bluuebunny/crossref_metadata_2025 huggingface dataset:
Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four
Authors: Fels M. E. , Renner A. G.
Published: December 2010

A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.
Reducibility for SUn and Generic Elliptic Representations
Authors: Goldberg David
Published: December 2010

We study reducibility of representations parabolically induced from discrete series representations of SUn(F) for F a p-adic field of characteristic zero. We use the approach of studying the relation between R-groups when a reductive subgroup of a quasi-split group and the full group have the same derived group. We use restriction to show the quotient of R-groups is in natural bijection with a group of characters. Applying this to SUn(F) ⊂ Un(F) we show the R group for SUn is the semidirect product of an R-group for Un(F) and this group of characters. We derive results on nonabelian R-groups and generic elliptic representations as well.
Geometric Study of Minkowski Differences of Plane Convex Bodies
Authors: Martinez-Maure Yves
Published: December 2010

In the Euclidean plane ℝ2, we define the Minkowski difference 𝒦–𝓛 of two arbitrary convex bodies 𝒦, 𝓛 as a rectifiable closed curve ℋh⊂ ℝ2that is determined by the differenceh=h𝒦–h𝓛of their support functions. This curve ℋhis called the hedgehog with support functionh. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ℝn+1, defined as (possibly singular and self-intersecting) hypersurfaces of ℝn+1. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their lengthmeasures and solve the extension of the Christoffel–Minkowski problemto plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ℝ2and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.
A Steinberg Cross Section for Non–Connected Affine Kac—Moody Groups
Authors: Mohrdieck Stephan
Published: December 2010

In this paper we generalise the concept of a Steinberg cross section to non–connected affine Kac–Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac–Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a “twisted Coxeter cell”, a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a 𝕔*-action.
The Generalized Cuspidal Cohomology Problem
Authors: Bart Anneke , Scannell Kevin P.
Published: December 2010

Let Γ ⊂ SO(3, 1) be a lattice. The well known bending deformations, introduced by Thurston and Apanasov, can be used to construct non-trivial curves of representations of Γ into SO(4, 1) when Γ\ℍ3 contains an embedded totally geodesic surface. A tangent vector to such a curve is given by a non-zero group cohomology class in H1(Γ, ℍ41). Our main result generalizes this construction of cohomology to the context of “branched” totally geodesic surfaces. We also consider a natural generalization of the famous cuspidal cohomology problem for the Bianchi groups (to coefficients in non-trivial representations), and perform calculations in a finite range. These calculations lead directly to an interesting example of a link complement in S3 which is not infinitesimally rigid in SO(4, 1). The first order deformations of this link complement are supported on a piecewise totally geodesic 2-complex.
Nonstandard Ideals from Nonstandard Dual Pairs for L1(ω) and l1(ω)
Authors: Read C. J.
Published: December 2010

The Banach convolution algebras l1(ω) and their continuous counterparts L1(ℝ+, ω) are much studied, because (when the submultiplicative weight function ω is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of “nice” weights ω, the only closed ideals they have are the obvious, or “standard”, ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in l1(ω). His proof was successfully exported to the continuous case L1(ℝ+, ω) by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in l1(ω) and L1(ℝ+, ω). The new proof is based on the idea of a “nonstandard dual pair” which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in L1(ℝ+, ω) containing functions whose supports extend all the way down to zero in ℝ+, thereby solving what has become a notorious problem in the area.
Karamata Renewed and Local Limit Results
Authors: Handelman David
Published: December 2010

Connections between behaviour of real analytic functions (with no negative Maclaurin series coefficients and radius of convergence one) on the open unit interval, and to a lesser extent on arcs of the unit circle, are explored, beginning with Karamata's approach. We develop conditions under which the asymptotics of the coefficients are related to the values of the function near 1; specifically, a(n) ∼ f(1 – 1/n)/αn (for some positive constant α), where f(t) = Σa(n)tn. In particular, if F = Σc(n)tn where c(n) ≥ 0 and Σc(n) = 1, then f defined as (1 – F)–1 (the renewal or Green's function for F) satisfies this condition if F′ does (and a minor additional condition is satisfied). In come cases, we can show that the absolute sum of the differences of consecutiveMaclaurin coefficients converges. We also investigate situations in which less precise asymptotics are available.
Gauge-Invariant Ideals in theC*-Algebras of Finitely Aligned Higher-Rank Graphs
Authors: Sims Aidan
Published: December 2010

We produce a complete description of the lattice of gauge-invariant ideals inC*(Λ) for a finitely alignedk-graph Λ. We provide a condition on Λ under which every ideal is gauge-invariant. We give conditions on Λ under whichC*(Λ) satisfies the hypotheses of the Kirchberg–Phillips classification theorem.
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
Authors: Develin Mike , Martin Jeremy L. , Reiner Victor
Published: December 2010

K. Ding studied a class of Schubert varieties Xƛ in type A partial flag manifolds, indexed by integer partitions ƛ and in bijection with dominant permutations. He observed that the Schubert cell structure of Xƛ is indexed by maximal rook placements on the Ferrers board Bƛ, and that the integral cohomology groups H*(Xƛ; ℤ), H*(Xμ; ℤ) are additively isomorphic exactly when the Ferrers boards Bƛ, Bμ satisfy the combinatorial condition of rook-equivalence.

