bluuebunny/crossref_metadata_2025 huggingface dataset:
Integrable Systems Associated to a Hopf Surface
Authors: Moraru Ruxandra
Published: December 2010

A Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.
Cohomology of Complex Torus Bundles Associated to Cocycles
Authors: Lee Min Ho
Published: December 2010

Equivariant holomorphic maps of Hermitian symmetric domains into Siegel upper half spaces can be used to construct families of abelian varieties parametrized by locally symmetric spaces, which can be regarded as complex torus bundles over the parameter spaces. We extend the construction of such torus bundles using 2-cocycles of discrete subgroups of the semisimple Lie groups associated to the given symmetric domains and investigate some of their properties. In particular, we determine their cohomology along the fibers.
Poisson Brackets and Structure of Nongraded Hamiltonian Lie Algebras Related to Locally-Finite Derivations
Authors: Su Yucai
Published: December 2010

Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.
Renormalized Periods on GL(3)
Authors: Beineke Jennifer , Bump Daniel
Published: December 2010

A theory of renormalization of divergent integrals over torus periods on GL(3) is given, based on a relative truncation. It is shown that the renormalized periods of Eisenstein series have unexpected functional equations.
Some Convexity Results for the Cartan Decomposition
Authors: Graczyk P. , Sawyer P.
Published: December 2010

In this paper, we consider the set = a(eXKeY) where a(g) is the abelian part in the Cartan decomposition of g. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of SL(3; F) where F= R, Cor H. In particular, we show that is convex.

We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.
Classification des représentations tempérées d'un groupe p-adique
Authors: Bettaïeb Karem
Published: December 2010

Soit G le groupe des points définis sur un corps p-adique d'un groupe réductif connexe. A l'aide des caractères virtuels supertempérés de G, on prouve (conjectures de Clozel) que toute représentation irréductible tempérée de G est irréductiblement induite d'une essentielle d'un sousgroupe de Lévi de G.
Admissible Majorants for Model Subspaces of H2, Part II: Fast Winding of the Generating Inner Function
Authors: Havin Victor , Mashreghi Javad
Published: December 2010

This paper is a continuation of [6]. We consider the model subspaces Kϴ = H2 ϴ ϴH2 of the Hardy space H2 generated by an inner function ϴ in the upper half plane. Our main object is the class of admissible majorants for Kϴ, denoted by Adm ϴ and consisting of all functions ω defined on ℝ such that there exists an f ≠ 0, f ∈ Kϴ satisfying |f(x)| ≤ ω(x) almost everywhere on ℝ. Firstly, using some simple Hilbert transformtechniques, we obtain a general multiplier theorem applicable to any Kϴ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions ϴ such that the unit vector ϴ(x) winds up fast as x grows from –∞ to ∞. In particular, we consider ϴ = B where B is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from ℝ. It is shown, among other things, that for any such B, any even ω decreasing on (0,∞) with a finite logarithmic integral is in Adm B (unlike the “vertical” case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm exp(iσz), σ > 0. Some oscillating ω's in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp(iσz), σ > 0, and to de Branges’ space ℋ(E).
Complex Uniform Convexity and Riesz Measures
Authors: Blower Gordon , Ransford Thomas
Published: December 2010

The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for LebesgueLpspaces and the von Neumann-Schatten trace ideals. Banach spaces that areq-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace idealscpare 2-uniformly PL-convex for 1 ≤p≤ 2.
Group Actions and Singular Martingales II, The Recognition Problem
Authors: Rosenblatt Joseph , Taylor Michael
Published: December 2010

We continue our investigation in [RST] of a martingale formed by picking a measurable set A in a compact group G, taking random rotates of A, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing A from a small amount of data from this martingale. This leads to problems in harmonic analysis on G, including an analysis of integrals of products of Gegenbauer polynomials.
Défaut de semi-stabilité des courbes elliptiques dans le cas non ramifié
Authors: Cali Elie
Published: December 2010

