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The date of publication of Helmholtz's first paper on the foundations of geometry is discussed.

That Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader.

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric which equals the Poincar\'e metric near the punctures and a holomorphic line bundle which polarizes the metric. We show that the Bergman kernel can be localized around the singularities and its local model is the Bergman kernel of the punctured unit disc endowed with the standard Poincar\'e metric. As a consequence, we obtain an optimal uniform estimate of the supremum norm of the Bergman kernel, involving a fractional growth order of the tensor power., Comment: 42 pages, 2 figures; v.2 is a final update to agree with the published paper

We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1 , under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ) . This domain includes resonance lines and, hence, large gaps among d-dimensional KAM tori. This phenomenon is known as Arnold diffusion. The method of proof follows closely the strategy of [28] , [29] . The main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions I ∈ I ⊂ R d . We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions I, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [28] , [29] . On the technical details of the proofs, we have taken advantage of the theory of the scattering map developed in [31] —notably the symplectic properties—which were not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [28] , [29] .

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λ φ initiated in the papers of Foralewski et al. (2008) [16] , [17] (cf. also Foralewski (2011) [11] , [12] ). As we will show in Example 1.1 , Example 1.2 , Example 1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ 2 λ , then λ φ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λ φ , a strong condition of δ 2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λ φ , of their subspaces of order continuous elements ( λ φ ) a as well as of finite dimensional subspaces λ φ n of λ φ are presented.

In this paper, we define a notion of AS-Gorenstein algebra for N -graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research., Comment: 14 pages

In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer–Meinhardt system on a compact two-dimensional Riemannian manifold without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity ϵ 2 is small enough. We show that for the threshold ratio D ∼ 1 ϵ 2 of the activator diffusivity ϵ 2 and the inhibitor diffusivity D, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o ( 1 ) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O ( 1 ) eigenvalues which all have negative part in this case.

In this paper we prove that, under an explicit integral pinching assumption between the L 2 -norm of the Ricci curvature and the L 2 -norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S 3 . The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the ''convex feasibility'' problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the ''angles'' between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the ''norm'' of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the ''linear regularity property'' of Bauschke and Borwein, the ''normal property'' of Jameson (as well as Bakan, Deutsch, and Li's generalization of Jameson's normal property), the ''strong conical hull intersection property'' of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.

A quasimode for a positive, symmetric and compact operator on a Hilbert space could be defined as a pair ( u , λ ), where u is a function approaching a certain linear combination of eigenfunctions associated with the eigenvalues of the operator in a “small interval” [ λ − r , λ + r ] . Its value in describing asymptotics for low and high frequency vibrations in certain singularly perturbed spectral problems, which depend on a small parameter e, has been made clear recently in many papers. In this paper, considering second order evolution problems, we provide estimates for the time t in which standing waves of the type e i λ t u approach their solutions u ( t ) when the initial data deal with quasimodes of the associated operators. We establish a general abstract framework and we extended it to the case where operators and spaces depend on the small parameter e: now λ and u can depend on e and also perform the estimates for t. We apply the results to vibrating systems with concentrated masses.

The purpose of this paper is to derive product representations for generalizations of the Rogers–Ramanujan series. Special cases of the results presented here were first stated by Ramanujan in the “Lost Notebook” and proved by George Andrews. The analysis used in this paper is based upon the work of Andrews and the broad contributions made by Mourad Ismail and Walter Hayman. Each series considered is related to an extension of the Rogers–Ramanujan continued fraction and corresponds to an orthogonal polynomial sequence generalizing classical orthogonal sequences. Using Ramanujan's differential equations for Eisenstein series and corresponding analogues derived by V. Ramamani, the coefficients in the series representations of each zero are expressed in terms of certain Eisenstein series.

