bluuebunny/crossref_metadata_2025_split huggingface dataset:
A Note on Permanents
Authors: Abramson Morton 
Published: December 2010

Let A = (aij) be an m × n matrix and let K = {s1, …, sk} be a k-subset from {1, 2, …, n}. For 0≤t≤k≤n define the (t, K)-permanent of A to be(1)the summation taken over all m-tuples (i1, i2, …, im) (repetitions allowed) of 1, 2, …, n each containing exactly t distinct entries from K and any number of distinct entries from the remaining n-k integers. For example, (4, 4, 7, 1, 1, 2), (4, 4, 6, 6, 6, 5) are 6-tuples, each containing exactly two distinct entries from K={2, 4, 5} for n ≥ 7.
On Integral Inequalities Related to Hardy's
Authors: Shum D. T. 
Published: December 2010

The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state asTheorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by1then2unless f≡0. The constant is the best possible.
A Compact Imbedding Theorem for Functions without Compact Support
Authors: Adams R. A. , Fournier John 
Published: December 2010

The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort1to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings2are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.
On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals
Authors: Koh Kwangil , Mewborn A. C. 
Published: December 2010

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.
A Generalization of the Concept of a Ring of Quotients
Authors: Luedeman John K. 
Published: December 2010

AbstractSanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.
Periodic Solutions For ẋ = Ax + G(x, t) + ∊p(t)
Authors: Ponzo Peter J. 
Published: December 2010

We wish to establish the existence of a periodic solution to1where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.
Remarks on a Problem of Moser
Authors: Chvátal V. 
Published: December 2010

Let M(n) be the set of all the points (x1, x2,…, xn)∈En such that xi∈{0,1,2} for each i= 1, 2,…, n and let f(n) be the cardinality of a largest subset of M(n) containing no three distinct collinear points. L. moser [4] asked for a proof of the inequality Let us consider the set Sn of those points (x1x2,…, xn)∈M(n) which satisfy |{i:Xi= 1}| = [(n +1)/3]. As Sn is a subset of the sphere with center at (1, 1,…, 1) and radius (n-[(n+1)/3])1/2, no three distinct points of Sn are collinear. Thus we have1
On Some Applications of Graph Theory III
Authors: Erdös P. , Meir A. , Sós V. T. , Turán P. 
Published: December 2010

In the first and second parts of this sequence we dealt with applications of graph theory to distance distribution in certain sets in euclidean spaces, to potential theory, to estimations of the transfinite diameter [1] and to value distribution of "triangle functional " (e.g. perimeter, area of triangles) [2]. The basic tool is provided in all these applications by the result formulated as Lemma 2. This, an essentially pure logical result, proves to be a very flexible and versatile instrument in applications.Here the same method is used in an abstract setting. First we deduce certain results for the density of a given family of subsets of an abstract set S in another family of subsets of the same S. Then we apply the results obtained to distance distribution in certain (e.g. totally bounded or compact) sets in metric spaces, in particular in a normed linear function space. Applications of this method to functional on Hilbert spaces were given by Katona [3].
R1, Pairwise Compact, and Pairwise Complete Spaces
Authors: Richardson G. D. 
Published: December 2010

The R1 axiom was first introduced by Davis in [1]. It is strictly weaker than the T2 axiom. Murdeshwar and Naimpally, in [4], have weakened the T2 hypothesis to R1 in some well-known theorems. We show that in many topological spaces the R1 axiom and regularity are equivalent. Also, the definition of local compactness given in [4] can be weakened to the usual definition and still get the same results.The notion of a bitopological space was first introduced by Kelley in [3]. Fletcher, Hoyle, and Patty discuss pairwise compactness for bitopological spaces in [2]. One of our main results is that a bitopological space (X, P, Q) is pairwise compact if and only if each ultrafilter v on X, containing a proper P closed set and a proper Q closed set, has a common P and Q limit.
On a Problem of P. Erdös
Authors: Ruzsa I. 
Published: December 2010

P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥11so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form2
Another Weak Stone-Weierstrass Theorem for C*-Algebras
Authors: Elliott George A. 
Published: December 2010

