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This textbook for the basic lecture course of the same name deals with selected topics of multidimensional analysis. It is also an introduction to the theory of ordinary differential equations and the Fourier theory, of importance in the application of image processing and acoustics.
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This is a proceedings of the international conference "Painlevé Equations and Related Topics" which was taking place at the Euler International Mathematical Institute, a branch of the Saint Petersburg Department of the Steklov Institute of Mathematics of the Russian Academy of Sciences, in Saint Petersburg on June 17 to 23, 2011. The survey articles discuss the following topics: General ordinary differential equations Painlevé equations and their generalizations Painlevé property Discrete Painlevé equations Properties of solutions of all mentioned above equations:- Asymptotic forms and asymptotic expansions- Connections of asymptotic forms of a solution near different points- Convergency and asymptotic character of a formal solution- New types of asymptotic forms and asymptotic expansions- Riemann-Hilbert problems- Isomonodromic deformations of linear systems- Symmetries and transformations of solutions- Algebraic solutions Reductions of PDE to Painlevé equations and their generalizations Ordinary Differential Equations systems equivalent to Painlevé equations and their generalizations Applications of the equations and the solutions
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The main classes of inverse problems for equations of mathematical physics and their numerical solution methods are considered in this book which is intended for graduate students and experts in applied mathematics, computational mathematics, and mathematical modeling.
Mathematical physics. --- Boundary value problems --- Physical mathematics --- Physics --- Numerical solutions. --- Mathematics --- Inverse problems (Differential equations) --- Differential equations, Partial --- Improperly posed problems. --- Evolution equation, ordinary differential equation, inverse problem, parameter estimation, partial differential equation.
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Matched asymptotic expansions and singular perturbations
517.928 --- #WSTE:STER --- Asymptotic methods in ordinary differential equation theory. Averaging methods. Invariant manifolds --- 517.928 Asymptotic methods in ordinary differential equation theory. Averaging methods. Invariant manifolds --- Asymptotic expansions. --- Differential equations --- Singular perturbations (Mathematics). --- Numerical solutions. --- Perturbation (Mathematics) --- Équations différentielles --- Perturbation (mathématiques) --- Singular perturbations (Mathematics) --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- 517.91 Differential equations --- Asymptotic theory --- Differential equations. --- Équations différentielles. --- Perturbation (mathématiques) --- Perturbations singulieres
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Differential equations --- Integral equations --- Numerical solutions --- Elasticity --- -Integral equations --- -531.2 --- Equations, Integral --- Functional equations --- Functional analysis --- Equations, Differential --- Bessel functions --- Calculus --- Elastic properties --- Young's modulus --- Mathematical physics --- Matter --- Statics --- Rheology --- Strains and stresses --- Strength of materials --- Statics. Forces. Equilibrium. Attraction --- Properties --- 531.2 Statics. Forces. Equilibrium. Attraction --- 517.91 Differential equations --- Equations intégrales --- Elasticity. --- Equations intégrales --- 531.2 --- Numerical analysis --- Partial differential equations --- 517.91 --- Numerical solutions. --- Elasticité --- Equations différentielles --- Solutions numériques --- Numerical solutions&delete& --- Differential equations - Numerical solutions --- Integral equations - Numerical solutions --- Boundary value problems --- Equilibrium (physics) --- Ordinary differential equation
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Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis.
integro–differential systems --- Cauchy matrix --- exponential stability --- distributed control --- delay differential equation --- ordinary differential equation --- asymptotic equivalence --- approximation --- eigenvalue --- oscillation --- variable delay --- deviating argument --- non-monotone argument --- slowly varying function --- Crank–Nicolson scheme --- Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- variable coefficients --- stability analysis --- Lane-Emden-Klein-Gordon-Fock system with central symmetry --- Noether symmetries --- conservation laws --- differential equations --- non-monotone delays --- fractional calculus --- stochastic heat equation --- additive noise --- chebyshev polynomials of sixth kind --- error estimate --- fractional difference equations --- delay --- impulses --- existence --- fractional Jaulent-Miodek (JM) system --- fractional logistic function method --- symmetry analysis --- lie point symmetry analysis --- approximate conservation laws --- approximate nonlinear self-adjointness --- perturbed fractional differential equations
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Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis.
