bluuebunny/crossref_metadata_2025_split huggingface dataset:
On extensions of inequalities of Kolmogoroff and others and some applications to almost periodic functions
Authors: Upton C. J. F. 
Published: May 2009

Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as,or, by putting ,The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).
A note on integral equations
Authors: Williams W. E. 
Published: May 2009

In a recent paper Cooke [1] obtained a solution of the integral equationby using the identityand the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equationsThe kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).
On Salié's sum
Authors: Mordell L. J. 
Published: May 2009

Let p be an odd prime and let f(x) be a complex-valued function such that f(x+p) = f(x) for all integers x. Write e(x) = exp(2πix/p), and define l/x by , where We consider the sumwhere is the Legendre symbol. The sum is zero if as is clear on replacing x by bjax. Salié has found a result which can be written in the formwhen h2 ≡ 4ab(modp).
An elementary proof of some character sum identities of Apostol
Authors: Berndt Bruce C. 
Published: May 2009

Let х denote a primitive character modulo k. Using two different representations for Dirichlet L-functions, Apostol [1] recently derived a representation forinvolving the sumswhere m is a positive integer. Furthermore, if х(r) = (r|p)the residue class character modulo the odd prime p, he derived a representation for Mm(х) involving the sums
A Fourier-Neumann series and its application to the reduction of triple cosine series
Authors: Tranter C. J. , Cooke J. C. 
Published: May 2009

The Jacobi expansionis well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+½)x is replaced by cos (n+½)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1].
Partitioned transversals
Authors: Mirsky L. 
Published: May 2009

Let E be a finite set and let= (Al, …, A2) be a family of subsets of E. A subset T of E iscalled atransversal ofif there exists a bijection Φ: T → {l, …, n} such thatx∈Aψ(x)for allx∈ T. If I ⊆ {1, …,n}, we shall, for brevity, write(and similarly for families denoted by other letters). The cardinal of a set S will be denoted by |S|. If λ is a non-negative integer, we define λS as S or Ø according as λ > 0 or λ = 0.
A local mean-value theorem for analytic functions with smooth boundary values
Authors: Novinger W. P. 
Published: May 2009

Let f be an analytic function on a connected open set Ώ in the complex plane. Then, for given a, b ∈ Ώ, the equationneed not have a solution z ∈ Ώ. As a matter of fact, this would happen with each locally one-to-one analytic function which is not one-to-one on Ώ. But if we fix a a ∈ Ώ, then, for all b sufficiently close to a, (1) is solvable for z. This is an easy consequence of the Open Mapping Theorem applied to f'. For, assuming that f' is non-constant (otherwise, (1) holds for all a, b, z ∈Ώ), the Open Mapping Theorem tells us that f'(Ώ), the image under f' of Ώ, is an open neighbourhood of f'(a); so it is a direct consequence of the definition of f'(a) that there exists δ > 0 such that 0 < |b – a| < δ implies (f(b) – f(a))/(b – a) ∈f'(Ώ). A stronger statement has been obtained by J. M. Robertson [1, p. 329], who has shown thatand, if f''(a) ≠ 0, then.
A characterization of determinants over topological fields
Authors: Odoni R. W. K. 
Published: May 2009

Let K be a topological field. On introducing the vector-space topology on the n × n matrices over K, it becomes clear that the determinant map φ enjoys the following properties:(A) φ is a continuous surjective homomorphism from GLn(K) to K*,(B) φ(μa) = μn φ(a) for each non-zero φ in K, and all a in GLn(K).
Group structure and properties of block ideals of the group algebra
Authors: Hamernik Wolfgang 
Published: May 2009

In this note relations between the structure of a finite group G and ringtheoretical properties of the group algebra FG over a field F with characteristic p > 0 are investigated. Denoting by J(R) the Jacobson radical and by Z(R) the centre of the ring R, our aim is to prove the following theorem generalizing results of Wallace [10] and Spiegel [9]:Theorem. Let G be a finite group and let F be an arbitrary field of characteristic p > 0. Denoting by BL the principal block ideal of the group algebra FG the following statements are equivalent:(i) J(B1) ≤ Z(B1)(ii) J(B1)is commutative,(iii) G is p-nilpotent with abelian Sylowp-subgroups.
Some expansions of hypergeometric functions in series of hypergeometric functions
Authors: Srivastava H. M. , Panda Rekha 
Published: May 2009

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.
On non-commutative regular local rings
Authors: Smith P. F. 
Published: May 2009

Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chainof ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.
On certain triple integral equations with trigonometric kernels
Authors: Stocks D. C. 
Published: May 2009

In this note we formally solve the following triple integral equations,where f1(x), f2(x) and f3(x) are integrable for 0<x<α, α<x<β and β<x<∞, respectively, and the function g(λ) is assumed to satisfy sufficient conditions for the Fourier sine transform to exist. A special case of this system arose in a problem concerned with transistors.
Transversally affine foliations
Authors: Furness P. M. D. , Fédida E. 
Published: May 2009

