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This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners.

Monte Carlo --- MCMC --- Markov chains --- computational statistics --- bayesian inference --- Non-Homogeneous Markov Systems --- Markov Set Systems --- limiting set --- tail expectation --- asymptotic bound --- quasi-asymptotic independence --- heavy-tailed distribution --- dominated variation --- copula --- branching process --- migration --- continuous time --- generating function --- period-life --- reliability --- redundant systems --- preventive maintenance --- multiple vacations --- process mining --- process modelling --- phase-type models --- process target compliance --- particle filter --- missing data --- single imputation --- impoverishment --- Markov Systems --- open population Markov chain models --- Semi-Markov processes --- controllable Markov jump processes --- compound Poisson processes --- diffusion limits --- stochastic control problem with incomplete information --- novel queuing models in applications --- semi-Markov model --- Markov model --- hybrid semi-Markov model --- manpower planning --- semi-Markov modeling --- occupancy --- first passage time --- duration --- non-homogeneity --- DNA sequences --- state space model --- Kalman filter --- constrained optimization --- two-sided components --- basketball --- Markov chain --- second order --- off-ball screens --- performance --- semi-Markov --- transient analysis --- asymptotic analysis --- n/a

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This book covers a broad range of research results in the field of Markov and Semi-Markov chains, processes, systems and related emerging fields. The authors of the included research papers are well-known researchers in their field. The book presents the state-of-the-art and ideas for further research for theorists in the fields. Nonetheless, it also provides straightforwardly applicable results for diverse areas of practitioners.

Research & information: general --- Mathematics & science --- Monte Carlo --- MCMC --- Markov chains --- computational statistics --- bayesian inference --- Non-Homogeneous Markov Systems --- Markov Set Systems --- limiting set --- tail expectation --- asymptotic bound --- quasi-asymptotic independence --- heavy-tailed distribution --- dominated variation --- copula --- branching process --- migration --- continuous time --- generating function --- period-life --- reliability --- redundant systems --- preventive maintenance --- multiple vacations --- process mining --- process modelling --- phase-type models --- process target compliance --- particle filter --- missing data --- single imputation --- impoverishment --- Markov Systems --- open population Markov chain models --- Semi-Markov processes --- controllable Markov jump processes --- compound Poisson processes --- diffusion limits --- stochastic control problem with incomplete information --- novel queuing models in applications --- semi-Markov model --- Markov model --- hybrid semi-Markov model --- manpower planning --- semi-Markov modeling --- occupancy --- first passage time --- duration --- non-homogeneity --- DNA sequences --- state space model --- Kalman filter --- constrained optimization --- two-sided components --- basketball --- Markov chain --- second order --- off-ball screens --- performance --- semi-Markov --- transient analysis --- asymptotic analysis --- Monte Carlo --- MCMC --- Markov chains --- computational statistics --- bayesian inference --- Non-Homogeneous Markov Systems --- Markov Set Systems --- limiting set --- tail expectation --- asymptotic bound --- quasi-asymptotic independence --- heavy-tailed distribution --- dominated variation --- copula --- branching process --- migration --- continuous time --- generating function --- period-life --- reliability --- redundant systems --- preventive maintenance --- multiple vacations --- process mining --- process modelling --- phase-type models --- process target compliance --- particle filter --- missing data --- single imputation --- impoverishment --- Markov Systems --- open population Markov chain models --- Semi-Markov processes --- controllable Markov jump processes --- compound Poisson processes --- diffusion limits --- stochastic control problem with incomplete information --- novel queuing models in applications --- semi-Markov model --- Markov model --- hybrid semi-Markov model --- manpower planning --- semi-Markov modeling --- occupancy --- first passage time --- duration --- non-homogeneity --- DNA sequences --- state space model --- Kalman filter --- constrained optimization --- two-sided components --- basketball --- Markov chain --- second order --- off-ball screens --- performance --- semi-Markov --- transient analysis --- asymptotic analysis

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Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soulé arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Arithmetical algebraic geometry. --- Shimura varieties. --- Varieties, Shimura --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Arithmetical algebraic geometry --- Number theory --- Abelian group. --- Addition. --- Adjunction formula. --- Algebraic number theory. --- Arakelov theory. --- Arithmetic. --- Automorphism. --- Bijection. --- Borel subgroup. --- Calculation. --- Chow group. --- Coefficient. --- Cohomology. --- Combinatorics. --- Compact Riemann surface. --- Complex multiplication. --- Complex number. --- Cup product. --- Deformation theory. --- Derivative. --- Dimension. --- Disjoint union. --- Divisor. --- Dual pair. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Elliptic curve. --- Endomorphism. --- Equation. --- Explicit formulae (L-function). --- Fields Institute. --- Formal group. --- Fourier series. --- Fundamental matrix (linear differential equation). --- Galois group. --- Generating function. --- Green's function. --- Group action. --- Induced representation. --- Intersection (set theory). --- Intersection number. --- Irreducible component. --- Isomorphism class. --- L-function. --- Laurent series. --- Level structure. --- Line bundle. --- Local ring. --- Mathematical sciences. --- Mathematics. --- Metaplectic group. --- Modular curve. --- Modular form. --- Modularity (networks). --- Moduli space. --- Multiple integral. --- Number theory. --- Numerical integration. --- Orbifold. --- Orthogonal complement. --- P-adic number. --- Pairing. --- Prime factor. --- Prime number. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Quadratic form. --- Quadratic residue. --- Quantity. --- Quaternion algebra. --- Quaternion. --- Quotient stack. --- Rational number. --- Real number. --- Residue field. --- Riemann zeta function. --- Ring of integers. --- SL2(R). --- Scientific notation. --- Shimura variety. --- Siegel Eisenstein series. --- Siegel modular form. --- Special case. --- Standard L-function. --- Subgroup. --- Subset. --- Summation. --- Tensor product. --- Test vector. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Trace (linear algebra). --- Triangular matrix. --- Two-dimensional space. --- Uniformization. --- Valuative criterion. --- Whittaker function.