We classify the varieties Xƛ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields
Authors: Okoh F. , Zorzitto F.
Published: December 2010

Purely simple Kronecker modules ℳ, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h: K ∪ ﹛∞﹜ → ﹛∞, 0, 1, 2, 3, … ﹜ is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X)[Y] called the regulator. When the module ℳ admits nontrivial endomorphisms, f must be linear or quadratic in Y. In that case ℳ is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End ℳ embeds in the quadratic function field K(X)[Y]/(f). For some height functions h of infinite support I, the search for a functional α for which (h, α) has regulator 0 comes down to having functions η : I → K such that no planar curve intersects the graph of η on a cofinite subset. If K has characterictic not 2, and the triplet (m, h, α) gives a purely-simple Kronecker module ℳ having non-trivial endomorphisms, then h attains the value ∞ at least once on K ∪ ﹛∞﹜ and h is finite-valued at least twice on K ∪ ﹛∞﹜. Conversely all these h form part of such triplets. The proof of this result hinges on the fact that a rational function r is a perfect square in K(X) if and only if r is a perfect square in the completions of K(X) with respect to all of its valuations.
Weak Semiprojectivity in Purely Infinite SimpleC*-Algebras
Authors: Lin Huaxin
Published: December 2010

LetAbe a separable amenable purely infinite simpleC*-algebra which satisfies the Universal Coefficient Theorem. We prove thatAis weakly semiprojective if and only ifKi(A) is a countable direct sum of finitely generated groups (i= 0, 1). Therefore, ifAis such aC*-algebra, for any ε > 0 and any finite subset ℱ ⊂Athere existδ> 0 and a finite subset⊂Asatisfying the following: for any contractive positive linear mapL:A→B(for anyC*-algebraB) with ∥L(ab) –L(a)L(b)∥ <δfora,b∈there exists a homomorphismh:A→Bsuch that ∥h(a) –L(a)∥ < ε fora∈ ℱ.
Distance from Idempotents to Nilpotents
Authors: MacDonald Gordon W.
Published: December 2010

We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of n × n matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.
Algèbres de Lie d'homotopie associées à une proto-bigèbre de Lie
Authors: Bangoura Momo
Published: December 2010

On associe à toute structure de proto-bigèbre de Lie sur un espace vectoriel F de dimension finie des structures d’algèbre de Lie d’homotopie définies respectivement sur la suspension de l’algèbre extérieure de F et celle de son dual F*. Dans ces algèbres, tous les crochets n-aires sont nuls pour n ≥ 4 du fait qu’ils proviennent d’une structure de proto-bigèbre de Lie. Plus généralement, on associe à un élément de degré impair de l’algèbre extérieure de la somme directe de F et F*, une collection d’applications multilinéaires antisymétriques sur l’algèbre extérieure de F (resp. F*), qui vérifient les identités de Jacobi généralisées, définissant les algèbres de Lie d’homotopie, si l’élément donné est de carré nul pour le grand crochet de l’algèbre extérieure de la somme directe de F et de F*.
Representations of the Fundamental Group of an L–Punctured Sphere Generated by Products of Lagrangian Involutions
Authors: Schaffhauser Florent
Published: December 2010

In this paper, we characterize unitary representations of π := π1(S2\{s1, … , sl}) whose generators u1, … , ul (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions uj = σjσj+1 with σl+1 = σ1. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space ℳᘓ := Homᘓ(π, U(n))/U(n). Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ℳᘓ. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ℳᘓ and give conditions on an involution defined on a quasi-Hamiltonian U-space (M, ω, μ: M → U) for it to induce an anti-symplectic involution on the reduced space M//U := μ–1({1})/U.
Admissibility for a Class of Quasiregular Representations
Authors: Currey Bradley N.
Published: December 2010

Given a semidirect product G = N ⋊ H where N is nilpotent, connected, simply connected and normal in G and where H is a vector group for which ad() is completely reducible and R-split, let τ denote the quasiregular representation of G in L2(N). An element ψ ∈ L2(N) is said to be admissible if the wavelet transform f ⟼ 〈 f, τ(·)ψ 〉 defines an isometry from L2(N) into L2(G). In this paper we give an explicit construction of admissible vectors in the case where G is not unimodular and the stabilizers in H of its action on are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of L2(G) into G-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.
The Chen–Ruan Cohomology of Weighted Projective Spaces
Authors: Jiang Yunfeng
Published: December 2010

