bluuebunny/crossref_metadata_2025_split huggingface dataset:
On Some Limit Theorems Involving the Empirical Distribution Function
Authors: Csörgo Miklós 
Published: December 2010

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that isFn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.Using a theorem of Manija [4], we proved among others the following statement in [1].
On the Conversion of the Determinant into the Permanent
Authors: Botta Peter 
Published: December 2010

Let Mm(F) be the vector space of m - square matrices over a field F. If X belongs to Mm(F), then xij will denote the element occuring in row i and column j of X.Let Sm be the symmetric group of degree m and ε: Sm → F the alternating character on Sm (i.e. ε(σ) = 1 or - 1 according as σ is an even or odd permutation).
On Banach Limit of Fourier Series and Conjugate Series I
Authors: Dayal S. 
Published: December 2010

Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:(1)Lim (axn + byn) = a Lim xn + b Lim yn(2)If xn ≥ 0 for every n, then Lim xn ≥ 0(3)Lim xn+1 = Lim xn(4)If xn = 1 for every n, then Lim xn = 1The existence of such limits is proved by Banach [1].
Almost Continuity of Mappings
Authors: Lin Shwu-Yeng T. 
Published: December 2010

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].
The No-Three-In-Line Problem
Authors: Guy Richard K. , Kelly Patrick A. 
Published: December 2010

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that
Local Near-Rings Of Cardinality P2
Authors: Maxson Carlton J. 
Published: December 2010

The main result of this paper is the determination of all nonisomorphic local near-rings <N, +, •> with <N, +> =<C(p) × C(p), +> which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)
On the Forced Lienard Equation
Authors: Stevens R.R. 
Published: December 2010

We consider the second order differential equation(1)with the assumptions that(2) f(x) is continuous (- ∞ < x < ∞) and p(t) is continuous and bounded: |p(t)| ≤ E, - ∞ < t < ∞.Also, throughout this paper, F(x) denotes an antiderivative of f(x).
A Note on an Equivalence Relation on a
Purely Inseparable Field Extension
Authors: Rygg P. , Lehman B. 
Published: December 2010

We assume F is a purely inseparable field extension of the field K. The characteristic of K is p ≠ 0, and we assume F and K are not perfect. For x ∈ F, the exponent of x over K is the smallestnon-negative integer e such that and will be denoted bye (x); ⨱ will denote . For any subset S of F, e(x; S) will denote the exponent of x over K(S); in case S = {y} we will write e(x; y) for e(x; S).
On A Conjecture of Conway
Authors: Marica John 
Published: December 2010

Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway:A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove that A - A always has more members than A + A, unless A is symmetrical about 0.First note that 'about 0' is redundant and erroneous.
A Remark on Separable Orders
Authors: Roggenkamp Klaus W. 
Published: December 2010

K = algebraic number field,R = algebraic integers in K,A = finite dimensional semi-simple K-algebra, A. = simple K-algebra,i = 1,…, n,Ki = center of Ai, = 1,…, n,G = R-order in A,Ri = G ∩ ki.All modules under consideration are finitely generated left modules. A G-lattice is a G-module which is R-torsion-free.
On Subsets with Intersections of Even Cardinality
Authors: Berlekamp E.R. 
Published: December 2010

This paper solves a question posed by P. Erdös:If A1, A2, …, A are distinct subsets of n elements and if |Ai ∩ Ai| ≡ 0 mod 2 (i ≠ j), thenand for each n, there exists a collection of subsets which achieves this bound with equality.
Sums of Complexes in Torsion-Free Abelian Groups
Authors: Tarwater J. D. , Entringer R. C. 
Published: December 2010

The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:If G is a torsion-free abelian group with finite subsets A and B with | B | ≥ 2, then at least two elements c of A + B admit exactly one representation c = a + b.
On the Rank of the Sum of two Rectangular Matrices
Authors: Meyer C.D. 
Published: December 2010

The purpose of this note is to present a short proof for the following theorem.Let A and B be two complex m x n matrices. If B*A = 0 and AB* = 0 then rank(A + B) = rank(A) + rank(B).Let A† and B† be the generalized inverses of A and B, respectively, in the sense of Penrose [ 1]. Now,
A Conformal Proof of a Jordan Curve Problem
Authors: Spoar G. , Lane N.D. 
Published: December 2010