Research & information: general --- Mathematics & science --- integro–differential systems --- Cauchy matrix --- exponential stability --- distributed control --- delay differential equation --- ordinary differential equation --- asymptotic equivalence --- approximation --- eigenvalue --- oscillation --- variable delay --- deviating argument --- non-monotone argument --- slowly varying function --- Crank–Nicolson scheme --- Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- variable coefficients --- stability analysis --- Lane-Emden-Klein-Gordon-Fock system with central symmetry --- Noether symmetries --- conservation laws --- differential equations --- non-monotone delays --- fractional calculus --- stochastic heat equation --- additive noise --- chebyshev polynomials of sixth kind --- error estimate --- fractional difference equations --- delay --- impulses --- existence --- fractional Jaulent-Miodek (JM) system --- fractional logistic function method --- symmetry analysis --- lie point symmetry analysis --- approximate conservation laws --- approximate nonlinear self-adjointness --- perturbed fractional differential equations --- integro–differential systems --- Cauchy matrix --- exponential stability --- distributed control --- delay differential equation --- ordinary differential equation --- asymptotic equivalence --- approximation --- eigenvalue --- oscillation --- variable delay --- deviating argument --- non-monotone argument --- slowly varying function --- Crank–Nicolson scheme --- Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- variable coefficients --- stability analysis --- Lane-Emden-Klein-Gordon-Fock system with central symmetry --- Noether symmetries --- conservation laws --- differential equations --- non-monotone delays --- fractional calculus --- stochastic heat equation --- additive noise --- chebyshev polynomials of sixth kind --- error estimate --- fractional difference equations --- delay --- impulses --- existence --- fractional Jaulent-Miodek (JM) system --- fractional logistic function method --- symmetry analysis --- lie point symmetry analysis --- approximate conservation laws --- approximate nonlinear self-adjointness --- perturbed fractional differential equations
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The aim of this book is to study harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. In several instances, the latter permits reduction of the original elliptic variational problem to the qualitative study of certain ordinary differential equations: the authors' primary objective is to provide representative examples to illustrate these reduction methods and their associated analysis with geometric and topological applications. The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.
Cartes harmoniques --- Harmonic maps --- Harmonische kaarten --- Immersies (Wiskunde) --- Immersions (Mathematics) --- Immersions (Mathématiques) --- Harmonic maps. --- Differential equations, Elliptic --- Applications harmoniques --- Immersions (Mathematiques) --- Équations différentielles elliptiques --- Numerical solutions. --- Solutions numériques --- Équations différentielles elliptiques --- Solutions numériques --- Differential equations [Elliptic] --- Numerical solutions --- Embeddings (Mathematics) --- Manifolds (Mathematics) --- Mappings (Mathematics) --- Maps, Harmonic --- Arc length. --- Catenary. --- Clifford algebra. --- Codimension. --- Coefficient. --- Compact space. --- Complex projective space. --- Connected sum. --- Constant curvature. --- Corollary. --- Covariant derivative. --- Curvature. --- Cylinder (geometry). --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential equation. --- Differential geometry. --- Elliptic partial differential equation. --- Embedding. --- Energy functional. --- Equation. --- Existence theorem. --- Existential quantification. --- Fiber bundle. --- Gauss map. --- Geometry and topology. --- Geometry. --- Gravitational field. --- Harmonic map. --- Hyperbola. --- Hyperplane. --- Hypersphere. --- Hypersurface. --- Integer. --- Iterative method. --- Levi-Civita connection. --- Lie group. --- Mathematics. --- Maximum principle. --- Mean curvature. --- Normal (geometry). --- Numerical analysis. --- Open set. --- Ordinary differential equation. --- Parabola. --- Quadratic form. --- Sign (mathematics). --- Special case. --- Stiefel manifold. --- Submanifold. --- Suggestion. --- Surface of revolution. --- Symmetry. --- Tangent bundle. --- Theorem. --- Vector bundle. --- Vector space. --- Vertical tangent. --- Winding number. --- Differential equations, Elliptic - Numerical solutions
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Modern biology is rapidly becoming a study of large sets of data. Understanding these data sets is a major challenge for most life sciences, including the medical, environmental, and bioprocess fields. Computational biology approaches are essential for leveraging this ongoing revolution in omics data. A primary goal of this Special Issue, entitled “Methods in Computational Biology”, is the communication of computational biology methods, which can extract biological design principles from complex data sets, described in enough detail to permit the reproduction of the results. This issue integrates interdisciplinary researchers such as biologists, computer scientists, engineers, and mathematicians to advance biological systems analysis. The Special Issue contains the following sections:•Reviews of Computational Methods•Computational Analysis of Biological Dynamics: From Molecular to Cellular to Tissue/Consortia Levels•The Interface of Biotic and Abiotic Processes•Processing of Large Data Sets for Enhanced Analysis•Parameter Optimization and Measurement
n/a --- inosine --- immune checkpoint inhibitor --- geometric singular perturbation theory --- simulation --- BioModels Database --- ADAR --- calcium current --- bifurcation analysis --- bacterial biofilms --- nonlinear dynamics --- explanatory model --- turning point bifurcation --- oscillator --- workflow --- bioreactor integrated modeling --- modeling methods --- elementary flux modes visualization --- multiscale systems biology --- evolutionary algorithm --- metabolic model --- differential evolution --- reduced-order model --- computational model --- gut microbiota dysbiosis --- canard-induced EADs --- computational biology --- metabolic modelling --- methods --- SREBP-2 --- mechanistic model --- systems modeling --- biological networks --- macromolecular composition --- provenance --- flux balance analysis --- immunotherapy --- compartmental modeling --- immuno-oncology --- metabolic network visualization --- mechanism --- bistable switch --- Clostridium difficile infection --- bioreactor operation optimization --- microRNA targeting --- CFD simulation --- biomass reaction --- RNA editing --- ordinary differential equation --- metabolic modeling --- mass-action networks --- hybrid model --- multiple time scales --- quantitative systems pharmacology (QSP) --- mathematical modeling --- microRNA --- cancer --- parameter optimization --- Hopf bifurcation --- breast
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This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks-a sort of potato-stamp method-Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.