In this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
Authors: Rodrigues B.
Published: December 2010

In this paper we study ruled surfaces which appear as exceptional surface in a succession of blowing-ups. In particular we prove that the e-invariant of such a ruled exceptional surface E is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of E). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of e to the study of the poles of the well-known topological, Hodge and motivic zeta functions.
Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series
Authors: Alayont Feryâl
Published: December 2010

Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.
k(n)-Torsion-Free H-Spaces and P(n)-Cohomology
Authors: Michael Boardman J. , Stephen Wilson W.
Published: December 2010

The H-space that represents Brown–Peterson cohomology BPk(–) was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum P(n), by constructing idempotent operations in P(n)–cohomology P(n)k(–) in the style of Boardman–Johnson–Wilson; this relies heavily on the Ravenel–Wilson determination of the relevant Hopf ring. The resulting (i – 1)-connected H-spaces Yi have free connective Morava K-homology k(n)*(Yi), and may be built from the spaces in the Ω-spectrum for k(n) using only vn-torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space X, the P(n)*-module P(n)*(X) is generated by elements of P(n)i(X) for i ≥ 0. This result is essential for the work of Ravenel–Wilson–Yagita, which in many cases allows one to compute BP–cohomology from Morava K-theory.
Calderón–Zygmund Operators Associated to Ultraspherical Expansions
Authors: Buraczewski Dariusz , Martinez Teresa , Torrea José L.
Published: December 2010

We define the higher order Riesz transforms and the Littlewood–Paley g-function associated to the differential operator Lλf(θ) = –f′′(θ)–2λ cot θ f′(θ) + λ2f(θ). We prove that these operators are Calderón–Zygmund operators in the homogeneous type space ((0, π), (sin t)2λdt). Consequently, Lp weighted, H1 – L1 and L∞ – BMO inequalities are obtained.
Generic Extensions and Canonical Bases for Cyclic Quivers
Authors: Deng Bangming , Du Jie , Xiao Jie
Published: December 2010

We use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the positive part U+of the quantum affine. This construction relies on analysis of quiver representations and the introduction of a new integral PBW–like basis for the Lusztig ℤ[v,v–1]-form of U+.
On an Elementary Problem in Number Theory
Authors: Erdös Paul
Published: December 2010

A question which Chalk and L. Moser asked me several years ago led me to the following problem: Let 0 < x ≤ y. Estimate the smallest f(x) so that there should exist integers u and v satisfying

1

I am going to prove that for every ∊ > 0 there exist arbitrarily large values of x satisfying

2

but that for a certain c > 0 and all x

3

A sharp estimation of f(x) seems to be a difficult problem. It is clear that f(p) = 2 for all primes p. I can prove that f(x)→ ∞ and f(x)/loglog x → 0 if we neglect a sequence of integers of density 0, but I will not give the proof here.
A Remark by Philip Hall
Authors: Robinson G. de B.
Published: December 2010

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.

As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.
Outline of an Introduction to Mathematical Logic I
Authors: Robinson Abraham
Published: December 2010

This is the first of (presumably) three articles on the subject mentioned in the title. The exposition is based on a course of fifteen lectures which formed part of the Edmonton (1957) Seminar of the Canadian Mathematical Congress. Limitations of space (and, originally, of time) compel us to be selective in two directions. First, while we shall refer to other branches of logic in passing, we shall be concerned principally with the two fundamental calculi - of propositions and of predicates (of the first order). Thus, except for a number of modern developments which are included here, our exposition will be similar in scope, though not in detail, to the first and third chapters of the well-known "Principles of Mathematical Logic" by D. Hilbert and W. Ackermann (English translation, Chelsea, New York, 1950) and this was in fact the recommended text for the Edmonton course. However, there exists a growing number of other good introductions to the field and some of these will be listed later.
A Note on the Compositions of an Integer
Authors: Narayana T. V. , Fulton G. E.
Published: December 2010

Given an integer n, we define an r-composition of n as follows:

An r-composition of n, (t1, ... , tr), is a set of ti where ti ≥ 1 is an integer for i = 1, ... , r such that

If r is an integer such that 1 ≤ r ≤ n, we have, obviously, distinct r - compositions of n.
On the Discriminant x'Ax.y'Ay-(x'Ay)2
Authors: Schwerdtfeger H.
Published: December 2010

Let x, y be column matrices of n real homogeneous coordinates xj, yj (j = 1,..., n) representing points in (n - 1) - dimensional real projective space Pn - 1. Let A be an n × n real symmetric matrix. The equation x' Ax = 0 represents a quadric in Pn - 1 and the equation
Outline of an Introduction to Mathematical Logic IV
Authors: Robinson Abraham
Published: December 2010