Let be an algebraic closure of ℚ2 and K be an unramified finite extension of ℚ2 included in . Let E be an elliptic curve defined over K with additive reduction over K, and having an integral modular invariant. Let us denote by Knr the maximal unramified extension of K contained in . There exists a smallest finite extension L of Knr over which E has good reduction. We determine in this paper the degree of the extension L/Knr.
Bump Functions with Hölder Derivatives
Authors: Gaspari Thierry
Published: December 2010

We study the range of the gradients of aC1,α-smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case ofC1-smooth bump functions. Finally, we give a sufficient condition on a subset ofX* so that it is the set of the gradients of aC1,1-smooth bump function. In particular, ifXis an infinite dimensional Banach space with aC1,1-smooth bump function, then any convex open bounded subset ofX* containing 0 is the set of the gradients of aC1,1-smooth bump function.
Fat Points in ℙ1× ℙ1and Their Hilbert Functions
Authors: Guardo Elena , Van Tuyl Adam
Published: December 2010

We study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
Authors: Lim Yongdo
Published: December 2010

We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold Sym(n, ℝ)++ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold Sym(p, ℝ)++ × Sym(q, ℝ)++ block diagonally embedded in Sym(n, ℝ)++ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when p ≤ 2 or q ≤ 2.
Semi-Classical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy
Authors: Michel Laurent
Published: December 2010

We study the semi-classical behavior ash→ 0 of the scattering amplitudef(θ, ω, λ,h) associated to a Schrödinger operatorwith short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potentialV(x) in a domain lying behind the barrier ﹛x:V(x) > λ﹜, the scattering amplitudef(θ, ω, λ,h) changes by a term of order(h∞). Under an escape assumption on the classical trajectories incoming with fixed direction ω, we obtain an asymptotic development off(θ, ω, λ,h) similar to the one established in the non-trapping case.
Kirillov Theory for a Class of Discrete Nilpotent Groups
Authors: Tandra Haryono , Moran William
Published: December 2010

This paper is concerned with the Kirillov map for a class of torsion-free nilpotent groups G. G is assumed to be discrete, countable and π-radicable, with π containing the primes less than or equal to the nilpotence class of G. In addition, it is assumed that all of the characters of G have idempotent absolute value. Such groups are shown to be plentiful.
On Log ℚ-Homology Planes and Weighted Projective Planes
Authors: Daigle Daniel , Russell Peter
Published: December 2010

We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that such a surface admits, up to conjugacy, one or two Ga-actions.
On Operators with Spectral Square but without Resolvent Points
Authors: Binding Paul , Strauss Vladimir
Published: December 2010

Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.
Jordan Structures of Totally Nonnegative Matrices
Authors: Fallat Shaun M. , Gekhtman Michael I.
Published: December 2010

An n × n matrix is said to be totally nonnegative if every minor of A is nonnegative. In this paper we completely characterize all possible Jordan canonical forms of irreducible totally nonnegative matrices. Our approach ismostly combinatorial and is based on the study of weighted planar diagrams associated with totally nonnegativematrices
Partial Euler Products on the Critical Line
Authors: Conrad Keith
Published: December 2010

The initial version of the Birch and Swinnerton-Dyer conjecture concerned asymptotics for partial Euler products for an elliptic curve L-function at s = 1. Goldfeld later proved that these asymptotics imply the Riemann hypothesis for the L-function and that the constant in the asymptotics has an unexpected factor of. We extend Goldfeld's theorem to an analysis of partial Euler products for a typical L-function along its critical line. The general phenomenon is related to second moments, while the asymptotic behavior (over number fields) is proved to be equivalent to a condition that in a precise sense seemsmuch deeper than the Riemann hypothesis. Over function fields, the Euler product asymptotics can sometimes be proved unconditionally.
On Local L-Functions and Normalized Intertwining Operators
Authors: Kim Henry H.
Published: December 2010

In this paper we make explicit all L-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local L-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for Re(s) ≥ 1/2 in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrumand determining poles of automorphic L-functions.
On the Structure of the Spreading Models of a Banach Space
Authors: Androulakis G. , Odell E. , Schlumprecht Th. , Tomczak-Jaegermann N.
Published: December 2010

