bluuebunny/crossref_metadata_2025_split huggingface dataset:
An Application of the Addition Theorem for Determinants
Authors: Jack Henry 
Published: January 2009

The integral evaluated in this note was suggested by the famous one connected with the Poincaré polynomials of the classical groups (see (1)).Let X be an n × n matrix whose elements depend on k parameters. Denote by a manifold in Euclidean space of dimension n2, with the property that if X ∈ , then so does XI−i for 1≦i≦n, where I−i is the unit matrix I altered by a minus sign in the (i, i)th place. Suppose further that there exists on a measure which is invariant under the transformation X→XI−i. Such manifolds and measures exist. For example (see (2), § 5), the set of all proper and improper n×n orthogonal matrices H is such a manifold, the H depending on ½n(n−1) parameters because of the orthogonality and normality of the columns of H. Since the set of all H is a compact topological group, an invariant measure exists.
Note on a Difference-Product Inequality
Authors: Aitken A. C. 
Published: January 2009

L. J. Mordell has recently considered (1) the squared modulus of a complex difference-product, namelyunder the conditionsand also under the quite different conditionHe proves that under (2) the maximum of δ is nn, and is attained when and only when the zr are vertices of a regular n-gon on the unit circle.
A nonselfadjoint dynamical system
Authors: Kreith Kurt 
Published: January 2009

This paper concerns criteria for assuring that every solution of a real fourth order nonselfadjoint differential equationis oscillatory at x = ∞. Our technique is a generalisation of that used by Whyburn (1) for the study of the selfadjoint equation,combined with the theory of H-oscillation of vector equations as introduced by Domšlak (2) and studied by Noussair and Swanson (3). Whyburn's technique consists of representing (1.2) as a dynamical system of the formand then studying (1.3) in terms of polar coordinates in the y, z-plane. In Section 2 below we show how to represent (1.1) as a dynamical system of the form
The number of primes representable as the sum of two square-free squares
Authors: Keates M. 
Published: January 2009

In this paper an asymptotic formula is obtained for the number of primes representable as the sum of two square-free squares. The precise result is:Theorem 1.Let N(x) be the number of primes not exceeding x represented by the quadratic formy2 + z2, where y and z are square-free. Let w be a fixed arbitrarily large number. Thenwhereand
A Harnack inequality for degenerate elliptic equations on minimal surfaces
Authors: Cooper Frances 
Published: January 2009

Harnack inequalities are known to be of great importance in the theory of quasilinear elliptic partial differential equations. In the case of such equations defined over a domain Ω in Rn, inequalities of this type have been proved for solutions of second-order equations in divergence form which are of either elliptic or degenerate elliptic structure. More recently Bombieri and Giusti (2) have proved a Harnack inequality for solutions of linear elliptic equations on a minimal surface in Rn+1. The equations are of the formwhere summation over i, j = 1, …, n+l is understood, and δ = (δ1, …, δn+1) is the tangential derivative on S. In (2), the inequality is used to give much simplified proofs of some classical results on minimal surfaces, and to generalise some more recent ones.
A note on “simplification and scaling”
Authors: Riley N. 
Published: January 2009

In a recent paper Segel (1) points out that the diverse techniques (which “comprise the core of the applied mathematicians art (or craft)”) of the applied mathematician, although in general reliably proven, are “rarely explicitly delineated but rather are transmitted indirectly and informally”. In his article Segel aims to clarify two such techniques, namely:(i) Scaling—or how to choose dimensionless variables in such a way that the relative size of the various terms in an equation is explicitly indicated by the magnitudes of the dimensionless parameters which precede them,(ii) Simplification—a procedure in which a term is neglected under the assumption that it is small, and the consistency of the assumption checked later.
Some inequalities for norm unitaries in Banach algebras
Authors: Crabb M. J. , Duncan J. 
Published: January 2009

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that
Numerical ranges of derivations
Authors: Kyle J. 
Published: January 2009

