DOCSLIB.ORG

José M. Vega-Guzmán

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
José M. Vega-Guzmán Jos´eM. Vega-Guzm´an Curriculum Vitae Contact Department of Mathematics Tel: (409) 880-8792 Information College of Arts and Sciences Fax: (409) 880-8794 Lamar University email: [email protected] 200-C Lucas Building URL: https://sites.google.com/site/profjmvg/ Beaumont, TX 77710 Education PhD (Applied Mathematics) 2008{2013 • Arizona State University, Tempe, AZ • Co-Advisors: Sergei K. Suslov and Carlos Castillo-Chavez • Dissertation Title: Solution Methods for Certain Evolution Equations Master in Natural Sciences (Mathematics) 2006{2008 • Arizona State University, Tempe, AZ B.S. (MathEd) 2000{2005 • Universidad de Puerto Rico, Cayey, PR • Advisor: Maria A. Avi~n´o Research Interests Nonlinear Wave Propagation. Quantum Mechanics. Integrability of Nonlinear Evolution Equations. Applications of ODE's and PDE's in the Natural Sciences. Mathematical Modeling. Publications 2017 Vega{Guzm´an,J.M.; Alqahtani, R.; Zhou, Q.; Mahmood, M.F.; Moshokoa, S.P.; Ullah, M.Z.; Biswas, A.; Belic, M. Optical solitons for Lakshmanan-Porsezian-Daniel model with spatio-temporal dispersion using the method of undetermined coefficients. Optik - International Journal for Light and Electron Optics. 144, 115{123. 2017 Wang, G.; Kara, A.H.; Vega{Guzm´an,J.M.; Biswas, A. Group analysis, nonlinear self-adjointness, conser- vation laws and soliton solutions for the mKdV systems. Nonlinear Analysis: Modeling and Control. 22(3), 334{346. 2017 Vega{Guzm´an,J.M.; Ullah, M.Z.; Asma, M.; Zhou, Q.; Biswas, A. Dispersive solitons in magneto-optic waveguides. Superlattices and Microstructures. 103, 161{170. 2017 Vega{Guzm´an,J.M.; Mahmood, M.F.; Zhou, Q.; Triki, H.; Arnous, A.H.; Biswas, A.; Moshokoa, S.P.; Belic, M. Solitons in nonlinear directional couplers with optical metamaterials. Nonlinear Dynamics. 87, 427{458. 2016 Vega{Guzm´an,J.M.; Zhou, Q; Mahmood, M.F.; Moraru, L.; Johnson, S.; Biswas, A. Singular optical soli- tons in parabolic and dual power law media. Journal of Computational and Theoretical Nanoscience. 13, 4825{4828. 2016 Banaja, M.A.; Alkhateeb, S.A.; Alshaery, A.A.; Hilal, E.M.; Vega-Guzman, J.M.; Moraru, L.; Zhou, Q; Biswas, A. Singular optical solitons in nonlinear directional couplers. Maejo International J. of Science and Technology. 13, 4660{4664. 2016 Wang, G.; Kara, A.H.; Fakhar, K.; Vega-Guzman, J.M.; Biswas, A. Group analysis, exact solutions and conservative laws of generalized fifth order KdV equation. Chaos, Solitons & Fractals. 86, 8{15. 2016 Brooks, S.D.; Tilghman, J.; Vega-Guzman, J.M.; Mahmood, M.F. A mathematical framework for analysis of medical adherence in the management of chronic diseases. Pioneer J. of Mathematics and Mathe- matical Science. 17(1), 7{22. 2016 Xu, Y.; Vega{Guzm´an,J.M.; Milovic, D.; Mirzazadeh, M.; Eslami, M.; Mahmood, M.F.; Biswas, A.; 1 of 7 Belic, M. Bright and exotic solitons in metamaterials by semi-inverse variational principle. Journal of Nonlinear Optical Physics Materials, 24(4), 1550042. 2016 Vega{Guzm´an,J.M.; Mahmood, M.F.; Zhou, Q; Zerrad, E.; Biswas, A.; Belic, M. Optical solitons with higher order dispersion and full nonlinearity in non-Kerr law media. Journal of Computational and Theoretical nano sciences, 24(4) 1550042. 