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space X. In particular we give an example of a reflexive X so that all spreading models of X contain ℓ1 but none of them is isomorphic to ℓ1. We also prove that for any countable set C of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of C. In certain cases this ensures that X admits, for each α < ω1, a spreading model such that if α < β then is dominated by (and not equivalent to) . Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
Authors: Burke Maxim R. , Miller Arnold W.
Published: December 2010

We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A ∩ C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.
The Square Sieve and the Lang–Trotter Conjecture
Authors: Cojocaru Alina Carmen , Fouvry Etienne , Murty M. Ram
Published: December 2010

Let E be an elliptic curve defined over ℚ and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which ℚ(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is OE(x17/18 log x), and unconditionally we show that this number is We also prove that the number of imaginary quadratic fields K, with −disc K ≤ x and of the form K = ℚ(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 Lang–Trotter conjecture.
Strictly Singular and Cosingular Multiplications
Authors: Lindström Mikael , Saksman Eero , Tylli Hans-Olav
Published: December 2010

Let L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sidedmultiplication operators S ↦ ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 < p < ∞and p ≠ 2. Our main result establishes that the multiplication S ↦ ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0, 1)) are strictly singular. We also discuss the case where X is a L1- or a L∞-space, as well as several other relevant examples.
A Semilinear Problem for the Heisenberg Laplacian on Unbounded Domains
Authors: Maad Sara
Published: December 2010

We study the semilinear equation

where is an unbounded domain of the Heisenberg group . The space is the Heisenberg analogue of the Sobolev space . The function is supposed to be odd in u, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator ΔH is the subelliptic Laplacian on the Heisenberg group. We give a condition on Ω which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais–Smale condition. This might be quite surprising since we deal with domains which are far frombounded. The technique we use rests on a compactness argument and the maximum principle.
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras
Authors: Saïd Salem Ben
Published: December 2010

This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.
Multiplicity Results for Nonlinear Neumann Problems
Authors: Filippakis Michael , Gasiński Leszek , Papageorgiou Nikolaos S.
Published: December 2010

In this paper we study nonlinear elliptic problems of Neumann type driven by the p-Laplacian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a C1-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-p-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative p-Laplacian with Neumann boundary condition.
Singular Integrals on Product Spaces Related to the Carleson Operator
Authors: Prestini Elena
Published: December 2010

We prove Lp(T2) boundedness, 1 < p ≤ 2, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.
Partie imaginaire des résonances de Rayleigh dans le cas d'une boule
Authors: Gamblin Didier
Published: December 2010

Résumé

Nous étudions les résonances de Rayleigh créées par une boule en dimension deux et trois. Nous savons qu’elles convergent exponentiellement vite vers l’axe réel. Nous calculons exactement les fonctions résonantes associées puis nous les estimons asymptotiquement en fonction de la partie réelle des résonances. L’application de la formule de Green nous donne alors le taux de décroissance exponentielle de la partie imaginaire des résonances.
Cohomology Pairings on the Symplectic Reduction of Products
Authors: Goldin R. F. , Martin S.
Published: December 2010

Let M be the product of two compact Hamiltonian T-spaces X and Y . We present a formula for evaluating integrals on the symplectic reduction of M by the diagonal T action. At every regular value of the moment map for X × Y, the integral is the convolution of two distributions associated to the symplectic reductions of X by T and of Y by T. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.
On Pointwise Estimates of Positive Definite Functions With Given Support
Authors: Kolountzakis Mihail N. , Révész Szilárd Gy.
Published: December 2010

The following problem has been suggested by Paul Turán. Let Ω be a symmetric convex body in the Euclidean space ℝd or in the torus . Then, what is the largest possible value of the integral of positive definite functions that are supported in Ω and normalized with the value 1 at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in ℝ. That is, under the same conditions and normalizations, the supremum of possible function values at z is to be found for any given point z ∈ Ω. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to ℝd and to non-convex domains as well.