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact intervals, only. This method, which is based on an approximate partition of unity, was introduced by Maz’ya in 1991 and has mainly been used for functions defined on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed.In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

This paper is a continuation of [2], where we complete our partial proof of the Deser-Schwimmer conjecture on the structure of ``global conformal invariants''. Our theorem deals with such invariants P(g^n) that locally depend only on the curvature tensor R_{ijkl} (without covariant derivatives). In [2] we developed a powerful tool, the ``super divergence formula'' which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator I_{g^n}(\phi) that measures the ``non-conformally invariant part'' of P(g^n). This paper resolves the problem of using this information we have obtained on the structure of I_{g^n}(\phi) to understand the structure of P(g^n)., Comment: 35 pages, final version, to appear in Advances in Mathematics

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307–321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold ( M , α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Levi-Civita connection must be of a special form ( ∇ β ) ( X , Y ) = ∥ β # ∥ 2 α ( X , Y ) − β ( X ) β ( Y ) In this paper we give a family of Riemannian manifolds admitting nontrivial solutions and it will be proved as a structure theorem that there are no further essentially different examples; in particular we consider the classical hyperbolic space together with a nontrivial solution. Examples of Wagnerian Finsler manifolds are given in the last section.

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results., Comment: 49 pages. To appear in the Advances in Mathematics

This paper gives the conditions necessary for weighted convergence of Hermite–Fejér interpolation for a general class of even weights which are of exponential decay on the real line or at the end points of (-1,1). The results of this paper guarantee that the conditions of Theorem 2.3 in [11] are optimal.

Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma and an element in \Pi determine a {\em subword complex}, as introduced in our paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented., Comment: 14 pages. Final version, to appear in Advances in Mathematics. This paper was split off from math.AG/0110058v2, whose version 3 is now shorter

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [4] in the framework of holonomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface SCP P of pairs ðx; yÞ; where x is a point in the n-dimensional complex projective space P belonging to the hyperplane yAP : As it turns out, a useful variant is obtained by considering the integral transform associated with the open complement U of S in P P : In the first part of this paper, we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the onedimensional case by the use of homogeneous coordinates. The second part of this paper applies the above result to the quantization of the Radon transform, in the sense of [7]. First we deal with line bundles. More precisely, let P 1⁄4 PðVÞ be the projective space of lines in the vector spaceV; denote by ð Þ 3 D R

Andrew M. Bruckner I and Brian S. Thomson 2 ~Department of Mathematics, University of California, Santa Barbara, California 93106, USA 2Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6 §1. Introduction. The Youngs 1 began to work on real functions in the earliest years of the 20th century. Then, as now, it was an unfashionable subject. Writing of this time Hardy [40, p. 224] (see also [1D says "these subjects were not popular, even in France, with conservatively minded mathematicians; in England they were regarded almost as a morbid growth in mathematics .... " Fashions are dictated, of course, by vested interests, pride and ignorance. It is hard to imagine, from our perspective at the beginning of the 21st century, a more profitable time to study this field than at the beginning of the previous century; Cantor's set theory was very much in the air and all of the important basic tools were being provided by Baire, Borel and Lebesgue. The whole field of what was then called "the theory of functions of a real variable" was reworked and rewritten in those first decades. The Youngs played a major role in this effort. It is beyond our ability to present a complete account of their influence on this field. Much of their work was "influential" in the sense that they popularized and made better known the seminal contributions of Cantor or the important work in integration theory that Lebesgue had produced or the category ideas of Baire. Many of their papers are extensions or applications of these themes with new proofs or new techniques. An indica- tion of their impact is evident in Hobson [43] which for many years was the main English language reference work on real functions: there are 139 citations of their works in these two volumes. This necessary professional work does not often lead to long lasting fame and renown and by now the sources have blurred considerably. A modern graduate course in real functions doubtless owes much to their activity but it is only infrequently explicit. 1 This essay on Youngs' influence on some aspects of real analysis was originally intended to accompany an edition of the collected works of the Youngs planned as a four volume work with essays covering various aspects of their contributions. Unfortunately this ambitious project had to be cancelled. In its place, a one volume edition of Selected Papers [1] was published by Presses Polytechniques et Universitaires Romandes in 2000, edited by S. D. Chatterji and H. Wefelscheid. Professor Chatterji has done a major service by summarizing in a short introduction many themes that run through the vast body of work that the Youngs produced in their careers. E-mail addresses: lbmckner@math.ucsb.edu 2thomson@sfu.ca 0732-0869/01/19/4-337 $ 15.00/0