2015 Vega{Guzm´an,J.M.; Mahmood, M.F.; Milovic, D.; Zerrad E.; Biswas, A.; Belic, M. Dark and singu- lar solitons of Kundu-Eckhaus equation for optical fibers. Journal of Optoelectronics and Advanced Materials-Rapid Communications, 9(11-12), 1353{1355. 2015 Younis, M.; Rizvi, S.; Mahmoud, A.; Vega{Guzm´an,J.M.; Zhou, Q.; Biswas, A.; Belic, M. Optical solitons in dual-core fibers with inter-modal dispersion. Journal of Optoelectronics and Advanced Materials-Rapid Communications, 9(9-10), 1126{1134. 2015 Vega{Guzm´an,J.M.; Hilal, E.M. ; Alshaery, A.A.; Bhrawy, A.H.; Mahmood, M.F.; Moraru, L.; Biswas, A. Thiring optical solitons with spatio-temporal dispersion. Proceedings of the Romanian Academy, Series A, 16(1), 41{46. 2015 Vega{Guzm´an,J.M.; Zhou, Q.; Alshaery, A.A.; Hilal, E.M.; Bhrawy, A.H.; Biswas, A. Optical solitons in cascaded systems with spatio-temporal dispersion by ansatz approach. Journal of Optoelectronics and Advanced Materials, 17(1-2), 165{171. 2014 Vega{Guzm´an,J.M.; Alshaery, A.A.; Hilal, E.M.; Bhrawy, A.H.; Mahmood, M.F.; Moraru, L.; Biswas, A. Optical soliton perturbation in magneto-optic waveguides with spatio-temporal dispersion. Journal of Optoelectronics and Advanced Materials 16(9-10), 1063-1070. 2014 Ebadi, G.; Mojaver, A.; Vega{Guzm´an,J.M.; Khan, K.R.; Mahmood, M.F.; Moraru, L.; Biswas, A.; Belic, M. Solitons in Optical Metamaterials by F-Expansion Scheme. Journal of Optoelectronics and Advanced Materials-Rapid Communications, 8(9-10), 828{832. 2014 Herrera{Valdez, M.; Suslov, S.; Vega{Guzm´an,J.M. A Graphical Approach to a Model of Neuronal Tree with Variable Diameter. Mathematics 2(3), 119{135. 2014 Suazo, E.; Suslov, S.; Vega{Guzm´an,J.M. The Riccati System and a Diffusion–type equation. Mathe- matics 2(2), 96{118. 2013 Kryuchkov, S.I.; Suslov, S.; Vega{Guzm´an,J.M. The minimum-uncertainty squeezed states for atoms and photons in cavity. Jornal of Physics B: Atoms and Molecules in Optical Physics 46, 104007 (15pp) **Selected by the Editorial Board of J. Phys. B as a Highlight of the year ** and **IOPselect** 2013 Vega{Guzm´an,J.M. Solution Methods for Certain Evolution Equations. Ph.D. Dissertation, ProQuest/UMI. 2013 Lopez, R.; Suslov, S.; Vega{Guzm´an,J.M. On a Hidden Symmetry of Quantum Harmonic Oscillators. Physica Scripta 19(4), 543{554. 2013 Lopez, R.; Suslov, S.; Vega{Guzm´an,J.M. Reconstructing the Schr¨odingerGroups. Physica Scripta 87(3), 038112 (6pp). 2011 Suazo, E.; Suslov, S.; Vega{Guzm´an,J.M. The Riccati Differential Equation and a Diffusion–type equa- tion. New York Journal of Mathematics. 17a, 225{244. 2009 Castro, F.N.; Rubio, I.; Vega, J.M. Divisibility of exponential sums and solvability of certain equations over finite fields. The Quarterly Journal of Mathematics. 60(2), 169{181. 