Here we present another approach to the problem, giving the solution in ℝd and for several cases in . Actually, we elaborate on the fact that the problemis essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for ℝd and that for is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.
Apolar Schemes of Algebraic Forms
Authors: Chipalkatti Jaydeep
Published: December 2010

This is a note on the classical Waring's problem for algebraic forms. Fix integers (n, d, r, s), and let ∧ be a general r-dimensional subspace of degree d homogeneous polynomials in n+1 variables. Let denote the variety of s-sided polar polyhedra of ∧. We carry out a case-by-case study of the structure of for several specific values of (n, d, r, s). In the first batch of examples, is shown to be a rational variety. In the second batch, is a finite set of which we calculate the cardinality.
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
Authors: Özlük A. E. , Snyder C.
Published: December 2010

In a previous article, we studied the distribution of “low–lying” zeros of the family of quadratic Dirichlet L–functions assuming the Generalized Riemann Hypothesis for all Dirichlet L–functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (–2, 2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic L-functions. In this note, wemake use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.
Holomorphic Generation of Continuous Inverse Algebras
Authors: Biller Harald
Published: December 2010

We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixedn-tuple of elements are dense. In particular, we characterize the compact subsets of ℂnwhich appear as joint spectra of suchn-tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.
Endomorphism Rings of Finite Global Dimension
Authors: Leuschke Graham J.
Published: December 2010

For a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal ℂ-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen–Macaulay local ring of finite Cohen–Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.
Zeta Functions of Supersingular Curves of Genus 2
Authors: Maisner Daniel , Nart Enric
Published: December 2010

We determine which isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to k-isomorphism, leading to the same zeta function.
On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials
Authors: Stoimenow A.
Published: December 2010

It is known that the Brandt–Lickorish–Millett–Ho polynomialQcontains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable fromQis an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degreed≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.
SLn, Orthogonality Relations and Transfer
Authors: Badulescu Alexandru Ioan
Published: December 2010

Let π be a square integrable representation of G′ = SLn(D), with D a central division algebra of finite dimension over a local field F of non-zero characteristic. We prove that, on the elliptic set, the character of π equals the complex conjugate of the orbital integral of one of the pseudocoefficients of π. We prove also the orthogonality relations for characters of square integrable representations of G′. We prove the stable transfer of orbital integrals between SLn(F) and its inner forms.
Radical Ideals in Valuation Domains
Authors: van den Berg John E.
Published: December 2010

An idealIof a ringRis called a radical ideal ifI= ℛ(R) where ℛ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that ifRis a valuation domain, then a proper idealIofRis a radical ideal if and only ifIis a distinguished ideal ofR(the latter property means that ifJandKare ideals ofRsuch thatJ⊂I⊂Kthen we cannot haveI/J≅K/Ias rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.
The Ground State Problem for a Quantum Hamiltonian Model Describing Friction
Authors: Bruneau Laurent
Published: December 2010

In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.
Quotients jacobiens : une approche algébrique
Authors: Reydy Carine
Published: December 2010

Le diagramme d’Eisenbud et Neumann d’un germe est un arbre qui représente ce germe et permet d’en calculer les invariants. On donne une démonstration algébrique d’un résultat caract érisant l’ensemble des quotients jacobiens d’un germe d’application (f, g) à partir du diagramme d’Eisenbud et Neumann de fg.
On a Pointwise Construction of the Lemniscate
Authors: Blum Richard
Published: December 2010

The writer of the following lines is aware of the possibility that the property discussed below is not a new discovery. He will, of course, be grateful to any reader who will provide him with bibliographical references.

It is well known-that the nine point circle of a triangle is tangent to the 4 contact (i.e. the inscribed and the escribed) circles. Given two tangent circles and we propose to find the locus of vertices of the triangles (Δ) which admit as their nine point circle and as one of their contact circles.
A Generalized Ring of Quotients I
Authors: Findlay G.D. , Lambek J.
Published: December 2010

