bluuebunny/crossref_metadata_2025_split huggingface dataset:
Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm
Authors: Forrest Brian E. , Runde Volker , Spronk Nico 
Published: December 2010

AbstractLet G be a locally compact group, and let Acb(G) denote the closure of A(G), the Fourier algebra of G, in the space of completely boundedmultipliers of A(G). If G is a weakly amenable, discrete group such that C*(G) is residually finite-dimensional, we show that Acb(G) is operator amenable. In particular, Acb() is operator amenable even though , the free group in two generators, is not an amenable group. Moreover, we show that if G is a discrete group such that Acb(G) is operator amenable, a closed ideal of A(G) is weakly completely complemented in A(G) if and only if it has an approximate identity bounded in the cb-multiplier norm.
Hp-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
Authors: Bu Shangquan , Merdy Christian Le 
Published: December 2010

AbstractWe consider maximal regularity in the Hp sense for the Cauchy problem u′(t) + Au(t) = f(t) (t ∈ ℝ), where A is a closed operator on a Banach space X and f is an X-valued function defined on ℝ. We prove that if X is an AUMD Banach space, then A satisfies Hp-maximal regularity if and only if A is Rademacher sectorial of type < . Moreover we find an operator A with Hp-maximal regularity that does not have the classical Lp-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces Hp(ℝ X), in the case when X is an AUMD Banach space.
On Gap Properties and Instabilities of p-Yang–Mills Fields
Authors: Chen Qun , Zhou Zhen-Rong 
Published: December 2010

AbstractWe consider the p-Yang–Mills functional (p ≥ 2) defined as . We call critical points of YMp( · ) the p-Yang–Mills connections, and the associated curvature R∇ the p-Yang–Mills fields. In this paper, we prove gap properties and instability theorems for p-Yang–Mills fields over submanifolds in ℝn+k and .
On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases
Authors: Ginzburg David , Lapid Erez 
Published: December 2010

AbstractWe prove two spectral identities. The first one relates the relative trace formula for the spherical variety GSpin(4, 3)/G2with a weighted trace formula forGL2. The second relates a spherical variety pertaining toF4to one of GSp(6). These identities are in accordance with a conjecture made by Jacquet, Lai, and Rallis, and are obtained without an appeal to a geometric comparison.
Restricted Partitions of Finite Sets
Authors: Miksa F.L. , Moser L. , Wyman M. 
Published: December 2010

In this paper we consider the following combinatorial problem. In how many ways can n distinguishable objects be placed into an unrestricted number of indistinguishable boxes, if each box can hold at most r objects? Let us denote this number by Gn, rSpecial cases of this problem have been the object of considerable study. In the case r = 2 we have the numbers Gn, 2 = Tn which have been treated by Rothe [12] as early as 1800. Tn is also the number of solutions of x2 = 1 in the symmetric group on n letters , and in this and related guises has been studied by Touchard [13], Chowla, Herstein and Moore [3] and two of the present authors [7].
Asymptotic Formulas for Some Arithmetic Functions
Authors: Erdős P. 
Published: December 2010

Let f(x) be an increasing function. Recently there have been several papers which proved that under fairly general conditions on f(x) the density of integers n for which (n, f(n)) = 1 is 6/π2 and that (d(n) denotes the number of divisors of n)In particular both of these results hold if f(x) = xα, 0 < α < 1 and the first holds if f(x) = [α x], α irrational.
On a Note by H. Schwerdtfeger
Authors: Scherk Peter 
Published: December 2010

Let x, y denote real column vectors with n elements. Let A be a regular symmetric real n×n matrix. Dashes indicate transposition.If x is fixed, x' Ax > 0, the discriminant of A at x is the quadratic formy' where S= S(x)= x'x'Ax ¨ Axx'A.In Can. Math. Bull. 1, pp. 175-179, Dr. Schwerdtfeger proved the equivalence of the following properties of A:(i)A is of the congruence type [+, -,..., -].(ii)y' Sy≤ 0 for all y, equality holding if and only if y is a multiple of x. His note is of particular interest because he also discusses the eigen-values of S. If only the quoted result is aimed at, the following procedure may be shorter.
On the Inversion of General Transformations*
Authors: Rooney P. G. 
Published: December 2010