2 of 7 Professional Employment 1. (Fall 2016 { ) Assistant Professor. Department of Mathematics, Lamar University. Beaumont, TX. • Courses Taught at Lamar University { Undergraduate: MATH 2311 (Precalculus I); MATH 2312 (Precalculus II); MATH 2413 (Calculus I); MATH 2413 (Calculus II); MATH 3435 (Calculus III); MATH 3328 (Linear Algebra I); MATH 4315 (Numerical Analysis); MATH 4318 (Applied Linear Algebra with Matrix Theory) { Graduate: MATH 5315 (Numerical Analysis); MATH 5328 (Applied Linear Algebra with Matrix Theory) 2. (Summer II 2016) Adjunct Professor. Department of Mathematics, Lamar University. Beaumont, TX. 3. (Fall 2013 { Summer 2016 ) Alliance Postdoc. Taught Calculus, College Algebra and Precalculus. Research: Nonlinear wave propagation in different medium[with M.F. Mahmood]. Department of Mathematics, Howard University. Washington, DC. 4. (Fall 2012 { Spring 2013) Research Assistant (RA). Simon A. Levin Mathematical Computational and Modeling Sciences Center. Arizona State University. Tempe, AZ. 5. (Fall 2010 { Srping 2011) Research Assistant (RA). Simon A. Levin Mathematical Computational and Modeling Sciences Center. Arizona State University, Tempe AZ. 6. (Fall 2009 { Spring 2010) Teacher Assistant (TA). Grader in Calculus for Engineers II, Modern Dif- ferential Equations, Linear Algebra, Applied Linear Algebra) Arizona State University. Tempe, AZ. Also, I taught a recitation period for the Calculus section. 7. (Summers 2008{2012) Graduate Research Assistant. Mathematical and Theoretical Biology Institute (MTBI) REU Summer Program. Arizona State University, Tempe AZ • Assisted in teaching a variety of mathematical biology topics at the level of calculus and differential equations. • Taught lessons on Calculus, Linear Algebra and Ordinary Differential Equations to undergraduate students. • Served as a Graduate Research Assistant to self directed undergraduate research for groups of 3{5 students. During Summer 2012 I additionally assisted the independent research of a foreign graduate student. 8. (Fall 2008 { Spring 2009) Research Assistant (RA). Simon A. Levin Mathematical Computational and Modeling Sciences Center. Arizona State University. Tempe, AZ. Presentations Seminar/Colloquium Talks 1. (October 2017) \Nonlinear Wave Solutions of Nonlinear Schrodinger-type equations". In: Mathematics Departmental Seminar, Lamar University. Beaumont, TX. 2. (September 2015) \Mathematical Modeling of Optical Solitons in Metamaterials: An application to In- formation Technology". In: Mathematics Departmental Colloquium, Howard University. Washington, D.C. 3. (March 2015) \Minimum Uncertainty Squeezed States for Simple Harmonic Oscillators". In: Evolution Equations and Dynamical Systems(EEDS) weekly seminar, Howard University. Washington, D.C. 4. (April 2015) \On the analytic solution of some nonautonomous evolution equations". In: Applied Mathematics Seminar, Mathematics Department, George Washington University, Washington D.C. Invited Conference Talks 3 of 7 1. (May 2017) \Minimum{Uncertainty Squeezed States for the Simple Harmonic Oscillator". In: WAVES 2017: 13TH International Conference on Mathematical and Numerical Aspects of Wave Propagation. University of Minnesota, MN. 2. (January 2016) “Schr¨odingergroups: An application in nonrelativistic quantum squeezing". In: Joint Mathematics Meeting (JMM 2016). Seattle, WA. 3. (April 2015) \The Minimum uncertainty squeezed states for quantum harmonic oscillator". In: The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory (9th IMACS 2015).