We discuss and examine Weierstrass’ main contributions to approximation theory. §1. Weierstrass This is a story about Karl Wilhelm Theodor Weierstrass (Weierstras), what he contributed to approximation theory (and why), and some of the consequences thereof. We start this story by relating a little about the man and his life. Karl Wilhelm Theodor Weierstrass was born on October 31, 1815 at Ostenfelde near Munster into a liberal (in the political sense) Catholic family. He was the eldest of four children, none of whom married. Weierstrass was a very successful gymnasium student and was subsequently sent by his father to the University of Bonn to study commerce and law. His father seems to have had in mind a government post for his son. However neither commerce nor law was to his liking, and he “wasted” four years there, not graduating. Beer and fencing seem to have been fairly high on his priority list at the time. The young Weierstrass returned home, and after a period of “rest”, was sent to the Academy at Munster where he obtained a teacher’s certificate. At the Academy he fortuitously came under the tutelage and personal guidance of C. Gudermann who was professor of mathematics at Munster and whose basic mathematical love and interest was the subject of elliptic functions and power series. This interest he was successful in conveying to Weierstrass. In 1841 Weierstrass received his teacher’s certificate, and then spent the next 13 years as a teacher (for 6 years he was a teacher in a pregymnasium in the town of Deutsch-Krone (West Prussia), then for another 7 years in a gymnasium in Braunsberg (East Prussia)). During this period he continued learning mathematics, mainly by studying the work of Abel. He also published some mathematical papers. However these appeared in school journals and were quite naturally not discovered at that time by any who could understand or appreciate them. (Weierstrass’ collected works contain 7 papers from before 1854, the first of which On the development of modular functions (49 pp.) was written in 1840.) In 1854 Weierstrass published the paper On the theory of Abelian functions in Crelle’s Journal fur die Reine und Angewandte Mathematik (the first mathematical research journal, founded in 1826, and now referred to without Crelle’s name in the formal title). It created a sensation within the mathematical community. Here was a 39 year old school teacher whom no one within the mathematical community had heard of. And he had written a masterpiece, not only in its depth, but also in its mastery of an area. Recognition was

The Hilbert function of a homogeneous ideal in R=k[x0 , ..., xn], k a field, is a much studied object. This is not surprising since the Hilbert function encodes important algebraic, combinatorial, and geometric information about the ideal. The fact that recent computer algebra developments have made the Hilbert function computable has not only sustained interest in them but sparked interest in many new questions about them. In this paper, we will concentrate on the Hilbert functions which are the Hilbert functions of points in P. From [11], we know that this is the same as studying 0-dimensional differentiable O-sequences (equivalently, the Hilbert functions of graded artinian quotients of k[x1 , ..., xn]). In our earlier paper [9], we began a discussion of n-type vectors and showed that they were in 1 1 correspondence with Hilbert functions of doi:10.1006 aima.1998.1889, available online at http: www.idealibrary.com on