In this paper we shall examine the relationship between the numerical range of aninner derivation, and that of its implementing element.Much of this paper is taken from the author's doctoral thesis (5) written at theUniversity of Newcastle upon Tyne with the helpful guidance of Professor B. E.Johnson. The author also acknowledges the financial support of the Science ResearchCouncil during the period of this research.
On central traces and groups of symmetries of order unit Banach spaces
Authors: Chu Cho-Ho 
Published: January 2009

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).
Asymptotic Behaviour of Eigen-Values of Certain Integral Equations
Authors: Reade J. B. 
Published: January 2009

The integral equations we consider areandwhere 0 < α < 1. The asymptotic behaviour of the eigen-values of the latter equation is already known (see (1) and (4)). The former equation has been studied by many authors but as yet no explicit statement seems to have been made about the behaviour of its eigen-values.
Centralisers in Wreath Products
Authors: Meldrum J. D. P. 
Published: January 2009

In this paper, the centraliser of an arbitrary element of a wreath product is determined. One application of this is to find the breadth of a wreath product (Theorems 21 and 22), a problem which was raised in discussion with Dr. I. D. Macdonald. Another application is to groups generated by elements generating their own centralisers (Theorem 20).Let A and B be two groups. DefineAB = {f : B → A; f(b) = e for all but a finite number of elements of B} to be a group by defining the product pointwise
Commutators in Banach algebras
Authors: Pták Vlastimil 
Published: January 2009

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the conditionfor all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition
Semi-embeddings of Banach space
Authors: Lotz Heinrich P. , Peck N. T. , Porta Horacio 
Published: January 2009

It is a most implausible fact that a one-to-one operator from c0 into a Banach space which maps the unit ball of c0 onto a closed set is necessarily an isomorphism.In this paper the term semi-embedding denotes a one-to-one operator from one Banach space into another, which maps the closed unit ball of the domain onto a closed set. In the first section we study semi-embeddings in conjunction with weak compactness; in the second section we apply our results to the case of semi-embeddings defined on C(X), X compact.
Small fractional parts of quadratic forms
Authors: Baker R. C. , Harman G. 
Published: January 2009

Let ‖x‖ denote the distance of x from the nearest integer. In 1948 H. Heilbronn proved [5] that for ε>0 and N>c1(ε) the inequalityholds for any real α. This result has since been generalised in many different directions, and we consider here extensions of the type: For ε>0, N>c2{ε, s) and a quadratic formQ(x1,…, xs) there exist integersn1,…,nsnot all zero with |n1|,…,|ns≦N and with
Angular momentum, convex Polyhedra and Algebraic Geometry
Authors: Atiyah M. F. 
Published: January 2009

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.
Pairs of additive equations III: quintic equations
Authors: Cook R. J. 
Published: January 2009

We consider R simultaneous equations of additive typewhere the coefficients aij are integers. Artin's conjecture, for additive forms, is that the equations (1) have a non-trivial solution in integers x1,…,xN provided that they have a non-trivial real solution, which is clearly satisfied when k is odd, and
Maximum idempotents in naturally ordered regular semigroups
Authors: McAlister D. B. , McFadden R. 
Published: January 2009

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense thatNo assumption is made about the reverse implication.
Polynomial approximation of an entire function and rate of growth of Taylor coefficients
Authors: Freund M. , Görlich E. 
Published: January 2009

The best uniform approximation of a functionfon [-1,1] by real algebraic polynomials satisfiesif and only if ƒ is the restriction to [- 1,1] of an entire function (Bernstein [2], p. 113, see also [12], pp. 83–85). For such functions ƒ the rate of best approximation has been characterized by Varga [24], Reddy [24], Shah [21], and Kapoor and Nautiyal [10] in terms of order and type of ƒ , lower order and type, and in terms of more general concepts of order. On the other hand, order and type of ƒ are connected with the Taylor coefficients, i.e. with the rate of growth of the sequence(see [23], p. 41 or [3], pp. 11/12; cf. also [19], [20], [6], [7], [8]) and this has been extended to iterated orders by Schonhage [17], Sato [16], Reddy [14], Juneja, Kapoor, and Bajpai [9] (also [22], [13]), and to generalized orders by Seremeta [18], Bajpai, Gautam, and Bajpai [1] as well as Kapoor and Nautiyal [10]. Combining the two kinds of characterizations (as done, e.g., by Reddy [15], p. 105) approximation theorems in terms of the sequenceare obtained. But in such results the rate ofbest approximation is always described by a limit relation, e.g. of the form, and this causes a considerable loss of precision, as will be discussed in more detail in Section 3 (in this respect cf. also the remark by Bernstein [2], pp. 114/115).
Boundary problems for Riccati and Lyapunov equations
Authors: Jódar Lucas 
Published: January 2009