Load more

Recommended publications
  • Introduction to the Modern Calculus of Variations
    MA4G6 Lecture Notes Introduction to the Modern Calculus of Variations Filip Rindler Spring Term 2015 Filip Rindler Mathematics Institute University of Warwick Coventry CV4 7AL United Kingdom [email protected] http://www.warwick.ac.uk/filiprindler Copyright ©2015 Filip Rindler. Version 1.1. Preface These lecture notes, written for the MA4G6 Calculus of Variations course at the University of Warwick, intend to give a modern introduction to the Calculus of Variations. I have tried to cover different aspects of the field and to explain how they fit into the “big picture”. This is not an encyclopedic work; many important results are omitted and sometimes I only present a special case of a more general theorem. I have, however, tried to strike a balance between a pure introduction and a text that can be used for later revision of forgotten material. The presentation is based around a few principles: • The presentation is quite “modern” in that I use several techniques which are perhaps not usually found in an introductory text or that have only recently been developed. • For most results, I try to use “reasonable” assumptions, not necessarily minimal ones. • When presented with a choice of how to prove a result, I have usually preferred the (in my opinion) most conceptually clear approach over more “elementary” ones. For example, I use Young measures in many instances, even though this comes at the expense of a higher initial burden of abstract theory. • Wherever possible, I first present an abstract result for general functionals defined on Banach spaces to illustrate the general structure of a certain result.
    [Show full text]
  • On the Ekeland Variational Principle with Applications and Detours
    Lectures on The Ekeland Variational Principle with Applications and Detours By D. G. De Figueiredo Tata Institute of Fundamental Research, Bombay 1989 Author D. G. De Figueiredo Departmento de Mathematica Universidade de Brasilia 70.910 – Brasilia-DF BRAZIL c Tata Institute of Fundamental Research, 1989 ISBN 3-540- 51179-2-Springer-Verlag, Berlin, Heidelberg. New York. Tokyo ISBN 0-387- 51179-2-Springer-Verlag, New York. Heidelberg. Berlin. Tokyo No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by INSDOC Regional Centre, Indian Institute of Science Campus, Bangalore 560012 and published by H. Goetze, Springer-Verlag, Heidelberg, West Germany PRINTED IN INDIA Preface Since its appearance in 1972 the variational principle of Ekeland has found many applications in different fields in Analysis. The best refer- ences for those are by Ekeland himself: his survey article [23] and his book with J.-P. Aubin [2]. Not all material presented here appears in those places. Some are scattered around and there lies my motivation in writing these notes. Since they are intended to students I included a lot of related material. Those are the detours. A chapter on Nemyt- skii mappings may sound strange. However I believe it is useful, since their properties so often used are seldom proved. We always say to the students: go and look in Krasnoselskii or Vainberg! I think some of the proofs presented here are more straightforward. There are two chapters on applications to PDE.
    [Show full text]
  • The Original Euler's Calculus-Of-Variations Method: Key
    Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems.
    [Show full text]
  • Subdifferential Test for Optimality
    Subdifferential Test for Optimality Florence JULES and Marc LASSONDE Universit´edes Antilles et de la Guyane 97159 Pointe `aPitre, France E-mail: fl[email protected], [email protected] Abstract. We provide a first-order necessary and sufficient condition for optimality of lower semicontinuous functions on Banach spaces using the concept of subdifferential. From the sufficient condition we derive that any subdifferential operator is monotone absorbing, hence maximal monotone when the function is convex. Keywords: lower semicontinuity, subdifferential, directional derivative, first-order condition, optimality criterion, maximal monotonicity. 2010 Mathematics Subject Classification: 49J52, 49K27, 26D10, 26B25. 1 Introduction First-order sufficient conditions for optimality in terms of derivatives or directional derivatives are well known. Typical such conditions are variants of Minty variational inequalities. Let us quote for example the following simple result (see [3]): Let f be a real-valued function which is continuous on a neighbourhood D centred at a in Rn and differentiable on D \ {a}. Then, f has a local minimum value at a if (x − a) ·∇f(x) > 0 for all x ∈ D \ {a}. For first-order conditions in terms of directional derivatives, we refer to [5]. The main objective of this note is to establish a necessary and sufficient condition for optimality of nonsmooth lower semicontinuous functions on Banach spaces using subdiffer- entials. To this end, we first provide a link between the directional derivative of a function and its dual companion represented by the subdifferential (Theorem 2.1). Then, we prove a sharp version of the mean value inequality for lower semicontinuous function using directional derivative (Lemma 3.1).
