the adequacy of the model. #### 2.1.8 ARIMA model forecasting An ARIMA model is particularly suited to extrapolation of a time series. The model equations are simply used for $t=n+1,n+2,\dots \text{}$ replacing the unknown future values of ${a}_{t}$ by zero. This produces future values of ${w}_{t}$, and if differencing has been used this process is reversed (the so-called integration part of ARIMA models) to construct future values of ${x}_{t}$. Forecast error limits are easily deduced. This process requires knowledge only of the model orders and parameters together with a limited set of the terms ${a}_{t-i},{e}_{t-i},{w}_{t-i},{x}_{t-i}$ which appear on the right-hand side of the models (3) and (4) (and the differencing equations) when $t=n$. It does not require knowledge of the whole series. We call this the state set. It is conveniently constituted after model estimation. Moreover, if new observations ${x}_{n+1},{x}_{n+2},\dots \text{}$ come to hand, then the model equations can easily be used to update the state set before constructing forecasts from the end of the new observations. This is particularly useful when forecasts are constructed on a regular basis. The new innovations ${a}_{n+1},{a}_{n+2},\dots \text{}$ may be compared with the residual standard deviation, ${\sigma }_{a}$, of the model used for forecasting, as a check that the model is continuing to forecast adequately. #### 2.1.9 Exponential smoothing Exponential smoothing is a relatively simple method of short term forecasting for a time series. A variety of different smoothing methods are possible, including; single exponential, Brown's double exponential, linear Holt (also called double exponential smoothing in some references), additive Holt–Winters and multiplicative Holt–Winters. The choice of smoothing method used depends on the characteristics of the time series. If the mean of the series is only slowly changing then single exponential smoothing may be suitable. If there is a trend in the time series, which itself may be slowly changing, then linear Holt smoothing may be suitable. If there is a seasonal component to the time series, e.g., daily or monthly data, then one of the two Holt–Winters methods may be suitable. For a time series ${y}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,n$, the five smoothing functions are defined by the following: • Single Exponential Smoothing $mt = α yt + 1-α mt-1 y^t+f =mt var y^t+f = varεt 1+ f-1 α2$ • Brown Double Exponential Smoothing $mt = α yt + 1-α mt-1 rt = α mt - mt-1 + 1-α rt-1 y^ t+f = mt + f-1 + 1 / α rt var y^ t+f = varεt 1+ ∑ i=1 f-1 2α+ i-1 α2 2$ • Linear Holt Smoothing $mt = α yt + 1-α mt-1 + ϕ rt-1 rt = γ mt - mt-1 + 1-γ ϕ rt-1 y^ t+f = mt + ∑ i=1 f ϕi rt var y^ t+f = var εt 1+ ∑ i=1 f-1 α + α γ ϕ ϕi-1 ϕ-1 2$ • Multiplicative Holt–Winters Smoothing $mt = α yt / s t-p + 1-α m t-1 +ϕ r t-1 rt = γ mt - m t-1 + 1-γ ϕ r t-1 st = β yt / mt + 1-β s t-p y^ t+f = mt + ∑ i=1 f ϕi rt × s t-p var y^ t+f = var εt ∑ i=0 ∞ ∑ j=0 p-1 ψ j+ip s t+f s t+f-j 2$ and $\psi$ is defined as in the additive Holt–Winters smoothing, where ${m}_{t}$ is the mean, ${r}_{t}$ is the trend and ${s}_{t}$ is the seasonal component at time $t$ with $p$ being the seasonal order. The $f$-step ahead forecasts are given by ${\stackrel{^}{y}}_{t+f}$ and their variances by $\mathrm{var}\left({\stackrel{^}{y}}_{t+f}\right)$. The term $\mathrm{var}\left({\epsilon }_{t}\right)$ is estimated as the mean deviation. The parameters, $\alpha$, $\beta$ and $\gamma$ control the amount of smoothing. The nearer these parameters are to one, the greater the emphasis on the current data point. Generally these parameters take values in the range $0.1$ to $0.3$. The linear Holt and two Holt–Winters smoothers include an additional parameter, $\varphi$, which acts as a trend dampener. For $0.0<\varphi <1.0$ the trend is dampened and for $\varphi >1.0$ the forecast function has an exponential trend, $\varphi =0.0$ removes the trend term from the forecast function and $\varphi =1.0$ does not dampen the trend. For all methods, values for $\alpha$, $\beta$, $\gamma$ and $\psi$ can be chosen by trying different values and then visually comparing the results by plotting the fitted values along side the original data. Alternatively, for single exponential smoothing a suitable value for $\alpha$ can be obtained by fitting an $\mathrm{ARIMA}\left(0,1,1\right)$ model. For Brown's double exponential smoothing and linear Holt smoothing with no dampening, (i.e., $\varphi =1.0$), suitable values for $\alpha$ and, in the case of linear Holt smoothing, $\gamma$ can be obtained by fitting an $\mathrm{ARIMA}\left(0,2,2\right)$ model. Similarly, the linear Holt method, with $\varphi \ne 1.0$, can be expressed as an $\mathrm{ARIMA}\left(1,2,2\right)$ model and the additive Holt–Winters, with no dampening, ($\varphi =1.0$), can be expressed as a seasonal ARIMA model with order $p$ of the form $\mathrm{ARIMA}\left(0,1,p+1\right)\left(0,1,0\right)$. There is no similar procedure for obtaining parameter values for the multiplicative Holt–Winters method, or the additive Holt–Winters method with $\varphi \ne 1.0$. In these cases parameters could be selected by minimizing a measure of fit using nonlinear optimization. #### 2.1.10 Change point analysis Given a time series ${y}_{1:n}=\left\{{y}_{j}:j=1,2,\dots ,n\right\}$, a change point $\tau$ is a place or time point such that segment of the series up to $\tau$, ${y}_{1:\tau }$, follows one distribution and the segment after $\tau$, ${y}_{\tau +1:n}$, follows a different distribution. This idea can be extended to $m$ change points, in which case $\tau =\left\{{\tau }_{i}:i=1,2,\dots ,m\right\}$ becomes a vector of ordered (strictly monotonic increasing) change points with $1\le {\tau }_{i}\le n$ and ${\tau }_{m}=n$. The $i$th segment therefore consists of ${y}_{{\tau }_{i-1}+1:{\tau }_{i}}$ where, for ease of notation, we define ${\tau }_{0}=0$. A change point problem is therefore twofold: estimating $m$ the number of change points (and hence the number of segments) and estimating $\tau$ the location of those change points. Given a cost function, $C\left({y}_{{\tau }_{i-1}+1:{\tau }_{i}}\right)$ one formulation of the change point problem can be expressed as the solution to: $minimize m,τ ∑ i=1 m Cyτi-1+1:τi + β$ (5) where $\beta$ is a penalty term used to control the number of change points. Two methods of solving equation (5) are available: the PELT algorithm and binary segmentation. The Pruned Exact Linear Time (PELT) algorithm of Killick et al. (2012) is a tree based method which is guaranteed to return the optimal solution to (5) as long as there exists a constant $K$ such that $C y u+1 : v + C y v+1 : w + K ≤ C y u+1 : w$ (6) for all $u. Unlike PELT, binary segmentation is an iterative method that only results in an approximate solution to (5). A description of the binary segmentation algorithm can be found in Section 3 in G13NDF and G13NEF. ### 2.2 Univariate Spectral Analysis In describing a time series using spectral analysis the fundamental components are taken to be sinusoidal waves of the form $R\mathrm{cos}\left(\omega t+\varphi \right)$, which for a given angular frequency $\omega$, $0\le \omega \le \pi$, is specified by its amplitude $R>0$ and phase $\varphi$, $0\le \varphi <2\pi$. Thus in a time series of $n$ observations it is not possible to distinguish more than $n/2$ independent sinusoidal components. The frequency range $0\le \omega \le \pi$ is limited to the shortest wavelength of two sampling units because any wave of higher frequency is indistinguishable upon sampling (or is aliased with) a wave within this range. Spectral analysis follows the idea that for a series made up of a finite number of sine waves the amplitude of any component at frequency $\omega$ is given to order $1/n$ by $R2= 1n2 ∑ t=1 n xt eiωt 2 .$ #### 2.2.1 The sample spectrum For a series ${x}_{1},{x}_{2},\dots ,{x}_{n}$ this is defined as $f* ω= 12nπ ∑ t=1 n xt eiωt 2 ,$ the scaling factor now being chosen in order that $2∫0πfωdω=σx2,$ i.e., the spectrum indicates how the sample variance (${\sigma }_{x}^{2}$) of the series is distributed over components in the frequency range $0\le \omega \le \pi$. It may be demonstrated that ${f}^{}\left(\omega \right)$ is equivalently defined in terms of the sample ACF ${r}_{k}$ of the series as $fω=12π c0+2∑k=1 n-1ckcos⁡kω ,$ where ${c}_{k}={\sigma }_{x}^{2}{r}_{k}$ are the sample autocovariance coefficients. If the series ${x}_{t}$ does contain a deterministic sinusoidal component of amplitude $R$, this will be revealed in the sample spectrum as a sharp peak of approximate width $\pi /n$ and height $\left(n/2\pi \right){R}^{2}$. This is called the discrete part of the spectrum, the variance ${R}^{2}$ associated with this component being in effect concentrated at a single frequency. If the series ${x}_{t}$ has no deterministic components, i.e., is purely stochastic being stationary with autocorrelation function (ACF) ${r}_{k}$, then with increasing sample size the expected value of ${f}^{}\left(\omega \right)$ converges to the theoretical spectrum – the continuous part $fω=12π γ0+2∑k=1∞γk cosωk ,$ where ${\gamma }_{k}$ are the theoretical autocovariances. The sample spectrum does not however converge to this value but at each frequency point fluctuates about the theoretical spectrum with an exponential distribution, being independent at frequencies separated by an interval of $2\pi /n$ or more. Various devices are therefore employed to smooth the sample spectrum and reduce its variability. Much of the strength of spectral analysis derives from the fact that the error limits are multiplicative so that features may still show up as significant in a part of the spectrum which has a generally low level, whereas they are completely masked by other components in the original series. The spectrum can help to distinguish deterministic cyclical components from the stochastic quasi-cycle components which produce a broader peak in the spectrum. (The deterministic components can be removed by regression and the remaining part represented by an ARIMA model.) A large discrete component in a spectrum can distort the continuous part over a large frequency range surrounding the corresponding peak. This may be alleviated at the cost of slightly broadening the peak by tapering a portion of the data at each end of the series with weights which decay smoothly to zero. It is usual to correct for the mean of the series and for any linear trend by simple regression, since they would similarly distort the spectrum. #### 2.2.2 Spectral smoothing by lag window The estimate is calculated directly from the sample autocovariances ${c}_{k}$ as $fω=12π c0+2∑k=1 M-1wkckcos⁡kω ,$ the smoothing being induced by the lag window weights ${w}_{k}$ which extend up to a truncation lag $M$ which is generally much less than $n$. The smaller the value of $M$, the greater the degree of smoothing, the spectrum estimates being independent only at a wider frequency separation indicated by the bandwidth $b$ which is proportional to $1/M$. It is wise, however, to calculate the spectrum at intervals appreciably less than this. Although greater smoothing narrows the error limits, it can also distort the spectrum, particularly by flattening peaks and filling in troughs. #### 2.2.3 Direct spectral smoothing The unsmoothed sample spectrum is calculated for a fine division of frequencies, then averaged over intervals centred on each frequency point for which the smoothed spectrum is required. This is usually at a coarser frequency division. The bandwidth corresponds to the width of the averaging interval. ### 2.3 Linear Lagged Relationships Between Time Series We now consider the context in which one time series, called the dependent or output series, ${y}_{1},{y}_{2},\dots ,{y}_{n}$, is believed to depend on one or more explanatory or input series, e.g., ${x}_{1},{x}_{2},\dots ,{x}_{n}$. This dependency may follow a simple linear regression, e.g., $yt=vxt+nt$ or more generally may involve lagged values of the input $yt=v0xt+v1xt- 1+v2xt- 2+⋯+nt.$ The sequence ${v}_{0},{v}_{1},{v}_{2},\dots \text{}$ is called the impulse response function (IRF) of the relationship. The term ${n}_{t}$ represents that part of ${y}_{t}$ which cannot be explained by the input, and it is assumed to follow a univariate ARIMA model. We call ${n}_{t}$ the (output) noise component of ${y}_{t}$, and it includes any constant term in the relationship. It is assumed that the input series, ${x}_{t}$, and the noise component, ${n}_{t}$, are independent. The part of ${y}_{t}$ which is explained by the input is called the input component ${z}_{t}$: $zt=v0xt+v1xt-1+v2xt-2+⋯$ so ${y}_{t}={z}_{t}+{n}_{t}$. The eventual aim is to model both these components of ${y}_{t}$ on the basis of observations of ${y}_{1},{y}_{2},\dots ,{y}_{n}$ and ${x}_{1},{x}_{2},\dots ,{x}_{n}$. In applications to forecasting or control both components are important. In general there may be more than one input series, e.g., ${x}_{1,t}$ and ${x}_{2,t}$, which are assumed to be independent and corresponding components ${z}_{1,t}$ and ${z}_{2,t}$, so $yt=z1,t+z2,t+nt.$ #### 2.3.1 Transfer function models In a similar manner to that in which the structure of a univariate series may be represented by a finite-parameter ARIMA model, the structure of an input component may be represented by a transfer function (TF) model with delay time $b$, $p$ autoregressive-like parameters ${\delta }_{1},{\delta }_{2},\dots ,{\delta }_{p}$ and $q+1$ moving-average-like parameters ${\omega }_{0},{\omega }_{1},\dots ,{\omega }_{q}$: $zt=δ1zt-1+δ2zt-2+⋯+δpzt-p+ω0xt-b-ω1xt-b-1-⋯-ωqxt-b-q.