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).
Existence and uniqueness in the theory of bending of elastic plates
Authors: Constanda Christian 
Published: January 2009

Kirchhoff's kinematic hypothesis that leads to an approximate two-dimensional theory of bending of elastic plates consists in assuming that the displacements have the form [1]In general, the Dirichlet and Neumann problems for the equilibrium equations obtained on the basis of (1.1) cannot be solved by the boundary integral equation method both inside and outside a bounded domain because the corresponding matrix of fundamental solutions does not vanish at infinity [2]. However, as we show in this paper, the method is still applicable if the asymptotic behaviour of the solution is suitably restricted.
On the strong maximum principle for parabolic differential equations
Authors: Walter Wolfgang 
Published: January 2009

In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equationor to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.
Multipliers of Bergman spaces into Lebesgue spaces
Authors: Luecking Daniel H. 
Published: January 2009

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0<p<+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0<p≦q.
PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces
Authors: Singerman David 
Published: January 2009

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentationthe subgroup PSL(2, ℤ) being generated by UV and VW.
A Method of Evaluating Certain Determinants
Authors: Burchnall J. L. 
Published: January 2009

1. Let [ast] (s, t=0, 1, … n) be a square matrix of order n+1 and determinant |ast| and suppose that by repeated “isolation” of the variables the corresponding bilinear form has been expressed aswhere, for all r,ThenNow (1) implies, and is implied by, the identitiesThus, from any known identity of the form (4), subject to the condition (2), we may at once infer, using (3). the value of the corresponding determinant |ars|.
Integral Functions with Gap Power Series
Authors: Erdös P. , Macintyee A. J. 
Published: January 2009

1. Letbe an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notationsdescribing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).
On Compact Riemannian Manifolds with Zero Ricci Curvature
Authors: Willmore T. J. 
Published: January 2009

In this note I prove the following result:—Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.
A Theorem on Algebras of Measures on Topological Groups
Authors: Williamson J. H. 
Published: January 2009

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.
The countable neighbourhood property and tensor products
Authors: Bonet José 
Published: January 2009

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.
On the relations between the numbers of standard tableaux
Authors: Rutherford D. E. 
Published: January 2009

§ 1. Corresponding to any partitionwhich we denote by [a1, . …, ak] or briefly by [a], of the integer n, we can construct a shape which has a1 spaces in the first row, a2 in the second row, . …, ak in the kth and last row. Thus the shape corresponding to the partition [5, 3, 3, 2] of 13 has the form:
On a type of canonical transformation
Authors: Davidson P. M. 
Published: January 2009

1. A well known theorem in classical dynamics tells us that if Q, P, and are written as functions of q, p and t by means of the equationswhere W (q, Q, t) is an arbitrary function, and if two functions F (Q, P, t) and f (q, p, t) are related bythen q and p are conjugate with respect to f(q, p, t) if Q and P are conjugate with respect to F (Q, P, t).
Conics on a Maschke Surface
Authors: Edge W. L. 
Published: January 2009

The six quaternary quartic formswere first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.
Some theorems on absolute Cesàro summability
Authors: Hyslop J. M. 
Published: January 2009

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper.Let k be any positive number and letwhere is the coefficient of xn in the formal expansion of (1 – x )–k–1, and letThen the series Σαn is said to be summable(C, k) if is convergent, that is, if tends to a limit, and absolutely summable (C, k), or summable |C, k|, if is absolutely convergent.
Matrices with finite period
Authors: Taussky Olga , Todd John 
Published: January 2009