Let k be the kernel of a “general transformation”; that is, k(x) / x ϵ L2(0,∞), and if x and y are positive1Then it is well known (see for example [8; Theorems 129 and 131]) that if the transform of fϵL2(0,∞) is g, that is, if2then the inverse transform is given by3In practice, the inversion formula (3) is often hard to use. For example, the integral may be too difficult to evaluate; moreover, since (2) requires a differentiation, it is not well suited for numerical calculation. Hence it seems worthwhile to find other methods for inverting the transformation.
Introduction to Von Neumann Algebras and Continuous Geometry
Authors: Halperin Israel 
Published: December 2010

What is a von Neumann algebra? What is a factor (i) of type I, (ii) of type II, (iii) of type III? What is a projection geometry? And finally, what is a continuous geometry?The questions recall some of the most brilliant mathematical work of the past 30 years, work which was done by John von Neumann, partly in collaboration with F. J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space.
An Enumeration of the Five Parallelohedra
Authors: Moser William 
Published: December 2010

A parallelohedron is a convex polyhedron, in real affine three-dimensional space, which can be repeated by translation to fill the whole space without interstices. It has centrally symmetrical faces [4, p. 120] and hence is centrally symmetrical.Let Fi denote the number of faces each having exactly i edges, Vi denote the number of vertices each incident with exactly i edges, E denote the number of edges, n denote the number of sets of parallel edges, F denote the total number of faces, V denote the total number of vertices.
Remarks on an Arithmetic Derivative
Authors: Barbeau E. J. 
Published: December 2010

Let D(n) denote a function of an integral variable n ≥ 0 such that(1)D(1) = D(0) = 0(2)D(p) = 1 for every prime p(3)D(n1n2) = n1D(n2) + n2D(n1) for every pair of non-negative integers n1, n2.
On the Problem of Steiner
Authors: Melzak Z.A. 
Published: December 2010

There is a well-known elementary problem:(S3) Given a triangle T with the vertices a1, a2, a3, to find in the plane of T the point p which minimize s the sum of the distances |pa1| + |pa2| + |pa3|.p, called the Steiner point of T, is unique: if an angle of T is ≥ 2π/3 then p is its vertex, otherwise p lies inside T and the sides of T subtend at p the angle 2π/3. In the latter case p is called the S-point of T, and it can be found by the following simple construction: let a12 be the third vertex of the equilateral triangle whose other two vertices are a1 and a2, and whose interior does not overlap that of T, let C be the circle through a1, a2 a12; then p is the intersection of C and the straight segment a12a3. It is easily proved that any one of the three ellipses through p with two of the vertices of T as foci is tangent at p to the circle through p about the third vertex of T.
On the Colourings of Graphs
Authors: Tutte W.T. 
Published: December 2010

A graph G is defined by a set V(G) of vertices, a set E(G) of edges, and a relation of incidence which associates with each edge two distinct vertices called its ends. We consider only the case in which V(G) and E(G) are both finite.An n-colouring of G is usually defined as a mapping f of V(G) into the set of integers { 1, 2,…, n} which maps the two ends of any edge onto distinct integers. The integers 1 to n are the n "colours". Much work has been done on n-colourings in recent years, especially by G. A. Dirac.
The Reversibility of a Differentiable Mapping
Authors: Atkinson F. V. 
Published: December 2010

Given n functions of n variables, in the real domain, by the equations1we have in various contexts to consider whether the equations are soluble for the xr when the yr are given. Such questions receive fairly complete answers in complex variable theory; a complex variable relation w = f(z) is of course brought under the heading of the real equations (1) by setting w = y1 + iy2, z = x1 + ix2. For example, if f(z) is a polynomial the fundamental theorem of algebra asserts that the equations are soluble, though not in general uniquely. Again, a basic theorem on conformal mapping gives conditions under which the equations are uniquely soluble, to the effect that a (1,1) mapping of the boundaries of domain and range implies a (1,1) mapping of the interiors.
A Note on the Tarry-Escott Problem
Authors: Melzak Z.A. 
Published: December 2010