The following theorem appears in [1].Let R be a closed simply connected region of the inversive plane bounded by a Jordan curve J, and let J be divided into three closed arcs A1, A2, A3. Then there exists a circle contained in R and having points in common with all three arcs.
Maximal Subsets of a given Set having No Triple in Common with a Steiner Triple System on the set
Authors: Sauer N. , Schönheim J. 
Published: December 2010

Let E be a finite set containing n elements, n ≡ 1, 3 (mod 6), S = S(E) a Steiner triple system on E, i.e. each unordered pair of elements of E is a subset of exactly one triple in S. Let T be a subset of E such that none of the triples of elements of T is a member of S. Erdös has asked (in a recent letter to the authors) for the maximal size of such a set T. Denote max |T| for fixed n and S by f(n, S). We prove in this note the following result:(i)(ii)for every n ≡ 1, 3 (mod 6) there exists a Steiner triple system S0 such that equality holds in i.
On Spectral Synthesis for One Point
Authors: Natzitz B. 
Published: December 2010

In [2, page 3831], Varopoulos proves that for any ∈ > 0 there exists a function on a neighbourhood of 0. Indeed, if 0 < ∈ < π/2, then f(x) defined to be equal to 1 - e when -eix, linear on [∈, 2π - ∈] and of period 2π, is an example of such a function.The above result can be used to give a direct proof of the following result without reference to the L2 theory [1, Theorem 2.6.4].
A Note on the Generalized (Positive) Cauchy Distribution
Authors: Rao B. Raja , Garg M. L. 
Published: December 2010

In this note, a certain generalization of the Cauchy distribution is obtained, using the result of Malik [2].The generalized gamma distribution having the density(1)is introduced by Stacy [1], who studied some of its properties. As remarked by Stacy [1], the familiar gamma, chi, chi-squared, exponential and Weibull distributions are special cases of (1), as are the distributions of certain functions of a normal variable - viz.
The Group of the Quadratic Residue Tournament
Authors: Goldberg Myron 
Published: December 2010

A tournament Tn is a set of n nodes a1a2, …, an such that every pair (ai, aj) of distinct nodes is joined by exactly one of the oriented edges or . If is in Tn, then we say that ai dominates aj and write ai → aj.The (automorphism) group G(Tn) of a tournament Tn is the group of all permutations ϕ of the nodes of Tn such that ϕ(a)→ϕ(b) if and only if a → b. It is known [9] that there exist tournaments whose group is abstractly isomorphic to a given group H if and only if H has odd order; thus all tournament groups are solvable, by the Feit-Thompson Theorem [7].
The Automorphisms of an Algebraically Closed Field
Authors: Charnow A. 
Published: December 2010

It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.
On Knock-Out Tournaments
Authors: Capell P. , Narayanan T. V. 
Published: December 2010

We define, as in [2], a random knock-out tournament with n players as a vector (m1, m2, …, mk) of positive integers satisfying1On the first round of the tournament 2m1 players, chosen at random, are paired off randomly; the remaining n — 2m1 players have a "bye". The m1 losers are knocked out, leaving a tournament n—m1 players with vector (m2, m3, …, mk).
A Remark on a Theorem of Lyapunov
Authors: Wong James S. W. 
Published: December 2010

Consider the linear ordinary differential equation1where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:Theorem (Lyapunov). The following three statements are equivalent:(I) The spectrum σ(A) of A lies in the negative half plane.(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies2where ∥ ∥ denotes the Euclidean norm.(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there
exist q1,q2>0 such that3satisfying4where I is the identity matrix.
Sur une Conjecture de R. M. Robinson
Authors: Giroux André 
Published: December 2010

Dans un récent article [2, p. 379], R. M. Robinson a émis une conjecture
intéressante sur le diamètre transfini des ensembles compacts et symétriques
du plan. La conjecture est la suivante:Soit E un ensemble compact de points du plan, symétrique par rapport
á l'axe réel R. Pour n ≥ 2, choisissons n points z1, …,
zn de E et posons xi = zi, pour i=1, 2,…, n.Soient:
On a Problem in Partial Difference Equations
Authors: Long Calvin T. 
Published: December 2010

The purpose of this paper is not to solve a problem but to pose one that may be of some interest, depth, and consequence.Given that the positive integer n has the canonical representation n = Πhi=1 piαi the problem of finding the number F(n) = f(α1, α2, … αn) of ordered factorizations of n into positive nontrivial integral factors is equivalent to that of finding the number of ordered partitions of the vector (α1, α2, … αn) into nonzero vectors with nonnegative integral components.
A Note on Undirected Graphs Realizable as P.O. Sets
Authors: Haff C. E. , Murty U. S. R. , Wilton R. C. 
Published: December 2010