According to a well-known theorem of algebra [3, p.47], an integral domain can be embedded in a field, called its field of quotients. Every freshman is familiar with the simplest form of this theorem concerning the integers and the rational numbers. Many generalisations have been given in which a "ring of quotients" is constructed for a given ring [e.g. 6, 7, 11, 12, 13]. Of course, we cannot expect a ring of quotients to be a field, or even a skew field. As Malcev [9] has shown, there even exist rings without proper divisors of zero which cannot be embedded isomorphically in a skew field. The most recent construction, by Utumi [12], gives a ring of quotients for any ring with zero left annihilator. We show in this paper that this construction can be extended to arbitrary rings, in fact, to arbitrary modules. The method used is more abstract: a fundamental relation between rings, defined by Utumi in terms of their elements, is here replaced by a corresponding relation between modules, defined by means of homomorphisms.
On the Index of a Quadratic Form
Authors: Wild Jonathan
Published: December 2010

Given a vector space V = {x, y, ...} over an arbitrary field. In V a symmetric bilinear form (x,y) i s given. A subspace W is called totally isotropic [t.i.] if (x,y) = 0 for every pair x W, y W.

Let Vn and Vm be two t.i. subspaces of V; n < m. Lower indices always indicate dimensions. It is a well known and fundamental fact of analytic geometry that there exists a t.i. subspace Wm of V containing Vn [cf. Dieudonné: Les Groupes classiques , P. 18]. As no simple direct proof seems to be available, we propose to supply one.
On a Normal Form of the Orthogonal Transformation III
Authors: Zassenhaus Hans
Published: December 2010

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equation

so that there corresponds the matrix B to the bilinear form

4.1

on the linear space

4.2

and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.

The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.
A Note on Uniqueness and Convergence of Successive Approximations
Authors: Brauer Fred
Published: December 2010

In a recent note in this Bulletin [3], W.A.J. Luxemburg has shown in two different ways that a condition of Krein and Krasnosel'skii [2] for the uniqueness of solutions of a differential equation also implies the convergence of the successive approximations. Here, a third proof of the uniqueness and the convergence of successive approximations, formulated for systems of differential equations, will be obtained. This third proof is modelled on the methods used in proving general uniqueness and convergence theorems [l]. The approach is suggested by Luxemburg's idea of breaking the argument into two stages and using one of the hypotheses in each stage. Since the proofs given here are hardly shorter than the earlier direct proofs, their main interest lies in the fact that they fit what appeared to be an isolated result into the framework of a general theory.
Abstract Definitions for the Mathieu Groups M11and M12
Authors: Moser W.O.J.
Published: December 2010

A list of known finite simple groups has been given by Dickson [3, 4]. With but five exceptions, all of them fall into infinite families. The five exceptional groups, discovered by Mathieu [8,9], were further investigated by Jordan [7], Miller [10], de Séguier [11], Zassenhaus [13], and Witt [12]. In Witt's notation they are M11, M12, M22, M23, M24. Generators for them may be seen in the book of Carmichael [1, pp. 151, 263, 288]; but only for the smallest of them, M11 of order 7920, has a set of defining relations been given.
A Property of Convex Pseudopolyhedra
Authors: Melzak Z.A.
Published: December 2010

In this note we prove one theorem and make a few conjectures, all of which are connected with the following problem raised by S. Mazur [l]: does there exist a closed convex surface whose plane sections give all plane closed convex curves, up to affinities? For our purposes we define a convex pseudopolyhedron to be the closed convex hull of a countable bounded nonplanar sequence of points in E3 with exactly one limit point.
A Note on the Hadamard Product
Authors: Marcus M. , Khan N.A.
Published: December 2010

Let A = (aij), B = (bjj), be two n-square matrices over the complex numbers. Then the n-square matrix H = (hjj) = ij(aijb) is called the Hadamard product of A and B, H = AoB, [l; p. 174]. Let the n2 - square matrix K = A⊗B denote the Kronecker product of A and B.
A Note on a Discriminant Inequality
Authors: Chalk J. H. H.
Published: December 2010

The following conjecture has been investigated recently by Mordell [1]. “Let z1 , …, zn be any set of n complex numbers. Then

1

the equality sign being necessary in the case when the z's are at the vertices of a regular n-sided polygon with center at the origin.”
Notes on Number Theory II : On a theorem of van der Waerden
Authors: Moser Leo
Published: December 2010

