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This Special Issue contains eight original papers with a high impact in various domains of set-valued analysis. Set-valued analysis has made remarkable progress in the last 70 years, enriching itself continuously with new concepts, important results, and special applications. Different problems arising in the theory of control, economics, game theory, decision making, nonlinear programming, biomathematics, and statistics have strengthened the theoretical base and the specific techniques of set-valued analysis. The consistency of its theoretical approach and the multitude of its applications have transformed set-valued analysis into a reference field of modern mathematics, which attracts an impressive number of researchers.

gauge multivalued integral --- scalarly defined multivalued integral --- decomposition of a multifunction --- Kuelbs–Steadman space --- Henstock–Kurzweil integrable function --- vector measure --- dense embedding --- completely continuous embedding --- Köthe space --- Banach lattice --- fractional differential inclusion --- maximal monotone operator --- Riemann–Liouville integral --- absolutely continuous in variation --- Vladimirov pseudo-distance --- fuzzy measure space --- fuzzy integration --- t-norm --- Chebyshev’s inequality --- Hölder’s inequality --- periodic boundary value inclusion --- Stieltjes derivative --- Stieltjes integrals --- Bohnenblust–Karlin fixed-point theorem --- regulated function --- solution set --- discontinuous function --- impulsive problem with variable times --- Riemann-Lebesgue integral --- interval valued (set) multifunction --- non-additive set function --- image processing --- b-metric space --- Hβ-Hausdorff–Pompeiu b-metric --- multi-valued fractal --- iterated multifunction system --- integral inclusion

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This Special Issue contains eight original papers with a high impact in various domains of set-valued analysis. Set-valued analysis has made remarkable progress in the last 70 years, enriching itself continuously with new concepts, important results, and special applications. Different problems arising in the theory of control, economics, game theory, decision making, nonlinear programming, biomathematics, and statistics have strengthened the theoretical base and the specific techniques of set-valued analysis. The consistency of its theoretical approach and the multitude of its applications have transformed set-valued analysis into a reference field of modern mathematics, which attracts an impressive number of researchers.

Research & information: general --- Mathematics & science --- gauge multivalued integral --- scalarly defined multivalued integral --- decomposition of a multifunction --- Kuelbs–Steadman space --- Henstock–Kurzweil integrable function --- vector measure --- dense embedding --- completely continuous embedding --- Köthe space --- Banach lattice --- fractional differential inclusion --- maximal monotone operator --- Riemann–Liouville integral --- absolutely continuous in variation --- Vladimirov pseudo-distance --- fuzzy measure space --- fuzzy integration --- t-norm --- Chebyshev’s inequality --- Hölder’s inequality --- periodic boundary value inclusion --- Stieltjes derivative --- Stieltjes integrals --- Bohnenblust–Karlin fixed-point theorem --- regulated function --- solution set --- discontinuous function --- impulsive problem with variable times --- Riemann-Lebesgue integral --- interval valued (set) multifunction --- non-additive set function --- image processing --- b-metric space --- Hβ-Hausdorff–Pompeiu b-metric --- multi-valued fractal --- iterated multifunction system --- integral inclusion

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Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

Differential geometry. Global analysis --- Bifurcation theory. --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Addition. --- Algebraic equation. --- Analytic function. --- Analytic manifold. --- Atmospheric pressure. --- Banach space. --- Bernhard Riemann. --- Bifurcation diagram. --- Boundary value problem. --- Bounded operator. --- Bounded set (topological vector space). --- Boundedness. --- Canonical form. --- Cartesian coordinate system. --- Codimension. --- Compact operator. --- Complex analysis. --- Complex conjugate. --- Complex number. --- Connected space. --- Coordinate system. --- Corollary. --- Curvature. --- Derivative. --- Diagram (category theory). --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Eigenvalues and eigenvectors. --- Elliptic integral. --- Embedding. --- Equation. --- Euclidean division. --- Euler equations (fluid dynamics). --- Existential quantification. --- First principle. --- Fredholm operator. --- Froude number. --- Functional analysis. --- Hilbert space. --- Homeomorphism. --- Implicit function theorem. --- Integer. --- Linear algebra. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linear space (geometry). --- Linear subspace. --- Linearity. --- Linearization. --- Metric space. --- Morse theory. --- Multilinear form. --- N0. --- Natural number. --- Neumann series. --- Nonlinear functional analysis. --- Nonlinear system. --- Numerical analysis. --- Open mapping theorem (complex analysis). --- Operator (physics). --- Ordinary differential equation. --- Parameter. --- Parametrization. --- Partial differential equation. --- Permutation group. --- Permutation. --- Polynomial. --- Power series. --- Prime number. --- Proportionality (mathematics). --- Pseudo-differential operator. --- Puiseux series. --- Quantity. --- Real number. --- Resultant. --- Singularity theory. --- Skew-symmetric matrix. --- Smoothness. --- Solution set. --- Special case. --- Standard basis. --- Sturm–Liouville theory. --- Subset. --- Symmetric bilinear form. --- Symmetric group. --- Taylor series. --- Taylor's theorem. --- Theorem. --- Total derivative. --- Two-dimensional space. --- Union (set theory). --- Variable (mathematics). --- Vector space. --- Zero of a function.

