bluuebunny/crossref_metadata_2025_split huggingface dataset:
Counterexample to a Conjecture on Positive Definite
Functions
Authors: Stewart James 
Published: December 2010

Cooper [1] called a complex-valued function f
on the real line Rpositive definite for F, where F is a set
of complex-valued functions on R, if the integralexists as a Lebesgue integral and is nonnegative for every
ϕ in F. Let us denote by
P(F) the set of functions positive definite for
F, by the set of
functions in LP(R) with compact
support, and by the set of
functions which are locally in LP(R), i.e.,
for every
compact subset K of R. Cooper showed that
for any
p ≧ 2 and that each function in is essentially bounded and hence equal almost
everywhere to an ordinary continuous positive definite function in the sense
of Bochner.
Domains of Paracompactness and Regularity
Authors: MacDonald S. , Willard S. 
Published: December 2010

Given a class of topological spaces and a class of maps of topological spaces, our interest is in characterization of the class () of topological spaces whose every -image lies in . The class () is referred to as the -resolvent of and is the largest class of spaces smaller than closed under -images (provided is closed under composition and includes identity maps).In the present paper, will be either the class of paracompact spaces or the class of regular spaces, and the conditions determining will always include separation axioms on the range, some of which can be dispensed with now by agreeing that all spaces are Hausdorff.
The Existence of Quadratic Differentials in Simply Connected Regions of the Complex Plane
Authors: Grassmann E. 
Published: December 2010

The general coefficient theorem [2] and the extended general coefficient theorem [3] state that the existence of certain quadratic differentials is a sufficient condition for a function to be a solution of certain extremum problems. The purpose of this paper is to show that in the case of simply connected regions this condition is also necessary.We shall do this by a variational method of the Schiffer-Golusin-type. The main difficulty is, that the class of admissible functions for the general coefficient theorem is restricted and we must therefore have a method of variation with restrictions.
On the Peano Derivatives
Authors: Bullen P. S. , Mukhopadhyay S. N. 
Published: December 2010

Let f be a real valued function defined in some neighbourhood of a point x. If there are numbers α1, α2, … αr-1, independent of h such thatthen the number αk is called the kth Peano derivative (also called kth de la Vallée Poussin derivative [6]) of f at x and we write αk = fk(x). It is convenient to write α0 = f0(x) = f(x). The definition is such that if the mth Peano derivative exists so does the nth for 0 ≦ n ≦ m.
Oscillation Criteria for y"" + p(t)f(y,y’) = 0 with f Homogeneous of Degree One
Authors: Gustafson G. B. 
Published: December 2010

Let be the class of functions f : R2 → R satisfying (1) f is continuous, (2) for all t, x, y ∈ R, (4) f is locally Lipschitzian. The classical Sturm theorems, the Leighton-Wintner oscillation theorem, and perturbation theorems have been established by Bihari in [1; 2; 3] for the equation , .This paper investigates the question of strong oscillation of(1.1)for p ∈ C[0, ∞ ), f ∈ . The equation is viewed as “almost linear” because of the homogeneity and sign conditions on f.
Covering Theorems for Classes of Univalent Functions
Authors: Aharonov Dov , Kirwan W. E. 
Published: December 2010

Let denote the class of functions f(z) = z + that are analytic and univalent in and will denote the collection of f ∈ that map U onto a domain that is respectively starlike with respect to the origin and convex.In [4, p. 85] Hayman used Steiner symmetrization to solve a problem, a special case of which is the following.
Quadratic Irrationals in the Lower Lagrange Spectrum
Authors: Davis Nancy , Kinney J. R. 
Published: December 2010

We let , where the xi are positive integers and i ∈ N, the set of all integers. We define , where where . We let = [0; x1, x2, … ] whereWe letand defineThe range of L(ξ) is known as the Lagrange spectrum and the range of M(ξ) as the Markov spectrum. It is known that both are closed and that the Markov spectrum includes the Lagrange spectrum.
Rigid and Finitely V-Determined Germs of C∞-Mappings
Authors: Bochnak Jacek , Kuo Tzee-Char 
Published: December 2010

Let (respectively ) denote the ring of germs at 0 ∈ Rn of all C∞ functions (respectively Cμ functions) from Rn to R. For a given where is the space of all germs of C∞ mappings Rn → Rp, let J(φ) denote the ideal in generated by φ1, … , φp and the Jacobian determinantswhere LetClearly, is an ideal in and where is the (unique) maximal ideal of .
Chromatic Sums for Rooted Planar Triangulations, III: The Case λ = 3
Authors: Tutte W. T. 
Published: December 2010