The extended completeness theorem of the predicate calculas of the first order. In section 12, we developed a deductive theory of the first order predicate calculus, while in section II we dealt with the semantic theory of that calculus. We now have to consider the connection between these two theories. We recall that a sentence X can be satisfied by a structure M only if X is defined in M. Given a sentence X (a set of sentences K) we shall say that the structure M is a model of X (of K) if X is (all the sentences of K are) satisfied by M.
Notes on Numerical Analysis I. Polynomial Iteration
Authors: Schwerdtfeger Hans
Published: December 2010

Let f(x) be a real analytic function of the real variable x and α a simple root of the equation f(x) = 0. It is well known that a function ϕ(x) can be associated with the equation in many different ways such that

(i)

α is a root of the equation ϕ(x), i.e. α is a fixed point (invariant point) of the function ϕ(x);

(ii)
On the Product of the Primes not Exceeding n
Authors: Moser Leo
Published: December 2010

One of the most elegant results of the elementary theory of the distribution of primes is that

1

where the product runs over primes. A very simple proof of (1) has recently been given by Erdös and Kalmar [1], [2].
The Half-Baked Method for Cooking of Experimental Data
Authors: Stanton Ralph G.
Published: December 2010

Students in Mathematics and Physics courses are commonly exposed to a method of treating data known as the "half-table method". The method is presented with encomiums by R. C. Dearle [1] and by Lucius Tuttle and John Satterly [2]. It can best be described by an illustration; since the determination of the period of a pendulum is one of the experiments most commonly used (and occurs in both [1] and [2]), let us borrow a set of data from [2] and some explanatory text from [1].
A Note on the Stochastic Rank of a Bipartite Graph
Authors: Dulmage A.L. , Mendelsohn N.S.
Published: December 2010

A bipartite graph is a system consisting of two sets of vertices S and T and a set of edges K, each edge joining a vertex of S to a vertex of T. A set U of edges of K is said to be independent if no two edges of U have a vertex in common. The largest possible number of independent edges has been variously called the exterior dimension [3], term rank [4, 5, 7], etc. This number is the same as the smallest number of vertices in a set W such that each edge of K has at least one of its vertices in W. The edges of a finite bipartite graph can be represented as a set of cells in a matrix as follows. If S = a1, a2, …, an T = b1, b2, … bm, the edges of K are represented by some of the cells of an n by m matrix as follows: if K contains the edge joining ai to bj then the (i, j)th cell of the matrix represents this edge. It is convenient sometimes to represent the set K by a matrix A with real entries aij where aij = 0 if ai is not joined to bj in K and aij > 0 if ai is joined to bj in K. Any non-null graph K will have infinitely many matrix representations.
On Continuous Linear Transformations of Integral Type
Authors: Ellis H.W. , Mott T.E.
Published: December 2010

Let (XxY, SxT, μxν) denote the completion of the Cartesian product of the σ-finite and complete "measure spaces (X, S, μ.) and (Y, T, ν) [3]. Let λx and λy denote arbitrary length functions defined on (X, S, μ.) and (Y, T, ν) respectively, the conjugate length functions [2]. We suppose that

1

is defined for every f(x, y) measurable (SxT). The Fubini theorem implies that f(x, y) is measurable (T) for almost all x. Thus λ xy(f) will be defined when λy(f) Is measurable (S). If Lλ y = Lp, 1 ≤ p < ∞, this is implied by the Fubini theorem. General conditions ensuring that λ y (f) is measurable (S) are given in [l, Theorem 3.2] When λxy(f) is defined for every f(x, y) measurable (SxT), it is a length function and Lλ xy is a Banach space [l, Theorem 3.l].
Self-Injective Rings
Authors: Wong E.T. , Johnson R.E.
Published: December 2010

Historically, the first example of a ring of quotients was the quotient field of an integral domain. Later on, conditions were found under which a noncommutative integral domain has a quotient division ring. More recently, R.E. Johnson [4], Y. Utumi [5], and G.D. Findlay and J. Lambek [3] have discussed the existence and structure of a maximal ring of quotients of any ring.

The present paper uses the methods of Findlay and Lambek to recast the results of Johnson on the quotient ring of a ring with zero singular ideal. It is also shown that such a ring has a unique left-right maximal ring of quotients.
Some Properties of Beatty Sequences I*
Authors: Connell Ian G.
Published: December 2010

Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences

(1)

(2)

(where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α.
A Non-Hamiltonian Graph
Authors: Tutte W. T.
Published: December 2010

There is a remarkable cubic graph of 28 vertices, discovered by Prof. H.S.M. Coxeter, which has no circuit of fewer than 7 edges and in which all oriented arcs made up of three edges are equivalent under the automorphism group.