The purpose of this article is to present a new generalization of the classical Stone-Weierstrass theorem for commutative C*-algebras.Under the assumption that B is a sub-C*-algebra of A separating the pure states of A and zero, Kaplansky has conjectured that B=A [4, p. 246]. He gave a proof for the case that A is postliminary ([4, Theorem 7.2]; see also [2, 11.1.8]). Glimm, Akemann, and Sakai have established the conjecture in the presence of various other additional hypotheses, most of which hold in the commutative case ([3], [1], [7]).
A Note on Unconditional Bases
Authors: Holub J. R. , Retherford J. R. 
Published: December 2010

A sequence (xi) in a Banach space X is a Schauder basis for X provided for each x∊X there is a unique sequence of scalars (ai) such that1.1convergence in the norm topology. It is well known [1] that if (xi) is a (Schauder) basis for X and (fi) is defined by1.2where then fi(xj) = δij and fi∊X* for each positive integer i.A sequence (xi) is a éasic sequence in X if (xi) is a basis for [xi], where the bracketed expression denotes the closed linear span of (xi).
A Fixed Point Theorem for Weakly Uniformly Strict Contractions(1)
Authors: Assad Nadim A. 
Published: December 2010

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:Given ∊ > 0, there exists δ > 0 such that(A)
On the Central Series of a Ring
Authors: Biggs R. G. 
Published: December 2010

The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.The series R = R0 ⊃ R1 ⊃ … ⊃ Rα = Rα+1is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ … ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1.The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α.
Densities and Measures of Linear Sets
Authors: Kelly B. G. A. 
Published: December 2010

If In, I∞ denote the intervals [0, n), [0, ∞) respectively, we propose to examine the properties of an upper and lower density(1)(1)of a set S belonging to the power set of I∞ where m*,m* denote the outer and inner (linear) Lebesgue measures.
Bernstein's Inequality in the Bivariate Case
Authors: Mullen Kenneth 
Published: December 2010

SummaryIf (Xl, X2,…, Xn), is a set of n independent random variables, such that EXi=0, Var and if t is a real positive number and , then Bernstein [2] has given an upper bound for Pr when the X's are bounded. The best English language discussion of Bernstein's work is probably by Bennett [1].
On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type
Authors: Zischka K. A. 
Published: December 2010

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].
Analytic Evaluation of Certain Characteristic Classes(1)
Authors: Jeffries Clark D. 
Published: December 2010

We have an alternative proof of the following result of Kervaire [2]:Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.
On the Solution of Moser's Problem in Four Dimensions
Authors: Chandra Ashok K. 
Published: December 2010

AbstractThe problem of finding the largest set of nodes in a d-cube of side 3 such that no three nodes are collinear was proposed by Moser. Small values of d (viz., d ≤,3) resulted in elegant symmetric solutions. It is shown that this does not remain the case in 4 dimensions where at most 43 nodes can be chosen, and these must not include the center node.
On the Absolute Nörlund Summability of a Fourier Series
Authors: Goel D. S. , Sahney B. N. 
Published: December 2010

Let be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write(1.1)If(1.2)as n→∞, we say that the series is summable by the Nörlund method (N,pn) to σ. The series is said to be absolutely summable (N,pn) or summable |N,pn| if σn is of bounded variation, i.e.,(1.3)
A Problem of Herstein on Group Rings
Authors: Formanek Edward 
Published: December 2010

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].
On Free Semigroups and Ramsey Numbers
Authors: Lallement Gerard 
Published: December 2010

AbstractIf the length of a word w in a free semigroup F(X) satisfies , then for every partition of F(X) into k classes, w has n consecutive factors of length ≥p in the same class. As a consequence, the diagonal Ramsey numbers R(pn+1, p+1, k) have 1+pnk as lower bound.
The Generalized Rayleigh Quotient
Authors: Pattabhiraman M. V. 
Published: December 2010

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient(1)considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that(2)where qi is a characteristic vector corresponding to the characteristic value λi
The Stability of Solutions of Generalized Emden-Fowler Equations
Authors: Teufel Hugo 
Published: December 2010

AbstractThis paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations*Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.
A Note on Contractive Selfmaps of ℓ∞
Authors: McCabe T. F. 
Published: December 2010

AbstractIn this note it is shown that a contractive matrix selfmap T of l∞ has T2 a contraction; hence the techniques used in showing ITNC for lp(1 ≤ p < ∞), c0, and c do not apply to l∞.
A Remark on a Theorem of M. Hall
Authors: Minc Henryk 
Published: December 2010