SummaryIn this paper we are chiefly concerned with the chromatic sums we have called l and h, with colour-number 3. In this case h can be interpreted as enumerating the rooted Eulerian triangulations with a given number of faces, and l as enumerating such triangulations with a given number of faces and a given valency for the root-vertex. The series h has been determined already, by summation from the formula enumerating even slicings [3]. However our formula for l does not seem to have been published before, though it could presumably be derived in a similar way.
Spectral Properties for Invertible Measure Preserving Transformations
Authors: Belley Jean-Marc 
Published: December 2010

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definitionfor all f , g in L2(I), the bar denoting complex conjugation.
A Necessary and Sufficient Condition for the Oscillation of an Even Order Nonlinear Delay Differential Equation
Authors: Singh Bhagat 
Published: December 2010

In this paper we study the oscillatory behavior of the even order nonlinear delay differential equation(1)where(i)denotes the order of differentiation with respect tot. The delay termsτiσiare assumed to be real-valued, continuous, non-negative, non-decreasing and bounded by a common constantMon the half line (t0, + ∞ ) for somet0≧ 0.
Finitely Generated Subgroups of HNN Groups
Authors: Burns R. G. 
Published: December 2010

In this paper we give sufficient conditions for an HNN group to have the following two properties:(1) any two finitely generated subgroups intersect in a finitely generated subgroup;(2) every finitely generated subgroup containing a non-trivial subnormal subgroup has finite index.The following result is a particular case of the main theorem.
On the Distribution of Square-Free Numbers
Authors: Hooley C. 
Published: December 2010

Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].
Entire Functions with Some Derivatives Univalent
Authors: Shah S. M. , Trimble S. Y. 
Published: December 2010

This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(np) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general
Characterization of a Class of Equicontinuous Sets of Finitely Additive
Measures with an Application to Vector Valued Borel Measures
Authors: Oberle Richard Alan 
Published: December 2010

Let V denote a ring of subsets of an abstract
space X, let R
+ denote the nonnegative reals, and let
N denote the set of positive integers. We denote by
C(V) the space of all
subadditive and increasing functions, from the ring V
into R
+, which are zero at the empty set. The space
C(V) is called the space
of contents on the ring V and elements are referred to
as contents.
A sequence of sets An
∊ V, n ∊
N is said to be dominated if there exists a set
B ∊ V such that
An
⊆ B, for n =
1, 2, A content p ∊
C(V) is said to be
Rickart on the ring V if lim
n

p(An
) = 0 for each dominated, disjoint sequence
An
∊ V, n ∊
N.
Additive Divisibility in Compact Topological Semirings
Authors: Karvellas P. H. 
Published: December 2010

A topological semiring (S, + , ·) is a nonempty Hausdorff space S on which are defined continuous and associative operations, termed addition (+) and multiplication (·), such that the multiplication distributes over addition from left and right. The additive semigroup (S, +) need not be commutative.We prove that the set A of additively divisible elements of a compact semiring S is a two-sided multiplicative ideal, containing the set E[+] of additive idempotents, with the property that (A.S) ∪ (S.A) ⊂ E[+].
On Gaussian and Geodesic Curvature of Riemannian Manifolds
Authors: Rummler Hansklaus 
Published: December 2010

In [1], S. S. Chern gave a very elegant and simple proof of the Gauss-Bonnet formula for closed (i.e. compact without boundary) oriented Riemannian manifolds of even dimension:Here, c is a suitable constant depending on the dimension of M and Ω is an n-form (n = dim M) which may be calculated from its curvature tensor. W. Greub gave a coordinate-free description of this integrand Ω (cf. [4]).
On the Selection of Compact Subsets of Positive Measure from Analytic Sets of Positive Measure
Authors: Larman D. G. 
Published: December 2010

An important but seemingly difficult problem is to decide whether or not an analytic set A of positive h-measure, for some continuous Hausdorff function h, contains a compact subset C of positive h-measure, in every complete separable metric space Ω.By extending some earlier work of R. O. Davies [1], M. Sion and D. Sjerve [8] proved that
(i)the selection of the set C is always possible in a σ-compact metric space Ω. More recently Davies [2] has shown that it is always possible to select C(ii)when h(t) = ts, t ≧ 0, for some fixed positive number s,(iii)when Ω is finite dimensional in the sense of [4],(iv)when A has σ-finite h-measure, and(v)when Ω is an ultra metric space.
On a Family of Generalized Numerical Ranges
Authors: Lin C.-S. 
Published: December 2010

Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].
Riesz Decompositions
Authors: Nagel Alexander , Rudin Walter 
Published: December 2010

All functions mentioned in this paper will be real-valued. If f1, f2, g are nonnegative functions on a set S that satisfy g ≦ f1 + f2, the Riesz decomposition problem associated with these data is to find functions gi on S such thatThe formulaalways furnishes a solution. The problem becomes more interesting if one asks under what conditions one can find solutions that are, roughly speaking, as smooth as the data.
A Bound for the Degree of H2(G, Zp)
Authors: Sharma Sneh 
Published: December 2010

Let G be a group and N a trivial G-module. We say an element ξ ∊ H2 (G, N) is of degree ≦ n if a 2-cocycle representative of ξ is a polynomial 2-cocycle of degree ≦ n [1], Let PnH2(G, N) denote the subgroup of H2(G, N) consisting of elements with degree ≦ n. Then we have a nitrationof H2(G, N). We say that the degree of H2(G, N) is ≦ n if PnH2(G, N) = H2(G, N).
The Structure of Quasi-Frobenius Rings
Authors: Müller Bruno J. 
Published: December 2010

Utilizing a matrix representation of semiperfect rings by a family of bimodules over local rings, we describe the structure of generalized quasi-Frobenius rings in two steps: a cyclic generalized quasi-Frobenius ring is a matrix ring over a cycle of Morita dualities between local rings, and an arbitrary generalized quasi-Frobenius ring is a matrix ring over a family of cyclic generalized quasi-Frobenius rings.Our results provide a complete classification of generalized quasi-Frobenius rings, modulo the classification of local rings with Morita duality, of certain bimodules over such rings, and of certain rest families of multiplication maps.
Topological Properties of Limits of Inverse Systems Of Measures
Authors: Mallory Donald J. 
Published: December 2010

It has been shown (Mallory and Sion [6]) that the problem of finding "limit" measures for inverse systems of measure spaces (Xi,μi)i∊Ican be successfully attacked by establishing the existence of a “limit” measureon the product space, then considering the restriction to the inverse limit set.In this paper we use a similar point of view to establish conditions under which a system of Radón measures has a “limit” measure which is also Radón.
Compact Operators in Reductive Algebras
Authors: Azoff Edward A. 
Published: December 2010

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].
Analogues of Entire Function Inequalities for an Analytic Function
Authors: Bajpai S. K. , Tanne Joseph 
Published: December 2010

1. Let be an analytic function with radius of convergence R (0 < R < ∞). Setand let the order p and lower order ⋋ of f(z) be defined bywhere x = Rr/(R — r). If 0 < ᑭ < ∞, we define the type T and lower type t of f(z) byAlso, if 0 < ᑭ < ∞, define the “growth numbers” 𝛄 and δ byThe purpose of our discussion will be to obtain some inequalities involving the growth constants defined above.
The Norm of the Lp-Fourier Transform, II
Authors: Russo Bernard 
Published: December 2010

Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and thenfor all .
Bounds for a Linear Diophantine Problem of Frobenius, II
Authors: Vitek Yehoshua 
Published: December 2010

Let A = ﹛a0, a1, … , a3﹜ be a set of relatively prime integers such that 0 < a0 < a1 < … < as = n. Let ϕ (A) denote the smallest integer such that, for N ≧ ϕ (A), the equationshould always have a solution in nonnegative integers.
Some Examples of Normal Moore Spaces
Authors: Ellen Rudin Mary , Starbird Michael 
Published: December 2010

A normal Moore space is non-metrizable only if it fails to be ƛ-collectionwise normal for some uncountable cardinal ƛ [1].For each uncountable cardinal X we present a class of normal, locally metrizable Moore spaces and a particular space Sλ in . If there is any space of class which is not X-collectionwise normal, then Sλ is such a space. The conditions for membership in make a space in behave like a subset of a product of a Moore space with a metric space. The class is sufficiently large to allow us to prove the following. Suppose F is a locally compact, 0-dimensional Moore space (not necessarily normal) with a basis of cardinality X and M is a metric space which is O-dimensional in the covering sense. If there is a normal, not X-collectionwise normal Moore space X where X ⊂ Y × M, then Sx is a normal, not λ-collectionwise normal Moore space.
Matrices Sous-Stochastiques et Fonctions Convexes
Authors: Fischer Pal , Holbrook John A. R. 
Published: December 2010