The structure of the graph is as follows. There are three disjoint heptagons A(a1a2a3a4a5a6a7a1), B(b1b5b2b6b3b7b4b1) and C(c1c6c4c2c7c5c3c1). We denote the seven remaining vertices by d1, d2,…, d7. For each suffix i the three edges incident with dijoin it to ai, bi and ci. In drawing a diagram of the graph it seems best to show only the heptagons A, B and C,
leaving the rest of the figure to the imagination.

The purpose of this note is to establish another property of the graph, that it has no Hamiltonian circuit.
A Note on Prime-Power Groups
Authors: Struik Ruth Rebekka
Published: December 2010

In this paper an unsuccessful attempt to classify prime - power groups is described. The attempt consisted in combining some ideas of P.Hall and O. N. Golovin.

Golovin [2] defined nilpotent products of groups and showed that every nilpotent group is a factor group of a nilpotent product of cyclic groups.
Elementary Proofs of the Extremal Properties of the Eigenvalues of the Sturm-Liouville Equation
Authors: Beesack Paul R.
Published: December 2010

If the Sturm-Liouville eigenvalue problem

1

is first approached from the standpoint of differential equations theory - as opposed, say, to the calculus of variations, or the theory of integral equations - the extremal properties of the eigenvalues seem to be generally regarded as lying beyond the scope of the theory. Thus, neither in the standard work of Bocher [l], nor in the recent work of Coddington and Levinson [2] is any mention made of this topic. Collatz [3, 166-8] gives an elementary proof of the minimum property of the least positive eigenvalue of (1.1), and a brief indication of how this argument can be extended to the higher eigenvalues. The purpose of this paper is to consolidate this elementary approach, and to extend it to cover the singular cases where either the interval is infinite, or one or more of the coefficients are singular at the end-points.
Sequence and Series Transformations
Authors: Macphail M. S.
Published: December 2010

The summability methods

where bnk=ank+an,k+1+ …,are regarded as the sequenceto-sequence and series-to-sequence forms of the same method, and if sk=uo + … + uk, we speak of the series or the sequence {sk} indifferently, as summable A or B.
A Note an a Group Defined by a Quadratic Form
Authors: Marcus Marvin , Khan Nisar A.
Published: December 2010

In a recent series of papers [3,4,5], H. Zassenhaus considered the structure of those linear transformations T on real 4-space, R4, into itself that preserve the quadratic form . That is,

1.1

Define a function ϕ on R4 to the space M 2 of 2-square matrices over the complex numbers as follows:

1.2

Let G2 be the vector space of matrices generated by all real linear combinations of

1.3
On Duo Rings
Authors: Thierrin G.
Published: December 2010

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.

In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.

It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.
On Schauder Bases for Spaces of Continuous Functions1)
Authors: Ellis H. W. , Kuehner D. G.
Published: December 2010

In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such that

In infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.
Odd and Even Permutations
Authors: Halperin Israel
Published: December 2010

This note gives a proof for the familiar elementary theorem that if a permutation of the integers 1, …, n (with n ≥ 2) is expressed as a product π1 of N1 transpositions and also as a product π2 of N2 transpositions, then N1 and N2 are both even or both odd (equivalently: N1 + N2 is even).

Here, a transposition means an interchange of two of the integers. If the interchange is between adjacent integers it is called an adjacent-trans position.
Hesse's Theorem for a Quadrilateral Whose Sides Touch a Conic
Authors: Brown William G.
Published: December 2010

Hesse's theorem states that “if two pairs of opposite vertices of a quadrilateral are respectively conjugate with respect to a given polarity, then the remaining pair of vertices are also conjugate ”.

In the real projective plane there cannot exist such a quadrilateral, all four sides of which are self-conjugate [1, §5.54]. We shall show that such a quadrilateral exists in PG(2, 3), and that any geometry in which such a quadrilateral exists contains the configuration 134 of PG(2,3). We shall thus provide a synthetic proof of Hesse1 s theorem for a quadrilateral of this type, which, together with [1, § 5.55], constitutes a complete proof of the theorem valid in general Desarguesian projective geometry. We shall also show analytically that a finite Desarguesian geometry which admits a Hessian quadrilateral all of whose sides touch a conic must be of type PG(2, 3n).
Tests for the Scale Parameter of the Truncated Normal
Authors: Guttman Irwin
Published: December 2010

This paper continues the work begun in [l] and examines the loss of power when using tests based on the assumption that the variable being sampled has a “complete” normal distribution, when in fact, sampling is from a symmetrically truncated distribution. The hypothesis considered here is the one - sided test for the variance of a normal distribution. Some tables have been computed and they show that appreciable losses in size occur. Some loss occurs in the power too, but this decreases with the alternative value of the variance and the degree of truncation.
On the Watson and Laplace Transformations
Authors: Rooney P. G. , Schubert C.
Published: December 2010