Let A = (aij) be an m×n (0, 1)-matrix, m≤n. The permanent of A is defined bywhere the summation is over all one-one functions σ:{1, 2, . . . , m}→{1, 2, . . . , n}. If A is regarded as the incidence matrix of a configuration of m subsets of an n-set, then Per (A) is the number of systems of distinct representatives in the configuration.
Sequences Realizable by Graphs with Hamiltonian Squares
Authors: Chungphaisan V. 
Published: December 2010

AbstractLet d=(d1,…,dn) be a sequence of positive integers. In this note we show that d is realizable by a graph whose square is hamiltonian if and only if (i) d is realizable by some graph, (ii) n≥3, and (iii) d1+…+dn≥2(n-1). In fact, we prove that if d is realizable by a connected graph, then d is realizable by a graph with a spanning caterpillar. From this it follows that if d is realizable by a connected graph, it is realizable by a graph whose square is pancyclic. We also prove that d is realizable by a graph with a spanning wreath if and only if d is realizable by some graph and d1+…+dn≥2n. (A wreath is a connected graph that has exactly one cycle and all vertices not in the cycle monovalent.)
Convergence of Averaged Occupation Times
Authors: Lamb✝ Charles W. 
Published: December 2010

Let X = {Xt, t ≥ 0} be a stationary Markov process with values in a measurable space (S, ℬ), transition function p, and initial distribution concentrated at a point x ∊ S. The occupation times of a set A ∊ ℬ are defined for t ≥ 0 bywhere 1A is the indicator function of A. The expected occupation times are given by
On Property of Families of Sets
Authors: Abbott H. L. 
Published: December 2010

A family of sets is said to have property if there exists a set B such that B ∩ F ≠ Ø and B ⊅ F for every F ∊ . Such a B will be called suitable with respect to F. It is known (see [3]) that for each positive integer n there exists a family of sets satisfying the following conditions:(a)|F| = n for each F ∊ (b)|F ∩ G| ≤ for F, G ∊ F ≠ G(c)does not have property
A Note on a Matrix Result of Ryser
Authors: Murphy I. S. 
Published: December 2010

The purpose of this note is to give a short proof of a generalisation of a theorem of Ryser, Theorem 10.2.3 of [1], concerning matrices that occur in the theory of symmetric block designs.The two main results of matrix theory required in the proof given below are:(1)If B, C are square matrices such that BC = zI where z is a non-zero complex number, then CB = zI(2)A matrix S which is both symmetric (i.e. S' = S) and skew-symmetric (i.e. S' = —S)is zero.
Separation Principles and Bounded Quantification
Authors: Dawes A. M. 
Published: December 2010

This note is concerned with the implication SepII(Q)→SepI(Q) where Q is a class of subsets of some set S.where cZ denotes S—Z.It is well-known that in general the above implication is false (e.g. let Q be the closed subsets of the reals).
A Property of Maximum Likelihood Estimators for Invariant Statistical Models
Authors: Tan Peter , Drossos Constantin 
Published: December 2010

AbstractThis paper generalizes some results on pivotal functions of maximum likelihood estimators of location and scale parameters and the related ancillary statistics obtained by Antle and Bain, and Fisher. It shows that the maximum likelihood estimator of the parameter in an invariant statistical model is an essentially equivariant estimator or a transformation variable in a structural model. In the latter case, ancillary statistics in the sense of Fisher used in conjunction with the maximum likelihood estimators can be easily recognized. It is also remarked that the values of maximum likelihood estimators from samples having the same “complexion” are simply related to those of other, perhaps simpler, transformation variables. In the development it also points out the importance of using the correct definition of the likelihood function originally proposed by Fisher.
A Uniqueness Theorem on the Degenerate Cauchy Problem(1)
Authors: Wang Chung-Lie 
Published: December 2010

In [4] Carroll and the author have treated the following problem(1)where Λ is a closed densely defined self-adjoint operator in a separable Hilbert space H with (Λu, u) ≥ c ‖u‖2, c > 0, Λ-1 ∊ L(H) (L(E, F) is the space of continuous linear maps from E to F; in particular, L(H) = L(H, H)), a(t) > 0 for t > 0 a(0) = 0 and S(t), R(t), B(t) ∈ L(H).
An Equational Spectrum Giving Cardinalities of Endomorphism Monoids1
Authors: Gould Matthew 
Published: December 2010