Dans cette note on se propose d'étendre certains résultats de L. Mirsky concernant des matrices doublement sous-stochastiques [5; 6; 7].On commence par rappeler des définitions et notations qu'on aura à utiliser par la suite.
Certain Integral Equalities which Imply Equimeasurability of Functions
Authors: Stephenson Kenneth 
Published: December 2010

1.1. Two complex measurable functions/ and g on complex measure spaces (X, η) and (Y, v) are equimeasurable, abbreviated ƒ ∼ g, iffor every Borel set E ⊆ C. If Φ is a continuous complex function on C, then we make the following standing hypothesis (HI) which relates Φ, f, and g:(HI) For all α, β ∊ C, we have
Finite Groups Admitting an Automorphism Trivial on a Sylow 2-Subgroup
Authors: Hayden John L. , Winter David L. 
Published: December 2010

In this paper we shall consider finite groups satisfying the following hypothesis.Hypothesis I. Let G be a finite group which admits an automorphism σ of primeorder p, (p, |G|) = 1. Assume the fixed point subgroup B = CG(σ) contains some Sylow 2-subgroup.
Linear Transformations on Matrices: The Invariance of a Class of General Matrix Functions
Authors: Ong Hock 
Published: December 2010

Let F be a field, F* be its multiplicative group and Mn(F) be the vector space of all n-square matrices over F. Let Sn be the symmetric group acting on the set {1, 2, … , n}. If G is a subgroup of Sn and λ is a function on G with values in F, then the matrix function associated with G and X, denoted by Gλ, is defined byand letℐ(G, λ) = { T : T is a linear transformation of Mn(F) to itself and Gλ(T(X)) = Gλ(X) for all X}.
Notes on Local Integral Extension Domains
Authors: Ratliff L. J. 
Published: December 2010

All rings in this paper are assumed to be commutative with identity, and the
undefined terminology is the same as that in [3].
In 1956, in an important paper [2], M. Nagata
constructed an example which showed (among other things): (i) a maximal chain of
prime ideals in an integral extension domain R' of a
local domain (R, M) need not contract in
R to a maximal chain of prime ideals; and, (ii) a
prime ideal P in R' may be
such that height P &lt; height
P ∩ R. In his example, Rf
was the integral closure of R and had two maximal
ideals. In this paper, by using Nagata's example, we show that there exists a
finite local integral extension domain of D =
R[X](M,X)
for which (i) and (ii) hold (see (2.8.1) and (2.10)).
Curvature Bounds for the Spectrum of Closed Einstein Spaces
Authors: Simon Udo 
Published: December 2010

The following is our main result.(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils
Planar Sublattices of a Free Lattice. II
Authors: Rival Ivan , Sands Bill 
Published: December 2010

In Planar sublattices of a free lattice, I [8] we verify Jonsson's conjecture for finite planar lattices; in particular we obtain a characterization of finite planar sublattices of a free lattice among all finite lattices. In the present paper we use arguments of a quite different flavour to obtain another characterization. Letbe the family of lattices illustrated in Figures 1, 2, 3, and 4. Our goal is to prove the following theorem: a finite lattice is a planar sublattice of a free lattice if andonly if it does not have a member of as a sublattice.
Some Radical Properties of Jordan Matrix Rings
Authors: Rich Michael 
Published: December 2010

Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].
On Global Inverse Theorems of Szász and Baskakov Operators
Authors: Ditzian Z. 
Published: December 2010

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.
A Semilinear Dirichlet Problem
Authors: Castro Alfonso 
Published: December 2010

Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem(I)where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem(II)has nontrivial weak solutions.
On Bernstein's Inequality
Authors: Giroux A. , Rahman Q. I. , Schmeisser G. 
Published: December 2010

1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])(1)Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdös and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by(2)On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then(3)
Biplanar Surfaces of Order Three
Authors: Bisztriczky Tibor 
Published: December 2010

0. Introduction. A surface of order three, F, in the real projective threespace P3 is met by every line, not in F, in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ1 and τ2. In the present paper, we classify and describe those biplanar F which contain the line τ1 ∩ τ2.We describe a surface by determining the tangent plane sections of the surface at the differentiable points. This approach was introduced in [1] and it is based upon A. Marchaud's definition of “surfaces of order three” in [4].
Ratio and Stochastic Ergodic Theorems for Superadditive Processes
Authors: Fong Humphrey 
Published: December 2010

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if(1.1)T is called sub-Markovian if(1.2)All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if(1.3)and(1.4)