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the formIf y′ is always affine in y, i.e.where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.
A note on Cauchy's root test
Authors: Murphy I. S. 
Published: May 2009

Throughout this note we deal with a series Σan of positive terms. The following tests for the convergence of this series are well-known.Test 1. (Ratio test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.Test 2. (Root test). LetThen, if K < 1, Σanconverges, while if K > 1, Σandiverges.
Support projections on Banach spaces
Authors: Spain P. G. 
Published: May 2009

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.
A Lyapunov inequality and forced oscillations in general nonlinear nth order differential-difference equations
Authors: Chen Lu-San 
Published: May 2009

The purpose of this paper is to consider the general nonlinear nth order differential-difference equationand derive an inequality of Lyapunov type. Later we use this inequality to find conditions to ensure that the oscillatory solutions of equation (1) tend to zero as t → ∞. The conditions that ensure that the oscillatory solutions of equation (1) tend to zero, also cause all solutions of equationto be non-oscillatory.
An acyclic analogue to Heawood's theorem
Authors: Albertson Michael O. , Berman David M. 
Published: May 2009

The concept of acyclic coloring was introduced by Grünbaum [5] and is a generalization of point arboricity.A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In [5], Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin [3] after the upper bound was improved three times in [7], [1] and [6]. In [2], we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the followingTheorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.It is not known for any n>0 whether a(n)>H(n), the Heawood number [8].
On the derivatives at the origin of entire harmonic functions
Authors: Armitage D. H. 
Published: May 2009

If f is an entire function in the complex plane such thatwhere 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).
On the approximation of π by special algebraic numbers
Authors: Braune Erhard 
Published: May 2009

Suppose that m0 is an integer, m0≥3, ρ = exp(2πi/m0), K = ℚ(ρ, i), v denotes the degree of K, ξ∈K has degree N over ℚ. The length, where is the (irreducible) minimal polynomial for with ξ relatively prime integer coefficients. Feldman [2, p. 49] proved that there is an absolute constant c0>0 such thatFrom [2, p. 49, Notes 1 and 2] we know that v = φ(m0) or v = 2φ(m0), and φ(m0)≥ c1m0(log log m0)−1 (c1 > an absolute constant), where φ(m0) denotes Euler's function.
A problem in lie rings
Authors: Wiegold James 
Published: May 2009

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t<p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.
Results on common fixed points on complete metric spaces
Authors: Fisher Brian 
Published: May 2009

The following theorem was proved in [1].Theorem 1. Let S and T be continuous, commuting mappings of a complete, bounded metric space (X, d) into itself satisfying the inequalityfor all x, y in X, where 0≤c<1 and p, p′, q, q′≥0 are fixed integers with p+p′, q+q′≥1. Then S and T have a unique common fixed point z. Further, if p′ or q′ = 0, then z is the unique fixed point of S and if p or q = 0, then z is the unique fixed point of T.
Subgroups of infinite index in the modular group III
Authors: Stothers W. W. 
Published: May 2009

As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that(i) each of r, s, t1 h0, h∞ is a non-negative integer,(ii) for each i, c(i) is a positive integer,(iii) if h∞ = 0 then ho = ∞,(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,(v) r + s + t1 + h0 + h∞ = ∞.
A note on the modulus of convexity
Authors: Plant Andrew T. 
Published: May 2009

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, whereIt is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.
On some generalisations of the Riemann-Liouville and Weyl fractional integrals and their applications
Authors: Lowndes J. S. 
Published: May 2009

1. For functions f ∈ LLoc[0, ∞) the Riemann-Liouville operator of fractional integration I∞ is defined byand its adjoint operator, the Weyl operator Kα, is defined byfor functions f ∈ LLoc[0, ∞) having a suitable behaviour at infinity.
On the fractional parts of αn2 and βn
Authors: Baker R. C. 
Published: May 2009

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such thatThis extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].
A confluent hypergeometric integral equation
Authors: Love E. R. , Prabhakar T. R. , Kashyap N. K. 
Published: May 2009

Recently there have appeared papers ([7], [8]; also see [9]) in which integral equations with kernels involving the confluent hypergeometric functionhave been studied. These equations are mainly Volterra equations of the first kind except that they have infinite domain (0, ∞). The rest are of the related type with integrals over (x, ∞) instead of (0, x); and all are convolution equations.
Arens regularity and retractions
Authors: Arikan Nilgün 
Published: May 2009