$ (7) If $p>0$ this represents an impulse response function (IRF) which is infinite in extent and decays with geometric and/or sinusoidal behaviour. The parameters ${\delta }_{1},{\delta }_{2},\dots ,{\delta }_{p}$ are constrained to satisfy a stability condition identical to the stationarity condition of autoregressive models. There is no constraint on ${\omega }_{0},{\omega }_{1},\dots ,{\omega }_{q}$. #### 2.3.2 Cross-correlations An important tool for investigating how an input series ${x}_{t}$ affects an output series ${y}_{t}$ is the sample cross-correlation function (CCF) ${r}_{xy}\left(\mathit{k}\right)$, for $\mathit{k}=0,1,\dots$ between the series. If ${x}_{t}$ and ${y}_{t}$ are (jointly) stationary time series this is an estimator of the theoretical quantity $ρxyk=corrxt,yt+k.$ The sequence ${r}_{yx}\left(\mathit{k}\right)$, for $\mathit{k}=0,1,\dots$, is distinct from ${r}_{xy}\left(k\right)$, though it is possible to interpret $ryxk=rxy-k.$ When the series ${y}_{t}$ and ${x}_{t}$ are believed to be related by a transfer function (TF) model, the CCF is determined by the impulse response function (IRF) ${v}_{0},{v}_{1},{v}_{2},\dots \text{}$ and the autocorrelation function (ACF) of the input ${x}_{t}$. In the particular case when ${x}_{t}$ is an uncorrelated series or white noise (and is uncorrelated with any other inputs): $ρxyk∝vk$ and the sample CCF can provide an estimate of ${v}_{k}$: $v~k=sy/sxrxyk$ where ${s}_{y}$ and ${s}_{x}$ are the sample standard deviations of ${y}_{t}$ and ${x}_{t}$, respectively. In theory the IRF coefficients ${v}_{b},\dots ,{v}_{b+p+q}$ determine the parameters in the TF model, and using ${\stackrel{~}{v}}_{k}$ to estimate ${\stackrel{~}{v}}_{k}$ it is possible to solve for preliminary estimates of ${\delta }_{1},{\delta }_{2},\dots ,{\delta }_{p}$, ${\omega }_{0},{\omega }_{1},\dots ,{\omega }_{q}$. #### 2.3.3 Prewhitening or filtering by an ARIMA model In general an input series ${x}_{t}$ is not white noise, but may be represented by an ARIMA model with innovations or residuals ${a}_{t}$ which are white noise. If precisely the same operations by which ${a}_{t}$ is generated from ${x}_{t}$ are applied to the output ${y}_{t}$ to produce a series ${b}_{t}$, then the transfer function relationship between ${y}_{t}$ and ${x}_{t}$ is preserved between ${b}_{t}$ and ${a}_{t}$. It is then possible to estimate $v~k=sb/sarabk.$ The procedure of generating ${a}_{t}$ from ${x}_{t}$ (and ${b}_{t}$ from ${y}_{t}$) is called prewhitening or filtering by an ARIMA model. Although ${a}_{t}$ is necessarily white noise, this is not generally true of ${b}_{t}$. #### 2.3.4 Multi-input model estimation The term multi-input model is used for the situation when one output series ${y}_{t}$ is related to one or more input series ${x}_{j,t}$, as described in Section 2.3. If for a given input the relationship is a simple linear regression, it is called a simple input; otherwise it is a transfer function input. The error or noise term follows an ARIMA model. Given that the orders of all the transfer function models and the ARIMA model of a multi-input model have been specified, the various parameters in those models may be (simultaneously) estimated. The procedure used is closely related to the least squares principle applied to the innovations in the ARIMA noise model. The innovations are derived for any proposed set of parameter values by calculating the response of each input to the transfer functions and then evaluating the noise ${n}_{t}$ as the difference between this response (combined for all the inputs) and the output. The innovations are derived from the noise using the ARIMA model in the same manner as for a univariate series, and as described in Section 2.1.6. In estimating the parameters, consideration has to be given to the lagged terms in the various model equations which are associated with times prior to the observation period, and are therefore unknown. The subroutine descriptions provide the necessary detail as to how this problem is treated. Also, as described in Section 2.1.7 the sum of squares criterion $S=∑at2$ is related to the quadratic form in the exact log-likelihood of the parameters: $-12logV-12wTV-1w.$ Here $w$ is the vector of appropriately differenced noise terms, and $wTV-1w=S/σa2,$ where ${\sigma }_{a}^{2}$ is the innovation variance parameter. The least squares criterion is therefore identical to minimization of the quadratic form, but is not identical to exact likelihood. Because $V$ may be expressed as $M{\sigma }_{a}^{2}$, where $M$ is a function of the ARIMA model parameters, substitution of ${\sigma }_{a}^{2}$ by its maximum likelihood (ML) estimator yields a concentrated (or profile) likelihood which is a function of $M1/NS.$ $N$ is the length of the differenced noise series $w$, and $\left\|M\right\|=\mathrm{det}M$. Use of the above quantity, called the deviance, $D$, as an objective function is preferable to the use of $S$ alone, on the grounds that it is equivalent to exact likelihood, and yields estimates with better properties. However, there is an appreciable computational penalty in calculating $D$, and in large samples it differs very little from $S$, except in the important case of seasonal ARIMA models where the number of whole seasons within the data length must also be large. You are given the option of taking the objective function to be either $S$ or $D$, or a third possibility, the marginal likelihood. This is similar to exact likelihood but can counteract bias in the ARIMA model due to the fitting of a large number of simple inputs. Approximate standard errors of the parameter estimates and the correlations between them are available after estimation. The model residuals ${\stackrel{^}{a}}_{t}$ are the innovations resulting from the estimation, and they are usually examined for the presence of either autocorrelation or cross-correlation with the inputs. Absence of such correlation provides some confirmation of the adequacy of the model. #### 2.3.5 Multi-input model forecasting A multi-input model may be used to forecast the output series provided future values (possibly forecasts) of the input series are supplied. Construction of the forecasts requires knowledge only of the model orders and parameters, together with a limited set of the most recent variables which appear in the model equations. This is called the state set. It is conveniently constituted after model estimation. Moreover, if new observations ${y}_{n+1},{y}_{n+2},\dots \text{}$ of the output series and ${x}_{n+1},{x}_{n+2},\dots \text{}$ of (all) the independent input series become available, then the model equations can easily be used to update the state set before constructing forecasts from the end of the new observations. The new innovations ${a}_{n+1},{a}_{n+2},\dots \text{}$ generated in this updating may be used to monitor the continuing adequacy of the model. #### 2.3.6 Transfer function model filtering In many time series applications it is desired to calculate the response (or output) of a transfer function (TF) model for a given input series. Smoothing, detrending, and seasonal adjustment are typical applications. You must specify the orders and parameters of a TF model for the purpose being considered. This may then be applied to the input series. Again, problems may arise due to ignorance of the input series values prior to the observation period. The transient errors which can arise from this may be substantially reduced by using ‘backforecasts’ of these unknown observations. ### 2.4 Multivariate Time Series Multi-input modelling represents one output time series in terms of one or more input series. Although there are circumstances in which it may be more appropriate to analyse a set of time series by modelling each one in turn as the output series with the remainder as inputs, there is a more symmetric approach in such a context. These models are known as vector autoregressive moving-average (VARMA) models. #### 2.4.1 Differencing and transforming a multivariate time series As in the case of a univariate time series, it may be useful to simplify the series by differencing operations which may be used to remove linear or seasonal trends, thus ensuring that the resulting series to be used in the model estimation is stationary. It may also be necessary to apply transformations to the individual components of the multivariate series in order to stabilize the variance. Commonly used transformations are the log and square root transformations. #### 2.4.2 Model identification for a multivariate time series Multivariate analogues of the autocorrelation and partial autocorrelation functions are available for analysing a set of $k$ time series, ${x}_{\mathit{i},1},{x}_{\mathit{i},2},\dots ,{x}_{\mathit{i},n}$, for $\mathit{i}=1,2,\dots ,k$, thereby making it possible to obtain some understanding of a suitable VARMA model for the observed series. It is assumed that the time series have been differenced if necessary, and that they are jointly stationary. The lagged correlations between all possible pairs of series, i.e., $ρ ijl = corr x i , t , x j , t + l$ are then taken to provide an adequate description of the statistical relationships between the series. These quantities are estimated by sample auto- and cross-correlations ${r}_{ijl}$. For each $l$ these may be viewed as elements of a (lagged) autocorrelation matrix. Thus consider the vector process ${x}_{t}$ (with elements ${x}_{it}$) and lagged autocovariance matrices ${\Gamma }_{l}$ with elements of ${\sigma }_{i}{\sigma }_{j}{\rho }_{ijl}$ where ${\sigma }_{i}^{2}=\mathrm{var}\left({x}_{i,t}\right)$. Correspondingly, ${\Gamma }_{l}$ is estimated by the matrix ${C}_{l}$ with elements ${s}_{i}{s}_{j}{r}_{ijl}$ where ${s}_{i}^{2}$ is the sample variance of ${x}_{it}$. For a series with short-term cross-correlation only, i.e., ${r}_{ijl}$ is not significant beyond some low lag $q$, then the pure vector $\text{MA}\left(q\right)$ model, with no autoregressive parameters, i.e., $p=0$, is appropriate. The correlation matrices provide a description of the joint statistical properties of the series. It is also possible to calculate matrix quantities which are closely analogous to the partial autocorrelations of univariate series (see Section 2.1.4). Wei (1990) discusses both the partial autoregression matrices proposed by Tiao and Box (1981) and partial lag correlation matrices. In the univariate case the partial autocorrelation function (PACF) between ${x}_{t}$ and ${x}_{t+l}$ is the correlation coefficient between the two after removing the linear dependence on each of the intervening variables ${x}_{t+1},{x}_{t+2},\dots ,{x}_{t+l-1}$. This partial autocorrelation may also be obtained as the last regression coefficient associated with ${x}_{t}$ when regressing ${x}_{t+l}$ on its $l$ lagged variables ${x}_{t+l-1},{x}_{t+l-2},\dots ,{x}_{t}$. Tiao and Box (1981) extended this method to the multivariate case to define the partial autoregression matrix. Heyse and Wei (1985) also extended the univariate definition of the PACF to derive the correlation matrix between the vectors ${x}_{t}$ and ${x}_{t+l}$ after removing the linear dependence on each of the intervening vectors ${x}_{t+1},{x}_{t+2},\dots ,{x}_{t+l-1}$, the partial lag correlation matrix. Note that the partial lag correlation matrix is a correlation coefficient matrix since each of its elements is a properly normalized correlation coefficient. This is not true of the partial autoregression matrices (except in the univariate case for which the two types of matrix are the same). The partial lag correlation matrix at lag $1$ also reduces to the regular correlation matrix at lag $1$; this is not true of the partial autoregression matrices (again except in the univariate case). Both the above share the same cut-off property for autoregressive processes; that is for an autoregressive process of order $p$, the terms of the matrix at lags $p+1$ and greater are zero. Thus if the sample partial cross-correlations are significant only up to some low lag $p$ then a pure vector $\text{AR}\left(p\right)$ model is appropriate with $q=0$. Otherwise moving-average terms will need to be introduced as well as autoregressive terms. Under the hypothesis that ${x}_{t}$ is an autoregressive process of order $l-1$, $n$ times the sum of the squared elements of the partial lag correlation matrix at lag $l$ is asymptotically distributed as a ${\chi }^{2}$ variable with ${k}^{2}$ degrees of freedom where $k$ is the dimension of the multivariate time series. This provides a diagnostic aid for determining the order of an autoregressive model. The partial autoregression matrices may be found by solving a multivariate version of the Yule–Walker equations to find the autoregression matrices, using the final regression matrix coefficient as the partial autoregression matrix at that particular lag. The basis of these calculations is a multivariate autoregressive model: $xt=ϕl,1xt-1+⋯+ϕl,lxt-l+el,t$ where ${\varphi }_{l,1},{\varphi }_{l,2},\dots ,{\varphi }_{l,l}$ are matrix coefficients, and ${e}_{l,t}$ is the vector of errors in the prediction. These coefficients may be rapidly computed using a recursive technique which requires, and simultaneously furnishes, a backward prediction equation: $xt-l-1=ψl,1xt-l+ψl,2xt-l+1+⋯+ψl,lxt-1+fl,t$ (in the univariate case ${\psi }_{l,i}={\varphi }_{l,i}$). The forward prediction equation coefficients, ${\varphi }_{l,i}$, are of direct interest, together with the covariance matrix ${D}_{l}$ of the prediction errors ${e}_{l,t}$. The calculation of these quantities for a particular maximum equation lag $l=L$ involves calculation of the same quantities for increasing values of $l=1,2,\dots ,L$. The quantities ${v}_{l}=\mathrm{det}{D}_{l}/\mathrm{det}{\Gamma }_{0}$ may be viewed as generalized variance ratios, and provide a measure of the efficiency of prediction (the smaller the better). The reduction from ${v}_{l-1}$ to ${v}_{l}$ which occurs on extending the order of the predictor to $l$ may be represented as $vl=vl-11-ρl2$ where ${\rho }_{l}^{2}$ is a multiple squared partial autocorrelation coefficient associated with ${k}^{2}$ degrees of freedom. Sample estimates of all the above quantities may be derived by using the series covariance matrices ${C}_{\mathit{l}}$, for $\mathit{l}=1,2,\dots ,L$, in place of ${\Gamma }_{l}$. The best lag for prediction purposes may be chosen as that which yields the minimum final prediction error (FPE) criterion: $FPEl=vl× 1+lk2/n 1-lk2/n .