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Quantum mechanics, discovered by Werner Heisenberg and Erwin Schrödinger in 1925-1926, is famous for its radical implications for our conception of physics and for our view of human knowledge in general. While these implications have been seen as scientifically productive and intellectually liberating to some, Niels Bohr and Heisenberg, among them, they have been troublesome to many others, including Schrödinger and, most famously, Albert Einstein. The situation led to the intense debate that started in the wake of its discovery and has continued into our own time, with no end appearing to be in sight. Epistemology and Probability aims to contribute to our understanding of quantum mechanics and of the reasons for its extraordinary impact by reconsidering, under the rubric of "nonclassical epistemology," the nature of epistemology and probability, and their relationships in quantum theory. The book brings together the thought of the three figures most responsible for the rise of quantum mechanics Heisenberg and Schrödinger, on the physical side, and Bohr, on the philosophical side in order to develop a deeper sense of the physical, mathematical, and philosophical workings of quantum-theoretical thinking. Reciprocally, giving a special emphasis on probability and specifically to the Bayesian concept of probability allows the book to gain new insights into the thought of these figures. The book reconsiders, from this perspective, the Bohr-Einstein debate on the epistemology of quantum physics and, in particular, offers a new treatment of the famous experiment of Einstein, Podolsky, and Rosen (EPR), and of the Bohr-Einstein exchange concerning the subject. It also addresses the relevant aspects of quantum information theory and considers the implications of its epistemological argument for higher-level quantum theories, such as quantum field theory and string and brane theories. One of the main contributions of the book is its analysis of the role of mathematics in quantum theory and in the thinking of Bohr, Heisenberg, and Schrödinger, in particular an examination of the new (vis-à-vis classical physics and relativity) type of the relationships between mathematics and physics introduced by Heisenberg in the course of his discovery of quantum mechanics. Although Epistemology and Probability is aimed at physicists, philosophers and historians of science, and graduate and advanced undergraduate students in these fields, it is also written with a broader audience in mind and is accessible to readers unfamiliar with the higher-level mathematics used in quantum theory.

Theory of knowledge --- Philosophy of science --- Operational research. Game theory --- Probability theory --- Quantum mechanics. Quantumfield theory --- Elementary particles --- elementaire deeltjes --- quantumfysica --- waarschijnlijkheidstheorie --- stochastische analyse --- kwantumleer --- wetenschapsfilosofie --- kennisleer --- fysica --- kansrekening --- Causality (Physics) --- Complementarity (Physics) --- Heisenberg uncertainty principle --- Knowledge, Theory of --- Physics --- Quantum theory --- Schrödinger equation --- Wave-particle duality --- Dualism, Wave-particle --- Duality principle (Physics) --- Wave-corpuscle duality --- Electromagnetic waves --- Matter --- Radiation --- Wave mechanics --- Equation, Schrödinger --- Schrödinger wave equation --- Differential equations, Partial --- Particles (Nuclear physics) --- WKB approximation --- Epistemology --- Philosophy --- Psychology --- Indeterminancy principle --- Uncertainty principle --- Causality --- Nuclear physics --- History --- Mathematics --- Constitution --- Bohr, Niels, --- Heisenberg, Werner --- Schrödinger, Erwin, --- Schredinger, Ervin, --- Schrödinger, E. --- Schroedinger, Erwin, --- Boer, Niersi, --- Boerh, Niersi, --- Bohr, N. --- Bohr, Niels Henrik David, --- Bor, Nil's,

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This book presents new achievements and results in the theory of conjugate duality for convex optimization problems. The perturbation approach for attaching a dual problem to a primal one makes the object of a preliminary chapter, where also an overview of the classical generalized interior point regularity conditions is given. A central role in the book is played by the formulation of generalized Moreau-Rockafellar formulae and closedness-type conditions, the latter constituting a new class of regularity conditions, in many situations with a wider applicability than the generalized interior point ones. The reader also receives deep insights into biconjugate calculus for convex functions, the relations between different existing strong duality notions, but also into several unconventional Fenchel duality topics. The final part of the book is consecrated to the applications of the convex duality theory in the field of monotone operators.

Convex functions. --- Duality theory (Mathematics). --- Mathematical optimization. --- Convex functions --- Duality theory (Mathematics) --- Mathematical optimization --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Monotone operators. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Functions, Convex --- Mathematics. --- Operations research. --- Decision making. --- Mathematical analysis. --- Analysis (Mathematics). --- System theory. --- Management science. --- Operations Research, Management Science. --- Operation Research/Decision Theory. --- Optimization. --- Systems Theory, Control. --- Analysis. --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Operator theory --- Algebra --- Topology --- Functions of real variables --- Systems theory. --- Global analysis (Mathematics). --- Operations Research/Decision Theory. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- 517.1 Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Quantitative business analysis --- Statistical decision --- Philosophy --- Decision making

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This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology. Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.

Shimura varieties. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Varieties, Shimura --- Arithmetical algebraic geometry --- Accuracy and precision. --- Adjoint. --- Algebraic closure. --- Archimedean property. --- Automorphism. --- Base change map. --- Base change. --- Calculation. --- Clay Mathematics Institute. --- Coefficient. --- Compact element. --- Compact space. --- Comparison theorem. --- Conjecture. --- Connected space. --- Connectedness. --- Constant term. --- Corollary. --- Duality (mathematics). --- Existential quantification. --- Exterior algebra. --- Finite field. --- Finite set. --- Fundamental lemma (Langlands program). --- Galois group. --- General linear group. --- Haar measure. --- Hecke algebra. --- Homomorphism. --- L-function. --- Logarithm. --- Mathematical induction. --- Mathematician. --- Maximal compact subgroup. --- Maximal ideal. --- Morphism. --- Neighbourhood (mathematics). --- Open set. --- Parabolic induction. --- Permutation. --- Prime number. --- Ramanujan–Petersson conjecture. --- Reductive group. --- Ring (mathematics). --- Scientific notation. --- Shimura variety. --- Simply connected space. --- Special case. --- Sub"ient. --- Subalgebra. --- Subgroup. --- Symplectic group. --- Theorem. --- Trace formula. --- Unitary group. --- Weyl group.