The Tarry-Escott problem in Diophantine analysis is the following: consider the system of equations1for the unknowns a1, …, ak, b1, …, bk, what is the smallest integer K = K(n) in the set of all k! s for which the system (1) possesses a non - trivial solution in integers ? By an integer we mean a rational integer and a solution is called trivial if the sets { a1, …, ak} and {b1, …, bk} are permutations of each other. By the Tarry-Escott problem one may also mean the problem of finding an actual solution of (1) or finding all such solutions but we shall be concerned with the bound K(n) only. However, the method given here is constructive, that is to say, with each estimate K for K it leads to finding actual non - trivial solutions of (1) with k = K1.
How to Program an Infinite Abacus
Authors: Lambek Joachim 
Published: December 2010

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.
Infinite Doubly Stochastic Matrices
Authors: Isbell J.R. 
Published: December 2010

This note proves two propositions on infinite doubly stochastic matrices,
both of which already appear in the literature: one with an unnecessarily
sophisticated proof (Kendall [2]) and the other with the incorrect assertion
that the proof is trivial (Isbell [l]). Both are purely algebraic; so we
are, if you like, in the linear space of all real doubly infinite matrices A
= (aij).Proposition 1. Every extreme point of the convex set of ail doubly
stochastic matrices is a permutation matrix.Kendall's proof of this depends on an ingenious choice of a topology and the
Krein-Milman theorem for general locally convex spaces [2]. The following
proof depends on practically nothing: for example, not on the axiom of
choice.
On Approximation by Fejér Means to Periodic Functions Satisfying
a Lipschitz Condition
Authors: Lorch Lee 
Published: December 2010

S. M. Nikolski [4, Theorem 1; cf. 3, esp. pp. 144 and 148] considered the
remainder term in the approximation by the n-th Fejér mean,
σn(x), to a function, f(x), of period 2π satisfying a Lipschitz
condition of order α, 0<α≤1. In this connection, he introduced the
quantity1where the maximum is taken over all x and the supremum is taken over all
functions of period 2π, bounded by 1 (a notational convenience only) and
satisfying a Laps chitz condition of order α.
On Maximal Rings of Right Quotients
Authors: Christensen Joanne 
Published: December 2010

Utumi has shown [3, Claim 5.1] that for a certain class of rings the
associated maximal rings of right quotients are isomorphic to the
endomorphism rings of modules over division rings. We shall prove a
generalization of this theorem and then show how it is obtained as a
corollary. The following proofs do not depend on Utumi's paper; instead,
they make extensive use of results proved in [1]. The terminology and
notations employed here are the same as in [1].I wish to thank Dr. B. Banaschewski for his suggestions and helpful
criticism.LEMMA: If J is a left ideal with zero left annihilator in a ring R then a
maximal ring of right quotients of R is also a maximal ring of right
quotients of J.
On the Gram Matrix
Authors: Halperin Israel 
Published: December 2010

The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).
A Remark on a Result of Marvin Marcus
Authors: Dulmage A.L. , Mendelsohn N.S. 
Published: December 2010

Marcus [2] has proved the following theorem.Suppose A is a non-negative normal matrix satisfying p(A) = 0 in which p(λ)
is a monic polynomial no two of whose non-zero roots have the same modulus.
Then there exists a permutation matrix P such that PAP* is a direct sum,
PAP* = A1 ⊕ A2 ⊕ … ⊕ Am, in which each
Ai is either O or primitive.This note gives a generalisation of this result, dropping the non-negative
assumption and weakening the normality assumption.
Order in Affine and Projective Geometry
Authors: Lipman Joe 
Published: December 2010

In this note we give a characterisation of ordered skew fields by certain
properties of the negative elements.Properties of negative elements of a skew field K are then interpreted as
statements about "betweenness" in any affine geometry over K, or as
statements about "separation" in any projective geometry over K.In this way, our axioms for ordered fields permit of immediate translation
into postulates of order for affine or projective geometry (over a field).
The postulates so obtained seem simple enough to be of interest, (cf. [4],
p. 22)
On the Sum of Consecutive Primes
Authors: Moser L. 
Published: December 2010