In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H withGD U.p;qIC/ andHD U.1IC/ U.p 1;qIC/. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroupH . We obtain the Plancherel formula by a special method which is also suitable for other problems, for example, for quantization in the spirit of Berezin. © 2000 Editions scientifiques et medicales Elsevier SAS Keywords: Plancherel formula, Spherical distributions, Complex hyperbolic spaces In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H with GD U.p;qIC/ and H D U.1IC/ U.p 1;qIC/, i.e. we present the decomposition of L 2 into irreducible represen- tations of the group G of class . In order to leave aside the well-known case of a hyperboloid with compact stabilizer subgroup, see (14), we assumep>1 ;q >0. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroup H. We obtain the Plancherel formula by Molchanov's method, see (9). Namely, we follow the detailed scheme in (1), Sections 4, 7. This method deals with the spectral resolution of the radial part of the Laplace operator. The essential step is setting the boundary conditions at certain special points. Those conditions are prescribed by the behaviour of spherical distributions. Finally, it is necessary to perform various analytic continuations. This method is also suitable for other problems, for example, for quantization in the spirit of Berezin, namely, for the decomposition of the Berezin form. It is therefore why this method has to be preferred to the existing methods, described in (3). We use our results from (13). There we define -spherical distributions, study their asymptotic behaviour and express them by means of hypergeometric functions. We describe the irreducible unitary representations of the group G ,o f class associated with an isotropic cone. We give constructions for the Fourier and Poisson transform, define intertwining operators and diagonalize them. Some of those results are presented in Section 1.

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).

1. Introduction and results The main purpose of this paper is to complete one step in our previous paper [6]. That is, we passed too rapidly and incorrectly in obtaining (4.37) and (4.38) of [6]. Let us preserve all notations in [6]. In particular, we consider the following stochastic differential equation (1) { dz(t) = cJ(s(t))dw(t) + b(x(t))dt z(0) = 20 where o : R” -+ R” x R” and b : R” + R” are smooth functions with bounded derivatives of all orders, w(t) is the r-dimensional standard Brownian motion, realized on the classical Wiener space (X, H, ,u) as the coordinate process. As in [6] we consider the Stroock-Varadhan approximation series of (1) (2)

In this paper we study sequences of linear operators which are "almost positive" outside sets of small Jordan measure. For them, we prove Korovkin-type theorems in terms of a modification of the R-convergence used previously by W. Dickmeis, K. Mevissen, R. J. Nessel, and E. Van Wickeren and the test families of functions which the author introduced in a previous paper.

One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper, Jacobi (1836). In this note the context for this theorem and its proof are presented as well as a discussion of the ‘error’ corrected by Clausen.

In previous papers the first author extended the classical equiconvergence theorem of Walsh by the application of summability methods in order to enlarge the disk of equiconvergence to regions of equisummability. The aim of this paper is to study equisummability of sequences which arise from Hadamard products of a fixed power series with Lagrange polynomial interpolants.