$ An alternative method of estimating the sample partial autoregression matrices is by using multivariate least squares to fit a series of multivariate autoregressive models of increasing order. #### 2.4.3 VARMA model estimation The cross-correlation structure of a stationary multivariate time series may often be represented by a model with a small number of parameters belonging to the VARMA class. If the stationary series ${w}_{t}$ has been derived by transforming and/or differencing the original series ${x}_{t}$, then ${w}_{t}$ is said to follow the VARMA model: $wt=ϕ1wt-1+⋯+ϕpwt-p+εt-θ1εt-1-⋯-θqεt-q,$ where ${\epsilon }_{t}$ is a vector of uncorrelated residual series (white noise) with zero mean and constant covariance matrix $\Sigma$, ${\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{p}$ are the $p$ autoregressive (AR) parameter matrices and ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{q}$ are the $q$ moving-average (MA) parameter matrices. If ${w}_{t}$ has a nonzero mean $\mu$, then this can be allowed for by replacing ${w}_{t},{w}_{t-1},\dots \text{}$ by ${w}_{t}-\mu ,{w}_{t-1}-\mu ,\dots \text{}$ in the model. A series generated by this model will only be stationary provided restrictions are placed on ${\varphi }_{1},{\varphi }_{2},\dots ,{\varphi }_{p}$ to avoid unstable growth of ${w}_{t}$. These are stationarity constraints. The series ${\epsilon }_{t}$ may also be usefully interpreted as the linear innovations in ${w}_{t}$, i.e., the error if ${w}_{t}$ were to be predicted using the information in all past values ${w}_{t-1},{w}_{t-2},\dots \text{}$, provided also that ${\theta }_{1},{\theta }_{2},\dots ,{\theta }_{q}$ satisfy what are known as invertibility constraints. This allows the series ${\epsilon }_{t}$ to be generated by rewriting the model equation as $εt=wt-ϕ1wt-1-⋯-ϕpwt-p+θ1εt-1+⋯+θqεt-q.$ The method of maximum likelihood (ML) may be used to estimate the parameters of a specified VARMA model from the observed multivariate time series together with their standard errors and correlations. The residuals from the model may be examined for the presence of autocorrelations as a check on the adequacy of the fitted model. #### 2.4.4 VARMA model forecasting Forecasts of the series may be constructed using a multivariate version of the univariate method. Efficient methods are available for updating the forecasts each time new observations become available. ### 2.5 Cross-spectral Analysis The relationship between two time series may be investigated in terms of their sinusoidal components at different frequencies. At frequency $\omega$ a component of ${y}_{t}$ of the form $Ryωcosωt-ϕyω$ has its amplitude ${R}_{y}\left(\omega \right)$ and phase lag ${\varphi }_{y}\left(\omega \right)$ estimated by $Ryωeiϕyω=1n∑t=1nyteiωt$ and similarly for ${x}_{t}$. In the univariate analysis only the amplitude was important – in the cross analysis the phase is important. #### 2.5.1 The sample cross-spectrum This is defined by $fxyω=12πn ∑t=1nyteiωt ∑t=1nxte-iωt .$ It may be demonstrated that this is equivalently defined in terms of the sample cross-correlation function (CCF), ${r}_{xy}\left(k\right)$, of the series as $fxyω=12π ∑-n-1 n-1 cxykeiωk$ where ${c}_{xy}\left(k\right)={s}_{x}{s}_{y}{r}_{xy}\left(k\right)$ is the cross-covariance function. #### 2.5.2 The amplitude and phase spectrum The cross-spectrum is specified by its real part or cospectrum $c{f}^{}\left(\omega \right)$ and imaginary part or quadrature spectrum $q{f}^{}\left(\omega \right)$, but for the purpose of interpretation the cross-amplitude spectrum and phase spectrum are useful: $Aω = fxy ω , ϕω = arg f xy ω .$ If the series ${x}_{t}$ and ${y}_{t}$ contain deterministic sinusoidal components of amplitudes ${R}_{y},{R}_{x}$ and phases ${\varphi }_{y},{\varphi }_{x}$ at frequency $\omega$, then ${A}^{}\left(\omega \right)$ will have a peak of approximate width $\pi /n$ and height $\left(n/2\pi \right){R}_{y}{R}_{x}$ at that frequency, with corresponding phase ${\varphi }^{}\left(\omega \right)={\varphi }_{y}-{\varphi }_{x}$. This supplies no information that cannot be obtained from the two series separately. The statistical relationship between the series is better revealed when the series are purely stochastic and jointly stationary, in which case the expected value of ${f}_{xy}^{}\left(\omega \right)$ converges with increasing sample size to the theoretical cross-spectrum $fxy ω = 12π ∑ -∞ ∞ γxy k eiωk$ where ${\gamma }_{xy}\left(k\right)=\mathrm{cov}\left({x}_{t},{y}_{t+k}\right)$. The sample spectrum, as in the univariate case, does not converge to the theoretical spectrum without some form of smoothing which either implicitly (using a lag window) or explicitly (using a frequency window) averages the sample spectrum ${f}_{xy\left(\omega \right)}^{}$ over wider bands of frequency to obtain a smoothed estimate ${\stackrel{^}{f}}_{xy}\left(\omega \right)$. #### 2.5.3 The coherency spectrum If there is no statistical relationship between the series at a given frequency, then ${f}_{xy}\left(\omega \right)=0$, and the smoothed estimate ${\stackrel{^}{f}}_{xy}\left(\omega \right)$, will be close to $0$. This is assessed by the squared coherency between the series: $W^ω=f^xyω2f^xxωf^yyω$ where ${\stackrel{^}{f}}_{xx}\left(\omega \right)$ is the corresponding smoothed univariate spectrum estimate for ${x}_{t}$, and similarly for ${y}_{t}$. The coherency can be treated as a squared multiple correlation. It is similarly invariant in theory not only to simple scaling of ${x}_{t}$ and ${y}_{t}$, but also to filtering of the two series, and provides a useful test statistic for the relationship between autocorrelated series. Note that without smoothing, $fxyω2=fxxωfyyω,$ so the coherency is $1$ at all frequencies, just as a correlation is $1$ for a sample of size $1$. Thus smoothing is essential for cross-spectrum analysis. #### 2.5.4 The gain and noise spectrum If ${y}_{t}$ is believed to be related to ${x}_{t}$ by a linear lagged relationship as in Section 2.3, i.e., $yt=v0xt+v1xt-1+v2xt-2+⋯+nt,$ then the theoretical cross-spectrum is $fxyω =Vω fxxω$ where $Vω=Gωeiϕω=∑k=0∞vkeikω$ is called the frequency response of the relationship. Thus if ${x}_{t}$ were a sinusoidal wave at frequency $\omega$ (and ${n}_{t}$ were absent), ${y}_{t}$ would be similar but multiplied in amplitude by $G\left(\omega \right)$ and shifted in phase by $\varphi \left(\omega \right)$. Furthermore, the theoretical univariate spectrum $fyyω=G ω 2fxxω+fnω$ where ${n}_{t}$, with spectrum ${f}_{n}\left(\omega \right)$, is assumed independent of the input ${x}_{t}$. Cross-spectral analysis thus furnishes estimates of the gain $G^ ω= f^ xy ω / f^xx ω$ and the phase $ϕ^ω=argf^xyω .$ From these representations of the estimated frequency response $\stackrel{^}{V}\left(\omega \right)$, parametric transfer function (TF) models may be recognized and selected. The noise spectrum may also be estimated as $f^ y∣x ω= f^yy ω 1-W^ ω$ a formula which reflects the fact that in essence a regression is being performed of the sinusoidal components of ${y}_{t}$ on those of ${x}_{t}$ over each frequency band. Interpretation of the frequency response may be aided by extracting from $\stackrel{^}{V}\left(\omega \right)$ estimates of the impulse response function (IRF) ${\stackrel{^}{v}}_{k}$. It is assumed that there is no anticipatory response between ${y}_{t}$ and ${x}_{t}$, i.e., no coefficients ${v}_{k}$ with $k=-1$ or $-2$ are needed (their presence might indicate feedback between the series). #### 2.5.5 Cross-spectrum smoothing by lag window The estimate of the cross-spectrum is calculated from the sample cross-variances as $f^xyω=12π ∑-M+S M+Swk-Scxykeiωk.$ The lag window ${w}_{k}$ extends up to a truncation lag $M$ as in the univariate case, but its centre is shifted by an alignment lag $S$ usually chosen to coincide with the peak cross-correlation. This is equivalent to an alignment of the series for peak cross-correlation at lag $0$, and reduces bias in the phase estimation. The selection of the truncation lag $M$, which fixes the bandwidth of the estimate, is based on the same criteria as for univariate series, and the same choice of $M$ and window shape should be used as in univariate spectrum estimation to obtain valid estimates of the coherency, gain, etc., and test statistics. #### 2.5.6 Direct smoothing of the cross-spectrum The computations are exactly as for smoothing of the univariate spectrum except that allowance is made for an implicit alignment shift $S$ between the series. ### 2.6 Kalman Filters #### 2.6.1 Linear State Space Models Kalman filtering provides a method for the analysis of multidimensional time series. The underlying model is: $Xt+1=AtXt+BtWt$ (8) $Yt=CtXt+Vt$ (9) where ${X}_{t}$ is the unobserved state vector, ${Y}_{t}$ is the observed measurement vector, ${W}_{t}$ is the state noise, ${V}_{t}$ is the measurement noise, ${A}_{t}$ is the state transition matrix, ${B}_{t}$ is the noise coefficient matrix and ${C}_{t}$ is the measurement coefficient matrix at time $t$. The state noise and the measurement noise are assumed to be uncorrelated with zero mean and covariance matrices: $E Wt WtT = Qt and E Vt VtT = Rt .$ If the system matrices ${A}_{t}$, ${B}_{t}$, ${C}_{t}$ and the covariance matrices ${Q}_{t},{R}_{t}$ are known then Kalman filtering can be used to compute the minimum variance estimate of the stochastic variable ${X}_{t}$. The estimate of ${X}_{t}$ given observations ${Y}_{1}$ to ${Y}_{t-1}$ is denoted by ${\stackrel{^}{X}}_{t\mid t-1}$ with state covariance matrix $E\left\{{\stackrel{^}{X}}_{t\mid t-1}{\stackrel{^}{X}}_{t\mid t-1}^{\mathrm{T}}\right\}={P}_{t\mid t-1}$ while the estimate of ${X}_{t}$ given observations ${Y}_{1}$ to ${Y}_{t}$ is denoted by ${\stackrel{^}{X}}_{t\mid t}$ with covariance matrix $E\left\{{\stackrel{^}{X}}_{t\mid t}{\stackrel{^}{X}}_{t\mid t}^{\mathrm{T}}\right\}={P}_{t\mid t}$. The update of the estimate, ${\stackrel{^}{X}}_{t+1\mid t}$, from time $t$ to time $t+1$, is computed in two stages. First, the update equations are $X^ t∣t = X^ t∣t-1 + Kt rt, Pt∣t= I- Kt Ct Pt∣t-1$ where the residual ${r}_{t}={Y}_{t}-{C}_{t}{X}_{t\mid t-1}$ has an associated covariance matrix ${H}_{t}={C}_{t}{P}_{t\mid t-1}{C}_{t}^{\mathrm{T}}+{R}_{t}$, and ${K}_{t}$ is the Kalman gain matrix with $Kt= Pt∣t-1 CtT Ht-1 .$ The second stage is the one-step-ahead prediction equations given by $X^ t+1 ∣ t = At X^ t ∣ t , Pt+1 ∣ t = At Pt ∣ t AtT + Bt Qt BtT .$ These two stages can be combined to give the one-step-ahead update-prediction equations $X^ t+1∣t = At X^ t∣t-1 + At Kt rt .$ The above equations thus provide a method for recursively calculating the estimates of the state vectors ${\stackrel{^}{X}}_{t\mid t}$ and ${\stackrel{^}{X}}_{t+1\mid t}$ and their covariance matrices ${P}_{t\mid t}$ and ${P}_{t+1\mid t}$ from their previous values. This recursive procedure can be viewed in a Bayesian framework as being the updating of the prior by the data ${Y}_{t}$. The initial values ${\stackrel{^}{X}}_{1\mid 0}$ and ${P}_{1\mid 0}$ are required to start the recursion. For stationary systems, ${P}_{1\mid 0}$ can be computed from the following equation: $P1∣0=A1P1∣0 A1T +B1Q1 B1T ,$ which can be solved by iterating on the equation. For ${\stackrel{^}{X}}_{1\mid 0}$ the value $E\left\{X\right\}$ can be used if it is available. ##### 2.6.1.1 Computational methods To improve the stability of the computations the square root algorithm is used. One recursion of the square root covariance filter algorithm which can be summarised as follows: $Rt1/2 CtSt 0 0 AtSt BtQt1/2 U= Ht1/2 0 0 Gt St+1 0$ where $U$ is an orthogonal transformation triangularizing the left-hand pre-array to produce the right-hand post-array, ${S}_{t}$ is the lower triangular Cholesky factor of the state covariance matrix ${P}_{t+1\mid t}$, ${Q}_{t}^{1/2}$ and ${R}_{t}^{1/2}$ are the lower triangular Cholesky factor of the covariance matrices $Q$ and $R$ and ${H}^{1/2}$ is the lower triangular Cholesky factor of the covariance matrix of the residuals. The relationship between the Kalman gain matrix, ${K}_{t}$, and ${G}_{t}$ is given by $AtKt=Gt Ht1/2 -1.