In this paper a characterisation of the regularity of a normed algebra A is given in terms of retractions onto A** from A4*. The second dual A** of a normed algebra A possesses two natural Banach algebra multiplications, say ° and *. Each of ° and * extends the original algebra multiplication on A; see (2). An algebra A is called regular if and only if F * G = F ° G for all F, G ∈ A**. See (1). The existing results in the Arens regularity theory can be found in a recent survey (2). Denoting the nth dual of A by An*, and en the natural embedding of An* in its second dual A(n+2)*, we can naturally represent the second dual A** of A as a Banach space retract of A4* in two different ways:Our main results say that A** is in fact a Banach algebra retract of A4* (i.e. the maps involved are homomorphisms) in either of these cases if and only if A is regular.
Representations of infinite soluble groups
Authors: Musson Ian M. 
Published: May 2009

The purpose of this paper is to study the following two questions.(1) When does the group algebra of a soluble group have infinite dimensional irreducible modules?(2) When is the group algebra of a torsion free soluble group primitive?In relation to the first question, Roseblade [13] has proved that if G is a polycyclic group and k an absolute field then all irreducible kG-modules are finite dimensional. Here we prove a converse.
A Zeuthen Segre formula for even dimensional submanifolds of real projective space
Authors: Bruce J. W. 
Published: May 2009

In this paper we generalise results of Craveiro de Carvalho ([3]) in two ways. First we prove the following fact.PROPOSITION 1. Given any smooth submanifold M of real projective space ℙn, for L in an open dense subset of the space of codimension 2 subspaces of ℙnwe have(a) L meets M transversally and(b) the pencil of hyperplanes through L have at worst Morse (A1) contact with M.
Semilocal semigroup rings
Authors: Okniński Jan 
Published: May 2009

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.Theorem A[4,10]. Let K be a field and G a group.(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].
On the monodromy groups of lifted Euler equations
Authors: Kuiken Kathryn , Masterson John T. 
Published: May 2009

In [13], Poincaré asked the following question: Which abstract groups can appear as monodromy groups [14] of second order, linear, homogeneous differential equations with meromorphic coefficients (which might depend on one or more parameters) on ℂ? In the present paper, we initiate a classification of monodromy groups of differential equations on compact Riemann surfaces of genus 1. We proceed as follows: Letbe the general Euler equation [1] which has two regular singular points at 0 and λ in the extended complex plane C. Further, let yu(z)(v – 1,2) be an arbitrary but fixed pair of linearly independent solutions to (1) valid in a neighborhood of some ordinary point. Analytic continuation of each solution along a closed loop A in C–{0, λ}, starting and ending at some fixed base point, produces a new solution yv,A (v = 1, 2) which can be expressed aswhere the constants avlx (v1 ν = l,2) in C depend on the homotopy class [A] of A. Clearly, y1,A(z)/y2,A(z) = To(y1(z)/y2(z)) where the Möbius transformation T:w→ (a11W + a12)/(a21W + a22) depends on [A]. The set of all Mobius transformations T belonging to every possible closed loop A in C–{0,λ} forms a group G, called the monodromy group of the Euler equation (1). G is generated by the Mobius transformation belonging to a simple, closed loop Ao encircling 0. Hence, G is cyclic.
The motions of algebraic differential equations
Authors: Rubel Lee A. 
Published: May 2009

We confine ourselves, for simplicity, to first-order algebraic differential equations (ADE's), although analogous considerations may be made for higher-order ADE's:P(x, y(x), y'(x)) = 0. (*)A motion of (*) is a change of independent variable that takes solutions to solutions, that is, a suitable map <p of the underlying interval I into itself so that if y is a solution of (*) then y ° φ is a solution of (*), i.e.
On the joint spectra of doubly commuting n-tuples of semi-normal operators
Authors: Chō Muneo , Dash A. T. 
Published: May 2009

Let H be a complex Hilbert space. For any operator (bounded linear transformation) T on H, we denote the spectrum of T by σ(T). Let T = (T1, …, Tn) be an n-tuple of commuting operators on H. Let Sp(T) be the Taylor joint spectrum of T. We refer the reader to [8] for the definition of Sp(T). A point v = (v1, …, vn) of ℂn is in the joint approximate point spectrum σπ(T) of T if there exists a sequence {xk} of unit vectors in H such that.A point v = (v1, …, vn) of ℂn is in the joint approximate compression spectrum σs(T) of T if there exists a sequence {xk} of unit vectors in H such thatA point v=(v1, …, vn) of ℂn is in the joint point spectrum σp(T) of T if there exists a non-zero vector x in H such that (Ti-vi)x = 0 for all i, 1 ≤ j ≤ n.
Integral Harnack inequality
Authors: Rao Murali 
Published: May 2009

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.
Representations of finite abelian groups Cnm,p
Authors: Saeed-ul-Islam M. 
Published: May 2009

Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further letand
On the asymptotic behaviour of nonlinear systems of ordinary differential equations
Authors: Athanassov Zhivko S. 
Published: May 2009

In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).
Algebraic points on quartic curves over function fields
Authors: Bremner Andrew 
Published: May 2009

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.