Only a few short papers on probability and error theory by Peter Gustav Lejeune Dirichlet are printed in his Werke . However, during his Berlin period, Dirichlet quite frequently gave courses on probability theory or the method of least squares. Unpublished lecture notes reveal that he presented original methods, especially when deriving probabilistic limit theorems; e.g., the use of his discontinuity-factor. The following article discusses some central ideas in Dirichlet's printed papers and unpublished lectures on probability and error theory. These include his deduction of the approximately normal distribution of medians connected with a criticism of least squares as well as his improvement of Laplace's method of approximations relating not only to Stirling's formula but also to the treatment of the central limit theorem. Moreover, the study attempts to place the methods Dirichlet used in probability and error calculus within the broader context of his work in analysis. Nur einige kurze Artikel von Peter Gustav Lejeune Dirichlet zur Wahrscheinlichkeitsund Fehlerrechnung sind in seinen Werken gedruckt. Dirichlet hat aber wahrend seiner Berliner Zeit recht haufig Vorlesungen fiber Wahrscheinlichkeitsrechnung oder Methode der kleinsten Quadrate gehalten. Aus unveroffentlichten Vorlesungsmitschriften geht hervor, dab er gerade bei der Herleitung von Grenzwertsatzen eigene and neue Methoden, z.B. die Verwendung seines Diskontinuitatsfaktors vorgestellt hat. In folgender Arbeit werden Schwerpunkte der gedruckten Arbeiten and der Vorlesungen von Dirichlet fiber Wahrscheinlichkeitsrechnung beschrieben: Grenzwertsatz fur Mediane and Kritik der Methode der kleinsten Quadrate, Fortentwicklung der Laplaceschen Methode fur Approximationen im Hinblick auf die Stirlingsche Formel and die Behandlung des zentralen Grenzwertsatzes. Auberdem wird versucht, die von Dirichlet in Wahrscheinlichkeits- and Fehlerrechnung verwendeten Methoden in den Zusammenhang seiner analytischen Arbeiten zu stellen. Les oeuvres publiees de Peter Gustav Lejeune Dirichlet ne contiennent que quelques rares et courts articles sur le calcul des probabilitds et les erreurs. Pourtant, alors qu'il etait a Berlin, cc mathdmaticien avait assez frequement donne des cours sur les probabilitds ou la methode des moindres carrels: des manuscrits inedits de ses cours en temoignent et permettent de voir qu'a l'occasion de theoremes sur les limites, il introduisit des methodes nouvelles et originales, telle, par exemple, celle qui repose sur l'utilisation du facteur de discontinuity qui porte maintenant son nom. Dans le present article, nous nous proposons de presenter certains aspects fondamentaux des travaux de Dirichlet publies ou non et relatifs au calcul des probabilites, notamment: -la question de la determination de la distribution approximative des medianes, associee a une critique de la methode des moindres carrels; -l'amelioration de la methode d'approximation de Laplace relativement a la formule de Stirling et au traitement du theoreme limite central. Nous nous efforcerons aussi de resituer les methodes probabilistes de Dirichlet dans le contexte de ses travaux analytiques.

This paper gives a detailed analysis of the scientific interaction between Cantor and Dedekind, which was a very important aspect in the history of set theory during the 19th century. A factor that hindered their relationship turns out to be the tension which arose in 1874, due to Cantor's publication of a paper based in part on letters from his colleague. In addition, we review their two most important meetings (1872, 1882) in order to establish the possible exchange of ideas connected with set theory. The one-week meeting in Harzburg (September 1882) was particularly rich in consequences, among other things Dedekind's proof of the Cantor-Bernstein equivalence theorem. But the analysis of this episode will corroborate the lack of collaboration between both mathematicians.

In recent years several papers have been published in quantum mechanical computation, kinetic theory, statistical applications and so forth, dealing with the so-called non-classical orthogonal polynomials. It happens that all these polynomials share a lot of properties, they belong to what we call semi-classical polynomials, i.e. orthogonal sequences whose derivative sequence is quasi-orthogonal of a certain order. This paper concentrates on describing the semi-classical linear functionals of class one. In the first place, we review the characterizations of semi-classical orthogonal polynomials, secondly, we produce the eight canonical forms of the functional equations satisfied by linear functionals of class one, thirdly, after rescaling the parameters (the rescaling suggests an obvious similarity with the classical situation) we establish the eight irreducible canonical functional equations (ICFE) of class one. As solutions to these ICFE we propose integral representations of the linear functionals of class one, and we point out the cases encountered in application fields. Finally, we give the non-linear systems satisfied by the coefficients of the three-term recurrence relation of semi-classical polynomials of class one.

The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p -adic numbers, finite fields, and number fields.