$ To improve the efficiency of the computations when the matrices ${A}_{t},{B}_{t}$ and ${C}_{t}$ do not vary with time the system can be transformed to give a simpler structure. The transformed state vector is ${U}^{}X$ where ${U}^{}$ is the transformation that reduces the matrix pair $\left(A,C\right)$ to lower observer Hessenberg form. That is, the matrix ${U}^{}$ is computed such that the compound matrix $CUT UAUT$ is a lower trapezoidal matrix. The transformations need only be computed once at the start of a series, and the covariance matrices ${Q}_{t}$ and ${R}_{t}$ can still be time-varying. ##### 2.6.1.2 Model fitting and forecasting If the state space model contains unknown parameters, $\theta$, these can be estimated using maximum likelihood (ML). Assuming that ${W}_{t}$ and ${V}_{t}$ are normal variates the log-likelihood for observations ${Y}_{\mathit{t}}$, for $\mathit{t}=1,2,\dots ,n$, is given by $constant-12∑t=1nlndetHt-12∑t=1t rtT Ht-1rt.$ Optimal estimates for the unknown model parameters $\theta$ can then be obtained by using a suitable optimizer routine to maximize the likelihood function. Once the model has been fitted forecasting can be performed by using the one-step-ahead prediction equations. The one-step-ahead prediction equations can also be used to ‘jump over’ any missing values in the series. ##### 2.6.1.3 Kalman filter and time series models Many commonly used time series models can be written as state space models. A univariate $\mathrm{ARMA}\left(p,q\right)$ model can be cast into the following state space form: $xt =Axt-1+Bεt wt =Cxt$ $A= ϕ1 1 ϕ2 1 . . . . ϕr-1 1 ϕr 0 0 . . 0 , B= -1 -θ1 -θ2 . . -θr-1 and CT= 1 0 0 . . 0 ,$ where $r=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(p,q+1\right)$. The representation for a $k$-variate $\mathrm{ARMA}\left(p,q\right)$ series (VARMA) is very similar to that given above, except now the state vector is of length $kr$ and the $\varphi$ and $\theta$ are now $k×k$ matrices and the 1s in $A$, $B$ and $C$ are now the identity matrix of order $k$. If $p or $q+1 then the appropriate $\varphi$ or $\theta$ matrices are set to zero, respectively. Since the compound matrix $C A$ is already in lower observer Hessenberg form (i.e., it is lower trapezoidal with zeros in the top right-hand triangle) the invariant Kalman filter algorithm can be used directly without the need to generate a transformation matrix ${U}^{}$. #### 2.6.2 Nonlinear State Space Models A nonlinear state space model, with additive noise, can, at time $t$, be described by: $xt+1 =Fxt+vt yt =Hxt+ut$ (18) where ${x}_{t}$ represents the unobserved state vector of length ${m}_{x}$ and ${y}_{t}$ the observed measurement vector of length ${m}_{y}$. The process noise is denoted ${v}_{t}$, which is assumed to have mean zero and covariance structure ${\Sigma }_{x}$, and the measurement noise by ${u}_{t}$, which is assumed to have mean zero and covariance structure ${\Sigma }_{y}$. The two nonlinear functions, $F$ and $H$ may be time dependent. Two methods are commonly used to analyse nonlinear state space models: the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF). The EKF solves the nonlinear state space model by first linearising the set of equations given in (18) using a first order taylor expansion around ${\stackrel{^}{x}}_{t}$ (the estimate of the state vector at time $t$ given the full data: ${y}_{1},{y}_{2},\dots ,{y}_{t}$) in the case of $F$ and around ${\stackrel{^}{x}}_{t}^{_}$ (the estimate of the state vector at time $t$ given the partial data: ${y}_{1},{y}_{1},\dots ,{y}_{t-1}$) in the case of $H$. This leads to the linear state space model: $xt+1 ≈F′⁢xt+vt+F′-Fx^t⁢x^t yt-Hx^t_+H′⁢x^t_ ≈H′⁢xt+ut$ where $F′ =∂Fx∂xx=x^t H′ =∂Hx∂xx=x^t_$ This linear state space model can then be solved using the standard Kalman Filter. See Haykin (2001) for more details. Unlike the EKF, the UKF of Julier and Uhlmann (1997) does not attempt to linearise the problem, rather it uses a minimal set of carefully chosen points, called sigma points, to capture the mean and covariance of the underlying Gaussian random variables. These points are then propagated through the nonlinear functions giving an estimate of the transformed mean and covariance. A brief description of the UKF can be found in Section 3 in G13EKF. ### 2.7 GARCH Models #### 2.7.1 ARCH models and their generalizations Rather than modelling the mean (for example using regression models) or the autocorrelation (by using ARMA models) there are circumstances in which the variance of a time series needs to be modelled. This is common in financial data modelling where the variance (or standard deviation) is known as volatility. The ability to forecast volatility is a vital part in deciding the risk attached to financial decisions like portfolio selection. The basic model for relating the variance at time $t$ to the variance at previous times is the autoregressive conditional heteroskedastic (ARCH) model. The standard ARCH model is defined as $yt ∣ ψt-1 ∼ N 0,ht , ht = α0+ ∑ i=1 q αi ε t-i 2 ,$ where ${\psi }_{t}$ is the information up to time $t$ and ${h}_{t}$ is the conditional variance. In a similar way to that in which autoregressive (AR) models were generalized to ARMA models the ARCH models have been generalized to a GARCH model; see Engle (1982), Bollerslev (1986) and Hamilton (1994) $ht = α0 + ∑ i=1 q αi ε t-i 2 + ∑ i=1 p β ht-i .$ This can be combined with a regression model: $yt=b0+∑i= 1k bi xit+εt,$ where ${\epsilon }_{t}\mid {\psi }_{t-1}\sim N\left(0,{h}_{t}\right)$ and where ${x}_{\mathit{i}t}$, for $\mathit{i}=1,2,\dots ,k$, are the exogenous variables. The above models assume that the change in variance, ${h}_{t}$, is symmetric with respect to the shocks, that is, that a large negative value of ${\epsilon }_{t-1}$ has the same effect as a large positive value of ${\epsilon }_{t-1}$. A frequently observed effect is that a large negative value ${\epsilon }_{t-1}$ often leads to a greater variance than a large positive value. The following three asymmetric models represent this effect in different ways using the parameter $\gamma$ as a measure of the asymmetry. Type I AGARCH($p,q$) $ht = α0 + ∑ i=1 q αi εt-i +γ 2 + ∑ i=1 p βi ht-i .$ Type II AGARCH($p,q$) $ht = α0 + ∑ i=1 q αi εt-i + γ εt-i 2+ ∑ i=1 p βi ht-i .$ GJR-GARCH($p,q$), or Glosten, Jagannathan and Runkle GARCH (see Glosten et al. (1993)) $ht = α0 + ∑ i=1 q αi + γ It-1 ε t-1 2 + ∑ i=1 p βi ht-i ,$ where ${I}_{t}=1$ if ${\epsilon }_{t}<0$ and ${I}_{t}=0$ if ${\epsilon }_{t}\ge 0$. The first assumes that the effects of the shocks are symmetric about $\gamma$ rather than zero, so that for $\gamma <0$ the effect of negative shocks is increased and the effect of positive shocks is decreased. Both the Type II AGARCH and the GJR GARCH (see Glosten et al. (1993)) models introduce asymmetry by increasing the value of the coefficient of ${\epsilon }_{t-1}^{2}$ for negative values of ${\epsilon }_{t-1}$. In the case of the Type II AGARCH the effect is multiplicative while for the GJR GARCH the effect is additive. Coefficient ${\epsilon }_{t-1}<0$ ${\epsilon }_{t-1}>0$ Type II AGARCH ${\alpha }_{i}{\left(1-\gamma \right)}^{2}$ ${\alpha }_{i}{\left(1+\gamma \right)}^{2}$ GJR GARCH ${\alpha }_{i}+\gamma$ ${\alpha }_{i}$ (Note that in the case of GJR GARCH, $\gamma$ needs to be positive to inflate variance after negative shocks while for Type I and Type II AGARCH, $\gamma$ needs to be negative.) A third type of GARCH model is the exponential GARCH (EGARCH). In this model the variance relationship is on the log scale and hence asymmetric. $lnht = α0 + ∑ i=1 q αi zt-i + ∑ i=1 q ϕi zt-i - E zt-i + ∑ i=1 p βi ln ht-i ,$ where ${z}_{t}=\frac{{\epsilon }_{t}}{\sqrt{{h}_{t}}}$ and $E\left[\left\|{z}_{t-i}\right\|\right]$ denotes the expected value of $\left\|{z}_{t-i}\right\|$. Note that the ${\varphi }_{i}$ terms represent a symmetric contribution to the variance while the ${\alpha }_{i}$ terms give an asymmetric contribution. Another common characteristic of financial data is that it is heavier in the tails (leptokurtic) than the Normal distribution. To model this the Normal distribution is replaced by a scaled Student's $t$-distribution (that is a Student's $t$-distribution standardized to have variance ${h}_{t}$). The Student's $t$-distribution is such that the smaller the degrees of freedom the higher the kurtosis for degrees of freedom $\text{}>4$. #### 2.7.2 Fitting GARCH models The models are fitted by maximizing the conditional log-likelihood. For the Normal distribution the conditional log-likelihood is $12∑i=1T loghi+εi2hi .$ For the Student's $t$-distribution the function is more complex. An approximation to the standard errors of the parameter estimates is computed from the Fisher information matrix. ### 2.8 Inhomogeneous Time Series If we denote a generic univariate time series as a sequence of pairs of values $\left({z}_{\mathit{i}},{t}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots$ where the $z$'s represent an observed scalar value and the $t$'s the time that the value was observed, then in a standard time series analysis, as discussed in other sections of this introduction, it is assumed that the series being analysed is homogeneous, that is the sampling times are regularly spaced with ${t}_{i}-{t}_{i-1}=\delta$ for some value $\delta$. In many real world applications this assumption does not hold, that is, the series is inhomogeneous. Standard time series analysis techniques cannot be used on an inhomogeneous series without first preprocessing the series to construct an artificial homogeneous series, by for example, resampling the series at regular intervals. Zumbach and Müller (2001) introduced a series of operators that can be used to extract robust information directly from the inhomogeneous time series. In this context, robust information means that the results should be essentially independent of minor changes to the sampling mechanism used when collecting the data, for example, changing a number of time stamps or adding or removing a few observations. The basic operator available for inhomogeneous time series is the exponential moving average (EMA). This operator has a single parameter, $\tau$, and is an average operator with an exponentially decaying kernel given by: $e -t/τ τ .$ This gives rise to the following iterative formula: $EMA τ;z ti = μ ⁢ EMA τ;z ti-1 + ν-μ ⁢ zi-1 + 1-ν ⁢ zi$ where $μ = e-α and α = ti - ti-1 τ .$ The value of $\nu$ depends on the method of interpolation chosen. Three interpolation methods are available: 1 Previous point: $\nu =1$. 2 Linear: $\nu =\left(1-\mu \right)/\alpha$. 3 Next point: $\nu =\mu$. Given the EMA, a number of other operators can be defined, including: (i) $\mathbf{m}$-Iterated Exponential Moving Average, defined as $EMA τ,m;z = EMA τ ; EMA τ,m-1 ; z where EMA τ,1;z = EMA τ ; z .$ (ii) Moving Average (MA), defined as $MA τ, m1, m2; z ti = 1 m2 - m1 +1 ∑ j=m1 m2 EMA τ~, j; z ti where τ~= 2⁢τ m2+m1$ (iii) Moving Norm (MNorm), defined as $MNorm τ,m,p;z = MA τ,1,m; z p 1 / p$ (iv) Moving Variance (MVar), defined as $MVar τ,m,p;z = MA τ,1,m; z - MA τ,1,m;z p$ (v) Moving Standard Deviation (MSD), defined as $MSD τ,m,p;z = MA τ,1,m; z - MA τ,1,m;z p 1 / p$ (vi) Differential ( $\mathbf{\Delta }$), defined as $Δτ,α,β,γ;z = γ⁢ EMAα⁢τ,1;z + EMAα⁢τ,2;z - 2⁢ EMAα⁢β⁢τ,4;z$ (vii) Volatility, defined as $Volatilityτ,τ′,m,p;z = MNorm τ/2,m,p;Δτ′;z$ A discussion of each of these operators, their use and in some cases, alternative definitions, are given in Zumbach and Müller (2001). ## 3 Recommendations on Choice and Use of Available Routines ### 3.1 Univariate Analysis The availability of routines for each of these four steps is given below. #### 3.1.1 ARMA-type Models ARMA-type modelling usually follows the methodology made popular by Box and Jenkins. It consists of four steps: identification, model fitting, model checking and forecasting. (a) Model identification The routine G13AUF may be used in obtaining either a range-mean or standard deviation-mean plot for a series of observations, which may be useful in detecting the need for a variance-stabilizing transformation. G13AUF computes the range or standard deviation and the mean for successive groups of observations and G01AGF may then be used to produce a scatter plot of range against mean or of standard deviation against mean. The routine G13AAF may be used to difference a time series. The $N=n-d-s×D$ values of the differenced time series which extends for $t=1+d+s×D,\dots ,n$ are stored in the first $N$ elements of the output array. The routine G13ABF may be used for direct computation of the autocorrelations. It requires the time series as input, after optional differencing by G13AAF. An alternative is to use G13CAF, which uses the fast Fourier transform (FFT) to carry out the convolution for computing the autocovariances. Circumstances in which this is recommended are (i) if the main aim is to calculate the smoothed sample spectrum; (ii) if the series length and maximum lag for the autocorrelations are both very large, in which case appreciable computing time may be saved. For more precise recommendations, see Gentleman and Sande (1966). In this case the autocorrelations ${r}_{k}$ need to be obtained from the autocovariances ${c}_{k}$ by ${r}_{k}={c}_{k}/{c}_{0}$. The routine G13ACF computes the partial autocorrelation function (PACF) and prediction error variance estimates from an input autocorrelation function (ACF). Note that G13DNF, which is designed for multivariate time series, may also be used to compute the PACF together with ${\chi }^{2}$ statistics and their significance levels. Finite lag predictor coefficients are also computed by the routine G13ACF. It may have to be used twice, firstly with a large value for the maximum lag $L$ in order to locate the optimum final prediction