Symmetry (Mathematics) --- Symmetry (Art) --- Abstract algebra. --- Addition. --- Algorithm. --- Antisymmetry. --- Arc length. --- Boundary value problem. --- Cartesian coordinate system. --- Circular motion. --- Circumference. --- Coefficient. --- Complex analysis. --- Complex multiplication. --- Complex number. --- Complex plane. --- Computation. --- Coordinate system. --- Coset. --- Cyclic group. --- Derivative. --- Diagonal. --- Diagram (category theory). --- Dihedral group. --- Division by zero. --- Domain coloring. --- Dot product. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein integer. --- Epicycloid. --- Equation. --- Euler's formula. --- Even and odd functions. --- Exponential function. --- Fourier series. --- Frieze group. --- Function (mathematics). --- Function composition. --- Function space. --- Gaussian integer. --- Geometry. --- Glide reflection. --- Group (mathematics). --- Group theory. --- Homomorphism. --- Horocycle. --- Hyperbolic geometry. --- Ideal point. --- Integer. --- Lattice (group). --- Linear interpolation. --- Local symmetry. --- M. C. Escher. --- Main diagonal. --- Mathematical proof. --- Mathematical structure. --- Mathematics. --- Mirror symmetry (string theory). --- Mirror symmetry. --- Morphing. --- Natural number. --- Normal subgroup. --- Notation. --- Ordinary differential equation. --- Parallelogram. --- Parametric equation. --- Parametrization. --- Periodic function. --- Plane symmetry. --- Plane wave. --- Point group. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pythagorean triple. --- Quantity. --- Quotient group. --- Real number. --- Reciprocal lattice. --- Rectangle. --- Reflection symmetry. --- Right angle. --- Ring of integers. --- Rotational symmetry. --- Scientific notation. --- Special case. --- Square lattice. --- Subgroup. --- Summation. --- Symmetry group. --- Symmetry. --- Tetrahedron. --- Theorem. --- Translational symmetry. --- Trigonometric functions. --- Unique factorization domain. --- Unit circle. --- Variable (mathematics). --- Vector space. --- Wallpaper group. --- Wave packet. --- Abstract algebra. --- Addition. --- Algorithm. --- Antisymmetry. --- Arc length. --- Boundary value problem. --- Cartesian coordinate system. --- Circular motion. --- Circumference. --- Coefficient. --- Complex analysis. --- Complex multiplication. --- Complex number. --- Complex plane. --- Computation. --- Coordinate system. --- Coset. --- Cyclic group. --- Derivative. --- Diagonal. --- Diagram (category theory). --- Dihedral group. --- Division by zero. --- Domain coloring. --- Dot product. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein integer. --- Epicycloid. --- Equation. --- Euler's formula. --- Even and odd functions. --- Exponential function. --- Fourier series. --- Frieze group. --- Function (mathematics). --- Function composition. --- Function space. --- Gaussian integer. --- Geometry. --- Glide reflection. --- Group (mathematics). --- Group theory. --- Homomorphism. --- Horocycle. --- Hyperbolic geometry. --- Ideal point. --- Integer. --- Lattice (group). --- Linear interpolation. --- Local symmetry. --- M. C. Escher. --- Main diagonal. --- Mathematical proof. --- Mathematical structure. --- Mathematics. --- Mirror symmetry (string theory). --- Mirror symmetry. --- Morphing. --- Natural number. --- Normal subgroup. --- Notation. --- Ordinary differential equation. --- Parallelogram. --- Parametric equation. --- Parametrization. --- Periodic function. --- Plane symmetry. --- Plane wave. --- Point group. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pythagorean triple. --- Quantity. --- Quotient group. --- Real number. --- Reciprocal lattice. --- Rectangle. --- Reflection symmetry. --- Right angle. --- Ring of integers. --- Rotational symmetry. --- Scientific notation. --- Special case. --- Square lattice. --- Subgroup. --- Summation. --- Symmetry group. --- Symmetry. --- Tetrahedron. --- Theorem. --- Translational symmetry. --- Trigonometric functions. --- Unique factorization domain. --- Unit circle. --- Variable (mathematics). --- Vector space. --- Wallpaper group. --- Wave packet.
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