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This book presents the latest findings on one of the most intensely investigated subjects in computational mathematics--the traveling salesman problem. It sounds simple enough: given a set of cities and the cost of travel between each pair of them, the problem challenges you to find the cheapest route by which to visit all the cities and return home to where you began. Though seemingly modest, this exercise has inspired studies by mathematicians, chemists, and physicists. Teachers use it in the classroom. It has practical applications in genetics, telecommunications, and neuroscience. The authors of this book are the same pioneers who for nearly two decades have led the investigation into the traveling salesman problem. They have derived solutions to almost eighty-six thousand cities, yet a general solution to the problem has yet to be discovered. Here they describe the method and computer code they used to solve a broad range of large-scale problems, and along the way they demonstrate the interplay of applied mathematics with increasingly powerful computing platforms. They also give the fascinating history of the problem--how it developed, and why it continues to intrigue us.

Traveling salesman problem. --- TSP (Traveling salesman problem) --- Combinatorial optimization --- Graph theory --- Vehicle routing problem --- AT&T Labs. --- Accuracy and precision. --- Addition. --- Algorithm. --- Analysis of algorithms. --- Applied mathematics. --- Approximation algorithm. --- Approximation. --- Basic solution (linear programming). --- Best, worst and average case. --- Bifurcation theory. --- Big O notation. --- CPLEX. --- CPU time. --- Calculation. --- Chaos theory. --- Column generation. --- Combinatorial optimization. --- Computation. --- Computational resource. --- Computer. --- Connected component (graph theory). --- Connectivity (graph theory). --- Convex hull. --- Cutting-plane method. --- Delaunay triangulation. --- Determinism. --- Disjoint sets. --- Dynamic programming. --- Ear decomposition. --- Engineering. --- Enumeration. --- Equation. --- Estimation. --- Euclidean distance. --- Euclidean space. --- Family of sets. --- For loop. --- Genetic algorithm. --- George Dantzig. --- Georgia Institute of Technology. --- Greedy algorithm. --- Hamiltonian path. --- Hospitality. --- Hypergraph. --- Implementation. --- Instance (computer science). --- Institute. --- Integer. --- Iteration. --- Linear inequality. --- Linear programming. --- Mathematical optimization. --- Mathematics. --- Model of computation. --- Neuroscience. --- Notation. --- Operations research. --- Optimization problem. --- Order by. --- Pairwise. --- Parameter (computer programming). --- Parity (mathematics). --- Percentage. --- Polyhedron. --- Polytope. --- Pricing. --- Princeton University. --- Processing (programming language). --- Project. --- Quantity. --- Reduced cost. --- Requirement. --- Result. --- Rice University. --- Rutgers University. --- Scientific notation. --- Search algorithm. --- Search tree. --- Self-similarity. --- Simplex algorithm. --- Solution set. --- Solver. --- Source code. --- Special case. --- Stochastic. --- Subroutine. --- Subsequence. --- Subset. --- Summation. --- Test set. --- Theorem. --- Theory. --- Time complexity. --- Trade-off. --- Travelling salesman problem. --- Tree (data structure). --- Upper and lower bounds. --- Variable (computer science). --- Variable (mathematics).

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The impacts of climate change on water resource management, as well as increasingly severe natural disasters over the last decades, have caught global attention. Reliable and accurate hydrological forecasts are essential for efficient water resource management and the mitigation of natural disasters. While the notorious nonlinear hydrological processes make accurate forecasts a very challenging task, it requires advanced techniques to build accurate forecast models and reliable management systems. One of the newest techniques for modeling complex systems is artificial intelligence (AI). AI can replicate the way humans learn and has great capability to efficiently extract crucial information from large amounts of data to solve complex problems. The fourteen research papers published in this Special Issue contribute significantly to the uncertainty assessment of operational hydrologic forecasting under changing environmental conditions and the promotion of water resources management by using the latest advanced techniques, such as AI techniques. The fourteen contributions across four major research areas: (1) machine learning approaches to hydrologic forecasting; (2) uncertainty analysis and assessment on hydrological modeling under changing environments; (3) AI techniques for optimizing multi-objective reservoir operation; (4) adaption strategies of extreme hydrological events for hazard mitigation. The papers published in this issue will not only advance water sciences but also help policymakers to achieve more sustainable and effective water resource management.

Research & information: general --- water resources management --- landslide --- dammed lake --- flood risk --- time-varying parameter --- GR4J model --- changing environments --- temporal transferability --- western China --- cascade hydropower reservoirs --- multi-objective optimization --- TOPSIS --- gravitational search algorithm --- opposition learning --- partial mutation --- elastic-ball modification --- Snowmelt Runoff Model --- parameter uncertainty --- data-scarce deglaciating river basin --- climate change impacts --- generalized likelihood uncertainty estimation --- Yangtze River --- cascade reservoirs --- impoundment operation --- GloFAS-Seasonal --- forecast evaluation --- small and medium-scale rivers --- highly urbanized area --- flood control --- whole region perspective --- coupled models --- flood-risk map --- hydrodynamic modelling --- Sequential Gaussian Simulation --- urban stormwater --- probabilistic forecast --- Unscented Kalman Filter --- artificial neural networks --- Three Gorges Reservoir --- Mahalanobis-Taguchi System --- grey entropy method --- signal-to-noise ratio --- degree of balance and approach --- interval number --- multi-objective optimal operation model --- feasible search space --- Pareto-front optimal solution set --- loss–benefit ratio of ecology and power generation --- elasticity coefficient --- empirical mode decomposition --- Hushan reservoir --- data synthesis --- urban hydrological model --- Generalized Likelihood Uncertainty Estimation (GLUE) --- Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) --- uncertainty analysis --- NDVI --- Yarlung Zangbo River --- machine learning model --- random forest --- Internet of Things (IoT) --- regional flood inundation depth --- recurrent nonlinear autoregressive with exogenous inputs (RNARX) --- artificial intelligence --- machine learning --- multi-objective reservoir operation --- hydrologic forecasting --- uncertainty --- risk