    [Show full text]
  • 8 the Variational Principle
    8 The Variational Principle 8.1 Approximate solution of the Schroedinger equation If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise hΦ|Hˆ |Φi E[a ] = n hΦ|Φi with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used. The proof is as follows, if we expand the normalised wavefunction 1/2 |φ(an)i = Φ(an)/hΦ(an)|Φ(an)i in terms of the true (unknown) eigenbasis |ii of the Hamiltonian, then its energy is X X X ˆ 2 2 E[an] = hφ|iihi|H|jihj|φi = |hφ|ii| Ei = E0 + |hφ|ii| (Ei − E0) ≥ E0 ij i i ˆ where the true (unknown) ground state of the system is defined by H|i0i = E0|i0i. The inequality 2 arises because both |hφ|ii| and (Ei − E0) must be positive. Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φi will be to |i0i. If the trial wavefunction consists of a complete basis set of orthonormal functions |χ i, each P i multiplied by ai: |φi = i ai|χii then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set.
    [Show full text]
  • Lectures on Partial Differential Equations
    Lectures on Partial Differential Equations Govind Menon1 Dec. 2005 Abstract These are my incomplete lecture notes for the graduate introduction to PDE at Brown University in Fall 2005. The lectures on Laplace’s equation and the heat equation are included here. Typing took too much work after that. I hope what is here is still useful. Andreas Kl¨ockner’s transcript of the remaining lectures is also posted on my website. Those however have not been proofread. Comments are wel- come by email. Contents 1 Laplace’s equation 3 1.1 Introduction............................ 3 1.1.1 Minimal surfaces and Laplace’s equation . 3 1.1.2 Fields and Laplace’s equation . 5 1.1.3 Motivation v. Results . 5 1.2 Notation.............................. 5 1.3 Themeanvalueinequality. 6 1.4 Maximumprinciples ....................... 8 1.5 Thefundamentalsolution . 10 1.6 Green’s function and Poisson’s integral formula . 13 1.7 The mean value property revisited . 17 1.8 Harmonic functions are analytic . 17 1.9 Compactness and convergence . 21 1.10 Perron’s method for the Dirichlet problem . 22 1.11 Energy methods and Dirichlet’s principle . 26 1.12 Potentials of measures . 28 1.13 Lebesgue’sthorn .. .. .. .. .. .. .. 30 1.14 The potential of a compact set . 33 1.15 Capacity of compact sets . 37 1 2 1.16 Variational principles for capacity . 39 2 The heat equation 43 2.1 Motivation ............................ 43 2.2 Thefundamentalsolution . 43 2.3 Uniquenessofsolutions . 47 2.4 Theweakmaximumprinciple . 48 2.5 Themeanvalueproperty . 49 2.6 Thestrongmaximumprinciple . 53 2.7 Differenceschemes . .. .. .. .. .. .. 54 2.8 Randomwalks .......................... 56 2.9 Brownianmotion ........................
    [Show full text]
  • Variational Principles Part 1A Mathematics Tripos
    Preprint typeset in JHEP style - PAPER VERSION Easter 2018 Variational Principles Part 1A Mathematics Tripos Prof. P.K. Townsend DAMTP, CMS Wilberforce Road, Cambridge CB3 0WA E-mail: [email protected] Abstract: See schedules. There are two example sheets. RECOMMENDED BOOKS: (1) D.S. Lemons, Perfect Form, Princeton University Press. (2) I.M. Gelfand and S.V. Fomin, Calculus of Variations, Dover Contents 1. Variational problems and variational principles 2 1.1 Calculus for functions of many variables 4 2. Convex functions 6 2.1 First-order conditions 7 2.1.1 An alternative first-order condition 8 2.2 The Hessian and a second-order condition 9 3. Legendre transform 10 3.1 Application to Thermodynamics 13 4. Constrained variation and Lagrange multipliers 15 5. Functionals and the Euler-Lagrange equation 20 5.1 Geodesics of the Euclidean plane 22 6. First integrals and Fermat's principle 24 7. Constrained variation of functionals 27 7.1 Sturm-Liouville problem 28 7.2 Function constraints; geodesics on surfaces 30 8. Hamilton's principle 30 8.1 Central force fields 32 8.2 The Hamiltonian and Hamilton's equations 34 9. Symmetries and Noether's theorem 35 9.0.1 A shortcut 37 9.1 Application to Hamiltonian mechanics 38 10. PDEs from variational principles 39 10.1 Minimal surfaces 39 10.2 Small amplitude oscillations of a uniform string 42 10.3 Maxwell's equations from Hamilton's principle 42 11. The second variation 43 12. The Jacobi condition 47 { 1 { 1. Variational problems and variational principles We often want to know how to maximize or minimize some quantity by varying other quantities on which it depends.
