Herbart's influence on bernhard Riemann

B. Riemann (1826–1866) knew a great deal of the thought of the German philosopher J. Fr. Herbart (1776–1841). During his studies of the philosopher's work he copied out numerous excerpts and made a few notes which are preserved (at least partially) in the Riemann Archiv at Göttingen. This material reveals that Herbart influenced Riemann much more in his epistemology and the comprehension of science than in his particular philosophy of space and spatial thinking. Thus the relationship between Herbart and Riemann has to be looked upon as an example of an influence of German Bildungsphilosophie on the mathematics of the 19th century.     

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

Witten Solution for the Gelfand–Dikii Hierarchy

We derive formulas making it possible to calculate the Taylor expansion coefficients of the string solution for the Gelfand–Dikii hierarchy. According to the Witten conjecture, these coefficients coincide with the Mumford–Morita–Miller intersection numbers (correlators) of stable cohomology classes for the moduli space of n-spin bundles on Riemann surfaces with punctures.     

Pseudoconvexity and Analytic Discs

The purpose of this paper is to investigate the geometry of pseudoconvex hypersurfaces and their interrelation with analytic discs. The method of analytic discs is a very powerful tool in the study of pseudoconvexity. One way to use this tool is via the so-called type function introduced by Dwilewicz and Hill for arbitrary Cauchy–Riemann manifolds. In the hypersurface case, and moreover, under the assumption of pseudoconvexity, it has additional properties which are given in this paper. As an application, boundary estimates of plurisubharmonic functions are given.     

Boundary-Value Problems for Second-Order Equations and for Systems of First-Order Equations

The oblique derivative problem for harmonic functions under violation of the Shapiro–Lopatinsky condition is considered as well as some multi-dimensional analogues of the Cauchy–Riemann system. These problems are reduced to nonelliptic pseudo-differential equations. A method generalizing the regularization of singular integral equations is also presented.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

Large sieve inequalities for -forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL(n) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL(n) forms by computing the ramified factors for GL(n)×GL(n) Rankin–Selberg integrals.     

Riemann integral operator for nonstationary processes in the axisymmetric case

A generalization of the Riemann operator method is proposed, which can be used to analyze in a unified framework the linear equations of nonstationary processes in the axisymmetric case. An integral representation of the solutions of a hyperbolic and a parabolic equation is constructed. The use of the apparatus of special functions produces a simple representation of solutions of partial differential equations, which is convenient for analysis.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 16–22, 1990.     

Boundary problems of the theory of analytic functions on Riemann surfaces

The survey includes papers reviewed in RZh Matematika from 1954–1979. We consider the Riemann boundary problem on a compact Riemann surface and in the class of piecewise-meromorphic automorphic functions; singular integral equations with automorphic kernels and in the form of Abelian integrals; the method of symmetry in solving the problems of Hilbert (linear and nonlinear), Schwarz, Carleman, etc., in the case of a Riemann surface with boundary and in the case of a planar domain, bounded by an algebraic curve; and boundary problems on open Riemann surfaces.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 3–66, 1980.     

Periodic problem for nonlinear Ablowitz model

On the basis of the gradient method of N. N. Bogolyubov, Jr. and the method of finite-zoned integration of S. P. Novikov the authors give a large class of quasiperiodic exact solutions of the Ablowitz dynamical system with the help of the Riemann theta-functions. It is proved that the system admits a standard Lax representation and is a completely integrable Hamiltonian system in the sense of Liouville on an infinite-dimensional smooth manifold.Translated from Matematicheskie Metody i Fiziko-Mechanicheskie Polya, No. 25, pp. 7–13, 1987.     

Factorization of univalent analytic functions on compact Riemann surfaces with boundary

The paper is devoted to the theory of factorization of univalent analytic functions on a compact Riemann surface with boundary. The main component of the theory is a new method for the removal of the periods of certain functions (the natural analogues of the Schwarz integral and Blaschke products) along the boundary curves and bundle sections.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 285–313, 1989.     

Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.     

λ-Equidistributed sequences of partitions and a theorem of the De Bruijn-Post type

Summary The notion of uniform distribution of a sequence is generalized to sequences of partitions in a separable metric space X. Results concern Riemann integrability with respect to a probability on X, and Riemann approximations of Lebesgue integrals.Lavoro presentato al terzo Convegno nazionale «Analisi reale e teoria della misura» (Capri, 12–16 settembre 1988).     

Curvature Symmetries

An alternative construction of Riemann curvature appeared in Acta Appl. Math. 59 (1999), 215–227, with a promise of a short direct proof of its symmetries. The present Section 5 repairs a flaw in the original Section 5, with the promised proof.     

Multidimensional nonlinear integrable systems and methods for constructing their solutions

A new method for constructing multidimensional nonlinear integrable systems and their solutions by means of a nonlocal Riemann problem is presented. This is the natural generalization of the method of the local Riemann problem to the case of several space variables and includes the well-known Zakharov-Shabat method of dressing by Volterra operators.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 77–91, 1984.     

On a class of perturbed conservation laws

We investigate behavior of the support and decay for large times of nonnegative entropy solutions of the Cauchy problem: , u=u₀ in{0, ∞}× R Here m > 1, p > 1, and u₀ has compact support. Sharp estimates are given studying the Riemann problem and using comparison results.     

Weierstrass Multiplicative Points on a Compact Riemann Surface

We introduce Weierstrass multiplicative points and develop the theory of Weierstrass multiplicative points for multiplicative meromorphic functions and Prym differentials on a compact Riemann surface. We prove some analogs of the Weierstrass and Noether theorems on the gaps of multiplicative functions. We obtain two-sided estimates for the number of Weierstrass multiplicative points and q-points. We propose a method for studying the Weierstrass and Noether gaps and Weierstrass multiplicative points by means of filtrations in the Jacobi variety of a compact Riemann surface.     

Interpolation approximations based on Gauss–Lobatto–Legendre–Birkhoff quadrature

We derive in this paper the asymptotic estimates of the nodes and weights of the Gauss–Lobatto–Legendre–Birkhoff (GLLB) quadrature formula, and obtain optimal error estimates for the associated GLLB interpolation in Jacobi weighted Sobolev spaces. We also present a user-oriented implementation of the pseudospectral methods based on the GLLB quadrature nodes for Neumann problems. This approach allows an exact imposition of Neumann boundary conditions, and is as efficient as the pseudospectral methods based on Gauss–Lobatto quadrature for PDEs with Dirichlet boundary conditions.     

Real Polarizable Hodge Structures Arising from Foliations

We construct real polarizable Hodge structures on the reduced leafwisecohomology of Kähler–Riemann foliations by complex manifolds. As inthe classical case one obtains a hard Lefschetz theorem for thiscohomology. Serre's Kählerin analogue of the Weil conjectures carriesover as well. Generalizing a construction of Looijenga and Lunts oneobtains possibly infinite-dimensional Lie algebras attached toKähler–Riemann foliations.Finally using ( , K)-cohomology wediscuss a class of examples obtained by dividing a product of symmetricspaces by a cocompact lattice and considering the foliations coming fromthe factors.     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.