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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Dirichlet series. --- Weyl groups. --- Weyl's groups --- Group theory --- Series, Dirichlet --- Series --- BZL pattern. --- Class I. --- Eisenstein series. --- Euler product. --- Gauss sum. --- Gelfand-Tsetlin pattern. --- Kashiwara operator. --- Kashiwara's crystal. --- Knowability Lemma. --- Kostant partition function. --- Riemann zeta function. --- Schur polynomial. --- Schützenberger involution. --- Snake Lemma. --- Statement A. --- Statement B. --- Statement C. --- Statement D. --- Statement E. --- Statement F. --- Statement G. --- Tokuyama's Theorem. --- Weyl character formula. --- Weyl denominator. --- Weyl group multiple Dirichlet series. --- Weyl vector. --- Whittaker coefficient. --- Whittaker function. --- Yang-Baxter equation. --- Yang–Baxter equation. --- accordion. --- adele group. --- affine linear transformation. --- analytic continuation. --- analytic number theory. --- archimedean place. --- basis vector. --- bijection. --- bookkeeping. --- box-circle duality. --- boxing. --- canonical indexings. --- cardinality. --- cartoon. --- circling. --- class. --- combinatorial identity. --- concurrence. --- critical resonance. --- crystal base. --- crystal graph. --- crystal. --- divisibility condition. --- double sum. --- episode. --- equivalence relation. --- f-packet. --- free abelian group. --- functional equation. --- generating function. --- global field. --- ice-type model. --- inclusion-exclusion. --- indexing. --- involution. --- isomorphism. --- knowability. --- maximality. --- nodal signature. --- nonarchimedean local field. --- noncritical resonance. --- nonzero contribution. --- p-adic group. --- p-adic integral. --- p-adic integration. --- partition function. --- polynomial. --- preaccordion. --- prototype. --- reduced root system. --- representation theory. --- residue class field. --- resonance. --- resotope. --- row sums. --- row transfer matrix. --- short pattern. --- six-vertex model. --- snakes. --- statistical mechanics. --- subsignature. --- tableaux. --- type. --- Γ-equivalence class. --- Γ-swap.

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Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist.Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles.The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.

Mathematical physics. --- Problem solving. --- MATHEMATICS / General. --- Methodology --- Psychology --- Decision making --- Executive functions (Neuropsychology) --- Physical mathematics --- Physics --- Mathematics --- Addition. --- Analytic function. --- Angular acceleration. --- Angular velocity. --- Axle. --- Calculation. --- Capacitor. --- Cartesian coordinate system. --- Cauchy's integral formula. --- Center of mass (relativistic). --- Center of mass. --- Centroid. --- Ceva's theorem. --- Clockwise. --- Complex analysis. --- Complex number. --- Conservation of energy. --- Convex curve. --- Curvature. --- Curve. --- Cylinder (geometry). --- Derivative. --- Diameter. --- Differential geometry. --- Dimension. --- Division by zero. --- Dot product. --- Eigenvalues and eigenvectors. --- Electric current. --- Equation. --- Euler's formula. --- Euler–Lagrange equation. --- Fermat's principle. --- Friction. --- Fundamental theorem of calculus. --- Gaussian curvature. --- Generating function. --- Geodesic curvature. --- Geometry. --- Gravity. --- Green's theorem. --- Heat flux. --- Hinge. --- Hooke's law. --- Horizontal plane. --- Hypotenuse. --- Inductance. --- Instant. --- Kinetic energy. --- Line integral. --- Linear map. --- Mathematics. --- Mechanics. --- Moment of inertia. --- Newton's laws of motion. --- Normal (geometry). --- Ohm's law. --- Optics. --- Partial derivative. --- Potential energy. --- Proportionality (mathematics). --- Pythagorean theorem. --- Quadratic function. --- Quantity. --- Rectangle. --- Resistor. --- Right angle. --- Right triangle. --- Second law of thermodynamics. --- Semicircle. --- Series and parallel circuits. --- Sign (mathematics). --- Slinky. --- Snell's law. --- Soap bubble. --- Soap film. --- Special case. --- Spring (device). --- Stiffness. --- Summation. --- Surface area. --- Surface tension. --- Tangent space. --- Tangent. --- Telescope. --- Theorem. --- Thought experiment. --- Tractrix. --- Trapezoid. --- Trigonometric functions. --- Two-dimensional gas. --- Uncertainty principle. --- Unit circle. --- Unit vector. --- Vacuum. --- Variable (mathematics). --- Vector field. --- Voltage drop. --- Voltage. --- Wavefront.