bluuebunny/crossref_metadata_2025_split huggingface dataset:
From Matrices to Graphs
Authors: Tutte W. T. 
Published: December 2010

All the matrices considered in this paper have their elements in the field of residues mod 2.Two non-singular matrices are equivalent if each row of either matrix is a linear combination of rows of the other. The matrices then have equal numbers of rows and equal numbers of columns.A nodal matrix is a non-singular matrix in which no column has more than two l's. A graphic matrix is a non-singular matrix equivalent to a nodal matrix.
Free Lattices Generated by Partially Ordered Sets and Preserving Bounds
Authors: Dean R. A. 
Published: December 2010

A construction of the free lattice generated by a partially ordered set P and preserving every least upper bound (lub) and greatest lower bound (glb) of pairs of elements existing in P has been given by Dilworth (2, pp. 124-129) and, when P is finite, by Gluhov (5).The results presented here construct the free lattice FL generated by the partially ordered set P and preserving(1) the ordering of P(2) those lub's of a family of finite subsets of P which possess lub's in P
Complemented Banach Algebras
Authors: Olubummo A. 
Published: December 2010

Let A be a complex Banach algebra and Lr (Ll) be the lattice of all closed right (left) ideals in A. Following Tomiuk (5), we say that A is a right complemented algebra if there exists a mapping I —> IP of Lτ into Lr such that if I ∊ Lr, then I ∩ Ip = (0), (Ip)p = I, I ⴲ Ip = A and if I1, I2 ∊ Lr with I1 ⊆ I2 then .If in a Banach algebra A every proper closed right ideal has a non-zero left annihilator, then A is called a left annihilator algebra. If, in addition, the corresponding statement holds for every proper closed left ideal and r(A) = (0) = l(A), A is called an annihilator algebra (1).
Complete Reductibility of Infinite Groups
Authors: Dixon John D. 
Published: December 2010

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).
Randomly Packed and Solidly Packed Spheres
Authors: Gilbert E. N. 
Published: December 2010

In the classical packing problem unit spheres are placed without overlapping in D-dimensional space. When D = 2, the densest packing is a familiar regular arrangement of circles, each circle touching six others. In this packing, circles cover a fraction π/√12 = 0.9069 . . . of the area of the plane. The densest packing is not known for D ≥ 3.Most of the packings to be considered here use spheres of many different sizes. In this way greater densities are obtainable; small spheres can fill up some of the space left over after large spheres have been packed.
Criteria for Solvability of Certain Congruences
Authors: Muskat Joseph B. 
Published: December 2010

Lower case italics will denote rational integers, while lower case Greek letters will denote algebraic integers. The Law of Quadratic Reciprocity can be formulated as follows:If p and a are distinct odd primes, then a is a quadratic residue(modp)if and only if ( — 1)(p-1)/2p is a quadratic residue(modq).LetXpdenote the non-principal quadratic character (modp) and letζpbe a primitivepth. root of unity. Then
Trace Formulas for Powers of a Sturm-Liouville Operator
Authors: Gilbert Richard C. , Kramer Vernon A. 
Published: December 2010

Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.
On the Decomposition of A Class of Functions of Bounded Variation
Authors: Laha R. G. 
Published: December 2010

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so thatthe above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,
Linear Homogeneous Equations Over Finite Rings
Authors: Stevens Harlan 
Published: December 2010

The intent of this paper is to apply the following theorem in several particular instances:Theorem 1. For any finite ring of q elements, let {} be a collection of s subsets of , each containing hi (i = 1, 2, . . . , s) members, and let denote the set of all differences d′ — d″ with d′ and d″ from including d′ = d″. Furthermore, suppose that
Relations Between the Digits of Numbers and Equal Sums of Like Powers
Authors: Roberts J. B. 
Published: December 2010

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.
Derivative of Singular Set-Functions
Authors: Anvari Morteza 
Published: December 2010