AbstractBy determining the spectrum of a particular set of equations of type 〈2, 2, 0, 0,0〉 it is shown that a positive integer n is the cardinality of the endomorphism monoid of a universal algebra of the form if and only if n is square.
A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi
Authors: Bogin Joseph 
Published: December 2010

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).
Rank k Vectors in Symmetry Classes of Tensors
Authors: Lim Ming-Huat 
Published: December 2010

Let F be a field, G a subgroup of Sm, the symmetric group of degree m, and χ a linear character on G, i.e., a homomorphism of G into the multiplicative group of F. Let V1,...,Vm be vector spaces over F such that Vi = Vσ(i) for i=1,…,m and for all σ∈G. If W is a vector space over F, then a m-multilinear function is said to be symmetric with respect to G and χ iffor any σ ∊ G and for arbitrary xi ∊ Vi.
The Gibbs Phenomenon for [S, αn] Means and [T, αn] Means
Authors: Swaminathan V. 
Published: December 2010

The Gibbs phenomenon of the Fourier series for different summability methods has been investigated by various authors. In this note, we study the same for the [S, αn] method of summability introduced by Meir and Sharma [3]. The corresponding result for the [T, αn] method of summability due to Powell [5] can be worked out in exactly the same way.The elements cnk of the [S, αn] matrix are defined by the relations:1where 0<αn<1. The [S, αn] matrix is regular if and only if
Cover-Avoidance Properties in Finite Soluble Groups
Authors: Tomkinson M. J. 
Published: December 2010

AbstractWe give a general method for constructing subgroups which either cover or avoid each chief factor of the finite soluble group G. A strongly pronorrnal subgroup V, a prefrattini subgroup W, an -normalizer D and intersections and products of V, W, and D axe all constructable. The constructable subgroups can be characterized by their cover-avoidance property and a permutability condition as in the results of J. D. Gillam [4] for prefrattini subgroups and -normalizers.
On Criticalr, λ-Systems
Authors: Vranch J. 
Published: December 2010

SummaryCriticalr, λ-systems are introduced such that any nontrivialr, λ-system must be an extension of some critical system. It is shown that parametric values for which critical systems can exist are restricted to λ(v-1) ≤r(r-1) and, further, to λ(v-1)r(r-1) if the critical system is extendible.
On Commutator Equalities and Stabilizers
in Free Groups
Authors: Burns R. G. , Edmunds C. C. , Farouqi I. H. 
Published: December 2010

AbstractA simple proof is given of a result of Hmelevskiï on the solutions of the equation [x, y] = [u, υ] over a free group for any specified u, υ. To illustrate, the equation is solved explicitly for (u, υ) = (a, b), (a2, b), ([a, b], c) (where a, b, c freely generate the free group) and thence stabilizers of the corresponding commutators in the automorphism group of this free group are determined.
Remarque Sur un Article de T. M. Apostol
Authors: de Koninck Jean-Marie , Mercier Armel 
Published: December 2010

Dans un récent article [1], T. M. Apostol obtient une identité liant et , où f est une fonction complètement multiplicative, |f(n)| ≤ l, et p est un nombre premier. Il utilise cette identité pour prouver quepour tout nombre premier p. Par la suite, T. M. Apostol obtenait d'autres identités [2].
Conjugacy of Elements in a Normal Ring
Authors: Lim Taw-Pin 
Published: December 2010

Let (R, *) be a ring R with an involution *, i.e., * is a map RϵR such that for all a, bϵRThe trace and norm of an element a in (R, *) are respectively(R, *) is said to be a normal ring if for all aϵRor equivalently,
Density of Non Quasi-Analytic Classes of Functions
Authors: Pham-Gia Thu 
Published: December 2010

In the study of quasi-analytic classes (see [3], pp. 372-379), a class C{Mn} is shown to have the following properties:(a)If M0= 1 and (i.e. {Mn} is log convex), C{Mn} forms an algebra.(b)C{Mn} is invariant under affine transformations.(c)C{Mn} is quasi-analytic iff it contains non non-trivial function with compact support.