Let k be the kernel of a Watson transformation; that is k(x)/x ϵ L2(0,∞), and if x and y are positive,

1

Then if g is the transform of F ϵ L2(0,∞), that is if

2

it i s known that g 2(0,∞), that

3

and that

4

For these results see [1; theorem 79].
On Geodesics of a Modified Riemannian Manifold
Authors: Sen D. K.
Published: December 2010

In Riemannian geometry the autoparallels associated with the affine connexion coincide with the geodesies which arise from the metric. This is not the case in a modification of Riemannian geometry suggested by Lyra. A sufficient condition that the two classes of curves coincide is obtained.
On the Analytic Continuation of the ζ -Function
Authors: Prowse Ebenezer
Published: December 2010

It i s well known that the Riemannian ζ-function can be extended into the whole region s≠1 by means of the Euler-MacLaurin summation formula. The author wishes to present a proof of this fact which i s rather more elementary than those he i s acquainted with.
A Generalization of Cox's Chain of Theorems
Authors: Al-Dhahir M. W.
Published: December 2010

In [5, p. 105] attention has been called to a set of propositions, due to H. Cox [3, p. 67], which are related to another set, due to Clifford [2, p. 145; 4, p. 447], concerning points and circles in the plane or on the sphere. One may state Cox's chain of theorems as follows:

In a projective 3-space, S3, let (1), (2), (3), (4) be four points lying in a plane α such that no three of them are collinear. Every two determine a line; let one plane such as [12], pass through each line. There are six such planes. The planes [12], [23], [13] determine a point (123); there are four such points. The first theorem of the chain states that they all lie in one plane [1234], It is not difficult to see that this is, in fact, a rewording of Möbius's theorem on mutually inscribed pairs of tetrahedra [4, p. 444].
A Note on the Construction of a Set of Fundamental Circuits on a Surface of Genus p
Authors: de Buda R. G.
Published: December 2010

On an orientable surface of genus p, a set of 2p fundamental circuits can be selected which all pass through a single point A. After cutting along the 2p circuits, the surface can be unfolded into a flat region bounded by a 4p-gon so that: the set of vertices corresponds to the one point A; and the 2p pairs of edges to the 2p fundamental circuits; and the interior of the polygon to the remainder of the surface. If the edges of the polygon are directed, the 2 edges which correspond to one fundamental circuit will be directed in opposite sense, since the surface is orientable [1]. The sequence and direction of the edges is the same as the sequence of the fundamental circuits.
A Note on Normal Matrices
Authors: Marcus Marvin , Khan Nisar
Published: December 2010

In 1954 A.J. Hoffman and O. Taussky [1] showed that if A is an n-square complex matrix with eigenvalues λ = (λ1, …, λn ) and P is a permutation matrix for which αA + βA* has eigenvalues for some αβ ≠ 0 then A is normal. Here is the conjugate vector of λ. As a companion result they also proved that if the eigenvalues of AA* are , i = 1, …, n then A is normal.
Weighted Compositions
Authors: Moser L. , Whitney E. L.
Published: December 2010

By a composition of a positive integer n is meant a representation of n as a sum of one or more positive integers where the order of the summands is taken into account. Thus for example 4 has the eight compositions 4 = 3 + 1 = 1 +3 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1. Now n can be written in the form 1 + 1 + … + 1 with n - 1 plus signs. Deletion of any subset of these plus signs breaks n into parts which form a composition of n. Conversely, any composition of n corresponds to a subset of plus signs, so that the number of compositions of n is the number of subsets of a set with n - 1 elements, namely 2n - 1. In this note we obtain a number of generalizations of this rather obvious remark by making use of the notion of a weighted composition and the method of generating series.
A Determinantal Inequality for Positive Definite Matrices
Authors: Thompson R. C.
Published: December 2010

Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, j is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H i, j|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.

Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if Hi, j = 0 whenever i ≠ j.

The case n = 2 of this theorem is contained in [1].
On Proving the Absence of Zero-Divisors for Semi-Group Rings
Authors: Banaschewski Bernhard
Published: December 2010

For any semi-group S and any ring Λ with unit 1 (always taken to be distinct from 0, the neutral element of Λ under addition) there is known to exist a ring Λ[S] ⊇S which is a A-bimodule such that (i) S is a sub semi-group of the multiplicative semi-group of Λ[S], (ii) λs = sλ, (iii) λ(st) = (λs)t = s(λt) ( s, t ∊ S and λ∊Λ) and (iv) Sis a Λ -basis of Λ[S]. This ring is uniquely determined by these conditions and is usually called the semi-group ring of S over Λ. It may be described explicitly as consisting of the functions f: S → Λ which vanish at all but finitely many places, with functional addition (f+g) (s) = f(s) + g(s) and convolution (fg) (s) = Σf(u) g(v) (uv = s) as the ring operations, the functional A-bimodule operations (λf) (s) = λf(s) and (fλ) (s) - f(s)λ, and each s ∊ S identified with the characteristic function of { s} with values in Λ.
A Norm Inequality for Linear Transformations
Authors: Moyls B.N. , Khan N.A.
Published: December 2010