In this paper we prove the results announced in [13]. Let G be a semi- simple, simply connected algebraic group defined over an algebraically closed field k. Let T be a maximal torus, B a Bore1 subgroup, B 3 T. Let W be the Weyl group of G. Let R (resp. R+ ) be the set of roots (resp. positive roots) relative to T (resp. B). Let S be the set of simple roots in R+. Let P be a maximal parabolic subgroup in G with associated fundamental weight w. Let W, be the Weyl group of P, and Wp be the set of minimal representatives of W/W,. For w E Wp, let e(w) be the point and X(w) the Schubert variety in G/P associated to w. In this paper we deter- mine the multiplicity m,(w) of X(w) at e(z), where e(z) E X(w), for all minuscule P’s and also for P = Pgn, G being of type C, (here Pun denotes the maximal parabolic subgroup obtained by omitting a,). The determina- tion of m,(w) is done as follows. Let L be the ample generator of Pic(G/P). A basis has been constructed for @(X(w), L”) in terms of standard monomials on X(w) (cf. [ 16, 11 I). Let U; be the unipotent subgroup of G generated by U-,, /?ET(R+ - Rp+) (here R, denotes the set of roots of P and U, denotes the unipotent subgroup of G, associated to tl E R). Then U; e(r) gives an affme neighborhood of e(z) in G/P. Let A, be the affine algebra of U, e(z) and A,.. = A,/&, where & is the ideal of elements of A, that vanish on X(w) n U; e(z). Let M,,, be the maximal ideal in A, H, corresponding to e(r). Then using the results of [ 16, 111, we obtain a basis of M;, &f:,+,,’ . This enables us to obtain an inductive formula for F,,,, the Hilbert polynomial of X(w) at e(T) (cf. Corollaries 3.8 and 4.11), and also express m, (w) in terms of m, (w’)‘s, X(w’)‘s being the Schubert divisors in X(w) such that e(T)E X(w’) (cf. Theorems 3.7 and 4.10). Using this we

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+\lambda\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^\lambda$ contain an interval for all $x\in(0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017)., Comment: 21 pages, 1 figure. To appear in Indag. Math

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.

Ergodic theory is a relatively new branch of mathematics which from a mathematical point of view may be regarded as generated by the interaction of measure theory and the theory of transformation groups. Its basic concept of "metric transitivity" or "ergodicity" was introduced in 1928 in a paper of Paul Smith and G. D. Birkhoff on dynamical systems. However, the significance of this concept was not appreciated until late 1931 when J. yon Neumann and G. D. Birkhoff proved the celebrated mean and pointwise ergodic theorems, and one may regard the nearly simultaneous appearance of these papers as marking the birth of the subject. Birkhoff's proof of the much more difficult pointwise ergodic theorem was stimulated by yon Neumann's theorem and yon Neumann, in turn, was stimulated by a key observation of B. O. Koopman. Let De be a surface of constant energy E in the phase space D of some Hamiltonian dynamical system. Let V,(~o) denote the point of phase space representing the "state" of the system t time units after it was represented by ~o. Then, for each t, oJ -. Vt(oJ ) is a one-to-one transformation of f2 e onto itself which conserves the natural volume element ~e in f2e induced in £2 e by the Liouville measure dql .." dqn dpl "'" d p n . Moreover, Vq+t~ = VqVt~ for all real numbers t 1 and tz. Koopman's observation (not so obvious 40 years ago as now) was that we may obtain a unitary representation't --+ Ut of the additive group of the real line in the Hilbert space 5°2(f2e, ~e) by defining Ut( f ) (co ) = f (V, ( , -o) ) .

The original Lagrange inversion formula, which gives explicitly the inverse under composition of a formal series, was obtained by Lagrange in 1770 through formal computations involving logarithms of inlmite products. See Lagrange [ 121. There exists an extensive literature on various versions of the inversion formula obtained by algebraic, analytic, and combinatorial methods. Comtet [3] gives a survey and an ample bibliography. During the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula. In 1974 Abhyankar obtained an inversion formula using algebraic methods. See Bass, Connell, and Wright [ 11. Several authors have related Lagrange inversion with Rota’s Finite Operator Calculus [ 151, for example, Joni [8], Garsia and Joni [4], Hofhauer [7], and Roman and Rota [14]. Joyal and Labelle used a combinatorial theory of formal series to obtain inversion formulas (see [9-l I] ) and Viskov [ 161 used ideas from Lie algebra theory to prove formulas similar to Abhyankar’s. In the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of Abhyankar, Joni, and Viskov. We get our results studying an algebra with involution of linear operators on the algebra of formal Laurent series in several indeterminates. We also show that by means of an anti-isomorphism of operator algebras, which we call the operator Bore1 transform, the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal Laurent series. Our approach may be considered as an algebraic analogue of the complex variable methods used by Good [5] to obtain the Lagrange inversion formula as a consequence of the change of variables theorem for the