error (FPE) lag, then again with $L$ reset to this lag. The routine G13DXF may be used to check that the autoregressive (AR) part of the model is stationary and that the moving-average (MA) part is invertible. (b) Model estimation The routine G13ADF is used to compute preliminary estimates of the ARIMA model parameters, the sample autocorrelations of the appropriately differenced series being input. The model orders are required. The main routine for parameter estimation for ARIMA models is G13AEF, and an easy-to-use version is G13AFF. Both these routines use the least squares criterion of estimation. In some circumstances the use of G13BEF or G13DDF, which use maximum likelihood (ML), is recommended. The routines require the time series values to be input, together with the ARIMA orders. Any differencing implied by the model is carried out internally. They also require the maximum number of iterations to be specified, and return the state set for use in forecasting. G13AEF should be preferred to G13AFF for: (i) more information about the differenced series, its backforecasts and the intermediate series; (ii) greater control over the output at successive iterations; (iii) more detailed control over the search policy of the nonlinear least squares algorithm; (iv) more information about the first and second derivatives of the objective function during and upon completion of the iterations. G13BEF is primarily designed for estimating relationships between time series. It is, however, easily used in a univariate mode for ARIMA model estimation. The advantage is that it allows (optional) use of the exact likelihood estimation criterion, which is not available in G13AEF or G13AFF. This is particularly recommended for models which have seasonal parameters, because it reduces the tendency of parameter estimates to become stuck at points on the parameter space boundary. The model parameters estimated in this routine should be passed over to G13AJF for use in univariate forecasting. The routine G13DDF is primarily designed for fitting vector ARMA models to multivariate time series but may also be used in a univariate mode. It allows the use of either the exact or conditional likelihood estimation criterion, and allows you to fit non-multiplicative seasonal models which are not available in G13AEF, G13AFF or G13BEF. (c) Model checking G13ASF calculates the correlations in the residuals from a model fitted by either G13AEF or G13AFF. In addition the standard errors and correlations of the residual autocorrelations are computed along with a portmanteau test for model adequacy. G13ASF can be used after a univariate model has been fitted by G13BEF, but care must be taken in selecting the correct inputs to G13ASF. Note that if G13DDF has been used to fit a non-multiplicative seasonal model to a univariate series then G13DSF may be used to check the adequacy of the model. (d) Forecasting using an ARIMA model Given that the state set produced on estimation of the ARIMA model by either G13AEF or G13AFF has been retained, G13AHF can be used directly to construct forecasts for ${x}_{n+1},{x}_{n+2},\dots \text{}$, together with probability limits. If some further observations ${x}_{n+1},{x}_{n+2},\dots \text{}$ have come to hand since model estimation (and there is no desire to re-estimate the model using the extended series), then G13AGF can be used to update the state set using the new observations, prior to forecasting from the end of the extended series. The original series is not required. The routine G13AJF is provided for forecasting when the ARIMA model is known but the state set is unknown. For example, the model may have been estimated by a procedure other than the use of G13AEF or G13AFF, such as G13BEF. G13AJF constructs the state set and optionally constructs forecasts with probability limits. It is equivalent to a call to G13AEF with zero iterations requested, followed by an optional call to G13AHF, but it is much more efficient. #### 3.1.2 Exponential smoothing A variety of different smoothing methods are provided by G13AMF, including; single exponential, Brown's double exponential, linear Holt (also called double exponential smoothing in some references), additive Holt–Winters and multiplicative Holt–Winters. The choice of smoothing method used depends on the characteristics of the time series. If the mean of the series is only slowly changing then single exponential smoothing may be suitable. If there is a trend in the time series, which itself may be slowly changing, then double exponential smoothing may be suitable. If there is a seasonal component to the time series, e.g., daily or monthly data, then one of the two Holt–Winters methods may be suitable. #### 3.1.3 Change point analysis Four routines are available for change point analysis, two implementing the PELT algorithm (G13NAF and G13NBF) and two binary segmentation (G13NDF and G13NEF). Of these, G13NAF and G13NDF have six pre-defined cost functions based on the log-likelihood of the Normal, Gamma, Exponential and Poisson distributions. In the case of the Normal distribution changes in the mean, standard deviation or both can be investigated. The remaining two routines, G13NBF and G13NEF take a user-supplied cost function. Binary segmentation only returns an approximate solution to the change point problem as defined in equation (5). It is therefore recommended that the PELT algorithm is used in most cases. However, for long time series the binary segmentation algorithm may give a marked improvement in terms of speed especially if the maximum depth for the iterative process (MDEPTH) is set to a low value. ### 3.2 Univariate Spectral Analysis Two routines are available, G13CAF carrying out smoothing using a lag window and G13CBF carrying out direct frequency domain smoothing. Both can take as input the original series, but G13CAF alone can use the sample autocovariances as alternative input. This has some computational advantage if a variety of spectral estimates needs to be examined for the same series using different amounts of smoothing. However, the real choice in most cases will be which of the four shapes of lag window in G13CAF to use, or whether to use the trapezium frequency window of G13CBF. The references may be consulted for advice on this, but the two most recommended lag windows are the Tukey and Parzen. The Tukey window has a very small risk of supplying negative spectrum estimates; otherwise, for the same bandwidth, both give very similar results, though the Parzen window requires a higher truncation lag (more autocorrelation function (ACF) values). The frequency window smoothing procedure of G13CBF with a trapezium shape parameter $p\simeq \frac{1}{2}$ generally gives similar results for the same bandwidth as lag window methods with a slight advantage of somewhat less distortion around sharp peaks, but suffering a rather less smooth appearance in fine detail. ### 3.3 Linear Lagged Relationships Between Time Series The availability of routines for each of four steps: identification, model fitting, model checking and forecasting, is given below. (a) Model identification Normally use G13BCF for direct computation of cross-correlations, from which cross-covariances may be obtained by multiplying by ${s}_{y}{s}_{x}$, and impulse response estimates (after prewhitening) by multiplying by ${s}_{y}/{s}_{x}$, where ${s}_{y},{s}_{x}$ are the sample standard deviations of the series. An alternative is to use G13CCF, which exploits the fast Fourier transform (FFT) to carry out the convolution for computing cross-covariances. The criteria for this are the same as given in Section 3.1.1 for calculation of autocorrelations. The impulse response function may also be computed by spectral methods without prewhitening using G13CGF. G13BAF may be used to prewhiten or filter a series by an ARIMA model. G13BBF may be used to filter a time series using a transfer function model. (b) Estimation of input-output model parameters The routine G13BDF is used to obtain preliminary estimates of transfer function model parameters. The model orders and an estimate of the impulse response function (see Section 3.2) are required. The simultaneous estimation of the transfer function model parameters for the inputs, and ARIMA model parameters for the output, is carried out by G13BEF. This routine requires values of the output and input series, and the orders of all the models. Any differencing implied by the model is carried out internally. The routine also requires the maximum number of iterations to be specified, and returns the state set for use in forecasting. (c) Input-output model checking The routine G13ASF, primarily designed for univariate time series, can be used to test the residuals from an input-output model. (d) Forecasting using an input-output model Given that the state set produced on estimation of the model by G13BEF has been retained, the routine G13BHF can be used directly to construct forecasts of the output series. Future values of the input series (possibly forecasts previously obtained using G13AHF) are required. If further observations of the output and input series have become available since model estimation (and there is no desire to re-estimate the model using the extended series) then G13BGF can be used to update the state set using the new observations prior to forecasting from the end of the extended series. The original series are not required. The routine G13BJF is provided for forecasting when the multi-input model is known, but the state set is unknown. The set of output and input series must be supplied to the routine which then constructs the state set (for future use with G13BGF and/or G13BHF) and also optionally constructs forecasts of the output series in a similar manner to G13BHF. In constructing probability limits for the forecasts, it is possible to allow for the fact that future input series values may themselves have been calculated as forecasts using ARIMA models. Use of this option requires that these ARIMA models be supplied to the routine. (e) Filtering a time series using a transfer function model The routine for this purpose is G13BBF. ### 3.4 Multivariate Time Series The availability of routines for each of four steps: identification, model fitting, model checking and forecasting, is given below. (a) Model identification The routine G13DLF may be used to difference the series. You must supply the differencing parameters for each component of the multivariate series. The order of differencing for each individual component does not have to be the same. The routine may also be used to apply a log or square root transformation to the components of the series. The routine G13DMF may be used to calculate the sample cross-correlation or cross-covariance matrices. It requires a set of time series as input. You may request either the cross-covariances or cross-correlations. The routine G13DNF computes the partial lag correlation matrices from the sample cross-correlation matrices computed by G13DMF, and the routine G13DPF computes the least squares estimates of the partial autoregression matrices and their standard errors. Both routines compute a series of ${\chi }^{2}$ statistics that aid the determination of the order of a suitable autoregressive model. G13DBF may also be used in the identification of the order of an autoregressive model. The routine computes multiple squared partial autocorrelations and predictive error variance ratios from the sample cross-correlations or cross-covariances computed by G13DMF. The routine G13DXF may be used to check that the autoregressive part of the model is stationary and that the moving-average part is invertible. (b) Estimation of VARMA model parameters The routine for this purpose is G13DDF. This routine requires a set of time series to be input, together with values for $p$ and $q$. You must also specify the maximum number of likelihood evaluations to be permitted and which parameters (if any) are to be held at their initial (user-supplied) values. The fitting criterion is either exact maximum likelihood (ML) or conditional maximum likelihood. G13DDF is primarily designed for estimating relationships between time series. It may, however, easily be used in univariate mode for non-seasonal and non-multiplicative seasonal ARIMA model estimation. The advantage is that it allows (optional) use of the exact maximum likelihood (ML) estimation criterion, which is not available in either G13AEF or G13AFF. The conditional likelihood option is recommended for those models in which the parameter estimates display a tendency to become stuck at points on the boundary of the parameter space. When one of the series is known to be influenced by all the others, but the others in turn are mutually independent and do not influence the output series, then G13BEF (the transfer function (TF) model fitting routine) may be more appropriate to use. (c) VARMA model checking G13DSF calculates the cross-correlation matrices of residuals for a model fitted by G13DDF. In addition the standard errors and correlations of the residual correlation matrices are computed along with a portmanteau test for model adequacy. (d) Forecasting using a VARMA model The routine G13DJF may be used to construct a chosen number of forecasts using the model estimated by G13DDF. The standard errors of the forecasts are also computed. A reference vector is set up by G13DJF so that should any further observations become available the existing forecasts can be efficiently updated using