Let A: a1 < a2 < … be a sequence of positive integers and let f(A:n) = f(n) denote the number of representations of n as the sum of one or more consecutive a's. Although f(n) will usually oscillate rapidly, an asymptotic estimate for its average valuecan be obtained for many sequences A.
Small Positive Values of Indefinite Quadratic Forms
Authors: Chalk J. H. H. 
Published: December 2010

Let ∧ denote the lattice of points X = (x1, …, xn) with integral coordinates. A basis of ∧ is a set of n points X1, …, Xn of ∧ such that every point of ∧ is expressible in the form where ui are integers. It is easy to see that points X1, …, Xn of ∧ form a basis if, and only if(1)where Xr = (x1(r), …, x1(r)).
A General Turán Expression for the Zeta Function
Authors: Gould H. W. 
Published: December 2010

In 1948 Gabor Szegő [9] gave four proofs of a remarkable inequality communicated to him by Paul Turán, who later published an original proof [10]. The Turán theorem states that if Pn(x) is the Le gendre polynomial, then1.1with equality holding only when |x| = 1.Since then many similar inequalities have been found for various special functions, particularly for the Legendre and Hermite polynomials. Reference may be had to the recent work of Danese [2] and Chatterjea [1]. Danese gives an extensive bibliography.
On a Combinatorial Problem of Erdös and Hajnal
Authors: Abbott H. L. , Moser L. 
Published: December 2010

In this note we consider some problems related to the following question:
What is the smallest integer m(n) for which there exists a family
Fn of sets A1, A2,…, Am(n)
with the following properties, (i) each member of Fn has n
elements and (ii) if S is a set which meets each member of Fn,
then S contains at least one member of Fn?Erdős and Hajnal [1] observed thatand that m(l) = 1, m(2) = 3, m(3) = 7.
A Generalization of Cauchy' s Double Alternant
Authors: Carlson David , Davis Chandler 
Published: December 2010

The subject of alternants and alternating functions was widely studied
during the last century (cf. Muir [6]). One of the best-known alternants is
actually a double alternant (rows and columns) defined by Cauchy [2] in
1841. Cauchy's result may be stated as follows:If D = [dpq] p, q
= l, …, n, where dpq =
(xp+yq)-1, then1.
Finite Projective Planes with Affine Subplanes
Authors: Ostrom T. G. , Sherk F. A. 
Published: December 2010

A well-known theorem, due to R. H. Bruck ([4], p. 398), is the following:If a finite projective plane of order n has a projective subplane of order m < n, then either n = m2 or n > m 2+ m.In this paper we prove an analagous theorem concerning affine subplanes of finite projective planes (Theorem 1). We then construct a number of examples; in particular we find all the finite Desarguesian projective planes containing affine subplanes of order 3 (Theorem 2).
Axiomatic Proof of J. Lambek's Homological Theorem
Authors: Leicht J. B. 
Published: December 2010

A category with zero-maps is called "quasi-exact" in the sense of D. Puppe (see [4], page 8, 2. 4), if it satisfies the following axioms:(Q1)Every may f is a product f=με of an epimorphisrn εfollowed by a monomorphism μ(Q2)a) Every epimorphism ε has a kernel k = ker εb) Every monomorphism μ has a cokernel γ = Coker ε, where Ker and Coker are characterized by the familiar universality properties (see [3], page 252, (1. 10) and (1. 11)).
The Geometry of Finite Markov Chains
Authors: Pullman N. 
Published: December 2010

The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history.
Some Properties of Compositions and their Application to the Ballot Problem
Authors: Mohanty S. G. 
Published: December 2010

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].
A Characterization of Group Rings as a Special Class of Hopf Algebras
Authors: Takahashi Shuichi 
Published: December 2010

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:such that A contains G and is freely generated by G over K, so thatThe ring A = KG has a co-multiplicationdefined byso that A is a Hopf algebra.
A Generalization of Divisibility and Injectivity in Modules
Authors: Sanderson D. F. 
Published: December 2010