A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a positive integer m such that in every distribution of the numbers 1,2, …, m into k classes, at least one class contains an arithmetic progression of t + 1 terms. Other proofs and generalizations of this theorem have been given by Griinwald [2], Witt [3] and Lukomskaya [4]. The last mentioned proof appears in the booklet of Khinchin “Three pearls of number theory” in which van der Waerden's theorem plays the role of the first pearl.
Interpolation and Curve Fitting by Sectionally Linear Functions
Authors: Schwerdtfeger Hans
Published: December 2010

Functions employed for interpolation and curve fitting are for the most part polynomials with numerical coefficients. Indeed these are functions whose values for numerically given arguments can be computed directly without resorting to non-algebraic designs. It is known, however, that there are cases where polynomial interpolation does not yield an adequate approximation to a given function (cf. [4], p. 34).
A Finite Analogue of the Goldbach Problem
Authors: Dixon J. D.
Published: December 2010

E. Cohen [1], [2] has considered an analogue of the famous Goldbach problem in certain finite rings. In the following another analogue is considered in the ring Jn of residue classes over the integers modulo n.

As 'primes' in Jn we take all residue classes coprime to n. That is, all elements p ϵ Jn which are not factors of zero in Jn. We denote this set by Rn.
A Geometric Model for the Generalized Symmetric Group
Authors: Johnson Norman W.
Published: December 2010

The generalized symmetric group S(n, m) consists of all permutations of mn symbols commutative with

Since each cycle Qi= (1i2i… mi) is of order m, there are mnpermutations within the n cycles, generating an invariant subgroup Q of order mn. Also, there are n! ways of permuting the cycles among themselves, by transformations

where i1, i2, …, inare the symbols 1, 2, …, n in some order [5, p. 39]. The permutations W* form a subgroup Sn* of order n!, isomorphic to the symmetric group Sn.
On an Inversion Operator for the Fourier Transformation
Authors: Rooney P. G.
Published: December 2010

In an earlier paper [1] we presented a representation theory for the Fourier transformation defined by

1.1

for functions f in certain function spaces. This theory made use of an operator

1.2

where k = 1, 2,…, and it was stated without proof that this operator is an inversion operator for the Fourier transformation; that is, that under certain conditions

1.3
A Lemma on Continuous Functions
Authors: Lipman J.
Published: December 2010

The point of this note is to get a lemma which is useful in treating homotopy between paths in a topological space [1].

As explained in the reference, two paths joining a given pair of points in a space E are homotopic if there exists a mapping F: I x I →E (I being the closed interval [0,1] ) which deforms one path continuously into the other. In practice, when two paths are homotopic and the mapping F is constructed, then the verification of all its required properties, with the possible exception of continuity, is trivial. The snag occurs when F is a combination of two or three functions on different subsets of I x I. Then the boundary lines between these subsets have to be given special consideration, and although the problems resulting are routine their disposal can involve some tedious calculation and repetition. In the development [l] of the fundamental group of a space, for example, this sort of situation comes up four or five times.
On an Extremal Problem in Fourier Series
Authors: Lorch Lee
Published: December 2010

Let f(x) be a bounded odd function, - π < x < π, |f(X)| ≤ 1, with non-negative Fourier coefficients bk, k = 1,2, ….

Otto Szász [l] proved anew the existence of a bounded set of numbers {βn}, n = 1,2,…, such that

where βn is the smallest constant satisfying the above inequality and added that 2/π ≤ βn ≤ 4/π. He pointed out [1, p. 170] that β1 = 4/π and raised the question of the value of βn for n > 1.
On Birkhoff's Problem 73 For Monoids
Authors: Brainerd Barron
Published: December 2010

Birkhoff in [2] poses the following problem:

“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.

This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.
Autoparallel Deviation in the Geometry of Lyra
Authors: Vanstone J. R.
Published: December 2010

One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.
On the Index of a Symmetric Form
Authors: Wild Jonathan
Published: December 2010

Let E be a finite dimensional vector space over a finite field of characteristic p > 0; dim E = n. Let (x,y) be a symmetric bilinear form in E. The radical Eo of this form is the subspace consisting of all the vectors x which satisfy (x,y) = 0 for every y ϵ E. The rank r of our form is the codimension of the radical.
On an Observation of M. Riesz
Authors: Scherk P.
Published: December 2010

On a recent visit to Toronto, Professor Riesz made an interesting remark on the invariance of a certain cross-ratio, I wish to present a simple proof of his result.