In 1949 Ky Fan [1] proved the following result: Let λ1…λn be the eigenvalues of an Hermitian operator H on an n-dimensional vector space Vn. If x1, …, xq is an orthonormal set in V1, and q is a positive integer such n that 1 ≤ q ≤ n, then

1
On Some Invariants of a Bilinear Form
Authors: Wild Jonathan
Published: December 2010

Let E be a finite dimensional vector space over an arbitrary field. In E a bilinear form is given. It associates with every sub s pa ce V its right orthogonal sub space V* and its left orthogonal subspace *V. In general we cannot expect that dim V* = dim *V. However this relation will hold in some interesting special cases.
An Informal Arithmetical Approach to Computability and Computation
Authors: Melzak Z.A.
Published: December 2010

In 1936 A. M. Turing published his analysis of the notion of effective computability. Very roughly speaking, its object was to distinguish between numbers defined by existential statements and those defined by purely constructive ones. Guided by his schematizing of what goes on when a person calculates using paper and pencil, Turing introduced the concept of an A-machine, eventually to be called a Turing machine. Such a machine consists of a box and a linear tape divided into squares, indefinitely long in both directions. The tape passes through the box and exactly one square of it is always within the box. The system changes only at certain discrete times: t = 0, 1, …. Each square can carry exactly one of the finite set of symbols: s1, s2, …, sn, one of which may be the blank.
Extensions Radicales et Quasi-Radicales Dans Les
Anneaux
Authors: Thierrin G.
Published: December 2010

Soient un anneau A et un sous-ensemble (non vide) B de A. L'anneau A est
dit:

Extension radicale de B, si pour tout a ∊ A, il existe un entier n(a)>0,
dépendant de a, tel que an(a)∊B.

Extension quasi-radicale de B, si pour tout a ∊ A, il existe un entier
n(a)>1, dépendant de a, tel que an(a)-a∊B.

En imposant certaines conditions à A ou B, on peut alors obtenir certains
renseignements sur la structure de A. Dans le cas où B est un sous-anneau de
A, plusieurs travaux ont été consacrés récemment à ce sujet. Notamment, on a
cherché quelles conditions, imposées à A ou B, entraînent la commutativité
de A. (Voir en particulier [2], [3], [4], [5] et [6].)
Almost all Tournaments are Irreducible
Authors: Moon J. W. , Moser L.
Published: December 2010

Given a set of n points, with each pair of distinct points joined by a line
that is oriented towards exactly one of the points, then the resulting
configuration is called a (roun-drobin) tournament. A tournament is
reducible if the points can be separated into two non-empty subsets, A and
B, such that every line that joins a point in A to a point in B is oriented
towards the point in B. If a tournament is not reducible it is called
irreducible. The object of this note is to derive an approximation for P(n),
the probability that a tournament on n point, chosen at random from the set
of possible ones, will be irreducible. p(1)=1, by
definition.
On the Fundamental Theorem of Affine Geometry
Authors: Scherk P.
Published: December 2010

The fundamental theorem of affine geometry is an easy corollary of the
corresponding projective theorem 2.26 in Artin's Geometric Algebra. However,
a simple direct proof based on Lipman's paper [this Bulletin, 4, 265−278]
and his axioms 1 and 2 may be of some interest.

Lipman's [desarguian] affine geometry G determined a left linear vector
space L={a, b,…} over a skew field F. We wish to construct 1−1
transformations γ of G onto itself such that γ and γ-1 map
straight lines onto straight lines preserving parallelism. Designate any
point 0 as the origin of G. Multiplying γ with a suitable translation, we
may assume γ0=0. Thus γ will then be equivalent to a 1−1 transformation Γ of
L onto itself which preserves linear dependence. Since Γ-1 will
have the same properties, Γ must also preserve linear independence.
A New Kind of Free Extension for Projective
Planes
Authors: Ditor Seymour
Published: December 2010

Marshall Hall [l] shows how projective planes of very general structure may
be constructed and at the same time exhibits an extensive class which are
non-Desarguesian. Here we shall indicate how his method of free extension
can be generalized to yield a class of planes which seem to be distinct from
those which he obtains.