The purpose of this paper is to prove that the general derivative of a completely additive singular set-function defined on certain measurable subsets of an abstract measure space is zero almost everywhere. As a corollary the celebrated Lebesgue decomposition theorem has been sharpened.This result is well known for set-functions defined on measurable subsets of an n-dimensional Euclidean space (2, p. 119). The proof in this setting depends on two things: Vitali's covering theorem and the fact that for every measurable set A there exists an open set O which contains A and the images of O and A under the set-function can be made arbitrarily close. Here the covering theorem is due to Trjitzinsky and the open set is replaced by an envelope, an entirely measure-theoretic concept.
A Class of Translation Lattices
Authors: Taylor John C. 
Published: December 2010

Let E be a completely regular space, let C* denote the Banach algebra of continuous bounded real-valued functions on E, and let A* denote a Banach subalgebra which contains the constant function 1. LetThe purpose of this note is to discuss some properties of the subsets L(A*) of the ring C of continuous real-valued functions on E. They arose out of an unsuccessful attempt to distinguish C* from its Banach subalgebras.
Uniform and Equicontinuous Schauder Bases of Subspaces
Authors: McArthur Charles W. , Retherford James R. 
Published: December 2010

A sequence {Mi} of non-trivial subspaces of a linear topological space X is a basis of subspaces for X if and only if corresponding to each x ∊ X there is a unique sequence {xi}, xi ∊ Mi, such thatCorresponding to a basis of subspaces {Mi} for X is a sequence of orthogonal projections {Ei} (Ei2 = Ei and EiEj = 0 if i ≠ j) defined by Ei(x) = xi if
On the Degenerate Cauchy Problem
Authors: Carroll R. W. , Wang C. L. 
Published: December 2010

The problem treated here is an abstract version of the Cauchy problem for an equation of mixed type in the hyperbolic region with initial data on the parabolic line (cf. 2, 3, 5, 11, 13, 14, 15, 16, 21, 27). A more complete bibliography may be found in (3, 5, 18). We begin with the equation (6)(1.1)
Refinement-Unbounded Interval Functions and Absolute Continuity
Authors: Appling William D. L. 
Published: December 2010

In this paper we prove the following characterization theorem (Section 3) :Theorem 1. If each of g and m is a real-valued non-decreasing function on the number interval [a, b], then the following two statements are equivalent: (1) If R is a real-valued, refinement-unbounded (Section 3) function of subintervals of [a, b], then the integral (Section 2)exists and is equal to g (b) — g (a), and(2) g is absolutely continuous with respect to m.
The Double Transitivity of a Class of Permutation Groups
Authors: Bercov Ronald D. 
Published: December 2010

Certain finite groups H do not occur as a regular subgroup of a uniprimitive (primitive but not doubly transitive) group G. If such a group H occurs as a regular subgroup of a primitive group G, it follows that G is doubly transitive. Such groups H are called B-groups (8) since the first example was given by Burnside (1, p. 343), who showed that a cyclic p-group of order greater than p has this property (and is therefore a B-group in our terminology).Burnside conjectured that all abelian groups are B-groups. A class of counterexamples to this conjecture due to W. A. Manning was given by Dorothy Manning in 1936 (3).
Structure of Semigroups
Authors: Hoehnke Hans-Jürgen 
Published: December 2010

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.
Inversion Theorems for the Laplace-Stieltjes Transform
Authors: Ditzian Z. , Jakimovski A. 
Published: December 2010

The Laplace-Stieltjes transform f(x) of the function α(t) is denned by1where α(t) is a function of bounded variation in each closed interval [0, R] (R > 0) and the right-hand side of (1) is supposed to be convergent for some x = X0. The Laplace transform f(x) of the function ϕ(t) is defined by2where ϕ(t) ∈ L1(0, R) for each R > 0 and the right-hand side of (2) is supposed to be convergent for some x = x0.
On the Integral Part of a Linear form with Prime Variables
Authors: Danicic I. 
Published: December 2010