Let x = ( x1, x2, x3, x4), … denote homogeneous coordinates in real projective three-space; x(t), … are differentiable curves in that space.
A Finite Packing Problem
Authors: Oler Norman
Published: December 2010

The maximum density of packings of a given type into the whole of a Euclidean space is defined to be the limit of the maximum density of such packings into a cube as the edge of the cube goes to infinity.

For E2 in particular, a number of well known results such as those due to A. Thue [1], L. Fejes-Toth [2], and C. A. Rogers [3] yield precise information about packings into the whole space. They are however of limited applicability to problems of finite packing in so-far as each requires some restriction upon the boundary of the configuration.
Density Matrices of n-Fermion Systems
Authors: Coleman A. J.
Published: December 2010

The energy of a system of indistinguishable particles in two-body interaction may be expressed in terms of the one and two-particle density matrices. In order to work directly with the density matrices we must know what restrictions are imposed on them by the statistics of the system. This report summarizes some results giving a partial answer to the question "What functions can arise as density matrices of a system of n fermions?".
On a Non-Analytic Perturbation Problem
Authors: Swanson C. A.
Published: December 2010

The purpose here is to study a type of perturbation problem, arising from a differential equation, which is not included in the realm of analytic or asymptotic perturbation theory [4], [6]. Such a problem arises when the domain of the differential operator has been subjected to a variation (rather than the formal operator). We propose to outline one simple problem of this type, concerned with a second order ordinary differential operator. Our purpose is to obtain asymptotic estimates for the characteristic values of a regular Sturm-Liouville problem on a closed interval [a, b] when b is near a singular point of the differential operator. Similar results have been obtained in [3], [8], and [9] by other methods.
On the Distribution of 4-Cycles in Random Bipartite
Tournaments
Authors: Moon J. W. , Moser L.
Published: December 2010

Let there be given two sets of points, P = {P1,…, Pm}1
m and Q={Q1,…, Qn}, such that joining each pair of
points (Pi, Qk), for i=1,…, m and k=1…, n, is a line
oriented towards one, and only one, of the pair. Such a configuration will
be called an m×n bipartite tournament. If the line joining Pi to
Qk is oriented towards Qk we may indicate this by
pi→Qk, and similarly if the line is oriented in the
other sense. The points Pi, Pj, Qk, and
Ql. will be said to form a 4-cycle if either
Pi→Qk→Pj→Ql→Pi or
Pi→Ql→Pj→Qk→Pi.
C(m, n), the number of 4-cycles in a given m×n bipartite tournament,
provides, in some sense, a measure of the degree of transitivity of the
relationship indicated by the orientation of the lines, and the complete
configuration may be thought of as representing the outcome of comparing
each member of one population with each member of a second population, and
making a decision, upon some basis, as to which component of each pair is
the preferred one.
Sur les Anneaux Partiellement Ordonnés
Authors: Thierrin G.
Published: December 2010

Un anneau partiellement ordonné A est un anneau sur lequel est définie une
relation d'ordre partiel telle que:

1.

a≥b entraîne a+c≥+c pour tout c∈A.

2.

a≥o et b≥o entraînent ab≥o.

Si la relation d'ordre est une relation d1 ordre total, l'anneau A est dit
un anneau totalement ordonné.

Il est bien connu (Théorème d'artin-Schreier [l]) que, pour qu'un corps
commutatif K puisse être totalement ordonné, il faut et il suffit que la
relation entraîne
a1=…=an=o. Ce théorème a été généralisé par T.
Szele [4] qui a montré que, pour qu'un corps quelconque K puisse être
totalement ordonné, il faut et il suffit que le demi-groupe additif et
multiplicatif S, engendré par les éléments de K qui sont des carrés non
nuls, ne contienne pas I'élément zéro de K. Ce résultat a été étendu au cas
d1 un anneau d'intégrité par R.E. Johnson [2] de la manière suivante.
Sur un Probléme de P. Scherk, Concernant la Somme des Carrés de
Deux Dérivées
Authors: Iosifescu Marius , Marcus Solomon
Published: December 2010

P. Scherk a posé le problème suivant (Canadian Mathematical Bulletin, vol.4,
no. 3, 1961, p. 306, problème 49):

Let x(t) and y(t) be two differentiable functions defined in some interval.
Does the function have the
Darboux property?