Let (P, ≥) be a p.o. set. The basis graph of (P, ≥) is defined to be the directed graph whose vertex set is P and in which the ordered pair 〈a, b〉 is an edge if and only if b covers a in (P, ≥).Let D be a directed graph. All graphs considered in this note are finite and are free of loops and multiple edges.
The Rank of the Sum of Two Rectangular Matrices
Authors: Murphy Ian S. 
Published: December 2010

In what follows, the transposed complex conjugate of a complex rectangular matrix D is denoted by D* and the rank of D by r(D). Meyer [1] proved the following result using generalized inverses:Theorem. Let A and B be complex m × n matrices such that AB*=B*A=0. Then r(A+B) = r(A)+r(B).Below we prove this result by repeated use of the fact that for every complex m × n matrix D we have r(D) = r(D*D) = r(DD*) (e.g. See [2] Theorem 5.5.4).
A Theorem on Involutions on Cyclic Peano Spaces
Authors: Hunt J. H. V. 
Published: December 2010

The purpose of this note is to prove that an involutionf on a cyclic Peano space S leaves some simple closed curve in Ssetwise invariant.We shall first define the required terms. A Peano space is a locally compact, connected and locally connected metric space. A connected space is called cyclic if it has no cut-point. An involution on a space is a periodic mapping whose period is 2; it is necessarily a homeomorphism. A mapping f: X → X is said to leave a subset E of S setwise invariant if f(E) = E. These definitions may be found, for example, in [2].
On Y. C. Wong's Conjecture
Authors: Datta D. K. 
Published: December 2010

Let M be an n-dimensional connected C∞ manifold with a linear connection Γ. M is said to be of recurrent curvature with respect to Γ if the corresponding curvature tensor R satisfies [1], [4]where Δ denotes covariant derivative with respect to Γ and W is a nonzero covector called the recurrence co-vector. Let T be the torsion of Γ.
On Free Groups of the Variety AN2 ∧ N2A
Authors: Gupta Chander Kanta 
Published: December 2010

Let R be a commutative ring with unity and let M(R) be the multiplicative group of 4 x 4 triangular matrices (aij) over R, where a11 is a unit element of R and aii = 1 for i = 2, 3, 4. If V(=AN2 ∧ N2A) denotes the variety of groups which are both abelian-by-class-2 and class-2-by-abelian, then it is routine to verify that M(R) ∊ V. Here we prove the following,Theorem. Let F(V) denote the free group of finite or countable infinite rank of the variety V. Then for a suitable choice of R, F(V) is embedded in M(R).
Orthogonal Recurrence Polynomials and Hamburger Moments
Authors: Law A. G. 
Published: December 2010

A three-term recurrence1where An (n ≥ 0), Bn (n ≥ 0) and Cn (n ≥ 0) are real numbers for which AnCn+1 ≠0 (n ≥ 0), generates a sequence {Pn} of real polynomials in which Pn has degree exactly n.
Elementary Remarks on Multiply Monotonic Functions and Sequences(1)
Authors: Muldoon M. E. 
Published: December 2010

A function f(x) is said to be completely monotonic on (0, ∞) if1Familiar examples of such functions are given by f(x) = exp(—αx) and f(x) = (x+β)-α, where α ≥ 0, β ≥ 0. A discussion of completely monotonic functions is given in [5, Ch. IV].
Paracompactness in Small Products
Authors: Willard S. 
Published: December 2010

All spaces in this note are regular, Hausdorff topological spaces.At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?
Note on the Number of Solutions of f(x1) = f(x2) = … =f(xr) over a Finite Field
Authors: Williams Kenneth S. 
Published: December 2010

Let GF(q) denote the finite field with q = pn elements and let(1)where each ai ∊ GF(q) and 1 < d <p. For r=2, 3, …, d we let nr denote the number of solutions (x1, …, xr) over GF(q) of(2)for which x1, …, xr are all different. Birch and Swinnerton-Dyer [1] have shown that(3)where each vr is a nonnegative integer depending on f and q and the constant implied by the O-symbol depends here, and throughout the paper, only on d.
Subdirectly Irreducible DQC Rings
Authors: Burgess W. , Chacron M. 
Published: December 2010