    [Show full text]
  • On the Variational Principle
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 47, 324-353 (1974) On the Variational Principle I. EKELAND UER Mathe’matiques de la DC&ion, Waiver& Paris IX, 75775 Paris 16, France Submitted by J. L. Lions The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11F’(uJj* < l , i.e., its derivative can be made arbitrarily small. Applications are given to Plateau’s problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infi- nite-dimensional manifolds, and to control theory. 1. A GENERAL RFNJLT Let V be a complete metric space, the distance of two points u, z, E V being denoted by d(u, v). Let F: I’ -+ 08u {+ co} be a lower semicontinuous function, not identically + 00. In other words, + oo is allowed as a value for F, but at some point w,, , F(Q) is finite. Suppose now F is bounded from below: infF > --co. (l-1) As no compacity assumptions are made, there need be no point where this infimum is attained. But of course, for every E > 0, there is some point u E V such that: infF <F(u) < infF + E.
    [Show full text]
  • Variational Inference and Learning
    Variational Inference and Learning Michael Gutmann Probabilistic Modelling and Reasoning (INFR11134) School of Informatics, University of Edinburgh Spring semester 2018 Recap I Learning and inference often involves intractable integrals I For example: marginalisation Z p(x) = p(x, y)dy y I For example: likelihood in case of unobserved variables Z L(θ) = p(D; θ) = p(u, D; θ)du u I We can use Monte Carlo integration and sampling to approximate the integrals. I Alternative: variational approach to (approximate) inference and learning. Michael Gutmann Variational Inference and Learning 2 / 36 History Variational methods have a long history, in particular in physics. For example: I Fermat’s principle (1650) to explain the path of light: “light travels between two given points along the path of shortest time” (see e.g. http://www.feynmanlectures.caltech.edu/I_26.html) I Principle of least action in classical mechanics and beyond (see e.g. http://www.feynmanlectures.caltech.edu/II_19.html) I Finite elements methods to solve problems in fluid dynamics or civil engineering. Michael Gutmann Variational Inference and Learning 3 / 36 Program 1. Preparations 2. The variational principle 3. Application to inference and learning Michael Gutmann Variational Inference and Learning 4 / 36 Program 1. Preparations Concavity of the logarithm and Jensen’s inequality Kullback-Leibler divergence and its properties 2. The variational principle 3. Application to inference and learning Michael Gutmann Variational Inference and Learning 5 / 36 log is concave I log(u) is concave log(au1 + (1 − a)u2) ≥ a log(u1) + (1 − a) log(u2) a ∈ [0, 1] I log(average) ≥ average (log) log(u) I Generalisation log(u) log E[g(x)] ≥ E[log g(x)] u with g(x) > 0 I Jensen’s inequality for concave functions.
