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Mathematical Notes, 29Originally published in 1983.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differential equations, Elliptic --- Schrödinger operator. --- Eigenfunctions. --- Functions, Proper --- Proper functions --- Boundary value problems --- Differential equations --- Integral equations --- Operator, Schrödinger --- Differential operators --- Quantum theory --- Schrödinger equation --- Numerical solutions. --- Numerical solutions --- Approximation. --- Ball (mathematics). --- Bounded function. --- Center of mass. --- Coefficient. --- Compact space. --- Complex number. --- Continuous function (set theory). --- Continuous function. --- Discrete spectrum. --- Distribution (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Equivalence class. --- Essential spectrum. --- Estimation. --- Existential quantification. --- Exponential decay. --- Function space. --- Fundamental theorem of calculus. --- Geometry. --- Ground state. --- Infimum and supremum. --- Lebesgue measure. --- Open set. --- Pointwise. --- Quadratic form. --- Quantity. --- Restriction (mathematics). --- Riemannian manifold. --- Robert Langlands. --- Schrödinger equation. --- Self-adjoint operator. --- Self-adjoint. --- Smoothness. --- Special case. --- Subset. --- Support (mathematics). --- Theorem. --- Upper and lower bounds. --- Weak solution. --- Without loss of generality.

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The description for this book, Order-Preserving Maps and Integration Processes. (AM-31), Volume 31, will be forthcoming.

Group theory. --- Integrals. --- Abelian group. --- Addition. --- Axiom. --- Baire function. --- Banach space. --- Big O notation. --- Binary operation. --- Binary relation. --- Borel set. --- Bounded function. --- Cartesian product. --- Characteristic function (probability theory). --- Circumference. --- Closure (mathematics). --- Coefficient. --- Combination. --- Commutative algebra. --- Compact space. --- Complete lattice. --- Continuous function (set theory). --- Continuous function. --- Contradiction. --- Corollary. --- Coset. --- Countable set. --- Directed set. --- Domain of a function. --- Elementary function. --- Empty set. --- Equation. --- Equivalence class. --- Estimation. --- Existential quantification. --- Finite set. --- Fubini's theorem. --- Hilbert space. --- I0. --- Infimum and supremum. --- Integer. --- L-function. --- Lattice (order). --- Lebesgue integration. --- Limit (mathematics). --- Limit superior and limit inferior. --- Linear map. --- Measure (mathematics). --- Monotonic function. --- Natural number. --- Order of operations. --- Parity (mathematics). --- Partially ordered group. --- Partially ordered set. --- Pointwise convergence. --- Pointwise. --- Polynomial. --- Projection (linear algebra). --- Quadratic function. --- Real number. --- Requirement. --- Riemann integral. --- Riemann–Stieltjes integral. --- Scalar multiplication. --- Scientific notation. --- Self-adjoint operator. --- Set (mathematics). --- Set function. --- Sign (mathematics). --- Special case. --- Subset. --- Subtraction. --- Summation. --- Theorem. --- Unification (computer science). --- Upper and lower bounds.

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Solomon Lefschetz pioneered the field of topology--the study of the properties of many�sided figures and their ability to deform, twist, and stretch without changing their shape. According to Lefschetz, "If it's just turning the crank, it's algebra, but if it's got an idea in it, it's topology." The very word topology comes from the title of an earlier Lefschetz monograph published in 1920. In Topics in Topology Lefschetz developed a more in-depth introduction to the field, providing authoritative explanations of what would today be considered the basic tools of algebraic topology. Lefschetz moved to the United States from France in 1905 at the age of twenty-one to find employment opportunities not available to him as a Jew in France. He worked at Westinghouse Electric Company in Pittsburgh and there suffered a horrible laboratory accident, losing both hands and forearms. He continued to work for Westinghouse, teaching mathematics, and went on to earn a Ph.D. and to pursue an academic career in mathematics. When he joined the mathematics faculty at Princeton University, he became one of its first Jewish faculty members in any discipline. He was immensely popular, and his memory continues to elicit admiring anecdotes. Editor of Princeton University Press's Annals of Mathematics from 1928 to 1958, Lefschetz built it into a world-class scholarly journal. He published another book, Lectures on Differential Equations, with Princeton in 1946.

Topology. --- Addition. --- Algebraic topology. --- Banach space. --- Barycentric coordinate system. --- C space. --- Centroid. --- Closed set. --- Compact space. --- Connected space. --- Continuous function. --- Contractible space. --- Convex set. --- Corollary. --- Diameter. --- Dimension (vector space). --- Existential quantification. --- General topology. --- Homology (mathematics). --- Homotopy. --- Intersection (set theory). --- K0. --- Local property. --- Locally compact space. --- Lowest common denominator. --- Manifold. --- Metric space. --- Metrization theorem. --- Notation. --- Parallelepiped. --- Polyhedron. --- Polytope. --- Retract. --- Simplex. --- Simplicial complex. --- Subset. --- Theorem. --- Topological space. --- Topology. --- Vector space.

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The description for this book, Contributions to the Theory of Nonlinear Oscillations (AM-20), Volume I, will be forthcoming.

Oscillations. --- Absolute value. --- Addition. --- Algebraic equation. --- Amplitude modulation. --- Angular frequency. --- Applied mathematics. --- Approximation. --- Boundary value problem. --- Coefficient. --- Complex analysis. --- Continuous function. --- Contradiction. --- Curve. --- Diagram (category theory). --- Differential equation. --- Dimensionless quantity. --- Discriminant. --- Eigenvalues and eigenvectors. --- Empty set. --- Equation. --- Experiment. --- Fourier. --- Frequency modulation. --- Homotopy. --- Implicit function theorem. --- Initial condition. --- Integer. --- Integral equation. --- Limit point. --- Linear map. --- Nonlinear system. --- Normal (geometry). --- Notation. --- Operator theory. --- Ordinary differential equation. --- Oscillation. --- Parameter. --- Periodic function. --- Phase space. --- Pure mathematics. --- Quantity. --- Rational function. --- Saddle point. --- Second derivative. --- Simply connected space. --- Singular perturbation. --- Solid torus. --- Special case. --- Suggestion. --- Summation. --- Tangent space. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Topology. --- Two-dimensional space. --- Uniqueness. --- Vacuum tube. --- Variable (mathematics). --- Vector field.

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This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

Partial differential equations --- Differential equations, Partial. --- Probabilities. --- Integration, Functional. --- Functional integration --- Functional analysis --- Integrals, Generalized --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- A priori estimate. --- Absolute continuity. --- Almost surely. --- Analytic continuation. --- Axiom. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Bounded function. --- Calculation. --- Cauchy problem. --- Central limit theorem. --- Characteristic function (probability theory). --- Chebyshev's inequality. --- Coefficient. --- Comparison theorem. --- Continuous function (set theory). --- Continuous function. --- Convergence of random variables. --- Cylinder set. --- Degeneracy (mathematics). --- Derivative. --- Differential equation. --- Differential operator. --- Diffusion equation. --- Diffusion process. --- Dimension (vector space). --- Direct method in the calculus of variations. --- Dirichlet boundary condition. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Exponential function. --- Feynman–Kac formula. --- Fokker–Planck equation. --- Function space. --- Functional analysis. --- Fundamental solution. --- Gaussian measure. --- Girsanov theorem. --- Hessian matrix. --- Hölder condition. --- Independence (probability theory). --- Integral curve. --- Integral equation. --- Invariant measure. --- Iterated logarithm. --- Itô's lemma. --- Joint probability distribution. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Limit (mathematics). --- Limit cycle. --- Limit point. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Markov chain. --- Markov process. --- Markov property. --- Maximum principle. --- Mean value theorem. --- Measure (mathematics). --- Modulus of continuity. --- Moment (mathematics). --- Monotonic function. --- Navier–Stokes equations. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Partial differential equation. --- Periodic function. --- Poisson kernel. --- Probabilistic method. --- Probability space. --- Probability theory. --- Probability. --- Random function. --- Regularization (mathematics). --- Schrödinger equation. --- Self-adjoint operator. --- Sign (mathematics). --- Simultaneous equations. --- Smoothness. --- State-space representation. --- Stochastic calculus. --- Stochastic differential equation. --- Stochastic. --- Support (mathematics). --- Theorem. --- Theory. --- Uniqueness theorem. --- Variable (mathematics). --- Weak convergence (Hilbert space). --- Wiener process.