G13DKF. On a call to G13DKF the reference vector itself is also updated so that G13DKF may be called again each time new observations are available. ### 3.5 Cross-spectral Analysis Two routines are available for the main step in cross-spectral analysis. To compute the cospectrum and quadrature spectrum estimates using smoothing by a lag window, G13CCF should be used. It takes as input either the original series or cross-covariances which may be computed in a previous call of the same routine or possibly using results from G13BCF. As in the univariate case, this gives some advantage if estimates for the same series are to be computed with different amounts of smoothing. The choice of window shape will be determined as the same as that which has already been used in univariate spectrum estimation for the series. For direct frequency domain smoothing, G13CDF should be used, with similar consideration for the univariate estimation in choice of degree of smoothing. The cross-amplitude and squared coherency spectrum estimates are calculated, together with upper and lower confidence bounds, using G13CEF. For input the cross-spectral estimates from either G13CCF or G13CDF and corresponding univariate spectra from either G13CAF or G13CBF are required. The gain and phase spectrum estimates are calculated together with upper and lower confidence bounds using G13CFF. The required input is as for G13CEF above. The noise spectrum estimates and impulse response function estimates are calculated together with multiplying factors for confidence limits on the former, and the standard error for the latter, using G13CGF. The required input is again the same as for G13CEF above. ### 3.6 Kalman Filtering #### 3.6.1 Linear state space models Two routines are available for analysing a linear state space model using Kalman filtering: G13EAF for time varying systems and G13EBF for time invariant systems. The latter will optionally compute the required transformation to lower observer Hessenberg form. Both these routines return the Cholesky factor of the residual covariance matrix, ${H}_{t}$, with the Cholesky factor of the state covariance matrix ${S}_{t+1}$ and the Kalman gain matrix, ${K}_{t}$ pre-multiplied by ${A}_{t}$; in the case of G13EBF these may be for the transformed system. To compute the updated state vector and the residual vector the required matrix-vector multiplications can be performed by F06PAF (DGEMV). #### 3.6.2 Nonlinear state space models Two routines are available for analysing a nonlinear state space model: G13EJF and G13EKF. The difference between the two routines is how the nonlinear functions, $F$ and $H$ are supplied, with G13EJF using reverse communication and G13EKF using direct communication. See Section 3.3.3 in How to Use the NAG Library and its Documentation for a description of the terms reverse and direct communication. As well as having the additional flexibility inherent in reverse communication routines G13EJF also offers an alternative method of generating the sigma points utilized by the Unscented Kalman Filter (UKF), potentially allowing for additional information to be propagated through the state space model. However, due to the increased complexity of the interface it is recommended that G13EKF is used unless this additional flexibility is definitely required. ### 3.7 GARCH Models The main choice in selecting a type of GARCH model is whether the data is symmetric or asymmetric and if asymmetric what form of asymmetry should be included in the model. A symmetric ARCH or GARCH model can be fitted by G13FAF and the volatility forecast by G13FBF. For asymmetric data the choice is between the type of asymmetry as described in Section 2.7. GARCH Type Fit Forecast Type I G13FAF G13FBF Type II G13FCF G13FDF GJR G13FEF G13FFF EGARCH G13FGF G13FHF All routines allow the option of including regressor variables in the model and the choice between Normal and Student's $t$-distribution for the errors. ### 3.8 Inhomogeneous Time Series The following routines deal with inhomogeneous time series, G13MEF, G13MFF and G13MGF. Both G13MEF and G13MFF calculate the $m$-iterated exponential moving average (EMA). In most cases G13MEF can be used, which returns $\text{EMA}\left[\tau ,m;z\right]$ for a given value of $m$, overwriting the input data. Sometimes it is advantageous to have access to the intermediate results, for example when calculating the differential operator, in which case G13MFF can be used, which can return $\text{EMA}\left[\tau ,\mathit{i};z\right]$, for $\mathit{i}=1,2,\dots ,m$. G13MFF can also be used if you do not wish to overwrite the input data. The last routine, G13MGF should be used if you require the moving average, (MA), moving norm (MNorm), moving variance (MVar) or moving standard deviation (MSD). Other operators can be calculated by calling a combination of these three routines and the use of simple mathematics (additions, subtractions, etc.). ### 3.9 Time Series Simulation There are routines available in Chapter G05 for generating a realization of a time series from a specified model: G05PHF for univariate time series and G05PJF for multivariate time series. There is also a suite of routines for simulating GARCH models: G05PDF, G05PEF, G05PFF and G05PGF. The routine G05PMF can be used to simulate data from an exponential smoothing model. ## 4 Functionality Index ARMA modelling, ACF G13ABF diagnostic checking G13ASF Dickey–Fuller unit root test G13AWF differencing G13AAF estimation (comprehensive) G13AEF estimation (easy-to-use) G13AFF forecasting from fully specified model G13AJF forecasting from state set G13AHF mean/range G13AUF PACF G13ACF update state set G13AGF Change point, detection, binary segmentation G13NDF binary segmentation, user supplied cost function G13NEF PELT G13NAF PELT, user supplied cost function G13NBF Exponential smoothing G13AMF GARCH, EGARCH, fitting G13FGF forecasting G13FHF GJR GARCH, fitting G13FEF forecasting G13FFF symmetric or type I AGARCH, fitting G13FAF forecasting G13FBF type II AGARCH, fitting G13FCF forecasting G13FDF Inhomogenous series, iterated exponential moving average, final value only returned G13MEF intermediate values returned G13MFF moving average G13MGF Kalman, filter, time invariant, square root covariance G13EBF time varying, square root covariance G13EAF unscented G13EKF unscented (reverse communication) G13EJF Spectral analysis Bivariate, Bartlett, Tukey, Parzen windows G13CCF cross amplitude spectrum G13CEF direct smoothing G13CDF gain and phase G13CFF noise spectrum G13CGF Univariate, Bartlett, Tukey, Parzen windows G13CAF direct smoothing G13CBF Transfer function modelling, cross-correlations G13BCF filtering G13BBF fitting G13BEF forecasting from fully specified model G13BJF forecasting from state set G13BHF preliminary estimation G13BDF pre-whitening G13BAF update state set G13BGF Vector ARMA, cross-correlations G13DMF diagnostic checks G13DSF differencing G13DLF fitting G13DDF forecasting G13DJF partial autoregression matrices G13DPF partial correlation matrices G13DNF squared partial autocorrelations G13DBF update forecast G13DKF zeros of ARIMA operator G13DXF ## 5 Auxiliary Routines Associated with Library Routine Arguments G13AFZ nagf_tsa_uni_arima_estim_sample_pivSee the description of the argument PIV in G13AEF. ## 6 Routines Withdrawn or Scheduled for Withdrawal The following lists all those routines that have been withdrawn since Mark 19 of the Library or are scheduled for withdrawal at one of the next two marks. WithdrawnRoutine Mark ofWithdrawal Replacement Routine(s) G13DCF 24 G13DDF ## 7 References Akaike H (1971) Autoregressive model fitting for control Ann. Inst. Statist. Math. 23 163–180 Bollerslev T (1986) Generalised autoregressive conditional heteroskedasticity Journal of Econometrics 31 307–327 Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day Engle R (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica 50 987–1008 Gentleman W S and Sande G (1966) Fast Fourier transforms for fun and profit Proc. Joint Computer Conference, AFIPS 29 563–578 Glosten L, Jagannathan R and Runkle D (1993) Relationship between the expected value and the volatility of nominal excess return on stocks Journal of Finance 48 1779–1801 Hamilton J (1994) Time Series Analysis Princeton University Press Haykin S (2001) Kalman Filtering and Neural Networks John Wiley and Sons Heyse J F and Wei W W S (1985) The partial lag autocorrelation function Technical Report No. 32 Department of Statistics, Temple University, Philadelphia Julier S J and Uhlmann J K (1997) A new extension of the Kalman Filter to nonlinear systems International Symposium for Aerospace/Defense, Sensing, Simulation and Controls (Volume 3) 26 Killick R, Fearnhead P and Eckely I A (2012) Optimal detection of changepoints with a linear computational cost Journal of the American Statistical Association 107:500 1590–1598 Tiao G C and Box G E P (1981) Modelling multiple time series with applications J. Am. Stat. Assoc. 76 802–816 Wei W W S (1990) Time Series Analysis: Univariate and Multivariate Methods Addison–Wesley Zumbach G O and Müller U A (2001) Operators on inhomogeneous time series International Journal of Theoretical and Applied Finance 4(1) 147–178 G13 Chapter Contents	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 710, "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": 0.9361021518707275, "perplexity": 3909.571736288417}, "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-2018-47/segments/1542039746398.20/warc/CC-MAIN-20181120130743-20181120152743-00517.warc.gz"}
https://math.stackexchange.com/questions/562846/two-questions-regarding-convergence-almost-everywhere	# Two questions regarding convergence almost everywhere: For the purpose of this post $\mathcal{L}$ denotes the Lebesgue $\sigma$ -algebra, $\mathcal{B}$ denotes the Borel $\sigma$ -algebra and $\lambda$ denotes the Lebesgue Measure. I'm trying prove the following two claims: 1. Let $A\subseteq\mathbb{R}$ be Lebesgue measurable with $\lambda\left(A\right)=\infty$ and let $f_{n}:\left(A,\mathcal{L}\right)\to\left(\mathbb{R},\mathcal{B}\right)$ be a sequence of borel-measurable functions such that $f_{n}$ converges to $f$ almost everywhere on $A$ (relative to $\lambda$). Show that for each $M>0$ there is a $\lambda$-measurable $A_{M}\subseteq A$ such that $\lambda\left(A_{M}\right)>M$ and $f_{n}$ converges uniformly to $f$ on $A_{M}$. My Thoughts: This is obviously similar to Egoroff's theorem and I have a feeling it could be used in the proof. I suppose from Egoroff's Theorem given $B\subseteq A$ such that $\lambda\left(B\right)<\infty$ for any $\varepsilon>0$ I can find a subset $C\subseteq B$ such that $\lambda\left(C\right)<\varepsilon$ and $f_{n}$ converges uniformly on $B\backslash C$ . I don't have a good idea on how to use this to prove the intended claim though. 2. Let $f_{n}:\left(\left[0,1\right],\mathcal{L}\right)\to\left(\mathbb{R},\mathcal{B}\right)$ be a sequence of Borel-measurable functions. Prove that there is a sequence $\left\{ c_{n}\right\} _{n=1}^{\infty}$ of positive real numbers such that $\frac{f_{n}}{c_{n}}\overset{n\to\infty}{\longrightarrow}0$ almost everywhere on $\left[0,1\right]$. [Hint: use Borel-Cantelli's Lemma] . My Thoughts: As per the hint my line of thinking was to try and find a sequence of sets $\left\{ B_{n}\right\} _{n=1}^{\infty}$ such ${\displaystyle \sum_{n=1}^{\infty}\lambda\left(B_{n}\right)<\infty}$ and $$\limsup B_{n}=\left\{ x\in\left[0,1\right]\:\|\:\lim\limits _{n\to\infty}\left(\frac{f_{n}\left(x\right)}{c_{n}}\right)\neq0\right\}$$ This given some choice of $\left\{ c_{n}\right\} _{n=1}^{\infty}$ which works. Unfortunately I have no idea what that choice might be. As for the $B_{n}$'s I imagine they would have to be something of the form $$\left\{ x\in\left[0,1\right]\,\|\,\left\|\frac{f_{n}\left(x\right)}{c_{n}}\right\|>\frac{1}{n}\right\}$$ but I'm not really sure if this works. ּ Proof of the second claim: To find an appropriate sequence $\{c_n\}$, fix $n$, and define $B_k=\{ x : \frac1k\|f_n(x)\|\geq\frac1n \}$. Then $B_k \searrow \emptyset$, and $\lambda(B_k)\rightarrow 0$. Therefore, there exists $k_n\in\mathbb N$ s.t. $\lambda(B_{k_n})<2^{-n}$. Letting $c_n=k_n$, and $A_n=B_{k_n}$, we have (by Borel-Cantelli) $$\lambda (\limsup A_n)=0.$$ Letting $$L:=\{x : \lim\frac1{c_n}f_n(x)=0\},$$ we note that $L^\complement\subseteq \limsup A_n$. Hence, $\lambda(L^\complement)=0$, and $\frac1{c_n} f_n\overset{\text{a.e.}}{\longrightarrow} 0$. Proof of the first claim: First we'll show that for all $M\in\mathbb{R}$ there is an $A_{M}\subseteq A$ such that $\lambda\left(A_{M}\right)>M$ , define:$\mathcal{A}=\left\{ \lambda\left(B\right)\;\|\: B\subseteq A\quad\lambda\left(B\right)<\infty\right\}$ Now given $N\in\mathbb{N}$ we know that $\lambda\left(A\cap\left[-N,N\right]\right)\leq2N$ and thus $A\cap\left[-N,N\right]\in\mathcal{A}$ and we can see that:$$\infty=\lambda\left(A\right)=\lim_{N\in\mathbb{N}}\mathcal{\lambda}\left(A\cap\left[-N,N\right]\right)\leq\sup\left(\mathcal{A}\right)$$ Thus for all $M\in\mathbb{R}$ there is an $A_{M}\subseteq A$ such that $M<\lambda\left(A_{M}\right)<\infty$ . Now given $M>0$ and $\varepsilon>0$ we have a subset $A_{M+\varepsilon}\subseteq A$ such that $M+\varepsilon<\lambda\left(A_{M}\right)<\infty$ . From Egoroff's Theorem there is a $B\subseteq A_{M+\varepsilon}$ such that $\lambda\left(B\right)<\varepsilon$ and $f_{n}$ converges uniformly to $f$ on $A_{M+\varepsilon}\backslash B$ and additionally $$\lambda\left(A_{M}\backslash B\right)=\lambda\left(A_{M}\right)-\lambda\left(B\right)>\left(M+\varepsilon\right)-\varepsilon=M$$ Thus $A_{M}=A_{M+\varepsilon}\backslash B$ meets the requirements of the claim.	{"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": 0.9979757070541382, "perplexity": 46.22428631950493}, "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/1558232256763.42/warc/CC-MAIN-20190522043027-20190522065027-00407.warc.gz"}