    [Show full text]
  • Variational Principle of Stationary Action for Fractional Nonlocal Media and Fields Vasily E
    Tarasov Pacific Journal of Mathematics for Industry DOI 10.1186/s40736-015-0017-1 ORIGINAL Open Access Variational principle of stationary action for fractional nonlocal media and fields Vasily E. Tarasov Abstract Derivatives and integrals of non-integer orders have a wide application to describe complex properties of physical systems and media including nonlocality of power-law type and long-term memory. We suggest an extension of the standard variational principle for fractional nonlocal media with power-law type nonlocality that is described by the Riesz-type derivatives of non-integer orders. As examples of application of the suggested variational principle, we consider an N-dimensional model of waves in anisotropic fractional nonlocal media, and a one-dimensional continuum (string) with power-law spatial dispersion. The main advantage of the suggested fractional variational principle is that it is connected with microstructural lattice approach and the lattice fractional calculus, which is recently proposed. PACS: 45.10.Hj; 04.20.Fy MSC: 26A33; 34A08; 49S05 1 Introduction Nonlocal continuum theory [8, 21] is based on the Derivatives and integrals of non-integer orders [11, 13, 23, assumption that the forces between particles of contin- 42, 43] have wide application in physics and mechanics uum have long-range type thus reflecting the long-range [6, 14, 22, 29, 30]. The tools of fractional derivatives and character of inter-atomic forces. If the nonlocality has integrals allow us to investigate the behavior of materials a power-law type then we can use the fractional calcu- and systems that are characterized by power-law nonlo- lus for nonlocal continuum mechanics.
    [Show full text]
  • Arxiv:1704.00088V1 [Math.OC]
    This is a preprint of a paper whose final and definite form is with “Discrete and Continuous Dynam- ical Systems – Series S” (DCDS-S), ISSN 1937-1632 (print), ISSN 1937-1179 (online), available at [https://www.aimsciences.org/journals/home.jsp?journalID=15]. Paper Submitted 01-Oct-2016; Revised 01 and 21 March-2017; Accepted 31-March-2017. NOETHER CURRENTS FOR HIGHER-ORDER VARIATIONAL PROBLEMS OF HERGLOTZ TYPE WITH TIME DELAY SIMAO˜ P. S. SANTOS, NATALIA´ MARTINS AND DELFIM F. M. TORRES Abstract. We study, from an optimal control perspective, Noether currents for higher-order problems of Herglotz type with time delay. Main result pro- vides new Noether currents for such generalized variational problems, which are particularly useful in the search of extremals. The proof is based on the idea of rewriting the higher-order delayed generalized variational problem as a first-order optimal control problem without time delays. 1. Introduction. This article is devoted to the proof of a second Noether type theorem for higher-order delayed variational problems of Herglotz. Such problems, which are invariant under a certain group of transformations, were first studied in 1918 by Emmy Noether for the particular case of first-order variational problems without time delay [22]. In her famous paper [22], Noether proved two remarkable theorems that relate the invariance of a variational integral with properties of its Euler–Lagrange equations. Since most physical systems can be described by using Lagrangians and their associated actions, the importance of Noether’s two theorems is obvious [3]. The first Noether’s theorem, usually simply called Noether’s theorem, ensures the existence of r conserved quantities along the Euler–Lagrange extremals when the variational integral is invariant with respect to a continuous symmetry transforma- tion that depend on r parameters [34].
    [Show full text]
  • Divergence Invariant Variational Problems
    Divergence Invariant Variational Problems Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 [email protected] http://www.math.umn.edu/∼olver In [3], Harvey Brown states: One of the most remarkable methodological trends in modern physics has been the a priori use of symmetry principles to constrain the action principles of the non-gravitational interactions in quantum electrodynamics (QED) and particle physics. In other words, the starting point of a modern physical theory is to postulate the relevant physical symmetries, and then construct an invariant variational principle (Lagrangian) whose Euler–Lagrange equations form the field equations whose solutions determine the associated physics. However, the theory of differential invariants, [10, 11], implies that there are infinitely many inequivalent invariant Lagrangians that could be employed for this purpose. What is not addressed is the question I pose in [12]: How does this affect the resulting physics governed by the ostensibly different field equations? ... So, either the underlying physics is, in some rather vague and as yet undefined sense, “the same”, in which case choosing the simplest invariant Lagrangian makes sense on purely practical grounds, or, more worryingly, the two physical theories are different, which then begs the question as to which invariant variational problem describes the correct physics — how does one decide among an infinite range of possibilities? The following note begins with a very preliminary investigation into this issue in the simplest possible case: the one-dimensional classical free particle. It ends with a new result characterizing divergence invariant Lagrangians and some speculation as to their role in fundamental physics.
    [Show full text]