The object of this paper is to prove the following:Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such thatIn particular [λp1 + μp2] represents infinitely many primes.Here [x] denotes the greatest integer not exceeding x.
Invariant Sub-Bundles of the Tangent Bundle of a Homogeneous Space
Authors: Tondeur Philippe 
Published: December 2010

Let M = G/H be the homogeneous space of a Lie group G and a closed subgroup H. Denote by p : G → G/H the canonical projection, e ∈ G the identity and x0 = p(e). Let W be a subspace of the tangent space Tx0(M).Definition. A lift W* of W is a subspace of the Lie algebra of G satisfying ∩ W* = ﹛0﹜ and p*W* = W, where p* : → Tx0(M) denotes the tangent map of p at e.Consider a G-invariant sub-bundle of the tangent bundle of M (4), i.e., a field of vector subspaces x ⊂ Tx(M) for every x ∈ M satisfying1Here μg : M → M denotes the diffeomorphism defined by g ∈ G and (μg)*x : Tx → Tμg(x) the induced tangent map at x.
Series Expansions of Generalized Temperature Functions in N Dimensions
Authors: Haimo Deborah Tepper 
Published: December 2010

The generalized heat equation is given by1.1where Δxf(x) = f″(x) + (2v/x)f′(x), v a fixed positive number. In a recent paper (5), the author established criteria for representing solutions of (1.1) in either the form1.2or1.3where Pn,v(x, t) is t he polynomial solution of (1.1) given explicitly by1.4and Wn,v(x, t) is its Appell transform; cf. (1). Our object is to generalize these results by extending them to higher dimensions. D. V. Widder (8) studied the problem for the ordinary heat equation.
Semi-Prime Modules
Authors: Feller E. H. , Swokowski E. W. 
Published: December 2010

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.
Construction of Principal Functions by Orthogonal Projection
Authors: Nakai Mitsuru , Sario Leo 
Published: December 2010

Given a point set E on an open Riemann surface V we denote by H(E) the space of functions u harmonic in open sets O(u) containing E. Let V0 be a regular region of V with border α, and consider restrictions f to α of functions in H(α). For V1 = V — ⊽0, an operator L from H(α) to H(⊽1) is, by definition, normal if12345
Summability of the Heine and Neumann Series of Legendre Polynomials
Authors: Jakimovski Amnon 
Published: December 2010

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series1where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.
Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary
Authors: Mitchell Josephine 
Published: December 2010

The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system1where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.
Group Algebra Modules. II
Authors: Gulick S. L. , Liu T. S. , Van Rooij A. C. M. 
Published: December 2010

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.
Concerning Non-Planar Circle-Like Continua
Authors: Ingram W. T. 
Published: December 2010

In this paper it is proved that if a circle-like continuum M cannot be embedded in the plane, then M is not a continuous image of any plane continuum (Theorem 5).Suppose that (S, ρ) is a metric space. A finite sequence of domains L1, L2, … , Ln is called a linear chain provided Li intersects Lj if and only if |i — j| ⩽ 1. If, in addition, there is a positive number ∊ such that, for each i, the diameter of Li is less than ∊, then the linear chain is called a linear ∊-chain. If for each positive number ∊ the continuum M can be covered by a linear ∊-chain, then M is said to be chainable (or snake-like) (2).
Sets of Disjoint Lines in PG(3, q)
Authors: Mesner Dale M. 
Published: December 2010

Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.
Embedding -Like Compacta in Manifolds
Authors: McCord Michael C. 
Published: December 2010

A compactum is a compact, metrizable space. A continuum is a connected compactum. All polyhedra will be finitely triangulable spaces. If a is an open cover of a compactum X, a map of X onto a compactum Y is called an α-map provided that the inverse image of each point in Y is contained in some member of α.If is a class of polyhedra, then, following Mardešić and Segal (10), we say a compactum X is -like provided that for each open cover α of X there exists an a-map of X onto some member of .
A Decomposition Theorem for Matrices
Authors: Pearl Martin H. 
Published: December 2010