The purpose of this paper is to develop a unified approach to the characterization of solutions of constrained and unconstrained approximation problems. Several papers have been written on the characterization of solutions of special approximation problems with particular types of constraints or without constraints. For uniform approximation a general theory has been obtained by using generalized weight functions. Recently a new approach via optimization theory has been presented in [I]. The idea is to show, first, that the local Kolmogoroff condition is satisfied. Assuming a convexity condition, it can be shown that the local Kolmogoroff condition implies the Kolmogoroff criterion. Hence best approximations are characterized by the local Kolmogoroff condition. An essential restriction in [I] is the assumption of linear equality constraints For uniform approximation problems with nonlinear equality constraints, the local Kolmogoroff condition has been deduced in [2] under the assumption cd a regularity condition that does not seem to be practical. By deleting inequality constraints a more satisfactory regularity condition has been studied in [3]. Our aim is to treat approximation problems with nonlinear equality and inequality constraints in a normed linear space and to present a new and satisfactory regularity condition. As in [I], we consider the problem as a particular type of optimization problem. Applying new kinds of differentiability, a new approach to optimization problems has been developed in [4]. A generalization of the well-known Lagrange multiplier theorem has been obtained that can be applied to convex optimization problems as well as to differentiable optimization problems. Here we shall apply this theorem to approximation problems with constraints. In particular we obtain new characterization theorems for constrained &-approximation problems of continuous functions.

The paper describes Sylow's discovery of the theorems named after him. He was led to this discovery by his study of Galois' work, in particular of Galois' criterion for the solvability of equations of prime degree. It is explained how Sylow used methods from Galois theory in his proofs. The paper also discusses relevant correspondence between Sylow, Jordan, and Petersen.ZusammenfassungDiese Arbeit behandelt Sylows Entdeckung der Sätze, die nach ihm benannt sind. Er gelangte zu ihnen durch seine Beschäftigung mit Galois' Arbeiten über die Auflösbarkeit von Gleichungen vom Primzahlgrad. Es wird beschrieben, wie Sylow im Beweis seiner Sätze Methoden aus der Galois-Theorie benutzt. Es wird weiterhin auf die diesbezügliche Korrespondenz zwischen Sylow, Jordan und Petersen eingegangen.

In 191, Kazhdan and Lusztig associated to a Coxeter group W certain polynomials P,,, (y, w E W). In the case where W is a Weyl group, these polynomials (or their values at 1, P,,,,(l)) give multiplicities of irreducible constituents in the Jordan-Holder series of Verma modules for a complex semisimple Lie algebra corresponding to W (see [ 3,4]). Also, in the case where W is an affine Weyl group, it has been conjectured by Lusztig [lo] that these polynomials or their values P,,,(l) enter the character formula of irreducible rational representations for a semisimple algebraic group (corresponding to W) over an algebraically closed field of prime characteristic. The purpose of this paper is to give some formula (4.2, see also 4.10) concerning the Kazhdan-Lusztig polynomials P,,, for affine Weyl groups which appear in this Lusztig conjecture; namely P,,, for y, w “dominant.” (This is a generalization of the q-analogue of Kostant’s weight multiplicity formula in [8].) As a corollary of the formula, we shall show that the Lusztig conjecture above is consistent with the Steinberg tensor product theorem [ 15 1. To be more precise, let P be the weight lattice of a root system R (II) R ‘, a positive root system). The Weyl group W of R acts on P. Hence we can define the semidirect product of W by P, p= W D( P. The element of @ corresponding to A E P is denoted by t,. The group @ contains an afline Weyl group W, = W D( Q as a normal subgroup, where Q is the root lattice. Although # is not a Coxeter group in general, we can define the Bruhat order >, the length function I: p+ La, and the Kazhdan-Lusztig polynomials P,., (y, w E I-V) as natural extensions of those for W,. The main result of this paper (under the specialization q --+ 1) is stated as follows (Corollary 4.10)

G. Freud had wide research interests which included Fourier series. In this survey paper we intend to show the great influence of his only paper [2] on strong approximation of Fourier series. This article has been the origin of a new subject called nowadays “converse-type results for strong approximation of Fourier series.” Before stating his initial result we outline briefly the background of the subject. After the classical result of Fejer in 1904 on the convergence of the arithmetical mean of the partial sums of a Fourier series of 2x-periodic functions, Hardy and Littlewood [3] began to investigate the problem of so-called strong summability. It turned out that under certain conditions not only the means

This paper is concerned with the question, “Is what a stored-program digital computer does thinking -in the full human sense of the term?” Several current controversies are examined, including the meaning and usefulness of the Turing test to determine “intelligence.” The Lucas controversy of the early 1960s is taken up, dealing with the philosophical issues related to the man-versus-machine debate, and Dreyfus' ideas against Machine Intelligence are explored. Searle's ideas in opposition to the validity of the Turing test are described, as are various interpretations of the Chinese room thought-experiment and its relation to real “thought”. Weizenbaum's opposition to the “information-processing model of man” is also developed. The paper concludes with a comparison of the 19th-century debates over Darwinian Evolution and those in this century over Artificial Intelligence.

In 1934, Walsh noted that the Taylor polynomial of degree n can be obtained by taking the limit as ϵ → 0 + of the net of n th degree polynomials which best approximate f in the closed discs ¦ z ¦ ⩽ ϵ . Later, this result was generalized to rational approximation. In a recent paper, Shisha and the first two authors generalized this idea to the idea of best local approximation . In this paper, using a different technique, we study this problem in the L 2 setting. Consequently, better results follow under weaker hypotheses.

The collection of Riemann's mathematical papers preserved in Gottingen University Library since 1895 includes none of Riemann's scientific correspondence nor any of his more personal papers. The present report gives an account of the documents (correspondence, lecture notes, etc.) discovered outside Gottingen in the course of a larger research project on Riemann, and briefly describes the history of the Riemann Nachlass. At the same time, readers are kindly requested to inform the author of the whereabouts of any further material relating to Riemann, so that it can be included in the collection of texts and sources currently in preparation.

For 1 d p < co, let L, denote the Banach space of pth power Lebesgue integrable functions on the interval [0, l] with /I f IID = (lh 1 f / p)“p. Let M, EL, denote the set of non-decreasing functions. Then M, is a closed convex lattice. For 1 < p < co, each f E L, has a unique best approximation from M,, while, for p = 1, existence of a best approximation from M, follows from Proposition 4 of [6]. Recently, there has been interest in characterizing best L, approximations from M, [ 1, 2, 3, 41. For example, in [ 1 ] it is shown that iff E L, and if each point in [0, l] is a Lebesgue point off [7], then the best L, approximation to f from M, is unique and continuous. In each of the papers mentioned above, the approach taken was measure theoretic, and the arguments were necessarily complicated. The purpose of this paper is to approach the best approximation problem from a duality viewpoint. This leads to considerable simplification in the derivation of the results, and allows for the omission of the assumption that f E L ~.

In a previous paper ( Numer. Math. 39 (1982), 1–14), M. Gasca and J. I. Maeztu used a geometrical method for the construction of the solutions of certain Hermite and Lagrange interpolation problems in R k . In the present paper, the method is generalized in two different ways: first, the interpolant is not assumed to be a polynomial, and second, a parameter is introduced in order to render the method more versatile.