La réponse est négative. En effet, Witold Wilkosz (Some properties of
derivative functions, Fundam. Math. vol.2, 1921, p. 145−154) a montré (p.
146−147 loc. cit.) que la fonction
Some Remarks on Limits in Categories
Authors: Maranda J. M.
Published: December 2010

The object of this paper is to give simple criteria for the existence of
direct limits in categories and for the permuting of a functor with direct
limits.

The notion of direct limit of a diagram that we shall use here is
essentially that of Kan (4), which is more general than the usual notion of
direct limit of a directed diagram.

Our treatment is based on the fact (Lemma 2) that the usual process for
constructing the direct limit of a diagram of modules, which consists in
taking a direct sum of the modules in the diagram and then considering a
certain homomorphic image of this direct sum (3, p. 220), is essentially,
once certain notions have been properly generalized, the only process for
constructing the direct limit of any diagram in any category.
On Algebraic Number Fields with Unique
Factorization
Authors: Connell Ian G.
Published: December 2010

In this note we obtain some simple criteria which show that in certain
algebraic number fields factorization of elements is not unique. All the
arguments depend only on the most elementary ideas (except in §6) so that
probably many of the results are not new. However the proofs are short and
direct and therefore should be of some interest.

We now list some basic facts, whose proofs can be found in any of the books
listed in the bibliography, at the same time fixing our notation.
An Expansion of the Laguerre Polynomials, Lαn(z)
Authors: Wyman Max
Published: December 2010

By a result due to Tricomi(1), it is known that the La guerre polynomials have an expansion in terms of the Bessel functions, Jv(z), of the form

1.1

where the coefficients Am are determined by

1.2

and

1.3
Matrices in Linear Mechanical Systems
Authors: Marcus Marvin
Published: December 2010

In a recent interesting paper [1] on the stability of linear mechanical systems S.P. Diliberto discusses certain reduction theorems for symmetric and skew-symmetric Hamiltonian matrices with respect to symplectic orthogonal similarity. In what follows it is shown that a unified simple argument will handle both of these problems together and then a s an example it is indicated how the argument can be also used to obtain the reduction theorem for symplectic orthogonal matrices.
Note on the Borel Method of Measure Extension
Authors: Fox G.
Published: December 2010

This note concerns a countably additive measure on a Boolean ring of subsets of an abstract set, this measure being real-valued, admitting ∞ as a possible value. We are interested only in unique extensions, so we suppose the measure to be σ- finite. The following well known result will be referred to as the "extension theorem": "Every σ-finite measure on a ring extends uniquely to a σ-finite measure on the generated σ-ring. "Besides the familiar proof using outer measure, there is a Borel-type proof using transfinite induction [4]. We attempt here to reduce the Borel-type proof to its ultimate simplicity, reducing the problem to the bounded case.
On Ordered Geometries
Authors: Scherk P.
Published: December 2010

In Theorem 2.20 of his Geometric Algebra, Artin shows that any ordering of a
plane geometry is equivalent to a weak ordering of its skew field. Referring
to his Theorem 1. 16 that every weakly ordered field with more than two
elements is ordered, he deduces his Theorem 2.21 that any ordering of a
Desarguian plane with more than four points is (canonically) equivalent to
an ordering of its field. We should like to present another proof of this
theorem stimulated by Lipman's paper [this Bulletin, vol.4, 3, pp. 265-278].
Our proof seems to bypass Artin's Theorem 1. 16; cf. the postscript.
An Inequality for Polynomials
Authors: Malik M.A.
Published: December 2010

Throughout this note let be a
polynomial of degree n. The following results are immediate.

For R ≧ 1

1

If p(z) has no zeros in |z| < 1, then

2