A partial plane is a system consisting of two distinct sets of elements, a
set of "points" P, Q,… and a set of "lines" l, m,…, and a
relation between these two sets, called "incidence", such that for any two
distinct points, there is at most one line incident with both (or,
equivalently, for any two distinct lines, there is at most one point
incident with both). A partial plane is complete if every two distinct
points are joined by a line and every two distinct lines intersect in a
point.
The Five and Six Dimensional Magic Hypercubes of Order
3
Authors: Hendricks John R.
Published: December 2010

A recent publication entitled "Magic Squares and Cubes", by W. S. Andrews,
published by the Dover Publications, New York, is an excellent authoritative
book on magic squares, and magic cubes. In fact, chapter XIV entitled "Magic
Octrahedroids", has shown examples of the extension into four-dimensional
space. Andrews' method seems to be that of extending symmetrical
considerations, which he calls reversions, in forming higher dimensional
forms of magic squares. Neither he, nor any other author, to the best of my
knowledge, has extended magic squares to higher than four dimensions.

La Hire must be given full credit for his method of breaking down magic
squares into component squares, and conversely constructing magic squares
from component squares. The approach of this author is to extend La Hire's
method for n-dimensional space by means of a relation:
Polynomials with Real Roots
Authors: Dixon J. D.
Published: December 2010

In a recent issue of this Bulletin a problem equivalent to the following is proposed by Moser and Pounder [1]:

If ax2+bx+c is a polynomial with real coefficients and real roots then a+b+c ≤9/4 max (a, b, c).

The object of this note is to prove the following theorems which generalise this result.

Theorem 1. Let αn be the smallest constant such that n for all polynomials

1
A Generalization of Certain Rings of A. L.
Foster
Authors: Yaqub Adil
Published: December 2010

The concept of a Boolean ring, as a ring A in which every element is
idempotent (i. e., a2 = a for all a in A), was first introduced
by Stone [4]. Boolean algebras and Boolean rings, though historically and
conceptually different, were shown by Stone to be equationally
interdefinable. Indeed, let (A, +, x) be a Boolean ring with unit 1, and let
(A, ∪, ∩, ') be a Boolean algebra, where ∩, ∪, ', denote "union", "
intersection", and "complement". The equations which convert the Boolean
ring into a Boolean algebra are:

I

Conversely, the equations which convert the Boolean algebra into a
Boolean ring are:

II
Sums of Reciprocal Powers of Terms in Arithmetic
Sequence
Authors: Whitney E. L.
Published: December 2010

A note by N. Kimura gives the sums

explicitly linearly interms of the sums for positive
integers p.

We note here that by a similar simple method, the Bernoulli numbers and
Euler numbers may be related similarly to these sums.
Triangular Dissections of N-Gons
Authors: Moon J. W. , Moser L.
Published: December 2010

Let f(n) denote the number of dissections of a regular n-gon into n-2 triangles by n-3 non-intersecting diagonals. It is known that

and that

1

for n = 3, 4, . . . , where f(2) = 1 by definition. (For pertinent references on this and related problems see, e. g. , Motzkin [2].) The object of this note is to obtain a simple expression for g(n) , the number of such dissections remaining when those which differ only by a rotation, reflection, or both are not considered as being different. For convenience we shall let g(2) = 1 and f(k) = 0 when k is not an integer.
The Hamilton-Jacobi Equations for a Relativistic Charged Particle
Authors: Vanstone J. R.
Published: December 2010

In the problem of finding the motion of a classical particle one has the choice of dealing with a system of second order ordinary differential equations (Lagrange's equations) or a single first order partial differential equation (the Hamilton-Jacobi equation, henceforth referred to as the H-J equation). In practice the latter method is less frequently used because of the difficulty in finding complete integrals. When these are obtainable, however, the method leads rapidly to the equations of the trajectories. Furthermore it is of fundamental theoretical importance and it provides a basis for quantum mechanical analogues.
Sur la Structure de Deux Classes D'anneaux
Authors: Barbeau A.
Published: December 2010

Dans un mémoire paru en 1960 [1], G. Thierrin a étudié la structure d'une classe d'anneaux, celle des anneaux bipotent à droite. Ce travail fait suite à ce mémoire et a pour objet d' étudier, dans un première partie, la structure d'une classe particulière d' anneaux bipotents a droite, celle des anneaux complètement bipotents à droite. Dans une deuxième partie, nous nous proposons d' étudier la structure df une autre classe dr anneaux, celle des anneaux d-bipotents à droite.
On the Principle of Superposition in Quantum Mechanics
Authors: Duff G.F.D.
Published: December 2010

The principle of superposition of states requires that the states of a dynamical system form a linear vector space. This hypothesis of linearity has usually been regarded as a fundamental postulate in quantum mechanics, of a kind that cannot be explained by classical concepts. Indeed, Dirac [2, p. 14] comments that "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation."