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Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.

Mathematics --- Mathématiques --- Congresses --- Congrès --- 51 --- -Math --- Science --- Congresses. --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Math --- Mathématiques --- Congrès --- A priori estimate. --- Addition. --- Additive group. --- Affine space. --- Algebraic geometry. --- Algebraic group. --- Atiyah–Singer index theorem. --- Bernoulli number. --- Boundary value problem. --- Bounded operator. --- C*-algebra. --- Canonical transformation. --- Cauchy problem. --- Characteristic class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Commutative property. --- Commutative ring. --- Complex manifold. --- Complex number. --- Complex vector bundle. --- Dedekind sum. --- Degenerate bilinear form. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Ellipse. --- Elliptic operator. --- Equation. --- Euler characteristic. --- Euler number. --- Existence theorem. --- Exotic sphere. --- Finite difference. --- Finite group. --- Fourier integral operator. --- Fourier transform. --- Fourier. --- Fredholm operator. --- Hardy space. --- Hilbert space. --- Holomorphic vector bundle. --- Homogeneous coordinates. --- Homomorphism. --- Homotopy. --- Hyperbolic partial differential equation. --- Identity component. --- Integer. --- Integral transform. --- Isomorphism class. --- John Milnor. --- K-theory. --- Lebesgue measure. --- Line bundle. --- Local ring. --- Mathematics. --- Maximal ideal. --- Modular form. --- Module (mathematics). --- Monoid. --- Normal bundle. --- Number theory. --- Open set. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Piecewise linear manifold. --- Poisson bracket. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal part. --- Projective space. --- Pseudo-differential operator. --- Quadratic form. --- Rational variety. --- Real number. --- Reciprocity law. --- Resolution of singularities. --- Riemann–Roch theorem. --- Shift operator. --- Simply connected space. --- Special case. --- Square-integrable function. --- Subalgebra. --- Submanifold. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Winding number. --- Mathematics - Congresses

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This volume is a collection of investigations involving the theory and applications of the various tools and techniques of mathematical analysis and analytic number theory, which are remarkably widespread in many diverse areas of the mathematical, biological, physical, chemical, engineering, and statistical sciences. It contains invited and welcome original as well as review-cum-expository research articles dealing with recent and new developments on the topics of mathematical analysis and analytic number theory as well as their multidisciplinary applications.

Research & information: general --- Mathematics & science --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem --- subordination --- functions with positive real part --- reciprocals --- univalent functions --- starlikeness --- convexity --- close-to-convexity --- hyper-Bessel functions --- Hardy space --- distribution --- fractional Laplacian --- Riesz fractional derivative --- delta sequence --- convolution --- subordinations --- starlike functions --- convex functions --- close-to-convex functions --- cardioid domain --- Hankel determinant --- m-fold symmetric functions --- harmonic univalent functions --- with symmetric conjecture point --- integral expressions --- coefficient estimates --- distortion --- umbral methods --- harmonic numbers --- special functions --- integral representations --- laplace and other integral transforms --- analytic functions --- quasi-Hadamard --- differential operator --- closure property --- riemann zeta function --- asymptotics --- exponential sums --- multivalent functions --- q-Ruschweyh differential operator --- q-starlike functions --- circular domain --- q-Bernardi integral operator --- Bessel functions --- Appell–Bessel functions --- generating functions --- Chebyshev polynomials --- Euler sums --- Catalan’s constant --- Trigamma function --- integral representation --- closed form --- ArcTan and ArcTanh functions --- partial fractions --- Lambert series --- cotangent sum --- modular transformation --- Dedekind sum --- lemniscate of Bernoulli Hankel determinant --- determinant --- inverse --- Mersenne number --- periodic tridiagonal Toeplitz matrix --- Sherman-Morrison-Woodbury formula --- Fibonacci number --- Lucas number --- Toeplitz matrix --- Hankel matrix --- univalent function --- second Hankel determinant --- bi-close-to-convex functions --- gamma function and its extension --- Pochhammer symbol and its extensions --- hypergeometric function and its extensions --- τ-Gauss hypergeometric function and its extensions --- τ-Kummer hypergeometric function --- Fox-Wright function --- p-valent analytic function --- Hadamard product --- q-integral operator --- generalized Lupaş operators --- q analogue --- Korovkin’s type theorem --- convergence theorems --- Voronovskaya type theorem --- starlike function --- subordinate --- Janowski functions --- conic domain --- q-convex functions --- q-close-to-convex functions --- theta-function identities --- multivariable R-functions --- Jacobi’s triple-product identity --- Ramanujan’s theta functions --- q-product identities --- Euler’s pentagonal number theorem --- Rogers-Ramanujan continued fraction --- Rogers-Ramanujan identities --- combinatorial partition-theoretic identities --- Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities --- Schur’s second partition theorem