https://deepai.org/publication/minimum-wasserstein-distance-estimator-under-finite-location-scale-mixtures	DeepAI # Minimum Wasserstein Distance Estimator under Finite Location-scale Mixtures When a population exhibits heterogeneity, we often model it via a finite mixture: decompose it into several different but homogeneous subpopulations. Contemporary practice favors learning the mixtures by maximizing the likelihood for statistical efficiency and the convenient EM-algorithm for numerical computation. Yet the maximum likelihood estimate (MLE) is not well defined for the most widely used finite normal mixture in particular and for finite location-scale mixture in general. We hence investigate feasible alternatives to MLE such as minimum distance estimators. Recently, the Wasserstein distance has drawn increased attention in the machine learning community. It has intuitive geometric interpretation and is successfully employed in many new applications. Do we gain anything by learning finite location-scale mixtures via a minimum Wasserstein distance estimator (MWDE)? This paper investigates this possibility in several respects. We find that the MWDE is consistent and derive a numerical solution under finite location-scale mixtures. We study its robustness against outliers and mild model mis-specifications. Our moderate scaled simulation study shows the MWDE suffers some efficiency loss against a penalized version of MLE in general without noticeable gain in robustness. We reaffirm the general superiority of the likelihood based learning strategies even for the non-regular finite location-scale mixtures. • 27 publications • 12 publications 09/17/2018 ### Homogeneity testing under finite location-scale mixtures The testing problem for the order of finite mixture models has a long hi... 06/26/2022 ### The Sketched Wasserstein Distance for mixture distributions The Sketched Wasserstein Distance (W^S) is a new probability distance sp... 08/22/2020 ### Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces The class of location-scale finite mixtures is of enduring interest both... 01/11/2015 ### Identifiability and optimal rates of convergence for parameters of multiple types in finite mixtures This paper studies identifiability and convergence behaviors for paramet... 09/22/2022 ### A unified study for estimation of order restricted location/scale parameters under the generalized Pitman nearness criterion We consider component-wise estimation of order restricted location/scale... 10/20/2020 ### Distributed Learning of Finite Gaussian Mixtures Advances in information technology have led to extremely large datasets ... 06/12/2018 ### Estimating finite mixtures of semi-Markov chains: an application to the segmentation of temporal sensory data In food science, it is of great interest to get information about the te... ## 1 Introduction Let be a parametric distribution family with density function with respect to some -finite measure. Denote by a distribution assigning probability on . A distribution with the following density function f(x\|G)=∫f(x\|\boldmathθ)dG(\boldmathθ)=K∑k=1wkf(x\|\boldmathθk) is called a finite mixture. We call the subpopulation density function, the subpopulation parameter, and the mixing weight of the th subpopulation. We use and for the cumulative distribution functions (CDF) of and respectively. Let GK={G:G=K∑k=1wk{\boldmathθk},0≤wk≤1,K∑k=1wk=1,\boldmathθk∈Θ} be a space of mixing distributions with at most support points. A mixture distribution of (exactly) order has its mixing distribution being a member of . We study the problem of learning the mixing distribution given a set of independent and identically distributed (IID) observations from a mixture . Throughout the paper, we assume the order of is known and is a known location-scale family. That is, f(x\|\boldmathθ)=1σf0(x−μσ) for some probability density function with with respect to Lebesgue measure where with . Finite mixture models provide a natural representation of heterogeneous population that is believed to be composed of several homogeneous subpopulations (Pearson, 1894; Schork et al., 1996). They are also useful for approximating distributions with unknown shapes which are particularly relevant in image generation (Kolouri et al., 2018), image segmentation (Farnoosh and Zarpak, 2008), object tracking (Santosh et al., 2013), and signal processing (Plataniotis and Hatzinak, 2000). In statistics, the most fundamental task is to learn the unknown parameters. In early days, the method of moments was the choice for its ease of computation (Pearson, 1894) under finite mixture models. Nowadays, the maximum likelihood estimate (MLE) is the first choice due to its statistical efficiency and the availability of an easy-to-use EM-algorithm. Under a finite location-scale mixture model, the log-likelihood function of is given by (1) At an arbitrary mixing distribution we have as . Hence, the MLE of is not well defined or is ill defined. Various remedies, such as penalized maximum likelihood estimate (pMLE), has been proposed to overcome this obstacle (Chen et al., 2008; Chen and Tan, 2009) . At the same time, MLE can be thought of a special minimum distance estimator. It minimizes a specific Kullback-Leibler divergence between the empirical distribution and the assumed model . Other divergences and distances have been investigated in the literature as in Choi (1969); Yakowitz (1969); Woodward et al. (1984); Clarke and Heathcote (1994); Cutler and Cordero-Brana (1996); Deely and Kruse (1968). Recently, the Wasserstein distance has drawn increased attention in machine learning community due to its intuitive interpretation and good geometric properties (Evans and Matsen, 2012; Arjovsky et al., 2017). The Wasserstein distance based estimator for learning finite mixture models is absent in the literature. Are there any benefits to learn finite location-scale mixtures by the minimum Wasserstein distance estimator (MWDE)? This paper answers this question from several angles. We find that the MWDE is consistent and derive a numerical solution under finite location-scale mixtures. We compare the robustness of the MWDE with pMLE in the presence of outliers and mild model mis-specifications. We conclude that the MWDE suffers some efficiency loss against pMLE in general without obvious gain in robustness. Through this paper, we better understand the pros and cons of the MWDE under finite location-scale mixtures. We reaffirm the general superiority of the likelihood based learning strategies even for the non-regular finite location-scale mixtures. In the next section, we first introduce the Wasserstein distance and some of its properties. This is followed by a formal definition of the MWDE, a discussion of its existence and consistency under finite location-scale mixtures. In Section 2.4, we give some algebraic results that are essential for computing -Wasserstein distance between the empirical distribution and the finite location-scale mixtures. We then develop a BFGS algorithm scheme for computing the MWDE of the mixing distribution. In addition, we briefly review the penalized likelihood approach and its numerical issues. In Section 3, we characterize the efficiency properties of the MWDE relative to pMLE in various circumstances via simulation. We also study their robustness when the data contains outliers, is contaminated or when the model is mis-specified. We then apply both methods in an image segmentation example. We conclude the paper with a summary in Section 4. ## 2 Wasserstein Distance and the Minimum Distance Estimator ### 2.1 Wasserstein Distance Wasserstein distance is a distance between probability measures. Let be a Polish space endowed with a ground distance and the space of Borel probability measures on . Let be a probability measure. If for some , ∫ΩDp(x,x0)η(dx)<∞, for some (and thus any) , we say has finite th moment. Denote by the space of probability measures with finite th moment. For any , we use to denote the space of the bivariate probability measures on whose marginals are and . Namely, Π(η,ν)={π∈P(Ω2):∫Ωπ(x,dy)=η(x), ∫Ωπ(dx,y)=ν(y)}. The -Wasserstein distance is defined as follows. ###### Definition 2.1 (p-Wasserstein distance). For any with , the th Wasserstein distance between and is Wp(η,ν)={infπ∈Π(η,ν)∫Ω2Dp(x,y)π(dx,dy)}1/p. Suppose and are two random variables whose distributions are and and induced probability measures are and . We regard the -Wasserstein distance between and also the distance between random variables or distributions: . The -Wasserstein distance is a distance on as shown by Villani (2003, Theorem 7.3). For any , it has the following properties: 1. Non-negativity: and if and only if ; 2. Symmetry: ; 3. Triangular inequality: . The Wasserstein distance has many nice properties. Let us denote for convergence in distribution or measure. Villani (2003, Theorem 7.1.2) shows that it has the following properties: • Property 1. For any , . • Property 2. as if and only if both • , and • for some (and thus any) . Computing the Wasserstein distance involves a challenging optimization problem in general but has a simple solution under a special case. Suppose is the space of real numbers, , and and are univariate distributions. Let and for be their quantile functions. We can easily compute the Wasserstein distance based on the following property. • Property 3. . ### 2.2 Minimum Wasserstein Distance Estimator Let be the -Wasserstein distance with ground distance for univariate random variables. Let be a set of IID observations from finite location-scale mixture of order and be the empirical distribution. We introduce the MWDE of the mixing distribution that is ^GMWDEN=arginfG∈GKWp(FN(⋅),F(⋅\|G))=arginfG∈GKWpp(FN(⋅),F(⋅\|G)). (2) As we pointed out earlier, the MLE is not well defined under finite location-scale mixtures. Is the MWDE well defined? We examine the existence or sensibility of the MWDE. We show that the MWDE exists when satisfies certain conditions. Assume that , is bounded, continuous, and has finite th moment. Under these conditions, we can see 0≤Wp(FN(⋅),F(⋅\|G))<∞ for any . When , the solution to (2) merits special attention. Let be a mixing distribution assigning probability on . When , each subpopulation in the mixture degenerates to a point mass at . Hence, as , Wp(FN(⋅),F(⋅\|Gϵ))→0. Since none of has zero-distance from , the MWDE does not exist unless we expand to include . To remove this technical artifact, in the MWDE definition we expand the space of to . We denote by a distribution with point mass at . With this expansion, is the MWDE when . Let . Clearly, . By definition, there exists a sequence of mixing distributions such that as . Suppose one mixing weight of has limit 0. Removing this support point and rescaling, we get a new mixing distribution sequence and it still satisfies . For this reason, we assume that its mixing weights have non-zero limits by selecting converging subsequence if necessary to ensure the limits exist. Further, when the mixing weights of assume their limiting values while keeping subpopulation parameters the same, we still have as . In the following discussion, we therefore discuss the sequence of mixing distributions whose mixing weights are fixed. Suppose the first subpopulation of has its scale parameter as . With the boundedness assumption on , the mass of this subpopulation will spread thinly over entire because uniformly. For any fixed finite interval, [], this thinning makes F(b\|\boldmathθ1)−F(a\|\boldmathθ1)→0 as . It implies that for any given , we have \|F−1(t\|\boldmathθ1)\|+\|F−1(1−t\|\boldmathθ1)\|→∞. This further implies for any , we have \|F−1(t\|Gm)\|+\|F−1(1−t\|Gm)\|→∞ as . In comparison, the empirical quantile satisfies for any . By Property 3 of , these lead to as . This contradicts the assumption . Hence, is not a possible scenario of nor for any . Can a subpopulation of instead have its location parameter ? For definitiveness, let this subpopulation correspond to . Note that at least -sized probability mass of is contained in the range . Because of this, when , we have for . Therefore, by Property 3. This contradicts . Hence, is not a possible scenario of either. For the same reason, we cannot have for any . After ruling out and , we find has a converging subsequence whose limit is a proper mixing distribution in . This limit is then an MWDE and the existence is verified. The MWDE may not be unique and the mixing distribution may lead to a mixture with degenerate subpopulations. We will show that the MWDE is consistent as the sample size goes to infinity. Thus, having degenerated subpopulations in the learned mixture is a mathematical artifact and also a sensible solution. In contrast, no matter how large the sample size becomes, there are always degenerated mixing distributions with unbounded likelihood values. ### 2.3 Consistency of MWDE We consider the problem when are IID observations from a finite location-scale mixture of order . The true mixing distribution is denoted as . Assume that is bounded, continuous, and has finite th moment. We say the location-scale mixture is identifiable if F(x\|G1)=F(x\|G2) for all given implies . We allow subpopulation scale . The most commonly used finite locate-scale mixtures, such as the normal mixture, are well known to be identifiable (Teicher, 1961). Holzmann et al. (2004) give a sufficient condition for the identifiability of general finite location-scale mixtures. Let be the characteristic function of . The finite location-scale mixture is identifiable if for any . . We consider the MWDE based on -Wasserstein distance with ground distance for some . The MWDE under finite location-scale mixture model as defined in (2) is asymptotically consistent. ###### Theorem 2.1. With the same conditions on the finite location-scale mixture and same notations above, we have the following conclusions. 