AbstractTwenty-five years ago McCoy published a characterization of commutative subdirectly irreducible rings. This result was generalized by Thierrin to duo rings with the word “field” which appeared in McCoy's theorem replaced by “division ring”. The purpose of this note is to give another generalization in which the words “division ring” will be replaced by “simple ring with 1 ”. The techniques resemble those of McCoy and Thierrin.
Certain Extensions of the Mehler Formula
Authors: Singhal J. P. 
Published: December 2010

For the Hermite polynomials Hn(z) defined by1it is easy to see from the Rodrigues formula that2where, as usual, Dz=d/dz.In recent papers ([1], [2]) Carlitz has proved the following formulae:34where, on the right-hand side of (4), the range of each summation is from i= 1 to i=k(k=1,2,…).
The Generalized Equations of Bisymmetry Associativity and Transitivity on Quasigroups
Authors: Taylor M. A. 
Published: December 2010

The generalized equations of bisymmetry, associativity and transitivity are, respectively,(1)(x1y)2(z3u) = (x4z)5(y6u)(2)(x1y)2z = x3(y4z)(3)(x1z)2(y3z) = x4y.The numbers 1, 2, 3,…, 6 represent binary operations and x, y, z and u are taken freely from certain sets.We shall be concerned with the cases in which x, y, z, and u are from the same set and each operation is a quasigroup operation. Under these conditions the solution of all three equations is known [1], [2]; equations (1) and (3) having been reduced to the form of (2) and a solution of (2) being given. We wish to present a new approach to these equations which we feel has the advantages that the equations may be resolved independently, the motivation behind the proof is clear, and the method lends itself to application on algebraic structures weaker than quasigroups. (Details of these generalizations will be given elsewhere.)
Invariance Theorems for First Passage Time Random Variables
Authors: Basu A. K. 
Published: December 2010

LetX1X2,… be i.i.d. r.v. withEX=μ>0, andE(X-μ)2= σ2<∞.LetSk=X1+…+Xkandvx=max{k:Sk≤x}, x≥0 andvx=0 ifX1>x. Billingsley [1] proved if X1≥0 thenconverges weakly to the Wiener measure W.Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove thatconverges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.
Remarks Concerning Uniformly Bounded Operators on Hilbert Space
Authors: Istrǎțescu I. 
Published: December 2010

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that1is similar to a unitary operator.The following problem is an extension of this result: If T and S are two operators such that1.sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)2.TS = STthen there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?Also, in [7] B. Sz.-Nagy has proved that every compact operator T such thatsup ‖Tn‖<∞ (n = 1, 2, 3,…)is similar to a contraction.
On Minimally Thin Sets in a Stolz Domain
Authors: Jackson H. L. 
Published: December 2010

Let D denote the open right half plane anda Stolz domain in D with vertex at the origin. If h is a minimal harmonic function on D with pole at the origin then E⊂D is minimally thin at the origin iff where is the reduced function of h on E in the sense of Brelot. We now definewhere s shall be fixed to be 1/e. For the set E∩In we shall let cn denote the outer ordinary capacity (see [1, pp. 320-321]), An the outer logarithmic capacity, and on the outer Green capacity with respect to D. If E⊂K, Mme. Lelong [3, p. 131] was able to prove that E is minimally thin at the origin Since one cannot easily relate the classical measure theoretic properties of a plane set with its Green capacity, it would appear desirable to find some other criteria for minimal thinness.
Discontinuity Conditions on Transformation Groups
Authors: Thompson D. V. 
Published: December 2010

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.
Almost Automorphic Integrals of Almost Automorphic Functions
Authors: Zaki M. 
Published: December 2010

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that(i) exits for every real t but no kind of uniformity of convergence is stipulated;(ii) exits for every t;(iii) for every t.
A Note on Group Rings of Certain Torsion-Free Groups
Authors: Burns R. G. , Hale V. W. D. 
Published: December 2010

AbstractAs a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.
The Group of Units of the Integral Group Ring ZS3
Authors: Hughes I. , Pearson K. R. 
Published: December 2010

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.
On Density of Fourier Coefficients
Authors: Siddiqi Rafat N. 
Published: December 2010

Letfbe anLintegrable real valued function of period 2π and let(1)be its Fourier series. It is known that iffis of bounded variation then allnanandnbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] whereV(f) is the total variation off. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.
Strong Oscillation of Elliptic Equations in General Domains
Authors: Swanson C. A. 
Published: December 2010

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation(1)in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.
Determinateness and the Pasch Axiom
Authors: Adler Andrew 
Published: December 2010

Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does.