The study of the primitive solutions of the equationwhere A = (aij) is an n × n matrix whose elements are rational integers, was begun a long time ago. In most cases this equation occurred incidentally in another theory; for instance Jordan encountered it in connection with linear differential equations having algebraic solutions, Minkowski in connection with quadratic forms and Turnbull in geometry. An important fact about these matrices is that any unimodular matrix can be represented as the product of matrices with finite period.
Expansions in terms of Parabolic Cylinder Functions
Authors: Cherry T. M. 
Published: January 2009

When the plane wave equation is expressed in terms of parabolic co-ordinates x, y, the variables are separable, and the elementary solutions have the formwhere x, y, μ are real. In this context, therefore, the functions Dν (z) which are directly significant are those where amp z = ± π/4 and ν + ½ is purely imaginary, rather than those where z is real and ν is a positive integer. The expansion of an arbitrary function in terms of the latter sort of D-function (substantially, in terms of Hermite polynomials) is well known. This paper is concerned with the expansion in terms of the former sort of D-function.
Enumeration of Indices of Given Altitude and Degree
Authors: Etherington I. M. H. 
Published: January 2009

This note is a sequel to the article by Mine (2) on the same problem.I described in (1) a notation for indices of powers in non-associative algebra, defined the degree † and altitude of a power or index, and observed that powers can be represented by bifurcating root-trees. For example, the power xx.x is denoted x2 + 1, with index 2 + 1, and is represented by the tree ; the degree (the number of factors, or free knots in the tree) is 3, and the altitude (the height of the tree) is 2. Multiplication being non-commutative or commutative, one maintains or ignores the distinction between left and right in the tree.
Asymptotic Formulæ for the Number of Partitions of a Multi-Partite Number
Authors: Robertson M. M. 
Published: January 2009

A multi-partite number of orders j is a j dimensional vector, the components of which are non-negative rational integers. A partition of (n1, n2, …, nj) is a solution of the vector equationin multi-partite numbers other than (0, 0, …, 0). Two partitions, which differ only in the order of the multi-partite numbers on the left-hand side of (1), are regarded as identical. We denote by P1(n1, …, nj) the number of different partitions of (n1 …, nj) and by p2(n1, …, nj) the number of those partitions in which no part has a zero component. Also, we write P3(n1, …, nj) for the number of partitions of (n1; …, nj) into different parts and p4(n1, …, nj) for the number of partitions into different parts none of which has a zero component.
On the Eigenvector belonging to the Maximal Root of a Non-negative Matrix
Authors: Ostrowski A. M. 
Published: January 2009

By a theorem of Perron, a non-negative irreducible (n × n) matrix A = (aμν) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we putσ lies between R and r. Since σ is not changed if A is transformed by a positive diagonal matrix D(p1,…pn, σ lies also between the expressionsBy a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = (x1 …, xn), satisfying
Asymptotic Representation of Certain Real Integrals with Parameter-dependent Limits of Integration
Authors: Ross E. W. 
Published: January 2009

In this paper we consider a function f(x) defined byAll quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.
On an Integral in the Distribution of Eigenvalues
Authors: McLeod J. B. 
Published: January 2009

In the case of a wave function with spherical symmetry, the wave equation can be separated using spherical polar coordinates, and the equation for the radial component becomeswhere λ is a constant parameter, proportional to the energy of the particle under consideration, q(r) is proportional to the potential energy, and l is a positive integer or zero.
An Extension of the Almost Isolated Singularity of Finite Exponential Order
Authors: Wilson R. 
Published: January 2009

Let f(z) be represented on its circle of convergence |z| = 1 by the Taylor seriesand suppose that its sole singularity on |z| = 1 is an almost isolated singularity at z = 1. In the neighbourhood of such a singularity f(z) is regular on a sufficiently small disk, centre z = 1, with the outward drawn radius along the positive real axis excised. If also in this neighbourhood |f(z)| e−(1/δ)ρ remains bounded for some finite ρ, where δ is the distance from the excised radius, then the singularity is said to be of finite exponential order.