1. For any sequence and , implies as . 2. The MWDE satisfies as almost surely. 3. The MWDE is consistent: as almost surely. ###### Proof. We present these three conclusions in the current order which is easy to understand. For the sake of proof, a different order is better. For ease presentation, we write and in this proof. We first prove the second conclusion. By the triangular inequality and the definition of the minimum distance estimator, we have Wp(F∗,F(⋅\|^GN))≤Wp(FN,F∗)+Wp(FN,F(⋅\|^GN))≤2Wp(FN,F∗). Note that is the empirical distribution and is the true distribution, we have uniformly in almost surely. At the same time, under the assumption that has finite th moment, also has finite th moment. The th moment of converges to that of almost surely. Given the ground distance , the th moment in Wasserstein distance sense is the usual moments in probability theory. By Property 2, we conclude as both conditions there are satisfied. Conclusion 3 is implied by Conclusions 1 and 2. With Conclusion 2 already established, we need only prove Conclusion 1 to complete the whole proof. By Helly’s lemma (Van der Vaart, 2000, Lemma 2.5) again, has a converging subsequence though the limit can be a sub-probability measure. Without loss of generality, we assume that itself converges with limit . If is a sub-probability measure, so would be . This will lead to Wp(F(⋅\|Gm),F(⋅\|G∗))→Wp(F(⋅\|~G),F(⋅\|G∗))≠0 which violates the theorem condition. If is a proper distribution in and Wp(F(⋅\|~G),F(⋅\|G∗))=0, then by identifiability condition, we have . This implies and completes the proof. ∎ The multivariate normal mixture is another type of location-scale mixture. The above consistency result of MWDE can be easily extended to finite multivariate normal mixtures. ###### Theorem 2.2. Consider the problem when are IID observations from a finite multivariate normal mixture distribution of order and is the minimum Wasserstein distance estimator defined by (2). Let the true mixing distribution be . The MWDE is consistent: as almost surely. The rigorous proof is long though the conclusion is obvious. We offer a less formal proof based on several well known probability theory results: 1. A multivariate random variable sequence converges in distribution to if and only if converges to for any unit vector ; 2. If is multivariate normal if and only if is normal for all ; 3. The normal distribution has finite moment of any order. Let be a random vector with distribution for some , , in a general mixture model setting. Suppose as , with the notation we introduced previously, Wp(Xm,X0)→0. Then for any unit vector , based on property 2 of the Wasserstein distance and the result (I), we can see that Wp(aτXm,aτX0)→0. Next, we apply this result to normal mixture so that becomes which stands for a finite multivariate normal mixture with mixing distribution . In this case, is a random vector with distribution . Let be generic subpopulation parameters. We can see that the distribution of , is a finite normal mixture with subpopulation parameters , and mixing weights the same as those of . Let the mixing distributions after projection be and . By the same argument in the proof of Theorem 2.1, Wp(Φ(⋅\|^GN),Φ(⋅\|G∗))→0 almost surely as . This implies Wp(Φa(⋅\|^GN),Φa(⋅\|G∗))→0 almost surely as for any . Hence, by Conclusion 1 of Theorem 2.1, almost surely for any unit vector . We therefore conclude the consistency result: almost surely. ### 2.4 Numerical Solution to MWDE Both in applications and in simulation experiments, we need an effective way to compute the MWDE. We develop an algorithm that leverages the explicit form of the Wasserstein distance between two measures on for the numerical solution to the MWDE. The strategy works for any -Wasserstein distance but we only provide specifics for as it is the most widely used. Let be a random variable with distribution . Denote the mean and variance of by and . Recall that . Let be the order statistics, , and be the th quantile of the mixture for . Let T(x)=∫x−∞tf0(t)dt and define ΔFnk ΔTnk =T(ξn−μkσk)−T(ξn−1−μkσk). When , we expand the squared distance, , between the empirical distribution and as follows: WN(G) = W22(FN(⋅),F(⋅\|G)) = ∫10{F−1N(t)−F−1(t\|G)}2dt = ¯¯¯¯¯x2+K∑k=1wk{μ2k+σ2k(μ20+σ20)+2μkσkμ0} −2∑kwk{μkN∑n=1x(n)ΔFnk+σkN∑n=1x(n)ΔTnk}. The MWDE minimizes with respect to . The mixing weights and subpopulation scale parameters in this optimization problem have natural constraints. We may replace the optimization problem with an unconstrained one by the following parameter transformation: σk=exp(τk), wk=exp(tk)/{K∑j=1exp(tj)} for . We may then minimize with respect to over the unconstrained space . Furthermore, we adopt the quasi-Newton BFGS algorithm (Nocedal and Wright, 2006, Section 6.1). To use this algorithm, it is best to provide the gradients of , which are given as follows: ∂∂μjWN=2wj{μj+σjμ0−N∑n=1x(n)ΔFnj}, ∂∂τjWN=2wj{σj(μ20+σ20)+μjμ0−N∑n=1x(n)ΔTnj}∂σj∂τj for where ∂∂wkWN= {μ2k+σ2k(μ20+σ20)+2μkσkμ0}−2N−1∑n=1{x(n+1)−x(n)}ξnF(ξn\|μk,σk) −2{μkN∑n=1x(n)ΔFnk+σkN∑n=1x(n)ΔTnk}. Since is non-convex, the algorithm may find a local minimum of instead of a global minimum as required for MWDE. We use multiple initial values for the BFGS algorithm, and regard the one with the lowest value as the solution. We leave the algebraic details in the Appendix. This algorithm involves computing the quantiles and repeatedly which may lead to high computational cost. Since , it can be found efficiently via a bisection method. Fortunately, has simple analytical forms under two widely used location-scale mixtures which make the computation of efficient: 1. When which is the density function of the standard normal, we have . In this case, we find T(x)=−f0(x). 2. For finite mixture of location-scale logistic distributions, we have f0(t)=exp(−x)(1+exp(−x))2 and T(x)=∫x−∞tf0(t)dt=x1+exp(−x)−log(1+exp(x)). (3) ### 2.5 Penalized Maximum Likelihood Estimator A well investigated inference method under finite mixture of location-scale families is the pMLE (Tanaka, 2009; Chen et al., 2008). Chen et al. (2008) consider this approach for finite normal mixture models. They recommend the following penalized log-likelihood function pℓN(G\|X)=ℓN(G\|X)−aN∑k{s2x/σ2k+logσ2k} for some positive and sample variance . The log-likelihood function is given in (1). They suggest to learn the mixing distribution via pMLE defined as ^GpMLEN=argsuppℓN(G\|X). The size of controls the strength of the penalty and a recommended value is . Regularizing the likelihood function via a penalty function fixes the problem caused by degenerated subpopulations (i.e. some ). The pMLE is shown to be strongly consistent when the number of components has a known upper bound under the finite normal mixture model. The penalized likelihood approach can be easily extended to finite mixture of location-scale families. Let be the density function in the location-scale family as before. We may replace the sample variance in the penalty function by any scale-invariance statistic such as the sample inter-quartile range. This is applicable even if the variance of is not finite. We can use the EM algorithm for numerical computation. Let be the membership vector of the th observation. That is, the th entry of is 1 when the response value is an observation from the th subpopulation and 0 otherwise. When the complete data are available, the penalized complete data likelihood function of is given by pℓcN(\|X)=N∑n=1K∑k=1znklog{wkσkf0(xi−μkσk)}−aN∑k{s2x/σ2k+log(σ2k)}. Given the observed data and proposed mixing distribution , we have the conditional expectation w(t)nk=E(znk\|X,G(t))=w(t)kf(xn\|μ(t)k,σ(t)k)∑Kj=1w(t)jf(xn\|μ(t)j,σ(t)j). After this E-step, we define Q(G\|G(t))= N∑n=1K∑k=1w(t)nklog{wkσkf0(xn−μkσk)}−aN∑k{s2x/σ2k+log(σ2k)}. Note that the subpopulation parameters are well separated in . The M-step is to maximize with respect to . The solution is given by the mixing distribution with mixing weights w(t+1)k=N−1N∑n=1w(t)nk and the subpopulation parameters \boldmathθ(t+1)k=argminθ{∑nw(t)nk{logσ−f(xn\|\boldmathθ)}+aN{s2x/σ2+logσ2}} (4) with the notational convention . For general location-scale mixture, the M-step (4) may not have a closed form solution but it is merely a simple two-variable function. There are many effective algorithms in the literature to solve this optimization problem. The EM-algorithm for pMLE increases the value of the penalized likelihood after each iteration. Hence, it should converge as long as the penalized likelihood function has an upper bound. We do not give a proof as it is a standard problem. ## 3 Experiments We now study the performance of MWDE and pMLE under finite location-scale mixtures. We explore the potential advantages of the MWDE and quantify its efficiency loss, if any, by simulation experiments. Consider the following three location-scale families (Chen et al., 2020): 1. Normal distribution: . Its mean and variance are given by and . 2. Logistic distribution: . Its mean and variance are given by and . 3. Gumbel distribution (type I extreme-value distribution): . Its mean and variance are given by and where is the Euler constant. We will also include a real data example to compare the image segmentation result of using the MWDE and pMLE. ### 3.1 Performance Measure For vector valued parameters, the commonly used performance metric of their estimators is the mean squared error (MSE). A mixing distribution with finite and fixed support points can be regarded as a real-valued vector in theory. Yet the mean squared errors of the mixing weights, the subpopulation means, and the subpopulation scales are not comparable in terms of the learned finite mixture. In this study, we use two performance metrics specific for finite mixture models. Let and be the learned mixing distribution and the true mixing distribution. We use distance between the learned mixture and the true mixture as the first performance metric. The distance between two mixtures and is defined to be L2(f(⋅\|G),f(⋅\|~G))={wτSGGw−2wτSG~G~w+~wτS~G~G~w}1/2 where and are three square matrices of size with their th elements given by ∫f(x\|\boldmathθn)f(x\|\boldmathθm)dx,∫f(x\|\boldmathθn)f(x\|~\boldmathθm)dx,∫f(x\|~\boldmathθn)f(x\|~% \boldmathθm)dx. Given an observed value of a unit from the true mixture population, by Bayes’ theorem, the most probably membership of this unit is given by k∗(x)=argmaxk{w∗kf∗(x\|\boldmathθ∗k)}. Following the same rule, if is the learned mixing distribution, then the most likely membership of the unit with observed value is ^k(x)=argmaxk{^wkf(x\|^\boldmathθk)}. We cannot directly compare and because the subpopulation themselves are not labeled. Instead, the adjusted rand index (ARI) is a good performance metric for clustering accuracy. Suppose the observations in a dataset are divided into clusters by one approach, and clusters by another. Let for , where is the number of units in set . The ARI between these two clustering outcomes is defined to be ARI=∑i,j(Nij2)−(N2)−1∑i,j(Ni2)(Mj2)12∑i(Ni2)+12∑j(Mj2)−(N2)−1∑i,j(Ni2)(Mj2). When the two clustering approaches completely agree with each other, the ARI value is . When data are assigned to clusters randomly, the expected ARI value is . ARI values close to 1 indicate a high degree of agreement. We compute ARI based on clusters formed by and . For each simulation, we choose or generate a mixing distribution , then generate a random sample from mixture . This is repeated times. Let be the learned based on the th data set. We obtain the two performance metrics as follows: 1. Mean distance: ML2=R−1R∑r=1L2(f(⋅\|^G(r)),f(⋅\|G∗(r))). 2. Mean adjusted rand index: MARI=R−1R∑r=1ARI(^G(r),G∗(r)). The lower the ML2 and the higher the MARI, the better the estimator performs. ### 3.2 Performance under Homogeneous Model The homogeneous location-scale model is a special mixture model with a single subpopulation . Both MWDE and MLE are applicable for parameter estimation. There have been no studies of MWDE in this special case in the literature. It is therefore of interest to see how MWDE performs under this model. Under three location-scale models given earlier, the MWDE has closed analytical forms. Using the same notation introduced, their analytical forms are as follows. 1. Normal distribution: ^μMWDE=¯x, ^σMWDE=N∑n=1x(n){f0(ξn−1)−f0(ξn)}. 2. Logistic distribution: ^μMWDE=¯x, ^σMWDE=3π2N∑n=1x(n){T(ξn)−T(ξn−1)} where is given in (3). 3. Gumbel distribution: ^μMWDE={1−γr}−1{¯x−γT}, ^σMWDE=T−r^μMWDE where T={γ2+π2/6}−1N∑n=1x(n)∫ξnξn−	{"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": 0.9865149855613708, "perplexity": 854.2101607718452}, "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/1674764500158.5/warc/CC-MAIN-20230205000727-20230205030727-00056.warc.gz"}

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