According to a classical theorem originally proved by L. Autonne (1; 3) in 1915, every m × n matrix of rank r with entries from the complex field can be decomposed aswhere U1 and U2 are unitary matrices of order m and n respectively and D is an m × n matrix having the form1where Δ is a non-singular diagonal matrix whose rank is r. If r = m, then the row of zero matrices of (1) does not actually appear. If r = n, then the column of zero matrices of (1) does not appear. The main purpose of this paper is to give a necessary and sufficient condition under which both U1 and U2 may be chosen to be real orthogonal matrices.
Notes on Hypersurfaces in a Riemannian Manifold
Authors: Yano Kentaro 
Published: December 2010

H. Liebmann (3) and W. Süss (7) provedTheorem A. The only convex closed hypersurface with constant mean curvature in a Euclidean space is a sphere.Y. Katsurada (1; 2) gave the following generalization.Theorem B. Let M be an orientable Einstein space which admits a proper conformai Killing vector field, that is, a vector field generating a local one-parameter group of conformai transformations which is not that of isometries, and S a closed orientable hypersurface in M whose first mean curvature is constant. If the inner product of the conformai Killing vector field and the normal to the hypersurface has fixed sign on S, then every point of S is umbilical.
On Unsolvable Groups of Degree p = 4q + 1, p and q Primes
Authors: Appel K. I. , Parker E. T. 
Published: December 2010

This paper presents two results. They are:Theorem 1. Let G be a doubly transitive permutation group of degree nq + 1 where a is a prime and n < g. If G is neither alternating nor symmetric, then G has Sylow q-subgroup of order only q.Result 2. There is no unsolvable transitive permutation group of degree p = 29, 53, 149, 173, 269, 293, or 317 properly contained in the alternating group of degree p.Result 2 was demonstrated by a computation on the Illiac II computer at the University of Illinois.
Directed Covers and Paracompact Spaces
Authors: Mack John 
Published: December 2010

A cover is directed (is well-ordered, totally ordered) if it is directed (is well-ordered, totally ordered, respectively) by set inclusion. Specifically is directed if, given U, V in , there exists W in such that U ⋃ V ⊂ W.Directed covers have been used implicitly in numerous theorems dealing with paracompactness and related properties. In particular, characterizations of paracompactness in terms of product spaces seem to require the use of directed covers (3; 11, §2; or 12). However, the utility of directed covers has not been fully exploited.
Representation of Groups by Generalized Normal Multiplication Tables
Authors: Ginzburg A. 
Published: December 2010

G will denote a finite (or infinite) group of order n. In a normal multiplication table (n.m.t.) of G (7, 8, 9, 12) all entries in one diagonal are equal to e (the identity of G), and if the entry on the intersection of the ith column and jth row is gi,j ∈ G, thenThe following is a n.m.t. of Z6 = {0, 1, 2, 3, 4, 5}:Remark. The cyclic groups Zn will always be written in additive notation. The table is uniquely defined by every one of its columns, in particular by the first.
On Some Applications of Graph Theory to Geometry
Authors: Erdös P. 
Published: December 2010

Let [Pn(k)] be the class of all subsets Pn(k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn(k).PutIn other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.
Axes of Points of Certain Linear Systems of Polarities
Authors: Schuster Seymour 
Published: December 2010

In this issue honouring Professor Coxeter, I am pleased to present some results of an investigation that was prompted by questions which he himself raised over a decade ago.With respect to a linear system of polarities in complex projective three-space, the polars of a fixed point Q form an axial pencil of planes. The axis of the pencil is called the axis of point Q with respect to the linear system of polarities. Since there are ∞3 axes and ∞4 lines in the space, not every line is an axis. The following discussion answers the questions of how many and which lines are axes with respect to the linear systems of polarities that have a fixed self-polar tetrahedron.
On The Generalized Cauchy Equation
Authors: Monroe Itrel 
Published: December 2010

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such thatThen ƒ is monotone and open and F is continuous.
Ramanujan Congruences for p-k(n)
Authors: Atkin A. O. L. 
Published: December 2010

Let12Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.