Classically, there has been, for obvious reasons, an intimate relation between the concepts "rings of quotients" and "divisible modules". Recently, however, their generalizations have appeared to diverge.For example, Hattori ([9]) and Levy ([15]) have generalized the concept of "divisibility" as follows: Hattori (respectively Levy) defines a left R-module M over a ring R to be divisible if and only if Ext1R(R/I, M)=0 for every principal left ideal I ⊂ R (respectively, every principal left ideal I ⊂ R which is generated by a regular element of R).
Stability on the basis of Orthogonal Trajectories
Authors: Burton T. A. 
Published: December 2010

We consider a system of differential equations of second order given by1(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.
Elementary Generalizations of Hilbert's Theorem 90
Authors: Connell Ian G. 
Published: December 2010

Let K, k be fields and K|k a finite galois extension with galois group G. The multiplicative group K* of K is a G-module, that is, a module over the integral group ring ZG, the module action of an element σ ϵ G being its effect as an automorphism. It is shown in [2, p. 158] that the first cohomology group vanishes:1
On a Problem by R.A. Hirschfeld: Counterexample
Authors: Edelstein M. 
Published: December 2010

The following question was asked by R. A. Hirschfeld in [1]: "E and F are Banach spaces, F reflexive, D is a subset of E and T: D→F a nonlinear contraction, i.e.,Can T be extended to a contraction (for E = F = Hilbert space the answer is yes)."
Curves on Surfaces of Constant Width
Authors: Armstrong William W. 
Published: December 2010

A surface S of constant width is the boundary of a convex set K of constant width in euclidean 3-dimensional space E3. (See [l] pp. 127–139. )Our first result concerns the interdependence of five properties which a curve on such a surface may possess. Let S be a surface of constant width D > 0 which satisfies the smoothness condition that it be a 2-dimensional submanifold of E3 of class C2.
Some Inequalities Related to the Wald-Wolfowitz-Noether Condition*
Authors: Mehra K. L. , Wong J. S. W. 
Published: December 2010

If {avα: or = 1, 2, …, N }, with Nv → ∞ as v → ∞, is a double sequence of real numbers with the property that , then1.1is known in statistical literature as the Wald- Wolfowitz- Noether condition and it plays an important role in the proofs of certain types of central limit theorems (see e. g., [ l ], [2] ).
On a Lemma of M. Abramson
Authors: Church C.A. 
Published: December 2010

Kaplansky's Lemma [3] states: the number of k-combinations of 1, 2,…, n with no two consecutive integers in any selection is Using this, Abramson [l; lemma 3] solves the problem: find the, number of k-combinations so that no two integers i and i+2 appear in any selection. (This is generalized by Abramson in [2].) An interesting solution, also using Kaplansky's lemma, is obtained as follows.If n = 2m, we choose s from among the m even integer s, no two consecutive, and k- s from among the m odd integers, no two consecutive.
On the Steiner Problem
Authors: Cockayne E. J. 
Published: December 2010

Let M be a metric space with metric ρ which has the following properties.1. M is finitely compact.2. There exists a geodesic in M joining each two points of M.3. For all a, b∈M, ρ(a, b) is equal to the length of a geodesic joining a and b.
A Note on Affine Pappus Conditions
Authors: Lane N. D. 
Published: December 2010

Let ℓ, m, n be three mutually distinct lines in the projective plane. The (ℓ, m, n)-Pappus condition can be described as follows.Let A, B, C, A', B', C' be any six mutually distinct points such that A, B, C lie on ℓ; A', B', C' lie on m; and none of these points lies on ℓ∩m, m∩n, or n∩ℓ. If the points AB'∩BA' and BC'∩CB' both lie on n, then the point AC'∩CA' also lies on n.
Exact Distribution of the Quotient of Independent Generalized Gamma Variables
Authors: Malik Henrick John 
Published: December 2010

Let X be a random variable whose frequency function is1.1Form (1.1) is Stacy′s [3] generalization of the gamma distribution. The familiar gamma, chi, chi-squared, exponential and Weibull variâtes are special cases, as are certain functions of normal variate - viz., its positive even powers, its modulus, and all positive powers of its modulus.